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On blow-up for the supercritical defocusing nonlinear wave equation

Published online by Cambridge University Press:  31 March 2025

Feng Shao
Affiliation:
School of Mathematical Sciences, Peking University, 5 Yiheyuan Road, Haidian District, Beijing, 100871, China; E-mail: [email protected]
Dongyi Wei
Affiliation:
School of Mathematical Sciences, Peking University, 5 Yiheyuan Road, Haidian District, Beijing, 100871, China; E-mail: [email protected]
Zhifei Zhang*
Affiliation:
School of Mathematical Sciences, Peking University, 5 Yiheyuan Road, Haidian District, Beijing, 100871, China
*
E-mail: [email protected] (corresponding author)

Abstract

In this paper, we consider the defocusing nonlinear wave equation $-\partial _t^2u+\Delta u=|u|^{p-1}u$ in $\mathbb {R}\times \mathbb {R}^d$. Building on our companion work (Self-similar imploding solutions of the relativistic Euler equations, arXiv:2403.11471), we prove that for $d=4, p\geq 29$ and $d\geq 5, p\geq 17$, there exists a smooth complex-valued solution that blows up in finite time.

Type
Differential Equations
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2025. Published by Cambridge University Press

1 Introduction

In this paper, we consider the defocusing nonlinear wave equation

(1.1) $$ \begin{align} \Box u=|u|^{p-1}u, \end{align} $$

where $u:\mathbb {R}^{1+d}\to \mathbb {C}$ is the unknown field, $ \Box =\partial ^{\alpha }\partial _{\alpha }=-\partial _t^2+\sum _{i=1}^d\partial _i^2$ is the d’Alembertian operatorFootnote 1 on Minkowski spacetime $\mathbb {R}^{1+d}$ with the standard Minkowski metric

$$\begin{align*}m_{00}=-1,\quad m_{ii}=1 \text{ for all }i\in\mathbb{Z}\cap[1,d],\quad m_{\mu\nu}=0\text{ if }\mu,\nu\in\mathbb{Z}\cap[0, d]\text{ with }\mu\neq \nu,\end{align*}$$

and we assume $p\in 2\mathbb {Z}_{+}+1$ for simplicity.

Given smooth initial data $(u|_{t=0}, \partial _tu|_{t=0})$ , there exists a local smooth solution on the maximal existence of interval $[0, T)$ ; $T<+\infty $ if and only if $\limsup _{t\uparrow T}\|u(t)\|_{L^\infty }=+\infty $ , see [Reference Sogge68, Reference Luk45]; moreover, there holds the energy conservation

(1.2) $$ \begin{align} E[u(t)]:=\int_{\mathbb{R}^d}\frac12|\partial_tu|^2+\frac12|\nabla_xu|^2+\frac1{p+1}|u|^{p+1}\,\mathrm dx. \end{align} $$

The class of solutions to (1.1) is invariant under the scaling

(1.3) $$ \begin{align} u(t,x)\mapsto u_\lambda(t,x):=\lambda^{\frac2{p-1}}u(\lambda t,\lambda x),\quad\lambda>0. \end{align} $$

This scaling symmetry preserves the critical norm invariant, i.e.,

$$\begin{align*}\|u_\lambda(t, \cdot)\|_{\dot H_x^{s_c}}=\|u(\lambda t, \cdot)\|_{\dot H_x^{s_c}}\quad\text{where}\quad s_c:=\frac d2-\frac2{p-1}.\end{align*}$$

We can split the range of parameters $(d,p)$ into three cases accordingly:

  • Subcritical case: $s_c<1 \Longleftrightarrow d\leq 2$ or $p<1+4/(d-2)$ for $d\geq 3$ .

  • Critical case: $s_c=1 \Longleftrightarrow p=1+4/(d-2)$ and $d\geq 3$ .

  • Supercritical case: $s_c>1 \Longleftrightarrow p>1+4/(d-2)$ and $d\geq 3$ .

For the subcritical case, the global well-posedness and propagation of regularity dated back to Jörgens [Reference Jörgens31] for $d=3$ ; see also [Reference Ginibre and Velo19, Reference Ginibre and Velo20] for the global well-posedness within the energy class $H^1\times L^2$ for all dimensions; the propagation of regularity holds at least for $d\leq 9$ [Reference Brenner and von Wahl3]. The critical case is much more difficult. The global regularity result was obtained firstly in [Reference Struwe71] for $d=3$ and spherically symmetric initial data, and then extended to $d\leq 9$ for general smooth data in [Reference Grillakis21, Reference Grillakis22, Reference Shatah and Struwe66], and all dimensions in [Reference Shatah and Struwe67] (in the energy class $H^1\times L^2$ ). For the long-time behavior of these global solutions, we refer to [Reference Yang75] and references therein.

For the supercritical case, it is known that the Cauchy problem is ill-posed in some low regularity spaces [Reference Christ, Colliander and Tao6], or even in the energy class [Reference Ibrahim, Majdoub and Masmoudi26], despite the global existence of weak solutions [Reference Strauss70], as well as the global well-posedness with scattering for small smooth data [Reference Lindblad and Sogge44]. The global well-posedness for general smooth data is a long-standing open problem [Reference Bourgain2, Reference Tao72]. In the breakthrough work [Reference Merle, Raphaël, Rodnianski and Szeftel51], Merle, Raphaël, Rodnianski and Szeftel construct radial and asymptotically self-similar blow-up solutions for the energy supercritical defocusing nonlinear Schrödinger equations (NLS). The goal of this paper is to extend a similar blow-up result for NLS to the defocusing supercritical wave equation.

Before stating our theorem, we recall Tao’s blow-up result [Reference Tao73] for the defocusing nonlinear wave system of the form $\Box u=(\nabla _{\mathbb {R}^m}F)(u)$ , where $u:\mathbb {R}^{1+d}\to \mathbb {R}^m$ is vector-valued, and $F:\mathbb {R}^m\to \mathbb {R}$ is a smooth potential which is positive and homogeneous of order $p+1$ outside of the unit ball for some $p>1$ (letting $m=2$ and $F(u)=|u|^{p+1}/(p+1)$ we recover (1.1)). Tao [Reference Tao73] proved that for any supercritical $(d,p)$ , and sufficiently large positive integer m, there exists a defocusing $F: \mathbb {R}^m\to \mathbb {R}$ such that the system $\Box u=(\nabla _{\mathbb {R}^m}F)(u)$ has no global smooth solution for some smooth compactly supported initial data. A similar result for the defocusing Schrödinger system was obtained in [Reference Tao74].

1.1 Main results

Roughly speaking, we prove that the defocusing supercritical nonlinear complex-valued wave equation for $d\ge 4$ admits finite time blow-up solutions arising from smooth initial data. The leading order term of blow-up solution is given by a self-similar blow-up solution of the relativistic compressible Euler equation, which is stated here as Assumption 1 (in Section 2). In our companion paper [Reference Shao, Wei and Zhang65], we have verified Assumption 1 for some $(d,p)$ .

Theorem 1.1. Let $d\in \mathbb {Z}\cap [4,+\infty )$ and $p\in 2\mathbb {Z}_{+}+1$ be such thatFootnote 2 $k>\ell +\sqrt \ell $ , where $k:=d-1$ and $\ell :=1+4/(p-1)$ . Assume that there exists $\beta \in (1, k/(\ell +\sqrt \ell ))$ such that Assumption 1 holds. Then there exist compactly supported smooth functions $u_0, u_1:\mathbb {R}^d\to \mathbb {R}^2(=\mathbb {C})$ such that there is no global smooth solution $u: [0,+\infty )\times \mathbb {R}^d\to \mathbb {R}^2(=\mathbb {C})$ to the defocusing nonlinear wave equation (1.1) with initial data $u(0)=u_0$ , $\partial _t u(0)=u_1$ .

Corollary 1.2. If $d=4$ , $p\in (2\mathbb {Z}+1)\cap [29,+\infty )$ or $d\geq 5, p\in (2\mathbb {Z}+1)\cap [17,+\infty )$ , then there exist compactly supported smooth functions $u_0, u_1:\mathbb {R}^d\to \mathbb {R}^2(=\mathbb {C})$ such that there is no global smooth solution $u: [0,+\infty )\times \mathbb {R}^d\to \mathbb {R}^2(=\mathbb {C})$ to the defocusing nonlinear wave equation (1.1) with initial data $u(0)=u_0$ , $\partial _t u(0)=u_1$ .

Several remarks are in order.

  1. 1. For the blow-up solution u we construct in Theorem 1.1, if u blows up at time $T_*\in (0, +\infty )$ , then according to our construction, we have the blow-up speed

    $$ \begin{align*} \|u(t,\cdot)\|_{L^\infty}\gtrsim(T_*-t)^{-\frac{2\beta}{p-1}},\quad \|(u(t),\partial_tu(t))\|_{\dot{H}_x^{s_c}\times\dot{H}_x^{s_c-1}}\gtrsim (T_*-t)^{(1-\beta)\frac d2}. \end{align*} $$
    As $\beta>1$ , our solution is unbounded in the critical space. This is compatible with the results in the literature, which state that the solutions for the supercritical defocusing wave equation that are bounded in the critical space $\dot {H}_x^{s_c}\times \dot {H}_x^{s_c-1}$ must be global and scattering (at least for real-valued solutions and some supercritical $(d,p)$ , see [Reference Bulut4, Reference Bulut5, Reference Duyckaerts and Yang18, Reference Killip and Visan35, Reference Killip and Visan36]).
  2. 2. As in the recent breakthrough work by Merle-Raphaël-Rodnianski-Szeftel [Reference Merle, Raphaël, Rodnianski and Szeftel51, Reference Merle, Raphaël, Rodnianski and Szeftel52, Reference Merle, Raphaël, Rodnianski and Szeftel53], the heart of proof of Theorem 1.1 is to study (1.1) in its hydrodynamical formulation, i.e., with respect to its phase and modulus variables, i.e. (2.1). After introducing a front re-normalization (2.2), (2.1) becomes (2.3). Taking the formal limit $b\to 0$ , we reveal the underlying relativistic compressible Euler dynamics (2.6). The relativistic Euler dynamics provides us with a self-similar blow-up solution, which has been constructed in our companion paper [Reference Shao, Wei and Zhang65] and which, in turn, acts as the leading order term of the blow-up solution of the defocusing supercritical wave equation (1.1).

  3. 3. In the proof, we first construct a good approximate solution, then solve the perturbation equation backward in time. This method is quite different from [Reference Merle, Raphaël, Rodnianski and Szeftel51], and simplifies our proof greatly. Unlike [Reference Merle, Raphaël, Rodnianski and Szeftel51], we do not need to analyze the stability of the linearized operator near the leading order profile constructed in [Reference Shao, Wei and Zhang65], hence we only use less information of the leading order profile (see Assumption 1). On the other hand, in [Reference Merle, Raphaël, Rodnianski and Szeftel51], the initial data for blow-up form a finite co-dimensional manifold in the class of radial smooth fast-decay functions, we only construct the blow-up solution for one initial data $(u_0, u_1)$ in Theorem 1.1. We believe that the blow-up should hold for a large class of initial data, just as in [Reference Merle, Raphaël, Rodnianski and Szeftel51]. This is left to the future work.

  4. 4. To prove Corollary 1.2, we just need to verify Assumption 1, which is related to the existence of a smooth global solution to a specific ODE (2.8). If $d=4, p\geq 29$ or $d=5, p\geq 17$ , Assumption 1 is verified in our companion paper [Reference Shao, Wei and Zhang65]. As a consequence, if one can find some other methods to verify Assumption 1 for smaller p, then one can also get the blow-up for that smaller p. The case $d>5$ follows from the result for $d=5$ and truncation, see Subsection 2.5.

  5. 5. We emphasize that if Assumption 1 is valid, then we must have $d>\beta (\ell +\sqrt \ell )+1$ , where $\ell :=1+4/(p-1)>1$ . Using $\beta>1$ , we get $d>3$ . As a result, the case of $d=3$ is not amenable to our analysis at present, and the existence of blow-up solutions for $d=3$ remains open. We point out that similar situation happens in [Reference Merle, Raphaël, Rodnianski and Szeftel51], where the construction fails for $3$ -D and $4$ -D defocusing supercritical NLS.

  6. 6. In this work, we can only construct the blow-up for the complex-valued solution. The blow-up for the scalar defocusing supercritical wave equation remains open at this point. We guess that the same blow-up result should hold for the scalar nonlinear wave equation, at least for $(d,p)$ satisfying the same hypothesis as in Theorem 1.1.

  7. 7. In this paper, we initiate our exploration of complex-valued blow-up solutions by employing the modulus-phase decomposition $u=w \mathrm {e}^{\mathrm {i}\Phi }$ , as detailed in Section 2. For the $\mathbb {R}$ -valued problem, an analogous approach appears promising. Specifically, we propose a decomposition of the form $u=wf(\Phi )$ , where $f:\mathbb {R}\to \mathbb {R}^+$ is an unknown real-valued function. Under this framework, the problem reduces to solving the following system of equations:

    $$\begin{align*}2\partial^\alpha w\partial_\alpha\Phi+w\Box \Phi=0,\quad f"(\Phi)=-f(\Phi)^p,\quad \Box w=f(\Phi)^{p-1}\left(w^p+w\partial^\alpha\Phi\partial_\alpha\Phi\right).\end{align*}$$
    When compared with (2.1), the above system exhibits a significantly higher level of complexity. Consequently, its thorough investigation is deferred to future work.

The road map of the proof of Theorem 1.1 and Corollary 1.2 can be found in Section 2. The proof is based on Propositions 2.4, 2.7 and 2.8. Our starting point is to introduce a front re-normalization (2.2), relying on a constant $b>0$ ; taking the limit $b\to 0$ , the defocusing wave equation becomes the relativistic compressible Euler equations.

We first write the desired solution to (1.1) in the form of a power series (see (2.4)) with respect to the constant $b>0$ . The non-degeneracy of the leading order approximation allows us to solve all high-order approximations $(\rho _n, \phi _n)$ , which is exactly the purpose of Proposition 2.4. The proof of Proposition 2.4 is rather technical and can be found in Section 5. One of the key ingredients used is the existence of smooth solutions to the second order ODEs having singular points with a parameter $\lambda $ , see Appendix B.

Since we do not have enough information on $(\rho _n, \phi _n)$ , especially the estimate uniform in n, we may not have the convergence of the formal series (2.4). To overcome this drawback, we truncate $(\rho _n, \phi _n)$ in the form of (2.23), and in Proposition 2.7 we prove that the truncated solution is a good approximate solution to the defocusing wave equation. The proof of Proposition 2.7 can be found in Section 3.

Finally, we construct a solution to (1.1) near the truncated approximation solution. This is exactly what Proposition 2.8 says. The proof of Proposition 2.8 can be found in Section 4, where we use the energy method to solve the wave equation in a time-backward direction, and we need to use a technical truncation to avoid the singularity at blow-up time. Such method of solving backward in time has been used in [Reference Krieger, Schlag and Tataru41, Reference Krieger, Schlag and Tataru42, Reference Perelman60]. Let’s emphasize that this part does not depend at all on our method of constructing the approximate solutions, and it includes the case $d=3$ and does not require Assumption 1 or the spherical symmetry of the approximate solutions either.

1.2 Blow-up phenomenon for related models

Let’s review some important results on the blow-up for other related equations.

It is more common to observe the blow-up phenomenon for the focusing nonlinear wave equation, i.e.,

(1.4) $$ \begin{align} \Box u=-|u|^{p-1}u. \end{align} $$

In fact, the spatial independent function $u(t)=C_p(T-t)^{-2/(p-1)}$ , where $C_p^{p-1}=2(p+1)/(p-1)^2$ , gives a blow-up solution to (1.4). This ODE-type solution can be further truncated to a smooth compactly supported blow-up solution to (1.4) by using the finite speed of propagation [Reference Alinhac1, Reference John30, Reference Levine43]. We will use similar ideas to prove Corollary 1.2 for the case $d>5$ . See also [Reference Donninger12, Reference Duyckaerts, Kenig and Merle15, Reference Duyckaerts, Kenig and Merle16, Reference Duyckaerts and Yang18, Reference Jendrej27, Reference Kenig32, Reference Kenig and Merle34, Reference Krieger and Schlag40, Reference Krieger, Schlag and Tataru42, Reference Martel, Yvan and Merle46, Reference Merle and Zaag58] for the construction and classification of blow-up (or global) solutions as well as recent breakthrough [Reference Duyckaerts, Jia, Kenig and Merle13, Reference Duyckaerts, Kenig and Merle17, Reference Jendrej and Lawrie29] on the soliton resolution conjecture.

Other related models such as the nonlinear Schrödinger equation, see [Reference Kenig and Merle33, Reference Merle and Raphaël48, Reference Merle, Raphaël and Rodnianski50, Reference Merle, Raphaël, Rodnianski and Szeftel51, Reference Merle, Raphaël and Szeftel54, Reference Merle, Raphaël and Szeftel55, Reference Perelman59, Reference Perelman61]; see [Reference Duyckaerts, Jia, Kenig and Merle14, Reference Krieger and Miao38, Reference Krieger, Miao and Schlag39, Reference Krieger, Schlag and Tataru41, Reference Raphaël and Rodnianski62, Reference Rodnianski and Sterbenz64] for the wave map; see [Reference Merle, Raphaël and Rodnianski49, Reference Perelman60] for the Schrödinger maps; see [Reference Collot, Raphaël and Szeftel7, Reference Collot, Raphaël and Szeftel8, Reference Cortázar, del Pino and Musso9, Reference del Pino, Musso and Wei11, Reference Harada23, Reference Matano and Merle47, Reference Merle, Raphaël and Szeftel56, Reference Merle and Zaag57] for the semilinear heat equation and [Reference Dávila, del Pino and Wei10, Reference Jendrej and Lawrie28, Reference Kim and Merle37, Reference Raphaël and Schweyer63] for the harmonic heat flow.

1.3 Notations and conventions

Unless stated otherwise, we adopt the following notations, abbreviations, and conventions:

  • Constants: $\mathrm i=\sqrt {-1}$ is the imaginary unit, $\mathrm e$ is the base of the natural logarithm.

  • For any $a\in \mathbb {R}$ , we denote $\mathbb {Z}_{\geq a}:=\mathbb {Z}\cap [a, +\infty )$ and $\mathbb {Z}_{>a}:=\mathbb {Z}\cap (a, +\infty )$ . Moreover, we denote $\mathbb {Z}_+:=\mathbb {Z}_{\geq 1}$ . Similarly, $\mathbb {R}_{\geq 0}:=\mathbb {R}\cap [0, +\infty )$ .

  • Greek indices run from $0$ to d, where $d\in \mathbb {Z}_{\geq 2}$ is the spatial dimension, Latin indices run from $1$ to d, and we use the Einstein’s summation convention: repeated indices appearing once upstairs and once downstairs are summed over their range.

  • $(t,x)=(t, x_1, \cdots , x_d)$ denotes coordinates in spacetime, $r=|x|=(\sum _{j=1}^dx_j^2)^{1/2}$ . We write $\partial _{0}=-\partial ^0=\partial _t=\frac {\partial }{\partial t}$ , $\partial _{j}=\partial ^j=\partial _{x_j}=\frac {\partial }{\partial x_j}$ for $j\in \mathbb {Z}\cap [1,d]$ , $\Box =\partial ^\alpha \partial _\alpha =-\partial _t^2+\sum _{j=1}^d\partial _j^2$ and $\Delta =\sum _{j=1}^d\partial _j^2$ , then $\Box =-\partial _t^2+\Delta $ .

  • We denote $\ell :=1+4/(p-1)>1$ , $k:=d-1\in \mathbb {Z}_{+}$ and $\gamma :=4\beta /(p-1)+2=\beta (\ell -1)+2$ .

  • For a (vector-valued) differentiable function $f=f(t,x)$ , we denote

    $$ \begin{align*} Df:=(\partial_tf, \partial_1f, \partial_2f,\cdots, \partial_df) &\quad\text{and}\quad D_xf:=(\partial_1f, \partial_2f,\cdots, \partial_df)=\nabla_x f, \end{align*} $$
    and $|Df|:=(|\partial _tf|^2+\sum _{j=1}^d|\partial _jf|^2)^{1/2}$ , $|D_xf|:=(\sum _{j=1}^d|\partial _jf|^2)^{1/2}$ . For all $j\in \mathbb {Z}_{+}$ we denote $D^jf:=DD^{j-1}f$ , $D_x^jf:=D_xD_x^{j-1}f$ , $D^0f=D_x^0f=f$ , noting that $D^{j-1}f$ and $D_x^{j-1}f$ are again vector-valued functions; moreover, $D^{\leq 1}f:=(f, Df)$ .
  • For $(t,x)\in [0, T)\times \mathbb {R}^d$ , we let $\tau :=-\ln (T-t)$ and $Z:=|x|/(T-t)\in [0, +\infty )$ .

  • For $N\geq 0$ , $H_x^N$ denotes the inhomogeneous Sobolev space with the norm $\|\cdot \|_{H_x^N}$ with respect to the spatial variables and $\dot H_x^N$ denotes the homogeneous Sobolev space with the norm $\|\cdot \|_{\dot H_x^N}$ . Moreover, we denote $L_x^2:=H_x^0$ .

  • A function space is a linear vector space if it is closed under addition and multiplication by a constant. A function space is a ring (algebra) if it contains all the constant functions and is closed under addition and multiplication. Then a ring is also a linear vector space.

2 A roadmap of the proof

We introduce the modulus-phase decomposition $u=w \mathrm {e}^{\mathrm {i}\Phi }$ , with $w:\mathbb {R}^{1+d}\to \mathbb {R}_{>0}$ and $\Phi :\mathbb {R}^{1+d}\to \mathbb {R}$ . Then

$$ \begin{align*} \Box u=(\Box w+2\mathrm{i}\partial^{\alpha} w\partial_{\alpha}\Phi+\mathrm{i} w\Box\Phi- w\partial^{\alpha}\Phi\partial_{\alpha}\Phi)\mathrm{e}^{\mathrm{i}\Phi}, \end{align*} $$

and (1.1) becomes

(2.1) $$ \begin{align} \Box w= w^{p}+ w\partial^{\alpha}\Phi\partial_{\alpha}\Phi,\qquad 2\partial^{\alpha} w\partial_{\alpha}\Phi+ w\Box\Phi=0. \end{align} $$

Let $b>0$ be a positive constant. We re-normalize according to

(2.2) $$ \begin{align} w(t,x)=b^{-\frac{1}{p-1}}\rho(t,x),\qquad \Phi(t,x)=b^{-\frac{1}{2}}\phi(t,x), \end{align} $$

then (2.1) becomes

(2.3) $$ \begin{align} b\Box\rho=\rho^{p}+\rho\partial^{\alpha}\phi\partial_{\alpha}\phi,\qquad 2\partial^{\alpha}\rho\partial_{\alpha}\phi+\rho\Box\phi=0. \end{align} $$

We seek solutions $(\rho , \phi )$ to (2.3) in the form of

(2.4) $$ \begin{align} \rho(t,x)=\sum_{n=0}^{\infty}\rho_n(t,x) b^n,\qquad \phi(t,x)=\sum_{n=0}^\infty \phi_n(t,x)b^n. \end{align} $$

Plugging (2.4) into (2.3), we obtain the following recurrence relation for $n\in \mathbb {Z}_{\geq 0}$ :

(2.5) $$ \begin{align} \begin{aligned} \Box\rho_{n-1}&=\sum_{n_1+n_2+\cdots+n_p=n}\rho_{n_1}\rho_{n_2}\cdots\rho_{n_p}+ \sum_{n_1+n_2+n_3=n}\rho_{n_1}\partial^\alpha\phi_{n_2}\partial_\alpha\phi_{n_3},\\ 0&=2\sum_{n_1+n_2=n}\partial^\alpha\rho_{n_1}\partial_\alpha\phi_{n_2}+\sum_{n_1+n_2=n}\rho_{n_1}\Box\phi_{n_2}, \end{aligned} \end{align} $$

where we have used the convention that $\rho _{-n'}=\phi _{-n'}=0$ for all $n'\in \mathbb {Z}_{+}$ . Here (2.4) is only a formal expansion and we will use cutoff functions to construct approximate solutions. Here b plays the role of deriving recurrence relation (2.5), the smallness lies in the functions $\rho _n,\phi _n$ as $t\uparrow T$ rather than $b^n$ . We will not let $b\downarrow 0$ , in fact, we will fix $b=1$ .

2.1 The leading order term of the blow-up solution

Letting $n=0$ in (2.5), we know that $(\rho _0,\phi _0)$ satisfies the systemFootnote 3

(2.6) $$ \begin{align} \rho_0^{p}+\rho_0\partial^{\alpha}\phi_0\partial_{\alpha}\phi_0=0,\qquad 2\partial^{\alpha}\rho_0\partial_{\alpha}\phi_0+\rho_0\Box\phi_0=0. \end{align} $$

For any $\beta>1$ , the system (2.6) is invariant under the scaling

$$\begin{align*}\phi_{0,\lambda}(t, x)=\lambda^{\beta-1}\phi_0(\lambda t, \lambda x),\qquad \rho_{0,\lambda}(t, x)=\lambda^{\frac{2\beta}{p-1}}\rho_0(\lambda t, \lambda x),\qquad\forall\ \lambda>0.\end{align*}$$

We seek radially symmetric self-similar blow-up solutions to (2.6) of the form

(2.7) $$ \begin{align} \phi_0(t, r)=(T-t)^{1-\beta}\widehat \phi_0(Z),\quad \rho_0(t,x)=(T-t)^{-\frac{2\beta}{p-1}}\widehat\rho_0(Z),\quad Z=\frac{r}{T-t}, \quad r=|x|, \end{align} $$

where $T>0$ is the blow-up time and $\beta>1$ is a constant.Footnote 4 Let $v=\partial _r\phi _0/\partial _t\phi _0$ , then $v=v(Z)$ solves the ODEFootnote 5

(2.8) $$ \begin{align} &\Delta_Z(Z,v){\mathrm dv}/{\mathrm dZ}=\Delta_v(Z,v),\quad \Delta_v(Z,v):=(1-v^2)[\beta\ell(1-v^2)Z-kv(1-Zv)], \nonumber\\ &\Delta_Z(Z,v):=Z\left[(1-Zv)^2-\ell(v-Z)^2\right],\end{align} $$

where $\ell :=1+4/(p-1)>1$ and $k:=d-1\in \mathbb {Z}_{\geq 1}$ . See Subsection A.1 for the derivation of (2.8).

Recall the following fact from [Reference Shao, Wei and Zhang65] (recalling footnote 4).

Lemma 2.1 ([Reference Shao, Wei and Zhang65], Lemma 2.1).

If $v(Z):[0,1]\to (-1,1)$ is a $C^1$ solution to (2.8) with $v(0)=0$ and $\ell>1, \beta >0, k>0$ , then $k>\beta (\ell +\sqrt {\ell })$ .

As a consequence, it is natural to restrict the parameters $(k,\ell ,\beta )$ in the following range:

(2.9) $$ \begin{align} \beta>1,\qquad \ell>1, \qquad k\in \mathbb{Z}\cap[3,+\infty),\qquad k>\beta(\ell+\sqrt{\ell}). \end{align} $$

Assumption 1. There exists a smooth function $v=v(Z)\in (-1,1)$ defined on $Z\in [0,+\infty )$ solving the ODE (2.8) with $v(0)=0$ and $v\in C_{\operatorname {o}}^{\infty }([0,+\infty ))$ .

Here we define (with $\mathbb {R}_{\geq 0}:=[0,+\infty ) $ )

(2.10) $$ \begin{align} C_{\text{e}}^\infty(\mathbb{R}_{\geq0}) &:=\left\{f\in C^\infty(\mathbb{R}_{\geq0}): \exists\ \widetilde f\in C^\infty(\mathbb{R}_{\geq0})\ \text{s.t. }f(Z)=\widetilde f(Z^2)\ \forall\ Z\in\mathbb{R}_{\geq0}\right\}, \end{align} $$
(2.11) $$ \begin{align} C_{\text{o}}^\infty(\mathbb{R}_{\geq0}) &:=\left\{f\in C^\infty(\mathbb{R}_{\geq0}): \exists\ \widetilde f\in C^\infty(\mathbb{R}_{\geq0})\ \text{s.t. }f(Z)=Z\widetilde f(Z^2)\ \forall\ Z\in\mathbb{R}_{\geq0}\right\}. \end{align} $$

Then $C_{\text {e}}^\infty (\mathbb {R}_{\geq 0}) $ is a ring and $C_{\text {o}}^\infty (\mathbb {R}_{\geq 0}) $ is a linear vector space.

Remark 2.2. Under Assumption 1 and (2.9), we can show that the solution $v(Z)$ satisfies

  • $v(Z)<Z$ and $Zv(Z)<1$ for all $Z\in (0,+\infty )$ .

  • $\Delta _Z(Z, v(Z))>0$ for $Z\in (0, Z_1)$ and $\Delta _Z(Z, v(Z))<0$ for $Z\in (Z_1, +\infty )$ , where $Z_1=\frac {k}{\sqrt {\ell }(k-\beta (\ell -1))}>0$ .

  • Let $\Delta _0(Z):=\Delta _Z(Z, v(Z))$ for $Z\in [0,+\infty )$ , then $\Delta _0'(Z_1)\neq 0$ .

See Subsection A.2 for the proof.

In view of Assumption 1, we can define that for $Z\in [0,+\infty )$

(2.12) $$ \begin{align} \begin{aligned} \widehat\phi_0(Z)&:=\frac1{\beta-1}\exp\left((\beta-1)\int_0^Z\frac{v(s)}{1-sv(s)}\,\mathrm ds\right),\\\widehat\rho_0(Z)&:=\frac{(\beta-1)^{\frac{2}{p-1}}\widehat\phi_0(Z)^{\frac{2}{p-1}}(1-v(Z)^2)^{\frac{1}{p-1}}} {(1-Zv(Z))^{\frac{2}{p-1}}}. \end{aligned} \end{align} $$

Then $\widehat \phi _0(Z)>0$ , $\widehat \rho _0(0)=1$ and $\widehat \rho _0(Z)>0$ for all $Z\in [0,+\infty )$ . As a consequence, $(\phi _0, \rho _0)$ defined by (2.7) solves (2.6) (see Lemma A.1), and $\widehat \phi _0,\ \widehat \rho _0\in C_{\text {e}}^\infty ([0,+\infty )) $ (see Lemma A.6). This is the leading order term of our blow-up solution $(\rho ,\phi )$ to (2.3).

2.2 Solving $(\rho _n, \phi _n)$ for $n\in \mathbb {Z}_{\geq 1}$

In Subsection 2.1, under Assumption 1, we construct the leading order blow-up solution $(\rho _0, \phi _0)$ . In view of the expansion (2.4), we construct $(\rho _n, \phi _n)$ for $n\in \mathbb {Z}_{\geq 1}$ . We rewrite the recurrence relation (2.5) for $n\in \mathbb {Z}_{\geq 1}$ as

(2.13) $$ \begin{align} \notag&(p\rho_0^{p-1}+\partial^\alpha\phi_0\partial_\alpha\phi_0)\rho_n+2\rho_0\partial^\alpha\phi_0\partial_\alpha\phi_n\\& \quad =\Box\rho_{n-1} -\sum_{\substack{n_1+\cdots+n_p=n\\ n_1, \cdots, n_p\leq n-1}}\rho_{n_1}\cdots\rho_{n_p}-\sum_{\substack{n_1+n_2+n_3=n\\ n_1, n_2, n_3\leq n-1}}\rho_{n_1}\partial^\alpha\phi_{n_2}\partial_\alpha\phi_{n_3}=:F_n,\end{align} $$
(2.14) $$ \begin{align} \notag&\rho_0\Box\phi_n+2\partial^\alpha\rho_0\partial_\alpha\phi_n+2\partial^\alpha\phi_0\partial_\alpha\rho_n+\Box\phi_0\rho_n \\&\quad =-2\sum_{\substack{n_1+n_2=n\\ n_1, n_2\leq n-1}}\partial^\alpha\rho_{n_1}\partial_\alpha\phi_{n_2}-\sum_{\substack{n_1+n_2=n\\ n_1, n_2\leq n-1}}\rho_{n_1}\Box\phi_{n_2}=:G_n. \end{align} $$

Using the equations for $(\rho _0, \phi _0)$ given by (2.6), the above recurrence relation becomes

(2.15) $$ \begin{align} &(p-1)\rho_0^{p-1}\rho_n+2\rho_0\partial^\alpha\phi_0\partial_\alpha\phi_n=F_n, \end{align} $$
(2.16) $$ \begin{align} &\partial^\alpha(\rho_0^2\partial_\alpha\phi_n)+2\partial^\alpha(\rho_0\partial_\alpha\phi_0\rho_n)=\rho_0 G_n. \end{align} $$

By (2.15), we have

(2.17) $$ \begin{align} \rho_n=\frac{\rho_0^{1-p}F_n}{p-1}-\frac2{p-1}\rho_0^{2-p}\partial^\alpha\phi_0\partial_\alpha\phi_n. \end{align} $$

Substituting the above identity into (2.16), we obtain the following linear equation for $\phi _n$ :

(2.18) $$ \begin{align} \partial^\alpha\left(\rho_0^2\partial_\alpha\phi_n-\frac4{p-1}\rho_0^{3-p}\partial_\alpha\phi_0\partial^{\widetilde\alpha}\phi_0\partial_{\widetilde\alpha}\phi_n\right)=\rho_0G_n-\frac2{p-1}\partial^\alpha\left(\rho_0^{2-p}\partial_\alpha\phi_0 F_n\right)=:H_n. \end{align} $$

We introduce the linearized operator

(2.19) $$ \begin{align} \mathscr{L}(\phi):=\partial^\alpha\left(\rho_0^2\partial_\alpha\phi-\frac4{p-1}\rho_0^{3-p}\partial_\alpha\phi_0\partial^{\widetilde\alpha}\phi_0\partial_{\widetilde\alpha}\phi\right),\qquad \phi=\phi(t,x)=\phi(t,r). \end{align} $$

Then our aim is to solve inductively $\mathscr {L}(\phi _n)=H_n$ for each $n\geq 1$ .

Indeed, we can show that $\mathscr {L}$ is surjective in some well-chosen functional spaces and then we solve $\mathscr {L}(\phi _n)=H_n$ in these spaces. Letting $\tau =\ln \frac 1{T-t}$ , we define (here $C_{\text {e}}^\infty ([0,+\infty )) $ is defined in (2.10))

(2.20) $$ \begin{align} {\mathscr X}_0&:=\left\{f(t,x)=\sum_{j=0}^n f_j(Z)\tau^j: n\in\mathbb{Z}_{\geq 0}, f_j\in C_{\text{e}}^\infty([0,+\infty))\ \ \forall\ j\in\mathbb{Z}\cap[0,n]\right\}, \end{align} $$
(2.21) $$ \begin{align} {\mathscr X}_\lambda:=&(T-t)^\lambda{\mathscr X}_0=\left\{f(t,x)=(T-t)^\lambda g(t,x)=\mathrm e^{-\lambda\tau}g(t,x):g\in{\mathscr X}_0\right\},\quad \forall\ \lambda\in\mathbb{C}. \end{align} $$

Then ${\mathscr X}_0$ is a ring (using that $\{f(t,x)=f(Z)\tau ^j:f_j\in C_{\text {e}}^\infty ([0,+\infty )),\ j\in \mathbb {Z}_{\geq 0}\} $ is closed under multiplication) and ${\mathscr X}_\lambda $ is a linear vector space.

We have the following properties for the functional spaces ${\mathscr X}_\lambda $ .

Lemma 2.3.

  1. (i) Let $\lambda ,\mu \in \mathbb {C}$ , $f\in {\mathscr X}_\lambda , g\in {\mathscr X}_\mu $ . Then $\partial _t f\in {\mathscr X}_{\lambda -1}$ , $\Delta f\in {\mathscr X}_{\lambda -2}$ , $\Box f\in {\mathscr X}_{\lambda -2}$ , $fg\in {\mathscr X}_{\lambda +\mu }$ , $\partial ^\alpha f\partial _\alpha g\in {\mathscr X}_{\lambda +\mu -2}$ , and $\partial ^\alpha (f\partial _\alpha g)\in {\mathscr X}_{\lambda +\mu -2}$ .

  2. (ii) Let $\lambda ,\mu \in \mathbb {R}$ and $j\in \mathbb {Z}_{\geq 0}$ be such that $\lambda>j+\mu $ . If $f\in {\mathscr X}_\lambda $ , then $(T-t)^{-\mu }D^jf\in L^\infty (\mathcal C)$ , where $\mathcal C$ is the light cone $\mathcal C:=\left \{(t,x)\in [0, T)\times \mathbb {R}^d: |x|<2(T-t)\right \}$ .

  3. (iii) Let $\lambda ,\mu \in \mathbb {R}$ and $j\in \mathbb {Z}_{\geq 0}$ be such that $\lambda \geq j+\mu $ . If $f(t,x)=(T-t)^\lambda \widehat f(Z)$ for some $\widehat f\in C_{\operatorname {e}}^\infty ([0,+\infty ))$ , then $(T-t)^{-\mu }D^jf\in L^\infty (\mathcal C)$ .

The proof of Lemma 2.3 can be found in Subsection 5.1.

Proposition 2.4. The linear operator $\mathscr {L}:{\mathscr X}_{\lambda }\to {\mathscr X}_{\lambda -\gamma }$ is surjective for all $\lambda \in \mathbb {C}$ , where $\gamma :=4\beta /(p-1)+2=\beta (\ell -1)+2$ .

See Section 5 for the proof of Proposition 2.4.

Let

(2.22) $$ \begin{align} \lambda_n:=(2n-1)(\beta-1),\qquad\mu_n:=2n(\beta-1)-\frac{2\beta}{p-1},\qquad\forall\ n\in\mathbb{Z}_{\geq 0}. \end{align} $$

Recall from (2.7) that

$$\begin{align*}\phi_0(t,r)=(T-t)^{\lambda_0}\widehat\phi_0(Z),\qquad \rho_0(t,r)=(T-t)^{\mu_0}\widehat\rho_0(Z).\end{align*}$$

As $\widehat \phi _0(Z), \widehat \rho _0(Z)\in C_{\text {e}}^\infty ([0,+\infty ))$ , by (2.20), (2.21) we have $\phi _0\in {\mathscr X}_{\lambda _0}$ and $\rho _0\in {\mathscr X}_{\mu _0}$ . Similarly, for $ a\in \mathbb {R}$ we have $\rho _0(t,r)^a=(T-t)^{a\mu _0}\widehat \rho _0(Z)^a $ and $\widehat \rho _0(Z)^a\in C_{\text {e}}^\infty ([0,+\infty ))$ , then $\rho _0^a\in {\mathscr X}_{a\mu _0}$ . Moreover, $\widehat \phi _0(Z), \widehat \rho _0(Z)$ are real-valued, so are $\phi _0$ , $\rho _0$ .

Lemma 2.5. Assume that $n\in \mathbb {Z}_{\geq 1}$ , $\phi _j\in {\mathscr X}_{\lambda _j}$ , $\rho _j\in {\mathscr X}_{\mu _j}$ are are real-valued for $j\in \mathbb {Z}\cap [0, n-1]$ . Let $F_n$ , $G_n$ be defined in (2.13), (2.14). Then there exist real-valued $\phi _n\in {\mathscr X}_{\lambda _n}$ and $\rho _n\in {\mathscr X}_{\mu _n}$ such that (2.15) and (2.16) hold.

Proof. By Lemma 2.3 (i) and the definition of $F_n$ , we have $F_n\in {\mathscr X}_{\mu _{n-1}-2}$ , where we have used the fact that $\mu _{n_1}+\cdots +\mu _{n_p}=\mu _{n-1}-2$ if $n_1+\cdots +n_p=n$ and $\mu _{n_1}+\lambda _{n_2}+\lambda _{n_3}-2=\mu _{n-1}-2$ if $n_1+n_2+n_3=n$ . Using Lemma 2.3 (i) and the definition of $G_n$ , we have $G_n\in {\mathscr X}_{(2n-1)(\beta -1)-2\beta /(p-1)-2}$ , where we have used the fact that $\mu _{n_1}+\lambda _{n_2}-2=(2n-1)(\beta -1)-2\beta /(p-1)-2$ if $n_1+n_2=n$ . It follows from Lemma 2.3 (i) that $\rho _0G_n\in {\mathscr X}_{\mu _0+(2n-1)(\beta -1)-2\beta /(p-1)-2}={\mathscr X}_{2n(\beta -1)-\beta \ell -1}$ (recall that $\rho _0\in {\mathscr X}_{\mu _0}$ , $\ell =1+\frac {4}{p-1}$ ). Since $\rho _0^{2-p}\in {\mathscr X}_{(2-p)\mu _0}$ , $F_n\in {\mathscr X}_{\mu _{n-1}-2}$ , by Lemma 2.3 (i) we get $\rho _0^{2-p}F_n\in {\mathscr X}_{\mu _{n-1}-2+(2-p)\mu _0}$ , then by $\phi _0\in {\mathscr X}_{\lambda _0}$ we have

$$\begin{align*}\partial^\alpha\left(\rho_0^{2-p}\partial_\alpha\phi_0 F_n\right)\in {\mathscr X}_{\mu_{n-1}-2+(2-p)\mu_0+\lambda_0-2}={\mathscr X}_{2n(\beta-1)-\beta\ell-1}.\end{align*}$$

Hence by the definition of $H_n$ in (2.18), we have $H_n\in {\mathscr X}_{2n(\beta -1)-\beta \ell -1}={\mathscr X}_{\lambda _n-\gamma }$ (recall that $\gamma =\beta (\ell -1)+2$ ). Moreover, $F_n, G_n, H_n$ are real-valued.

By Proposition 2.4, there exists (real-valued) $\phi _n\in {\mathscr X}_{\lambda _n}$ such that $\mathscr {L}(\phi _n)=H_n$ (otherwise take $\operatorname {Re}\phi _n$ ), then (2.18) holds. Let $\rho _n$ be defined by (2.17). Then $\rho _n$ is real-valued. Moreover, using (i) of Lemma 2.3, $\rho _0^{1-p}\in {\mathscr X}_{(1-p)\mu _0}$ , $\rho _0^{2-p}\in {\mathscr X}_{(2-p)\mu _0}$ , $F_n\in {\mathscr X}_{\mu _{n-1}-2}$ , $\phi _0\in {\mathscr X}_{\lambda _0}$ and $\phi _n\in {\mathscr X}_{\lambda _n}$ , we have

$$ \begin{align*}\rho_0^{1-p}F_n\in{\mathscr X}_{\mu_0(1-p)+\mu_{n-1}-2}={\mathscr X}_{\mu_n},\quad \rho_0^{2-p}\partial^\alpha\phi_0\partial_\alpha\phi_n\in{\mathscr X}_{\mu_0(2-p)+\lambda_0+\lambda_n-2}={\mathscr X}_{\mu_n},\end{align*} $$

hence $\rho _n\in {\mathscr X}_{\mu _n}$ . Now (2.15) follows from (2.17), and (2.16) follows from (2.17) and (2.18).

As $\phi _0\in {\mathscr X}_{\lambda _0}$ , $\rho _0\in {\mathscr X}_{\mu _0}$ and $\phi _0$ , $\rho _0$ are real-valued, by Lemma 2.5 and the induction, we have the following result.

Proposition 2.6. Let $\phi _0$ , $\rho _0$ be defined in (2.7). For each $n\in \mathbb {Z}_{\geq 1}$ , there exist real-valued $\phi _n\in {\mathscr X}_{\lambda _n}$ and $\rho _n\in {\mathscr X}_{\mu _n}$ such that (2.15) and (2.16) hold with $F_n$ , $G_n$ defined in (2.13), (2.14). Hence, (2.5) holds for $n\in \mathbb {Z}_{\geq 0}$ .

Now we briefly explain the ideas in the proof of Proposition 2.4. In the proof of Lemma 2.5, we see that we only need to use the surjectivity of $\mathscr {L}$ from ${\mathscr X}_{\lambda }$ to ${\mathscr X}_{\lambda -\gamma }$ for $\lambda \in \{\lambda _n:n\in \mathbb {Z}_{+}\}$ . However, this is not easy to solve the equation $\mathscr {L} f=g$ for $f\in {\mathscr X}_{\lambda }$ even in the simplest case $g=(T-t)^{\lambda -\gamma }\widehat g(Z)\in {\mathscr X}_{\lambda -\gamma }$ for some $\widehat g\in C_{\text {e}}^\infty ([0, +\infty ))$ (without the logarithm correction $\tau ^j$ for $j\in \mathbb {Z}_{+}$ ), in which process we need to check a non-degenerate property (nonzero of Wronski defined in (5.32)) on the coefficients of $\mathscr {L}_\lambda $ (defined in (5.23)), and it is difficult to check that all $\lambda _n$ satisfy the non-degenerate property, even for one $\lambda _{n_0}$ . To overcome this drawback, we solve the equation for all $\lambda \in \mathbb {C}$ , not merely for those $\lambda \in \{\lambda _n:n\in \mathbb {Z}_{+}\}$ . It turns out that the non-degenerate property holds for all but countably many $\lambda \in \mathbb {C}$ (these $\lambda $ are “bad” in some sense) and the solution depends analytically on $\lambda $ . In this way, we can show that the solution $f=f(\cdot ;\lambda )$ is a meromorphic function on $\lambda $ . For those countably many $\lambda \in \mathbb {C}$ not satisfying the non-degenerate property, the analytic property of f allows us to introduce a logarithm correction to solve the corresponding equation for “bad” $\lambda $ . See Section 5 and Appendix B for details.

2.3 The approximate solution

Let $(\rho _n,\phi _n)\in {\mathscr X}_{\mu _n}\times {\mathscr X}_{\lambda _n} (n\in \mathbb {Z}_{\geq 0})$ be given by Proposition 2.6. We fix a bump function $\eta \in C_c^\infty (\mathbb {R};[0,1])$ such that $\eta |_{[0,1]}=1$ and $\eta |_{[2,+\infty )}=0$ .

Proposition 2.7. Let $T=b=1$ . There exist $N_0\in \mathbb {Z}_+$ , $c_0\in (0,T)$ and a sequence $\{T_n\}_{n\geq 0}$ such that $T_n=T$ for $0\leq n<N_0$ , $0<T_n\leq T_{n-1}/4$ for all $n\geq N_0$ , and for functionsFootnote 6

(2.23) $$ \begin{align} \rho_*(t,x)&:=\sum_{n=0}^\infty \eta\left(\frac{T-t}{T_n}\right)\rho_n(t,x)b^n,\quad \phi_*(t,x):=\sum_{n=0}^\infty \eta\left(\frac{T-t}{T_n}\right)\phi_n(t,x)b^n,\end{align} $$
(2.24) $$ \begin{align} E_*&:=\rho_*^p+\rho_*\partial^\alpha\phi_*\partial_\alpha\phi_*-b\Box\rho_*,\quad J_*:=2\partial^\alpha\rho_*\partial_\alpha\phi_*+\rho_*\Box\phi_*, \end{align} $$

defined on $(t,x)\in [0,T)\times \mathbb {R}^d$ we have

(2.25) $$ \begin{align} &(T-t)^{\beta+j-1}D^j\phi_*\in L^\infty(\mathcal C),\quad (T-t)^{\frac{2\beta}{p-1}+j}D^j\rho_*\in L^\infty(\mathcal C),\qquad\forall\ j\in\mathbb{Z}_{\geq 0}, \end{align} $$
(2.26) $$ \begin{align} &(T-t)^{\beta}(\partial_t\phi_*-|D_x\phi_*|)\geq c_0,\quad (T-t)^{\frac{2\beta}{p-1}}\rho_*\geq c_0,\quad \forall\ (t,x)\in\mathcal C,\ T-t<c_0, \end{align} $$
(2.27) $$ \begin{align} &(T-t)^{-\lambda}(D^jE_*, D^jJ_*)\in L^\infty(\mathcal C),\qquad\forall\ \lambda>0,\ \forall\ j\in\mathbb{Z}_{\geq 0}. \end{align} $$

See Section 3 for the proof of Proposition 2.7.

2.4 Solving nonlinear wave equation

Proposition 2.8. Assume that $T=1$ , $w_*\in C^\infty (\mathcal C),\Phi _*\in C^\infty (\mathcal C)$ satisfy

(2.28) $$ \begin{align} &(T-t)^{\beta+j-1}D^j\Phi_*\in L^\infty(\mathcal C),\quad (T-t)^{\frac{2\beta}{p-1}+j}D^jw_*\in L^\infty(\mathcal C),\qquad\forall\ j\in\mathbb{Z}_{\geq 0}, \end{align} $$
(2.29) $$ \begin{align} &(T-t)^{\beta}(\partial_t\Phi_*-|D_x\Phi_*|)\geq c_0,\quad (T-t)^{\frac{2\beta}{p-1}}w_*\geq c_0,\quad \forall\ (t,x)\in\mathcal C,\ T-t<c_0, \end{align} $$

for some $c_0\in (0,T)=(0,1)$ . Suppose that (2.27) holds for $E_*$ , $J_*$ defined as

(2.30) $$ \begin{align} E_*&:=w_*^p+w_*\partial^\alpha\Phi_*\partial_\alpha\Phi_*-\Box w_*,\quad J_*:=2\partial^\alpha w_*\partial_\alpha\Phi_*+w_*\Box\Phi_*, \end{align} $$

Then there exist $c_1\in (0,c_0)$ and $u\in C^2((T-c_1,T)\times \mathbb {R}^d;\mathbb {C})$ such that $u(t,\cdot ),\partial _tu(t,\cdot )\in C_c^{\infty }(\mathbb {R}^d;\mathbb {C})$ for $t\in (T-c_1,T),$ $\Box u=|u|^{p-1}u$ for $t\in (T-c_1,T), |x|\leq T-t$ , and

(2.31) $$ \begin{align} C^{-1}(T-t)^{-\frac{2\beta}{p-1}}\leq |u(t,x)|\leq C(T-t)^{-\frac{2\beta}{p-1}},\qquad\forall\ t\in(T-c_1,T),\ |x|\leq T-t \end{align} $$

for some constant $C>0$ .

In fact, $u=(1+h) w_*\mathrm e^{\text {i}\Phi _*}$ , $h=O((T-t)^{\lambda })$ , $ \forall \ \lambda>0$ . To prove Proposition 2.8, it suffices to solve the equation for h (see (4.1)) and prove that h is small. In view of the singularity of (4.1) at blow-up time T, we take a sequence $\varepsilon _n\downarrow 0$ and then we solve (4.1) (with technical truncation) with zero initial data at $T-\varepsilon _n$ in a backward direction. We denote the solution for each $n\in \mathbb {Z}_{+}$ by $h_n$ . Using energy estimates and a bootstrap argument, we can show that $h_n$ lives in an interval with a positive lower bound independent of $n\in \mathbb {Z}_{+}$ . Taking the limit $n\to \infty $ we get a desired solution to (4.1) (in the light cone). See Section 4 for details.

2.5 Proof of main results

Let’s begin with the proof of Theorem 1.1.

Proof of Theorem 1.1.

Let $T=b=1$ and $(\rho _*, \phi _*)\in C^\infty (\mathcal C)$ , $c_0\in (0,T)$ be given by Proposition 2.7, and $w_*=\rho _*$ , $\Phi _*=\phi _*$ . Then $w_*, \Phi _*\in C^\infty (\mathcal C)$ , (2.28) is equivalent to (2.25), (2.29) is equivalent to (2.26). The definitions of $E_*$ , $J_*$ in (2.24) and (2.30) are the same, and (2.27) also follows from Proposition 2.7. It follows from Proposition 2.8 that there exist $c_1\in (0,c_0)\subset (0,1)$ , $\widetilde u\in C^2((T-c_1,T)\times \mathbb {R}^d;\mathbb {C})$ such that $\widetilde u(t,\cdot ),\partial _t\widetilde u(t,\cdot )\in C_c^{\infty }(\mathbb {R}^d;\mathbb {C})$ for $t\in (T-c_1,T),$ $\Box \widetilde u=|\widetilde u|^{p-1}\widetilde u$ for $t\in (T-c_1,T), |x|\leq T-t$ , and

(2.32) $$ \begin{align} C_1^{-1}(T-t)^{-\frac{2\beta}{p-1}}\leq |\widetilde u(t,x)|\leq C_1(T-t)^{-\frac{2\beta}{p-1}},\qquad\forall\ t\in(T-c_1,T),\ |x|\leq T-t \end{align} $$

for some constant $ C_1>0$ . Choose initial data $u_0, u_1: \mathbb {R}^d\to \mathbb {C}$ such that

$$\begin{align*}u_0(x)=\widetilde u(T-c_1/2, x), \quad u_1(x)=\partial_t\widetilde u(T-c_1/2, x),\qquad\forall\ x\in\mathbb{R}^d.\end{align*}$$

Then $u_0,u_1\in C_c^{\infty }(\mathbb {R}^d;\mathbb {C})$ . Moreover, let $u_*(t,x)=\widetilde u(t+T-c_1/2,x)$ for $t\in (-c_1/2,c_1/2)$ , $x\in \mathbb {R}^d$ then $u_*(0,x)=u_0(x)$ , $\partial _t u_*(0,x)=u_1(x)$ and $\Box u_*=| u_*|^{p-1} u_*$ for $t\in [0,c_1/2), |x|\leq c_1/2-t$ . Suppose for contradiction that Theorem 1.1 fails for this initial data $u_0, u_1$ , then there exists a smooth function $u: [0,+\infty )\times \mathbb {R}^d\to \mathbb {C}$ such that $\Box u=|u|^{p-1}u$ and $u(0, x)=u_0(x), \partial _tu(0, x)=u_1(x)$ for all $x\in \mathbb {R}^d$ . Finite speed of propagation shows that $u= u_*$ in the region $\{(t,x)\in [0, c_1/2)\times \mathbb {R}^d: |x|\leq c_1/2-t\}$ . Hence by (2.32) we have

$$\begin{align*}|u(t,0)|=|u_*(t,0)|=|\widetilde u(t+T-c_1/2,0)|\geq C_1^{-1}(c_1/2-t)^{-\frac{2\beta}{p-1}},\qquad\forall\ t\in[0, c_1/2).\end{align*}$$

On the other hand, since u is smooth on $[0, +\infty )\times \mathbb {R}^d$ , we have $|u(t,x)|\leq C$ for all $|x|\leq 2T$ and $t\in [0, c_1/2]$ , where $C>0$ is a constant. This reaches a contradiction.

The following result was proved in [Reference Shao, Wei and Zhang65] Theorem 2.2 and Lemma A.7 ( $ \beta>\ell +1$ in [Reference Shao, Wei and Zhang65] is equivalent to $\beta>1$ in this paper, recalling footnote 4).

Lemma 2.9. There exist $\ell ^*(3)=\frac {76-4\sqrt {154}}{23}\in (\frac {8}{7},\frac {7}{6})$ and $\ell _1(4)\in (5/4,4/3) $ such that if

(2.33) $$ \begin{align} k=4,\ 1<\ell<\ell_1(4)\ \ \text{ or }\ k=3,\ 1<\ell<\ell^*(3), \end{align} $$

Then there exists $\beta \in (1, k/(\ell +\sqrt \ell ))$ Footnote 7such that Assumption 1 holds for $d=k+1$ .

Proof of Corollary 1.2.

Let $T=1$ . If $d=4$ , $k=3$ , $p\geq 29$ , $\ell =1+\frac {4}{p-1}$ , then $1<\ell \leq 1+\frac {4}{29-1}=\frac {8}{7}<\ell ^*(3)$ . If $d=5$ , $k=4$ , $p\geq 17$ , $\ell =1+\frac {4}{p-1}$ , then $1<\ell \leq 1+\frac {4}{17-1}=\frac {5}{4}<\ell _1(4)$ . Thus, if $d=4$ , $p\geq 29$ or $d=5$ , $p\geq 17$ , then (2.33) holds for $k=d-1$ , $\ell =1+\frac {4}{p-1}$ and the result follows from Theorem 1.1.

The remaining case is $d>5$ , $p\geq 17$ . Then Assumption 1 holds with d replaced by $d'=5$ . By the proof of Theorem 1.1, there exists $c_1\in (0,1)$ , $\widetilde u\in C^2((T-c_1,T)\times \mathbb {R}^5;\mathbb {C})$ such that $\widetilde u(t,\cdot ),\partial _t\widetilde u(t,\cdot )\in C_c^{\infty }(\mathbb {R}^5;\mathbb {C})$ for $t\in (T-c_1,T),$ $\Box \widetilde u=|\widetilde u|^{p-1}\widetilde u$ for $t\in (T-c_1,T), |x|\leq T-t$ , and (2.32) holds for some constant $ C_1>0$ . Choose initial data $u_0, u_1: \mathbb {R}^d\to \mathbb {C}$ such that

$$\begin{align*}u_0(x)=\eta(|x|)\widetilde u(T-c_1/2, x_1,\cdots,x_5), \quad u_1(x)=\eta(|x|)\partial_t\widetilde u(T-c_1/2, x_1,\cdots,x_5),\end{align*}$$

for all $x=(x_1,\cdots ,x_d)\in \mathbb {R}^d$ . Then $u_0,u_1\in C_c^{\infty }(\mathbb {R}^d;\mathcal C)$ . Moreover, let $u_*(t,x)=\eta (|x|)\widetilde u(t+T-c_1/2,x_1,\cdots ,x_5)$ for $t\in (-c_1/2,c_1/2)$ , $x=(x_1,\cdots ,x_d)\in \mathbb {R}^d$ . Then $u_*(0,x)=u_0(x)$ , $\partial _t u_*(0,x)=u_1(x)$ and $\Box u_*=| u_*|^{p-1} u_*$ for $t\in [0,c_1/2), |x|\leq c_1/2-t$ . Here we used that $\eta (|x|)=1 $ for $|x|\leq 1$ and that if $t\in [0,c_1/2), |x|\leq c_1/2-t$ then $|x|\leq c_1/2<1$ . Suppose for contradiction that Corollary 1.2 fails for this kind of initial data $u_0, u_1$ , then there exists a smooth function $u: [0,+\infty )\times \mathbb {R}^d\to \mathbb {C}$ such that $\Box u=|u|^{p-1}u$ and $u(0, x)=u_0(x), \partial _tu(0, x)=u_1(x)$ for all $x\in \mathbb {R}^d$ , and we can get a contradiction as in the proof of Theorem 1.1.

3 The approximate solution

In this section, we prove Proposition 2.7, i.e., the construction of the approximate solution.

3.1 Construction of the approximate solution

Let $T=b=1$ and $(\rho _n,\phi _n)\in {\mathscr X}_{\mu _n}\times {\mathscr X}_{\lambda _n} (n\in \mathbb {Z}_{\geq 0})$ be given by Proposition 2.6. For $N\in \mathbb {Z}_+$ and $(t,x)\in [0,T)\times \mathbb {R}^d$ , let

$$ \begin{align*} &\rho_{(N)}(t,x):=\sum_{n=0}^{N}\rho_n(t,x) b^n,\qquad \phi_{(N)}(t,x):=\sum_{n=0}^N \phi_n(t,x)b^n,\\ &E_N:=\rho_{(N)}^{p}+\rho_{(N)}\partial^{\alpha}\phi_{(N)}\partial_{\alpha}\phi_{(N)}-b\Box\rho_{(N)},\\ &J_N:= 2\partial^{\alpha}\rho_{(N)}\partial_{\alpha}\phi_{(N)}+\rho_{(N)}\Box\phi_{(N)}. \end{align*} $$

It follows from (2.5) that

$$ \begin{align*} E_{N}(t,x)=\sum_{n=N+1}^{pN}E_{N,n}(t,x) b^n,\qquad J_{N}(t,x)=\sum_{n=N+1}^{2N} J_{N,n}(t,x)b^n \end{align*} $$

with

$$ \begin{align*} E_{N,n}&:=\sum_{\substack{n_1+\cdots+n_p=n\\ n_1, \cdots, n_p\leq N}}\rho_{n_1}\rho_{n_2}\cdots\rho_{n_p}+\sum_{\substack{n_1+n_2+n_3=n\\ n_1, n_2, n_3\leq N}}\rho_{n_1}\partial^\alpha\phi_{n_2}\partial_\alpha\phi_{n_3}-\Box\rho_{N}\mathbf{1}_{n=N+1},\\ J_{N,n}&:=2\sum_{\substack{n_1+n_2=n\\ n_1, n_2\leq N}}\partial^\alpha\rho_{n_1}\partial_\alpha\phi_{n_2}+\sum_{\substack{n_1+n_2=n\\ n_1, n_2\leq N}}\rho_{n_1}\Box\phi_{n_2}. \end{align*} $$

Then $E_{N,n}\in \mathscr {X}_{2n(\beta -1)-2p\beta /(p-1)}$ , $J_{N,n}\in \mathscr {X}_{(2n-1)(\beta -1)-2\beta /(p-1)-2}=\mathscr {X}_{(2n+1)(\beta -1)-2p\beta /(p-1)}$ . Here the proof is similar to Lemma 2.5.

Take $N_0\in \mathbb {Z}_+$ such that $2N_0(\beta -1)-2p\beta /(p-1)>3 $ . We fix such $N_0$ (which is the same as the one in Proposition 2.7) and a non-decreasing sequence $\{k_N\}_{N\in \mathbb {Z}_{\geq N_0}}\subset \mathbb {Z}_+$ such that

(3.1) $$ \begin{align} 2N(\beta-1)-2p\beta/(p-1)>3k_N \ \forall\ N\in\mathbb{Z}\cap[N_0,+\infty), \text{ and }\lim_{N\to\infty}k_N=+\infty. \end{align} $$

Then by Lemma 2.3 (ii), we have $(T-t)^{-2k_N}(D^jE_{N},D^jJ_{N})\in L^{\infty }(\mathcal C) $ for $0\leq j\leq k_N$ , $N\geq N_0$ . Or equivalently, for each $N\in \mathbb {Z}\cap [N_0,+\infty )$ there is a constant $A_N>0$ satisfying

$$ \begin{align*} |D^jE_{N}(t,x)|+|D^jJ_{N}(t,x)|\leq A_N(T-t)^{2k_N},\quad \forall\ 0\leq j\leq k_N,\ (t,x)\in \mathcal C. \end{align*} $$

In fact for every fixed $n>d/2$ , we can use $(\rho _{(N)},\phi _{(N)})$ as an approximate solution for N large enough (but fixed) to construct blow-up solutions of $H^n$ regularity. But to obtain a blow-up solution of $C^{\infty }$ initial data, we need to sum all the $(\rho _{n},\phi _{n}) $ with truncation as in (2.23). Note that for $T-t\in [2T_{N+1},T_N] $ , we have $E_{*}(t,x)=E_{N}(t,x) $ and $J_{*}(t,x)=J_{N}(t,x) $ . The following result extends the above estimate to the case $T-t\in [T_{N+1},T_N]$ (with a possible different $A_N$ ).

Lemma 3.1. Let $T=b=1$ . Then there exists a sequence $\{A_N\}_{N\in \mathbb {Z}_{\geq N_0-1}}$ such that for all $\{T_n\}_{n\in \mathbb {Z}_{\geq 0}}$ satisfying $T_n=T$ for $0\leq n<N_0$ , $0<T_n\leq T_{n-1}/4$ for all $n\geq N_0$ , if we define $ \rho _*$ , $ \phi _*$ , $E_*$ , $J_*$ by (2.23), (2.24), then for $j\in \mathbb {Z}\cap [0,k_N]$ we have

(3.2) $$ \begin{align} |D^jE_{*}(t,x)|+|D^jJ_{*}(t,x)|\leq A_N(T-t)^{2k_N},\quad \forall\ T-t\in [T_{N+1},T_N],\ (t,x)\in \mathcal C. \end{align} $$

Lemma 3.2. Let $T=b=1$ . There exists a sequence $\{\widetilde {T}_n\}_{n\geq 0}$ satisfying

$$ \begin{align*}\widetilde{T}_n=T \text{ for }0\leq n<N_0\quad\text{and}\quad 0<\widetilde{T}_n\leq \widetilde{T}_{n-1}/4 \text{ for }n\geq N_0,\end{align*} $$

such that for every sequence $\{{T}_n\}_{n\geq 0}$ with ${T}_n=T$ for $0\leq n<N_0$ and $0<{T}_n\leq \min (\widetilde T_n, T_{n-1}/4)$ for $n\geq N_0$ , for $ \rho _*$ , $ \phi _*$ defined in (2.23), we have

(3.3) $$ \begin{align} (T-t)^{j}D^j(\phi_*-\phi_0)\in L^\infty(\mathcal C),\ (T-t)^{\frac{2\beta}{p-1}+j-\beta+1}D^j(\rho_*-\rho_0)\in L^\infty(\mathcal C),\ \forall\ j\in\mathbb{Z}_{\geq0}.\end{align} $$

Lemma 3.3. Let $T=1$ . There exists $\widetilde {c}\in (0,T)$ such that

(3.4) $$ \begin{align} &(T-t)^{\beta-1+j}D^j\phi_0\in L^\infty(\mathcal C),\quad (T-t)^{\frac{2\beta}{p-1}+j}D^j\rho_0\in L^\infty(\mathcal C),\quad \forall\ j\in\mathbb{Z}_{\geq0}, \end{align} $$
(3.5) $$ \begin{align} &(T-t)^{\beta}(\partial_t\phi_0-|D_x\phi_0|)(t,x)\geq \widetilde{c},\quad (T-t)^{\frac{2\beta}{p-1}}\rho_0(t,x)\geq \widetilde{c},\quad \forall\ (t,x)\in\mathcal C. \end{align} $$

Let’s first prove Proposition 2.7 by admitting Lemma 3.1 $\sim $ Lemma 3.3 for the moment.

Proof of Proposition 2.7.

Step 1. Construction of the sequence $\{{T}_n\}_{n\geq 0}$ . Let $A_N>0$ be given by Lemma 3.1 and the sequence $\{\widetilde {T}_n\}_{n\geq 0}$ be given by Lemma 3.2. Let ${T}_n=T$ for $0\leq n<N_0$ and $T_n=\min (\widetilde {T}_n,A_n^{-1/k_n},T_{n-1}/4)$ for $n\geq N_0$ . Then $0<{T}_n\leq {T}_{n-1}/4$ , $T_n\leq \widetilde {T}_n$ , $A_n{T}_n^{k_n}\leq 1$ , for all $n\geq N_0$ , and $\lim \limits _{N\to +\infty }T_N=0$ . So, there hold (3.2) for $j\in \mathbb {Z}\cap [0,k_N]$ , (3.3), (3.4), and (3.5) with $\widetilde {c}\in (0,T)$ given by Lemma 3.3.

Step 2. Proof of (2.25). As $ \beta>1$ , $(T-t)^{\beta -1}\in L^\infty (\mathcal C)$ , we get by (3.3) that

$$ \begin{align*} &(T-t)^{\beta-1+j}D^j(\phi_*-\phi_0)=(T-t)^{\beta-1}(T-t)^{j}D^j(\phi_*-\phi_0)\in L^\infty(\mathcal C),\quad \forall\ j\in\mathbb{Z}_{\geq0},\\ &(T-t)^{\frac{2\beta}{p-1}+j}D^j(\rho_*-\rho_0)=(T-t)^{\beta-1}(T-t)^{\frac{2\beta}{p-1}+j-\beta+1}D^j(\rho_*-\rho_0)\in L^\infty(\mathcal C),\quad \forall\ j\in\mathbb{Z}_{\geq0}, \end{align*} $$

which, along with with (3.4), implies (2.25).

Step 3. Proof of (2.26). By (3.3), we have (for some $ C_1>0$ )

$$ \begin{align*} &(T-t)(|\partial_t(\phi_*-\phi_0)|+|D_x(\phi_*-\phi_0)|)+(T-t)^{\frac{2\beta}{p-1}-\beta+1}|\rho_*-\rho_0|\leq C_1\quad \text{in}\,\, \mathcal C. \end{align*} $$

Now we take $c_0\in (0,T)$ such that $c_0+c_0^{\beta -1}C_1\leq \widetilde {c}$ , where the existence of such a $c_0$ is ensured by $\beta>1$ and $ \widetilde {c}>0$ . Then for $(t,x)\in \mathcal C, T-t<c_0$ , we get by (3.5) that (as $T-t>0$ )

$$ \begin{align*} (T-t)^{\beta}(\partial_t\phi_*-|D_x\phi_*|) \geq& (T-t)^{\beta}(\partial_t\phi_0-|D_x\phi_0|)-(T-t)^{\beta}(|\partial_t(\phi_*-\phi_0)|+|D_x(\phi_*-\phi_0)|)\\ \geq& \widetilde{c}-(T-t)^{\beta-1}C_1\geq \widetilde{c}-c_0^{\beta-1}C_1\geq c_0, \end{align*} $$

and

$$ \begin{align*} (T-t)^{\frac{2\beta}{p-1}}\rho_*\geq (T-t)^{\frac{2\beta}{p-1}}&\rho_0-(T-t)^{\frac{2\beta}{p-1}}|\rho_*-\rho_0| \geq \widetilde{c}-(T-t)^{\beta-1}C_1\geq \widetilde{c}-c_0^{\beta-1}C_1\geq c_0. \end{align*} $$

Step 4. Proof of (2.27). We fix $ \lambda>0$ , $j\in \mathbb {Z}_{\geq 0} $ . As $\lim \limits _{N\to +\infty }k_N=+\infty $ , there exists $N_1\in \mathbb {Z}_{\geq N_0}$ such that $ k_N>\lambda +j$ for $N\in \mathbb {Z}_{\geq N_1}$ . Then by (3.2) and $A_n{T}_n^{k_n}\leq 1$ , we have

$$ \begin{align*} |D^jE_{*}(t,x)|+|D^jJ_{*}(t,x)|\leq A_N(T-t)^{2k_N}\leq A_N{T}_N^{k_N}(T-t)^{k_N}\leq (T-t)^{k_N}\leq (T-t)^{\lambda}, \end{align*} $$

for $T-t\in [T_{N+1},T_N]$ , $(t,x)\in \mathcal C$ , $N\in \mathbb {Z}_{\geq N_1}$ . As $\lim _{N\to +\infty }T_N=0$ , we have

(3.6) $$ \begin{align} |D^jE_{*}(t,x)|+|D^jJ_{*}(t,x)|\leq (T-t)^{\lambda},\quad \forall\ T-t\in (0,T_{N_1}],\ (t,x)\in \mathcal C. \end{align} $$

As $\rho _*, \phi _*\in C^\infty ([0, T)\times \mathbb {R}^d)$ (see footnote 6), we have $E_*, J_*\in C^\infty ([0,T)\times \mathbb {R}^d)$ by (2.24). Thus, there exists a constant $ C(j,T_{N_1})>0$ such that

$$ \begin{align*} |D^jE_{*}(t,x)|+|D^jJ_{*}(t,x)|\leq C(j,T_{N_1}),\quad \forall\ T-t\in [T_{N_1},T],\ |x|\leq 2T. \end{align*} $$

Then (recall that $\mathcal C=\left \{(t,x)\in [0, T)\times \mathbb {R}^d: |x|<2(T-t)\right \}$ )

$$ \begin{align*} |D^jE_{*}(t,x)|+|D^jJ_{*}(t,x)|\leq C(j,T_{N_1})T_{N_1}^{-\lambda}(T-t)^{\lambda},\quad \forall\ T-t\in [T_{N_1},T],\ (t,x)\in \mathcal C, \end{align*} $$

which along with with (3.6) implies (2.27).

3.2 Proof of main lemmas

We define the following auxiliary spaces

(3.7) $$ \begin{align} \mathscr Y_0&:=\left\{f\in C^\infty([0,+\infty)): f'=0\ \text{in}\ [0,1]\cup[2,+\infty)\right\}, \end{align} $$
(3.8) $$ \begin{align} {\mathscr X}_{\lambda}^*&:=\left\{f(t,x,s)=\sum_{j=0}^n f_j(t,x)\eta_j\left(\frac{T-t}{s}\right): n\in\mathbb{Z}_{\geq 0}, f_j\in {\mathscr X}_{\lambda}, \eta_j\in\mathscr Y_0,\ \forall\ j\right\}. \end{align} $$

Note that $ \eta \in \mathscr Y_0$ , $ \mathscr Y_0$ is a ring, and $ {\mathscr X}_{\lambda }^*$ is a linear vector space.

Lemma 3.4.

  1. (i) Let $\lambda ,\mu \in \mathbb {C}$ , $f\in {\mathscr X}_\lambda ^*, g\in {\mathscr X}_\mu ^*$ . Then $\Box f\in {\mathscr X}_{\lambda -2}^*$ , $fg\in {\mathscr X}_{\lambda +\mu }^*$ , $\partial ^\alpha f\partial _\alpha g\in {\mathscr X}_{\lambda +\mu -2}^*$ .

  2. (ii) Let $\lambda ,\mu \in \mathbb {R}$ and $j\in \mathbb {Z}_{\geq 0}$ be such that $\lambda>j+\mu $ . If $f\in {\mathscr X}_\lambda ^*$ , then $(T-t)^{-\mu }D^jf\in L^\infty (\mathcal C\times (0,1])$ .

Here the operators $\Box $ , $\partial _\alpha $ and D are only acted on $(t,x)$ and not on s.

Lemma 3.5.

  1. (i) Let $\lambda ,\mu \in \mathbb {C}$ , $f\in {\mathscr X}_\lambda ^*, g\in {\mathscr X}_\mu ^*$ . Then $\Delta f\in {\mathscr X}_{\lambda -2}^*$ , $\partial _t f\in {\mathscr X}_{\lambda -1}^*$ , $fg\in {\mathscr X}_{\lambda +\mu }^*$ .

  2. (ii) Let $\lambda ,\mu \in \mathbb {R}$ and $j\in \mathbb {Z}_{\geq 0}$ be such that $\lambda>j+\mu $ . If $f\in {\mathscr X}_\lambda ^*$ , then $(T-t)^{-\mu }D_x^jf\in L^\infty (\mathcal C\times (0,1])$ .

Proof. By the definition of ${\mathscr X}_{\lambda }^*$ , it suffices to prove the result for $f(t,x,s)=f_1(t,x)\eta _1\big (\frac {T-t}{s}\big )$ , $g(t,x,s)=g_1(t,x)\widetilde {\eta }_1\big (\frac {T-t}{s}\big )$ for some $f_1\in {\mathscr X}_{\lambda }, g_1\in {\mathscr X}_\mu $ , $\eta _1,\widetilde {\eta }_1\in \mathscr Y_0$ .

In this case, $\Delta f(t,x,s)=\Delta f_1(t,x)\eta _1\big (\frac {T-t}{s}\big )$ . By Lemma 2.3 (i), we have $\Delta f_1\in {\mathscr X}_{\lambda -2},$ thus $\Delta f\in {\mathscr X}_{\lambda -2}^*$ . We also have

$$ \begin{align*}\partial_t f(t,x,s)=\partial_t f_1(t,x)\eta_1\big((T-t)/{s}\big)+(T-t)^{-1}f_1(t,x)\eta_2\big({(T-t)/{s}}\big)\end{align*} $$

with $ \eta _2(z)=-z\eta _1'(z)\in \mathscr Y_0$ (as $\eta _2=0 $ , $\eta _2'=0 $ in $[0,1]\cup [2,+\infty ) $ ). By Lemma 2.3 (i), we have $\partial _t f_1\in {\mathscr X}_{\lambda -1}, (T-t)^{-1} \in {\mathscr X}_{-1}, (T-t)^{-1} f_1\in {\mathscr X}_{\lambda -1},$ thus $\partial _t f\in {\mathscr X}_{\lambda -1}^*$ .

In this case, $(fg)(t,x,s)=(f_1g_1)(t,x)(\eta _1\widetilde {\eta }_1)\big (\frac {T-t}{s}\big )$ . By Lemma 2.3 (i), we have $f_1g_1\in {\mathscr X}_{\lambda +\mu },$ as $\mathscr Y_0$ is a ring we have $\eta _1\widetilde {\eta }_1\in \mathscr Y_0$ , thus $fg\in {\mathscr X}_{\lambda +\mu }^*$ . This completes the proof of (i).

Assume that $\alpha _1,\cdots ,\alpha _d\in \mathbb {Z}_{\geq 0}$ are such that $\alpha _1+\cdots +\alpha _d=j$ , then we have

$$ \begin{align*} &(T-t)^{-\mu}\partial_{x_1}^{\alpha_1}\cdots\partial_{x_d}^{\alpha_d}f(t,x,s) =(T-t)^{-\mu}\partial_{x_1}^{\alpha_1}\cdots\partial_{x_d}^{\alpha_d} f_1(t,x)\eta_1(({T-t})/{s}). \end{align*} $$

Then (ii) follows from Lemma 2.3 (ii) and $ \eta _{1}\in L^{\infty }([0,+\infty ))$ .

Proof of Lemma 3.4.

By Lemma 3.5 (i), we have $fg\in {\mathscr X}_{\lambda +\mu }^*$ , $\Delta f\in {\mathscr X}_{\lambda -2}^*$ , $\partial _t f\in {\mathscr X}_{\lambda -1}^*$ , $\partial _t^2 f\in {\mathscr X}_{\lambda -2}^*$ , thus $\Box f=-\partial _t^2f+\Delta f\in {\mathscr X}_{\lambda -2}^*$ . As a consequence, we have $\Box (fg)\in \mathscr {X}_{(\lambda +\mu )-2}^* $ , $(\Box f)g\in \mathscr {X}_{(\lambda -2)+\mu }^*$ , $f\Box g\in \mathscr {X}_{\lambda +(\mu -2)}^* $ , hence $\partial ^\alpha f\partial _\alpha g=(\Box (fg)-(\Box f)g-f\Box g)/2\in \mathscr {X}_{\lambda +\mu -2}^*$ .

Assume that $\alpha _0,\alpha _1,\cdots ,\alpha _d\in \mathbb {Z}_{\geq 0}$ are such that $\alpha _0+\cdots +\alpha _d=j$ . By Lemma 3.5 (i), we have $\partial _t^{\alpha _0} f\in {\mathscr X}_{\lambda -\alpha _0}^*$ . Then by Lemma 3.5 (ii) and $\lambda -\alpha _0>j'+\mu $ (here $j'=\alpha _1+\cdots +\alpha _d=j-\alpha _0$ ), we have

$$ \begin{align*} &(T-t)^{-\mu}|\partial_t^{\alpha_0}\partial_{x_1}^{\alpha_1}\cdots\partial_{x_d}^{\alpha_d}f|\leq (T-t)^{-\mu}|D_x^{j'}\partial_t^{\alpha_0}f|\in L^\infty(\mathcal C\times(0,1]). \end{align*} $$

This completes the proof.

Now we are in a position to prove Lemma 3.1.

Proof of Lemma 3.1.

For $t\in [0, T), x\in \mathbb {R}^d, s\in (0,+\infty )$ and $N\in \mathbb {Z}_{\geq 0}$ , let

$$ \begin{align*} &\rho_{N*}(t,x,s):=\sum_{n=0}^{N}\rho_n(t,x) b^n+\eta\left(\frac{T-t}{s}\right)\rho_{N+1}(t,x) b^{N+1},\\ & \phi_{N*}(t,x,s):=\sum_{n=0}^N \phi_n(t,x)b^n+\eta\left(\frac{T-t}{s}\right)\phi_{N+1}(t,x) b^{N+1}. \end{align*} $$

Then by (2.23), for all $N\in \mathbb {Z}_{\geq N_0-1}$ , we have

$$ \begin{align*} &\rho_{*}(t,x)=\rho_{N*}(t,x,T_{N+1}),\quad \phi_{*}(t,x)=\phi_{N*}(t,x,T_{N+1}),\quad \forall\ T-t\in [T_{N+1},T_N],\ (t,x)\in \mathcal C. \end{align*} $$

Let

(3.9) $$ \begin{align} &E_{N*}=\rho_{N*}^{p}+\rho_{N*}\partial^{\alpha}\phi_{N*}\partial_{\alpha}\phi_{N*}-b\Box\rho_{N*},\quad J_{N*}= 2\partial^{\alpha}\rho_{N*}\partial_{\alpha}\phi_{N*}+\rho_{N*}\Box\phi_{N*}. \end{align} $$

Then by (2.24), for all $N\in \mathbb {Z}_{\geq N_0-1}$ , we have

$$ \begin{align*} &E_{*}(t,x)=E_{N*}(t,x,T_{N+1}),\quad J_{*}(t,x)=J_{N*}(t,x,T_{N+1}),\quad \forall\ T-t\in [T_{N+1},T_N],\ (t,x)\in \mathcal C. \end{align*} $$

Now (3.2) is reduced to the proof of

(3.10) $$ \begin{align} (T-t)^{-2k_N}(D^jE_{N*},D^jJ_{N*})\in L^{\infty}(\mathcal C\times(0,1]),\quad \forall\ j\in\mathbb{Z}\cap[0,k_N],\ N\in \mathbb{Z}_{\geq N_0-1}. \end{align} $$

For $t\in [0, T), x\in \mathbb {R}^d, s\in (0,+\infty )$ , let

$$ \begin{align*} &\rho_{N,n}(t,x,s):=\left\{\begin{array}{ll} \rho_{n}(t,x) & n\in\mathbb{Z}\cap[0,N] \\ \rho_{n}^*(t,x,s) &n=N+1 \\ 0 & n\in\mathbb{Z}_{\geq N+2} \end{array} \right.,\\ &\phi_{N,n}(t,x,s):=\left\{\begin{array}{ll} \phi_{n}(t,x) & n\in\mathbb{Z}\cap[0,N] \\ \phi_{n}^*(t,x,s) &n=N+1 \\ 0 & n\in\mathbb{Z}_{\geq N+2} \end{array} \right.. \end{align*} $$

where

(3.11) $$ \begin{align} &\rho_{n}^*(t,x,s):=\eta\left((T-t)/{s}\right)\rho_{n}(t,x) ,\ \phi_{n}^*(t,x,s):=\eta\left((T-t)/{s}\right)\phi_{n}(t,x),\quad \forall\ n\in\mathbb{Z}_{\geq 0}. \end{align} $$

As $\rho _n\in {\mathscr X}_{\mu _n}$ , $\phi _n\in {\mathscr X}_{\lambda _n}$ for all $n\in \mathbb {Z}_{\geq 0}$ , we have $\rho _n^*,\rho _{N,n}\in {\mathscr X}_{\mu _n}^*$ , $\phi _n^*,\phi _{N,n}\in {\mathscr X}_{\lambda _n}^*$ for all $n,N\in \mathbb {Z}_{\geq 0}$ . For $t\in [0, T), x\in \mathbb {R}^d, s\in (0,+\infty )$ and $N\in \mathbb {Z}_{\geq 0}$ , we have

$$ \begin{align*} \rho_{N*}(t,x,s):=\sum_{n=0}^{N+1}\rho_{N,n}(t,x,s) b^n,\quad \phi_{N*}(t,x,s):=\sum_{n=0}^{N+1}\phi_{N,n}(t,x,s)b^n. \end{align*} $$

Then by (3.9), (2.5) and $\rho _{N,n}(t,x,s)=\rho _{n}(t,x) $ for $n\in \mathbb {Z}\cap [0,N] $ , we have

(3.12) $$ \begin{align} E_{N*}(t,x,s)=\sum_{n=N+1}^{p(N+1)}E_{N,n}^*(t,x,s) b^n,\quad J_{N*}(t,x,s)=\sum_{n=N+1}^{2(N+1)} J_{N,n}^*(t,x,s)b^n, \end{align} $$

with (note that $\rho _{N,n}(t,x,s)=0$ for $n\in \mathbb {Z}_{\geq N+2} $ )

$$ \begin{align*} E_{N,n}^*&=\sum_{n_1+\cdots+n_p=n}\rho_{N,n_1}\rho_{N,n_2}\cdots\rho_{N,n_p}+\sum_{n_1+n_2+n_3=n}\rho_{N,n_1}\partial^\alpha\phi_{N,n_2}\partial_\alpha\phi_{N,n_3}-\Box\rho_{N,n-1},\\ J_{N,n}^*&=2\sum_{n_1+n_2=n}\partial^\alpha\rho_{N,n_1}\partial_\alpha\phi_{N,n_2}+\sum_{n_1+n_2=n}\rho_{N,n_1}\Box\phi_{N,n_2}. \end{align*} $$

By Lemma 3.4 (i), we have $E_{N,n}^*\in \mathscr {X}_{2n(\beta -1)-2p\beta /(p-1)}^*$ , ${ J_{N,n}^*}\in \mathscr {X}_{(2n+1)(\beta -1)-2p\beta /(p-1)}^*$ , where we have used the facts that $\mu _{n_1}+\cdots +\mu _{n_p}=\mu _{n-1}-2$ if $n_1+\cdots +n_p=n$ , $\mu _{n_1}+\lambda _{n_2}+\lambda _{n_3}-2=\mu _{n-1}-2=2n(\beta -1)-2p\beta /(p-1)$ if $n_1+n_2+n_3=n$ and $\mu _{n_1}+\lambda _{n_2}-2=(2n-1)(\beta -1)-2\beta /(p-1)-2=(2n+1)(\beta -1)-2p\beta /(p-1)$ if $n_1+n_2=n$ .

If $n\geq N+1$ and $j\in \mathbb {Z}\cap [0,k_N]$ , then we get by (3.1) that

$$ \begin{align*} &(2n+1)(\beta-1)-2p\beta/(p-1)-j>2n(\beta-1)-2p\beta/(p-1)-j\\ &>2N(\beta-1)-2p\beta/(p-1)-j>3k_N-j\geq 2k_N. \end{align*} $$

Thus, by Lemma 3.4 (ii), we have $(T-t)^{-2k_N}D^jE_{N,n}^*\in L^\infty (\mathcal C\times (0,1])$ for $n\in \mathbb {Z}\cap [N+1,p(N+1)]$ , and $(T-t)^{-2k_N}D^jJ_{N,n}^*\in L^\infty (\mathcal C\times (0,1])$ for $n\in \mathbb {Z}\cap [N+1,2(N+1)]$ , which along with (3.12) implies (3.10).

Next we prove Lemma 3.2.

Proof of Lemma 3.2.

For $(t,x)\in [0,T)\times \mathbb {R}^d$ , and any fixed $\{T_n\}_{n\geq 0}$ satisfying $T_n=T$ for $0\leq n<N_0$ and $0<T_n\le T_{n-1}/4$ for $n\ge N_0$ , we define $\rho _{*}, \phi _*$ by (2.23), then ( $ (\rho _n^*,\phi _n^*)$ is defined in (3.11))

$$ \begin{align*} &\rho_{*}(t,x)=\sum_{n=0}^{\infty}\rho_n^*(t,x,T_n) b^n, \quad \phi_{*}(t,x)=\sum_{n=0}^{\infty} \phi_n^*(t,x,T_n)b^n,\\ &(\rho_{*}-\rho_0)(t,x)=\sum_{n=1}^{\infty}\rho_n^*(t,x,T_n) b^n,\quad (\phi_{*}-\phi_{0})(t,x)=\sum_{n=1}^{\infty} \phi_n^*(t,x,T_n)b^n. \end{align*} $$

Recall that $\rho _n^*\in {\mathscr X}_{\mu _n}^*$ and $\phi _n^*\in {\mathscr X}_{\lambda _n}^*$ for all $n\in \mathbb {Z}_{\geq 0}$ . By (2.22), we have

$$ \begin{align*} \lambda_n-j&=(2n-1)(\beta-1)-j>(n-1)(\beta-1)-j,\quad \forall\ n\in\mathbb{Z}_{+},\\ \mu_n-j&=2n(\beta-1)-\frac{2\beta}{p-1}-j>n(\beta-1)-\frac{2\beta}{p-1}-j,\quad \forall\ n\in\mathbb{Z}_{+}. \end{align*} $$

Thus, by Lemma 3.4 (ii), for any $j,n\in \mathbb {Z}_{\geq 0}$ , there exists a constant $B_{n,j}>0$ , which is independent of the sequence $\{T_n\}_{n\geq 0}$ , such that for all $(t,x)\in \mathcal C$ , we have

$$ \begin{align*} (T-t)^{j-(n-1)(\beta-1)}|D^j\phi_n^*(t,x,T_n)|+(T-t)^{\frac{2\beta}{p-1}+j-n(\beta-1)}|D^j\rho_n^*(t,x,T_n)|\leq B_{n,j}, \end{align*} $$

which gives (recalling that $\eta ((T-t)/T_n))\neq 0$ implies $T-t\leq 2T_n$ )

$$ \begin{align*} (T-t)^{j}|D^j\phi_n^*(t,x,T_n)|+(T-t)^{\frac{2\beta}{p-1}+j-\beta+1}|D^j\rho_n^*(t,x,T_n)|\leq B_{n,j}(2T_n)^{(n-1)(\beta-1)}. \end{align*} $$

Let $\widetilde {T}_n:=T$ for $0\leq n<N_0$ and for $n\geq N_0$ we let

$$\begin{align*}\widetilde{B}_{n}:=2^n\max\limits_{0\leq j\leq n-N_0}B_{n,j}, \qquad\widetilde{T}_n:=\min\left(\widetilde{B}_{n}^{-1/[(n-1)(\beta-1)]}/2,\widetilde{T}_{n-1}/4\right).\end{align*}$$

Now we prove that $\{\widetilde T_n\}_{n\ge 0}$ is a desired sequence for Lemma 3.2.

Let $\{T_n\}_{n\geq 0}$ be such that $T_n=T$ for $0\le n<N_0$ and $0<T_n\leq \min (\widetilde T_n, T_{n-1}/4)$ for $n\geq N_0$ . Then $\widetilde {B}_{n}(2\widetilde {T}_n)^{(n-1)(\beta -1)}\leq 1 $ for $n\geq N_0$ . Fix $j\in \mathbb {Z}_{\geq 0}$ . For any $(t,x)\in \mathcal C$ , we have

$$ \begin{align*} &(T-t)^{j}|D^j(\phi_*-\phi_0)(t,x)|+(T-t)^{\frac{2\beta}{p-1}+j-\beta+1}|D^j(\rho_*-\rho_0)(t,x)|\\&\quad\leq \sum_{n=1}^{\infty}\left((T-t)^{j}|D^j\phi_n^*(t,x,T_n)|+(T-t)^{\frac{2\beta}{p-1}+j-\beta+1}|D^j\rho_n^*(t,x,T_n)|\right)\\&\quad\leq \sum_{n=1}^{\infty}B_{n,j}(2T_n)^{(n-1)(\beta-1)}\leq \sum_{n=1}^{\infty}B_{n,j}(2\widetilde{T}_n)^{(n-1)(\beta-1)}\\&\quad\leq \sum_{n=1}^{N_0+j-1}B_{n,j}(2\widetilde{T}_n)^{(n-1)(\beta-1)}+\sum_{n=N_0+j}^{\infty}2^{-n}\widetilde{B}_{n}(2\widetilde{T}_n)^{(n-1)(\beta-1)}\\&\quad\leq \sum_{n=1}^{N_0+j-1}B_{n,j}(2\widetilde{T}_n)^{(n-1)(\beta-1)}+\sum_{n=N_0+j}^{\infty}2^{-n} \leq \sum_{n=1}^{N_0+j-1}B_{n,j}(2\widetilde{T}_n)^{(n-1)(\beta-1)}+1, \end{align*} $$

which implies (3.3), as the right hand side is a finite constant independent of $(t,x)\in \mathcal C$ .

Finally, we prove Lemma 3.3.

Proof of Lemma 3.3.

By (2.7), Lemma A.6 and Lemma 2.3 (iii), we obtain (3.4). It suffices to prove (3.5). By (2.7), we have

$$ \begin{align*} \partial_t\phi_0=(T-t)^{-\beta}\left((\beta-1)\widehat\phi_0(Z)+Z\widehat\phi_0'(Z)\right),\quad \partial_r\phi_0=(T-t)^{-\beta}\widehat\phi_0'(Z). \end{align*} $$

It follows from (2.12) that

(3.13) $$ \begin{align} \widehat\phi_0'(Z)=\frac{(\beta-1)\widehat\phi_0(Z)v(Z)}{1-Zv(Z)},\quad Z\widehat\phi_0'(Z)+(\beta-1)\widehat\phi_0(Z)=\frac{(\beta-1)\widehat\phi_0(Z)}{1-Zv(Z)}. \end{align} $$

Hence,

(3.14) $$ \begin{align} (T-t)^\beta\left(\partial_t\phi_0-|D_x\phi_0|\right)=(\beta-1)\frac{\widehat\phi_0(Z)(1-{|v(Z)|})}{1-Zv(Z)}. \end{align} $$

Since $\beta>1$ , $\widehat \phi _0(Z)>0, v(Z)\in (-1,1), Zv(Z)<1$ for all $Z\in [0,+\infty )$ and $\widehat \phi _0, v\in C^\infty ([0,+\infty ))$ , we know that the right hand side of (3.14) is strictly positive and continuous. Thus, there exists $\widetilde c_1\in (0, T)$ such that

$$ \begin{align*} \inf_{Z\in[0,2]}(\beta-1)\frac{\widehat\phi_0(Z)(1-{|v(Z)|})}{1-Zv(Z)}>\widetilde c_1. \end{align*} $$

On the other hand, by (2.7), we have $(T-t)^{2\beta /(p-1)}\rho _0=\widehat \rho _0(Z)$ . As $\widehat \rho (Z)>0$ and $\widehat \rho \in C([0,+\infty ))$ , there exists $\widetilde c_2\in (0, T)$ such that $\inf _{Z\in [0,2]}\widehat \rho _0(Z)>\widetilde c_2$ . As a consequence, letting $\widetilde c:=\min (\widetilde c_1, \widetilde c_2)\in (0, T)$ , we have (3.5).

4 The blow-up solution of nonlinear wave equation

Fix $T=1$ . Recall that $\mathcal C=\{(t,x)\in [0, T)\times \mathbb {R}^d: |x|<2(T-t)\}$ . Let $w_*\in C^\infty (\mathcal C;\mathbb {R}), \Psi _*\in C^\infty (\mathcal C;\mathbb {R})$ be such that both (2.28) and (2.29) hold; moreover, (2.27) also holds for $E_*, J_*$ defined by (2.30).

4.1 Derivation of the error equation

We construct a blow-up solution u to $\Box u=|u|^{p-1}u$ of the form $u=(1+h)w_*\text {e}^{\text {i}\Phi _*}$ , where h is complex-valued. First of all, we deduce the equation for the error h.

Lemma 4.1. Assume that $u=(1+h)w_*\operatorname {e}^{\operatorname {i}\Phi _*}$ solves $\Box u=|u|^{p-1}u$ . Then h satisfies

(4.1) $$ \begin{align} \Box h+2\operatorname{i} \partial^\alpha\Phi_*\partial_\alpha h+2\frac{\partial^\alpha w_*}{w_*}\partial_\alpha h-(p-1)w_*^{p-1}h_{\operatorname{r}}=w_*^{p-1}\varphi_1(h)+\frac{E_*-\operatorname{i}J_*}{w_*}(1+h), \end{align} $$

where $h_{\operatorname {r}}=\mathrm {Re}\,h=(h+\bar h)/2$ and

$$ \begin{align*} \varphi_1(h) =(|1+h|^{p-1}-1-(p-1)h_{\operatorname{r}})(1+h)+(p-1)h_{\operatorname{r}}h=O(|h|^2). \end{align*} $$

The converse is also true.

Proof. This is a brute force computation. If $u=(1+h)w_*\text {e}^{\text {i}\Phi _*}$ , then for any $\alpha \in \mathbb {Z}\cap [0, d]$ , we have

$$ \begin{align*} \partial_\alpha u=\partial_\alpha hw_*\text{e}^{\text{i}\Phi_*}+(1+h)\partial_\alpha w_*\text{e}^{\text{i}\Phi_*}+\text{i}(1+h)w_*\text{e}^{\text{i}\Phi_*}\partial_\alpha\Phi_*. \end{align*} $$

Hence,

By the definition of $\varphi _1$ , we have $1+(p-1)h_{\text {r}}+h+\varphi _1(h)=(1+h)|1+h|^{p-1}$ . Thus,

$$ \begin{align*} &\Box u=|u|^{p-1}u\Longleftrightarrow\Box h+2\text{i}\partial^\alpha\Phi_*\partial_\alpha h+2\frac{\partial^\alpha w_*}{w_*}\partial_\alpha h+(1+h)\left(w_*^{p-1}+\frac{\text{i} J_*-E_*}{w_*}\right)\\ &\qquad\qquad\qquad\qquad\qquad\qquad =\big(1+(p-1)h_{\text{r}}+h+\varphi_1(h)\big)w_*^{p-1}\\ &\Longleftrightarrow\Box h+2\text{i}\partial^\alpha\Phi_*\partial_\alpha h+2\frac{\partial^\alpha w_*}{w_*}\partial_\alpha h-(p-1)w_*^{p-1}h_{\text{r}}=w_*^{p-1}\varphi_1(h)+\frac{E_*-\text{i} J_*}{w_*}(1+h). \end{align*} $$

This completes the proof of Lemma 4.1.

We fix a bump function $\xi \in C_c^\infty (\mathbb {R};[0,1])$ such that $\operatorname {supp}\xi \subset [-1,1]$ and $\xi _{[0,4/5]}=1$ . We define the vector fields $X, Y$ byFootnote 8

(4.2) $$ \begin{align} X(t,x):=D\Phi_*(t,x)\xi\left(\frac{3|x|}{5(T-t)}\right),\quad Y(t,x):=\frac{Dw_*(t,x)}{w_*(t,x)}\xi\left(\frac{3|x|}{5(T-t)}\right) \end{align} $$

for $(t, x)\in [0, T)\times \mathbb {R}^d$ . We also define the functions on $[0, T)\times \mathbb {R}^d$ by

(4.3) $$ \begin{align} V_*(t,x):=(p-1)w_*^{p-1}(t,x)\xi\left(\frac{3|x|}{5(T-t)}\right),\quad N_*(t,x):=\frac{E_*-\operatorname{i}J_*}{w_*}(t,x)\xi\left(\frac{3|x|}{4(T-t)}\right). \end{align} $$

Then $X, Y\in C^\infty ([0, T)\times \mathbb {R}^d; \mathbb {R}^{d+1})$ and $V_*\in C^\infty ([0, T)\times \mathbb {R}^d; \mathbb {R}), N_*\in C^\infty ([0, T)\times \mathbb {R}^d; \mathbb {C})$ . Moreover, we have

(4.4) $$ \begin{align} \operatorname{supp}_xN_*(t,\cdot)\subset \{x\in\mathbb{R}^d: |x|\leq 4(T-t)/3\}, \quad \forall\ t\in[0, T). \end{align} $$

Let $c_0\in (0, T)$ satisfy (2.29). Let $\mathcal C_1:=\{(t,x)\in [T-c_0, T)\times \mathbb {R}^d: |x|\leq 4(T-t)/3\}\subset \mathcal C$ . Using (2.28), (2.29) and (2.27), we have $X_0(t,x)>0, V_*(t,x)>0$ for all $(t,x)\in \mathcal C_1$ . The following lemma gives more useful properties.

Lemma 4.2. There exists a constant $M>0$ such that

(4.5) $$ \begin{align} &(T-t)\left(\frac{|DX|}{X_0}+|Y|+\frac{|DV_*|}{V_*}\right)\leq M, \end{align} $$
(4.6) $$ \begin{align} \frac1M(T-t)^{-\beta}\leq& X_0\leq M(T-t)^{-\beta},\quad \frac1M(T-t)^{-2\beta}\leq V_*\leq M(T-t)^{-2\beta} \end{align} $$

on $\mathcal C_1$ . Moreover, for any $j\in \mathbb {Z}_{\geq 0}$ ,

(4.7) $$ \begin{align} (T-t)^{\beta+j}|D^j X|+(T-t)^{1+j}|D^j Y|+(T-t)^{2\beta+j}|D^j V_*|+(T-t)^{1+j}|D^j N_*|\in L^{\infty}(\mathcal C_1). \end{align} $$

For any $j\in \mathbb {Z}_{\geq 0}$ and $\lambda>0$ , there exists a constant $M_{j,\lambda }>0$ such that

(4.8) $$ \begin{align} |D^jN_*|\leq M_{j,\lambda}(T-t)^\lambda\quad\text{on}\quad\mathcal C_1. \end{align} $$

Proof. On $\mathcal C_1$ , we have

(4.9) $$ \begin{align}X=D\Phi_*,\quad Y=Dw_*/w_*,\quad V_*=(p-1)w_*^{p-1}.\end{align} $$

By (2.28), we have $(T-t)^{\beta }X_0=(T-t)^{\beta }\partial _t\Phi _*\in L^\infty (\mathcal C_1)$ and $(T-t)^{2\beta }V_*\in L^\infty (\mathcal C_1)$ . By (2.29), we have $(T-t)^\beta X_0=(T-t)^{\beta }\partial _t\Phi _*\geq c_0, (T-t)^{2\beta }V_*=(p-1)(T-t)^{2\beta }w_*^{p-1}\geq (p-1)c_0^{p-1}$ on $\mathcal C_1$ . This proves (4.6).

It follows from (2.28) that $(T-t)^{\beta +1}|DX|=(T-t)^{\beta +1}|D^2\Phi _*|\in L^\infty (\mathcal C_1)$ , hence by $(T-t)^\beta X_0\geq c_0$ on $\mathcal C_1$ , we have $(T-t)|DX|/X_0\in L^\infty (\mathcal C_1)$ . Similarly, by using (2.28) and (4.6), we get $(T-t)(|Y|+|DV_*|/V_*)\in L^\infty (\mathcal C_1)$ . This proves (4.5).

Next we prove (4.7) and (4.8). Recall the product rule: for smooth $f,g$ and $(\alpha _0, \alpha _1,\cdots , \alpha _d)\in \mathbb {Z}_{\geq 0}^{d+1}$ , we have (see [Reference Hardy24])

$$ \begin{align*} \frac{\partial^{\alpha_0+\alpha_1+\cdots+\alpha_d}}{\partial_t^{\alpha_0}\partial_{x_1}^{\alpha_1}\cdots\partial_{x_d}^{\alpha_d}}(fg)= &\sum_{j=0}^{\alpha_0}\sum_{j_1=0}^{\alpha_1}\cdots\sum_{j_d=0}^{\alpha_d}\binom{\alpha_0}{j_0}\binom{\alpha_1}{j_1}\cdots\binom{\alpha_d}{j_d}\\ &\qquad\qquad\times\frac{\partial^{j_0+j_1+\cdots+j_d}f}{\partial_t^{j_0}\partial_{x_1}^{j_1}\cdots\partial_{x_d}^{j_d}}\cdot \frac{\partial^{\alpha_0-j_0+\alpha_1-j_1+\cdots+\alpha_d-j_d}g}{\partial_t^{\alpha_0-j_0}\partial_{x_1}^{\alpha_1-j_1}\cdots\partial_{x_d}^{\alpha_d-j_d}}. \end{align*} $$

Hence,

(4.10) $$ \begin{align} |D^n(fg)|&\lesssim_n\sum_{j=0}^n|D^jf||D^{n-j}g|,\quad |D_x^n(fg)|\lesssim_n\sum_{j=0}^n|D_x^jf||D_x^{n-j}g|,\quad \forall\ n\in\mathbb{Z}_{\geq 0}, \end{align} $$
(4.11) $$ \begin{align} |gD^nf|&\lesssim_n |D^n(fg)|+\sum_{j=0}^{n-1}|D^jf||D^{n-j}g|,\quad\forall\ n\in\mathbb{Z}_{+}. \end{align} $$

As $X=D\Phi _*$ on $\mathcal C_1$ , we get by (2.28) that

(4.12) $$ \begin{align} (T-t)^{\beta+j}|D^jX|\in L^\infty(\mathcal C_1),\quad\forall\ j\in\mathbb{Z}_{\geq 0}. \end{align} $$

Now we use the induction argument to prove that

(4.13) $$ \begin{align} (T-t)^{1+j}|D^jY|\in L^\infty(\mathcal C_1),\quad\forall\ j\in\mathbb{Z}_{\geq 0}. \end{align} $$

By (4.5), we know that (4.13) holds for $j=0$ . Assume that (4.13) holds for all $j\in \mathbb {Z}\cap [0, n-1]$ for some $n\in \mathbb {Z}_{+}$ . Note that $Dw_*=w_*Y$ on $\mathcal C_1$ , hence by (4.11) we have

$$\begin{align*}|w_*D^nY|\lesssim_n|D^n(Dw_*)|+\sum_{j=1}^{n}|D^jw_*||D^{n-j}Y|\quad\text{on}\quad \mathcal C_1.\end{align*}$$

Using (2.29), (2.28) and the induction assumption, we obtain

$$ \begin{align*} &(T-t)^{1+n}|D^nY|\leq \frac1{c_0}(T-t)^{1+n+\frac{2\beta}{p-1}}|w_*D^nY|\\& \quad\lesssim_n (T-t)^{1+n+\frac{2\beta}{p-1}}|D^{1+n}w_*|+\sum_{j=1}^{n}(T-t)^{\frac{2\beta}{p-1}+j}|D^jw_*|(T-t)^{1+n-j}|D^{n-j}Y|\in L^\infty(\mathcal C_1). \end{align*} $$

This proves (4.13).

Now we prove that

(4.14) $$ \begin{align} (T-t)^{\frac{2\beta}{p-1}m+j}\left|D^j(w_*^m)\right|\in L^\infty(\mathcal C_1),\quad \forall\ m\in\mathbb{Z}_{+},\ \forall\ j\in\mathbb{Z}_{\geq 0}. \end{align} $$

By (2.28), we know that (4.14) holds for $m=1$ . We assume that (4.14) holds for $m-1$ , where $m\in \mathbb {Z}\cap [2,+\infty )$ . By (4.10), for $j\in \mathbb {Z}_{\geq 0}$ we have

$$ \begin{align*} \left|D^j(w_*^m)\right|=\left|D^j(w_*^{m-1}w_*)\right|\lesssim_j \sum_{i=0}^j\left|D^i(w_*^{m-1})\right|\left|D^{j-i}w_*\right|, \end{align*} $$

which gives

$$ \begin{align*} (T-t)^{\frac{2\beta}{p-1}m+j}\left|D^j(w_*^m)\right|&\lesssim_j\sum_{i=0}^j(T-t)^{\frac{2\beta}{p-1}(m-1)+i}\left|D^i(w_*^{m-1})\right|(T-t)^{\frac{2\beta}{p-1}+j-i}\left|D^{j-i}w_*\right|\\ &\in L^\infty(\mathcal C_1). \end{align*} $$

By the induction argument, we have (4.14). Letting $m=p-1$ in (4.14), we get (using (4.9))

(4.15) $$ \begin{align} (T-t)^{2\beta+j}|D^jV_*|\in L^\infty(\mathcal C_1),\quad\forall\ j\in\mathbb{Z}_{\geq 0}. \end{align} $$

Finally, we estimate $N_*$ . Let $\widetilde \xi (t,x):=\xi (3|x|/(4(T-t)))$ . Then by Lemma 2.3 (iii), we have $(T-t)^{j}D^j\widetilde \xi \in L^\infty (\mathcal C)$ . Let $\widetilde N_*:=(E_*-\text {i} J_*)\widetilde \xi $ , then $N_*=\widetilde N_*/w_*$ . By (4.10), we have

$$ \begin{align*} \left|D^j\widetilde N_*\right|\lesssim_j \sum_{i=0}^j\left|D^i(E_*-\text{i} J_*)\right|\left|D^{j-i}\widetilde \xi\right|, \end{align*} $$

hence by (2.27), for all $\lambda>0$ and $j\in \mathbb {Z}_{\geq 0}$ we have

(4.16) $$ \begin{align} (T-t)^{-\lambda}\left|D^j\widetilde N_*\right|\lesssim_j\sum_{i=0}^j(T-t)^{-(\lambda+j-i)}\left|D^i(E_*-\text{i} J_*)\right|(T-t)^{j-i}\left|D^{j-i}\widetilde \xi\right|\in L^\infty(\mathcal C). \end{align} $$

Now we use the induction argument to prove that

(4.17) $$ \begin{align} (T-t)^{-\lambda}|D^jN_*|\in L^\infty(\mathcal C_1),\quad\forall\ j\in\mathbb{Z}_{\geq 0},\ \forall\ \lambda>0. \end{align} $$

For $j=0$ , (4.17) follows from (4.16) and (2.29). Assume that (4.17) holds for all $j\in \mathbb {Z}\cap [0, n-1]$ for some $n\in \mathbb {Z}_{+}$ . As $\widetilde N_*=w_*N_*$ , we get by (4.11) that

$$\begin{align*}\left|w_*D^nN_*\right|\lesssim_n \left|D^n\widetilde N_*\right|+\sum_{j=1}^{n}|D^jw_*||D^{n-j}N_*|\quad\text{on}\quad \mathcal C_1.\end{align*}$$

Using (2.27), (2.28), (2.29) and the induction assumption, for any $\lambda>0$ we obtain

$$ \begin{align*} &(T-t)^{-\lambda}|D^nN_*|\leq \frac1{c_0}(T-t)^{-\lambda+\frac{2\beta}{p-1}}\left|w_*D^nN_*\right|\\ \lesssim_n& (T-t)^{\frac{2\beta}{p-1}}(T-t)^{-\lambda}\left|D^n\widetilde N_*\right|+\sum_{j=1}^{n}(T-t)^{\frac{2\beta}{p-1}+j}\left|D^jw_*\right|(T-t)^{-(\lambda+j)}\left|D^{n-j}N_*\right| \in L^\infty(\mathcal C_1). \end{align*} $$

This proves (4.17) for $j=n$ . By induction, we have (4.17), which is equivalent to (4.8).

Taking $\lambda =1$ in (4.8), we get

(4.18) $$ \begin{align} (T-t)^{1+j}|D^jN_*|\leq T^{j+2}(T-t)^{-1}|D^jN_*|\in L^\infty(\mathcal C_1),\quad\forall\ j\in\mathbb{Z}_{\geq 0}. \end{align} $$

Therefore, (4.7) follows from (4.12), (4.13), (4.15) and (4.18).

4.2 Energy estimates for the linearized wave equation

Lemma 4.3. Let $T_*\in (0, c_0)$ and $h\in C_c^\infty ([T-T_*, T)\times \mathbb {R}^d;\mathbb {C})$ be such that $\operatorname {supp}_xh(t,\cdot )\subset \{x\in \mathbb {R}^d: |x|\leq 4(T-t)/3\}$ for all $t\in [T-T_*, T)$ . We define the linear operator

(4.19) $$ \begin{align} \mathcal L h:= \Box h+2\operatorname{i} Xh+2Yh-V_*h_{\operatorname{r}}, \end{align} $$

where $h_{\operatorname {r}}=(h+\bar h)/2$ and energy functionals

(4.20) $$ \begin{align} E_0[h](t)&:=\frac12\int_{\mathbb{R}^d}\left(|Dh(t,x)|^2+V_*(t,x)|h_{\operatorname{r}}(t,x)|^2\right)\,\mathrm dx,\quad\forall\ t\in[T-T_*, T), \end{align} $$
(4.21) $$ \begin{align} E_j[h]&:=E_0[D_x^jh],\quad\forall\ j\in\mathbb{Z}_{+}. \end{align} $$

Then there exist positive constants $M_1>1$ and $\{C_j\}_{j\in \mathbb {Z}_{\geq 0}}$ such that

(4.22) $$ \begin{align} \sqrt{E_j[h](t)}\leq C_j\int_t^T\left(\frac{T-t}{T-s}\right)^{M_1}\sum_{i=0}^j\frac{\left\|D_x^i\mathcal Lh(s)\right\|_{L^2}}{(T-s)^{(j-i)\beta}}\mathrm ds,\quad\forall\ t\in[T-T_*, T),\ \forall\ j\in\mathbb{Z}_{\geq 0}. \end{align} $$

Proof. Let $T_*\in (0, c_0)$ and $h\in C_c^\infty ([T-T_*, T)\times \mathbb {R}^d;\mathbb {C})$ be such that

$$ \begin{align*}\operatorname{supp}_xh(t,\cdot)\subset\{x\in\mathbb{R}^d: |x|\leq 4(T-t)/3\},\quad \forall\ t\in[T-T_*, T).\end{align*} $$

We define the energy momentum tensor $T[h]$ by

(4.23) $$ \begin{align} T[h]_{\mu\nu}:=\operatorname{Re}\left(\partial_\mu h\overline{\partial_\nu h}\right)-\frac12m_{\mu\nu}\left(\partial^\alpha h\overline{\partial_\alpha h}+V_*h_{\text{r}}^2\right), \quad\forall\ \mu,\nu\in\mathbb{Z}\cap[0, d], \end{align} $$

where we have used the Einstein’s convention in $\partial ^\alpha h\overline {\partial _\alpha h}$ . Then we have

(4.24) $$ \begin{align} E_0[h](t)=\int_{\mathbb{R}^d}T[h]_{00}(t,x)\,\mathrm dx,\quad \forall\ t\in[T-T_*, T). \end{align} $$

We define

(4.25) $$ \begin{align} P_\mu^X[h]:=T[h]_{\mu\nu}X^\nu,\quad \forall\ \mu\in\mathbb{Z}\cap[0,d]. \end{align} $$

Let’s first claim that there exists a constant $\widetilde c_0>0$ such that

(4.26) $$ \begin{align} P_0^X[h]\leq \widetilde c_0T[h]_{00}X^0\leq0\ \text{on}\ \mathcal C_*:=\{(t,x)\in[T-T_*, T)\times\mathbb{R}^d: |x|\leq 4(T-t)/3\}; \end{align} $$

and there exists a constant $C_*>0$ such that

(4.27) $$ \begin{align} \left|\partial^\mu P_\mu^X[h]\right|\leq C_*(T-t)^{-\beta}\left((T-t)^{-1}T[h]_{00}+\sqrt{T[h]_{00}}|\mathcal Lh|\right)\quad\text{on}\quad\mathcal C_*, \end{align} $$

and moreover, for all $j\in \mathbb {Z}_{\geq 0}$ ,

(4.28) $$ \begin{align} \left\|D_x^j(\mathcal LD_xh-D_x\mathcal Lh)\right\|_{L_x^2}\lesssim_j \sum_{i=0}^j(T-t)^{-\beta-1+i-j}\sqrt{E_i[h](t)}\quad\forall\ t\in[T-T_*, T), \end{align} $$

where the implicit constants only depend on $X, Y, V_*, N_*$ (and they are independent of h).

Now we prove (4.22) by the induction argument. We first consider $j=0$ . For all $t\in [T-T_*, T)$ , by (4.24), $X^0=-X_0$ , $\operatorname {supp}_xh(t,\cdot )\subset \{x\in \mathbb {R}^d: |x|\leq 4(T-t)/3\}$ , (4.26) and (4.6), we have

(4.29) $$ \begin{align} E_0[h](t)\leq \int_{\mathbb{R}^d}\frac{-P_0^X[h](t,x)}{\widetilde c_0X_0(t,x)}\,\mathrm dx\leq \frac{M}{\widetilde c_0}(T-t)^{\beta}\int_{\mathbb{R}^d}-P_0^X[h](t,x)\,\mathrm dx. \end{align} $$

Let

$$\begin{align*}\widetilde E_0[h](t)=\int_{\mathbb{R}^d}-P_0^X[h](t,x)\,\mathrm dx\geq0, \quad\forall\ t\in[T-T_*, T).\end{align*}$$

By the divergence theorem (recall that $\partial ^0=-\partial _0=-\partial _t$ ), we get

$$ \begin{align*} \frac{\mathrm d}{\mathrm dt}\widetilde E_0[h](t)=\int_{\mathbb{R}^d}\partial^0P_0^X[h](t,x)\,\mathrm dx=\int_{\mathbb{R}^d}\partial^\mu P_\mu^X[h](t,x)\,\mathrm dx, \quad\forall\ t\in[T-T_*, T). \end{align*} $$

Using (4.27), (4.24) and Cauchy’s inequality, we obtain

$$ \begin{align*} &\left|\frac{\mathrm d}{\mathrm dt}\widetilde E_0[h](t)\right|\leq C_*(T-t)^{-\beta}\left((T-t)^{-1}E_0[h](t)+\sqrt{E_0[h](t)}\|\mathcal Lh(t)\|_{L_x^2}\right),\quad\forall\ t\in[T-T_*, T). \end{align*} $$

Hence, by (4.29) and $h\in C_c^\infty ([T-T_*, T)\times \mathbb {R}^d;\mathbb {C})$ , for all $t\in [T-T_*, T)$ we have

$$ \begin{align*} E_0[h](t)&\leq \frac M{\widetilde c_0}(T-t)^\beta\widetilde E_0[h](t)\leq \frac M{\widetilde c_0}(T-t)^\beta\int_t^T\left|\frac{\mathrm d}{\mathrm dt}\widetilde E_0[h](s)\right|\,\mathrm ds\\&\leq \frac{MC_*}{\widetilde c_0}(T-t)^\beta\int_t^T(T-s)^{-\beta}\left((T-s)^{-1}E_0[h](s)+\sqrt{E_0[h](s)}\|\mathcal Lh(s)\|_{L_x^2}\right)\,\mathrm ds. \end{align*} $$

By Grönwall’s lemma, we have

$$ \begin{align*} \sqrt{E_0[h](t)}\leq\frac{MC_*}{2\widetilde c_0}\int_t^T\left(\frac{T-t}{T-s}\right)^{\frac{MC_0/\widetilde c_0+\beta}{2}}\|\mathcal Lh(s)\|_{L_x^2}\,\mathrm ds,\quad\forall\ t\in[T-T_*, T). \end{align*} $$

Letting $M_1:=\frac {MC_*/\widetilde c_0+\beta }{2}>0$ , we know that (4.22) holds for $j=0$ .

Let $n\in \mathbb {Z}_{+}$ . We assume that (4.22) holds for all $j\in \mathbb {Z}\cap [0,n-1]$ . Then by (4.22) for $j=n-1$ and (4.28), for $t\in [T-T_*, T)$ we have (also using (4.20) and (4.21))

$$ \begin{align*} &\sqrt{E_n[h](t)}=\sqrt{E_{n-1}[D_xh](t)}\lesssim_n\int_t^T\left(\frac{T-t}{T-s}\right)^{M_1}\sum_{j=0}^{n-1}\frac{\|D_x^j\mathcal LD_xh(s)\|_{L_x^2}}{(T-s)^{(n-1-j)\beta}}\,\mathrm ds\\&\quad\lesssim_n \int_t^T\left(\frac{T-t}{T-s}\right)^{M_1}\sum_{j=0}^{n-1}\frac{\|D_x^{j+1}\mathcal Lh(s)\|_{L_x^2}+\sum_{i=0}^j(T-s)^{-\beta-1+i-j}\sqrt{E_i[h](s)}}{(T-s)^{(n-1-j)\beta}}\,\mathrm ds\\&\quad\lesssim_n \int_t^T\left(\frac{T-t}{T-s}\right)^{M_1}\sum_{j=1}^{n}\frac{\|D_x^{j}\mathcal Lh(s)\|_{L_x^2}}{(T-s)^{(n-j)\beta}}\,\mathrm ds+I_n(t), \end{align*} $$

where

$$ \begin{align*} I_n(t):=\sum_{j=0}^{n-1}\sum_{i=0}^j\int_t^T\left(\frac{T-t}{T-s}\right)^{M_1}(T-s)^{-1+i-j-(n-j)\beta}\sqrt{E_i[h](s)}\,\mathrm ds. \end{align*} $$

For $T-T_*\leq t<s<T$ , $j\geq i\geq 0$ we have $0<T-s<T_*<c_0<T=1$ and $ (T-s)^{-1+i-j-(n-j)\beta }=(T-s)^{-1-(n-i)\beta +(j-i)(\beta -1)}\leq (T-s)^{-1-(n-i)\beta }$ (as $ \beta>1$ ). Then

$$ \begin{align*} &I_n(t)\leq n\sum_{i=0}^{n-1}\int_t^T\left(\frac{T-t}{T-s}\right)^{M_1}(T-s)^{-1-(n-i)\beta}\sqrt{E_i[h](s)}\,\mathrm ds. \end{align*} $$

Using the induction assumption and Fubini’s theorem, we have

$$ \begin{align*} I_n(t)&\lesssim_n\sum_{i=0}^{n-1}\sum_{j=0}^i \int_t^T\frac{\left((T-t)/(T-s)\right)^{M_1}}{(T-s)^{1+(n-i)\beta}} \int_s^T\left(\frac{T-s}{T-\tau}\right)^{M_1}\frac{\|D_x^j\mathcal Lh(\tau)\|_{L_x^2}}{(T-\tau)^{(i-j)\beta}}\,\mathrm d\tau\,\mathrm ds\\&=\sum_{i=0}^{n-1}\sum_{j=0}^i\int_t^T\left(\frac{T-t}{T-\tau}\right)^{M_1}\frac{\|D_x^j\mathcal Lh(\tau)\|_{L_x^2}}{(T-\tau)^{(i-j)\beta}}\int_t^\tau\frac{\mathrm ds}{(T-s)^{1+(n-i)\beta}}\,\mathrm d\tau\\&\leq\sum_{i=0}^{n-1}\sum_{j=0}^i\int_t^T\left(\frac{T-t}{T-\tau}\right)^{M_1}\frac{\|D_x^j\mathcal Lh(\tau)\|_{L_x^2}}{(T-\tau)^{(i-j)\beta}}\frac{1}{(T-\tau)^{(n-i)\beta}}\,\mathrm d\tau\\&\leq n\sum_{j=0}^{n-1}\int_t^T\left(\frac{T-t}{T-\tau}\right)^{M_1}\frac{\|D_x^j\mathcal Lh(\tau)\|_{L_x^2}}{(T-\tau)^{(n-j)\beta}}\,\mathrm d\tau. \end{align*} $$

Therefore, we obtain (4.22) for $j=n$ . This proves (4.22) for all $j\in \mathbb {Z}_{\geq 0}$ .

Thus, it remains to prove (4.26), (4.27) and (4.28). We start with

$$\begin{align*}P_0^X[h]=T[h]_{0\nu}X^\nu=T[h]_{00}X^0+\sum_{i=1}^dT[h]_{0i}X^i.\end{align*}$$

On $\mathcal C_*\subset \mathcal C_1$ , by (4.2), we have $-X^0=X_0=\partial _t\Phi _*$ and $X^i=X_i=\partial _i\Phi _*$ for $i\in \mathbb {Z}\cap [1,d]$ , hence by (4.23) and Cauchy’s inequality,

$$ \begin{align*} \left|\sum_{i=1}^dT[h]_{0i}X^i\right|&\leq\sum_{i=1}^d|\partial_th||\partial_ih||\partial_i\Phi_*|\leq |\partial_th||D_xh||D_x\Phi_*|\leq \frac{|\partial_th|^2+|D_xh|^2}{2}|D_x\Phi_*|\\ &\leq T[h]_{00}|D_x\Phi_*|. \end{align*} $$

On the other hand, by (2.28) and (2.29), there exists a constant $\widetilde c_0\in (0,1)$ such that

$$\begin{align*}\partial_t\Phi_*-|D_x\Phi_*|\geq c_0(T-t)^{-\beta}\geq \widetilde c_0\partial_t\Phi_*>0\quad\text{on}\quad \mathcal C_1.\end{align*}$$

Thus, we have $|X|\leq |\partial _t\Phi _*|+|D_x\Phi _*|\leq 2\partial _t\Phi _*=2X_0 $ and

$$\begin{align*}\left|\sum_{i=1}^dT[h]_{0i}X^i\right|\leq T[h]_{00}|D_x\Phi_*|\leq T[h]_{00}(1-\widetilde c_0)\partial_t\Phi_*=T[h]_{00}(1-\widetilde c_0)X_0,\end{align*}$$

hence

$$ \begin{align*} P_0^X[h]\leq T[h]_{00}X^0+T[h]_{00}(1-\widetilde c_0)X_0=\widetilde c_0T[h]_{00}X^0\leq0\quad\text{on}\quad \mathcal C_*. \end{align*} $$

This proves (4.26).

As for (4.27), we compute

$$ \begin{align*} \partial^\mu T[h]_{\mu\nu}&=\operatorname{Re}\left(\Box h\overline{\partial_\nu h}\right)+\operatorname{Re}\left(\partial_\mu h\partial^\mu\overline{\partial_\nu h}\right)-\frac12\partial_\nu\left(\partial^\alpha h\overline{\partial_\alpha h}+V_*h_{\text{r}}^2\right)\\ &=\operatorname{Re}\left(\Box h\overline{\partial_\nu h}\right)+\frac12\operatorname{Re}\partial_\nu\left(\partial_\mu h\overline{\partial^\mu h}\right)-\frac12\partial_\nu\left(\partial^\alpha h\overline{\partial_\alpha h}\right)-h_{\text{r}}\partial_\nu h_{\text{r}}V_*-\frac12 h_{\text{r}}^2\partial_\nu V_*\\ &=\operatorname{Re}\left(\Box h\overline{\partial_\nu h}\right)-h_{\text{r}}\partial_\nu h_{\text{r}}V_*-\frac12 h_{\text{r}}^2\partial_\nu V_* \end{align*} $$

for $\nu \in \mathbb {Z}\cap [0,d]$ . Hence,

$$ \begin{align*} \partial^\mu P_\mu^X[h]&=T[h]_{\mu\nu}\partial^\mu X^\nu+(\partial^\mu T[h]_{\mu\nu})X^\nu\\ &=T[h]_{\mu\nu}(\pi^X)^{\mu\nu}+\operatorname{Re}\left(\Box hX^\nu\overline{\partial_\nu h}\right)-h_{\text{r}}X^\nu\partial_\nu h_{\text r}V_*-\frac12h_{\text{r}}^2X^\nu\partial_\nu V_*\\ &=T[h]_{\mu\nu}(\pi^X)^{\mu\nu}+\operatorname{Re}\left(\Box h\overline{Xh}\right)-V_*h_{\text r}Xh_{\text r}-\frac12h_{\text r}^2XV_*, \end{align*} $$

where we have used the fact that $X^\nu $ is real-valued for $\nu \in \mathbb {Z}\cap [0, d]$ , $Xh=X^\nu \partial _\nu h$ and we define

(4.30) $$ \begin{align} (\pi^X)^{\mu\nu}:=\frac{\partial^\mu X^\nu+\partial^\nu X^\mu}2,\quad\forall\ \mu,\nu\in\mathbb{Z}\cap[0,d]. \end{align} $$

Hence, it follows from (4.19) that

(4.31) $$ \begin{align} \partial^\mu P_\mu^X[h]&=T[h]_{\mu\nu}(\pi^X)^{\mu\nu}+\operatorname{Re}\left(\Box h\overline{Xh}\right)-V_*h_{\text r}Xh_{\text r}-\frac12h_{\text r}^2XV_*\nonumber\\ &=T[h]_{\mu\nu}(\pi^X)^{\mu\nu}-\frac12h_{\text r}^2XV_*+\operatorname{Re}\left(\mathcal Lh\overline{Xh}\right)-2\operatorname{Re}\left(Yh\overline{Xh}\right). \end{align} $$

By (4.23), we have $|T[h]_{\mu \nu }|\leq T[h]_{00}$ for all $\mu ,\nu \in \mathbb {Z}\cap [0,d]$ and $|Dh|^2\leq 2T[h]_{00}$ . Thus, by (4.31), $|X|\leq 2 X_0$ , (4.5) and (4.6), on $\mathcal C_*$ we have (note that $|XV_*|\leq |X||DV_*|$ , $|Xh|\leq |X||Dh|$ , $|Yh|\leq |Y||Dh|$ , see footnote 8)

$$ \begin{align*} \left|\partial^\mu P_\mu^X[h]\right|&\lesssim T[h]_{00}|DX|+T[h]_{00}X_0\frac{|DV_*|}{V_*}+|\mathcal Lh||X||Dh|+|Y||Dh||X||Dh|\\ &\lesssim T[h]_{00}(T-t)^{-1}X_0+|\mathcal Lh|X_0\sqrt{T[h]_{00}}\\ &\lesssim (T-t)^{-\beta}\left((T-t)^{-1}T[h]_{00}+\sqrt{T[h]_{00}}|\mathcal Lh|\right), \end{align*} $$

which gives (4.27).

Finally, we prove (4.28). By (4.19), we have

$$ \begin{align*} \mathcal LD_xh-D_x\mathcal Lh=-2\text{i} D_xX^{\alpha}\partial_{\alpha}h-2D_xY^{\alpha}\partial_{\alpha}h+D_xV_*\cdot h_{\text r}. \end{align*} $$

Let $j\in \mathbb {Z}_{\geq 0}$ , by (4.10) and (4.7), for any $t\in [T-T_*, T)$ we have

$$ \begin{align*} \|D_x^j(D_xX^{\alpha}\partial_{\alpha}h)(t)\|_{L_x^2}&\lesssim_j\sum_{i=0}^j\|D_x^{j-i}D_xX(t)\|_{L_x^\infty}\|DD_x^{i}h(t)\|_{L_x^2}\\ &\lesssim_j\sum_{i=0}^j(T-t)^{-\beta-1+i-j}\sqrt{E_i[h](t)}. \end{align*} $$

Similarly, we have (recalling $\beta>1$ )

$$ \begin{align*} \|D_x^j(D_xY^{\alpha}\partial_{\alpha}h)(t)\|_{L_x^2}\lesssim_j \sum_{i=0}^j(T-t)^{-2+i-j}\sqrt{E_i[h](t)}\lesssim_j\sum_{i=0}^j(T-t)^{-\beta-1+i-j}\sqrt{E_i[h](t)}. \end{align*} $$

By (4.6) and (4.7), we have

$$ \begin{align*} \|D_x^j(D_xV_*\cdot h_r)\|_{L_x^2}&\lesssim_j\sum_{i=0}^j\|D_x^{j-i+1}V_*/\sqrt{V_*}\|_{L_x^\infty}\|\sqrt{V_*}D_x^ih_{\text{r}}\|_{L_x^2}\\ &\lesssim_j\sum_{i=0}^j(T-t)^{-\beta-1+i-j}\sqrt{E_i[h](t)}. \end{align*} $$

Hence, we get (4.28).

4.3 Solving the error equation

Lemma 4.4. There exists a constant $c_2\in (0, c_0)$ that depends only on $X, Y, V_*, N_*$ such that for any $f\in C^\infty _c([T-c_0, T)\times \mathbb {R}^d;\mathbb {C})$ satisfying $|D_x^jf|\leq |D_x^j N_*|$ for all $j\in \mathbb {Z}_{\geq 0}$ , there is a solution $h\in C_c^\infty ([T-c_2, T)\times \mathbb {R}^d;\mathbb {C})$ to the error equation

(4.32) $$ \begin{align} \Box h+2\operatorname{i}Xh+2Yh-V_*h_{\operatorname{r}}-\frac1{p-1}V_*\varphi_1(h)-N_*h=f. \end{align} $$

Moreover, $\operatorname {supp}_xh(t,\cdot )\subset \{x\in \mathbb {R}^d: |x|\leq 4(T-t)/3\}$ for all $t\in [T-c_2, T)$ , and there exists a constant $C_{\Box }>0$ that depends only on $X, Y, V_*, N_*$ ( $C_{\Box }$ does not depend on f) such that

(4.33) $$ \begin{align} |\Box h(t,x)|\leq C_\Box,\quad \forall\ t\in [T-c_2, T),\ \forall\ x\in\mathbb{R}^d, \end{align} $$

and for any $j\in \mathbb {Z}_{\geq 0}$ , $\lambda>0$ , there exists a constant $C_{j,\lambda }>0$ that depends only on $X, Y, V_*, N_*$ ( $C_{j,\lambda }$ does not depend on f) such that

(4.34) $$ \begin{align} |D_x^jh(t,x)|+|\partial_tD_x^jh(t,x)|\leq C_{j,\lambda}(T-t)^\lambda,\quad \forall\ t\in [T-c_2, T),\ \forall\ x\in\mathbb{R}^d. \end{align} $$

The proof is based on the following lemma.

Lemma 4.5. Let $f\in C_c^\infty ([T-c_0, T)\times \mathbb {R}^d;\mathbb {C})$ be such that $|D_x^jf|\leq |D_x^jN_*|$ for all $j\in \mathbb {Z}_{\geq 0}$ . Let $T_*\in (0, c_0)$ . Assume that $h\in C_c^\infty ([T-T_*, T)\times \mathbb {R}^d;\mathbb {C})$ solves (4.32) on $[T-T_*, T)\times \mathbb {R}^d$ ; moreover, $\operatorname {supp}_xh(t,\cdot )\subset \{x\in \mathbb {R}^d: |x|\leq 4(T-t)/3\}$ for all $t\in [T-T_*, T)$ and

(4.35) $$ \begin{align} \|h(t,\cdot)\|_{L^\infty(\mathbb{R}^d)}\leq (T-t)^{2\beta-1},\quad\forall\ t\in[T-T_*, T). \end{align} $$

Then there exists a constant $C_{\Box }>0$ that depends only on $X, Y, V_*, N_*$ ( $C_{\Box }$ does not depend on $f, T_*$ ) such that

(4.36) $$ \begin{align} |\Box h(t,x)|\leq C_\Box,\quad \forall\ t\in [T-T_*, T),\ \forall\ x\in\mathbb{R}^d, \end{align} $$

and for any $j\in \mathbb {Z}_{\geq 0}$ , $\lambda>0$ , there exists a constant $C_{j,\lambda }>0$ that depends only on $X, Y, V_*, N_*$ ( $C_{j,\lambda }$ does not depend on $f, T_*$ ) such that

(4.37) $$ \begin{align} |D_x^jh(t,x)|+|\partial_tD_x^jh(t,x)|\leq C_{j,\lambda}(T-t)^\lambda,\quad \forall\ t\in [T-T_*, T),\ \forall\ x\in\mathbb{R}^d. \end{align} $$

Now we present the proof of Lemma 4.4

Proof of Lemma 4.4.

Let $f\in C_c^\infty ([T-c_0, T)\times \mathbb {R}^d;\mathbb {C})$ be such that $|D_x^jf|\leq |D_x^jN_*|$ for all $j\in \mathbb {Z}_{\geq 0}$ . We assume that $\varepsilon \in (0, c_0)$ satisfies $f(t,x)=0$ for all $(t,x)\in (T-\varepsilon , T)\times \mathbb {R}^d$ . By the standard local well-posedness theory (Theorem 6.4.11 in [Reference Hörmander25]), there is a unique local solution $h\in C^\infty ((T-T_+, T)\times \mathbb {R}^d;\mathbb {C})$ to (4.32) with $(h, \partial _th)|_{t=T-\varepsilon /2}=(0, 0)$ , where $\varepsilon <T_+\leq c_0$ corresponds to the left life span of h; moreover, if $T_+<c_0$ , then

(4.38) $$ \begin{align} \limsup_{t\downarrow T-T_+}\|h(t,\cdot)\|_{L^\infty(\mathbb{R}^d)}=+\infty. \end{align} $$

By the uniqueness and $f(t,x)=0$ for all $(t,x)\in (T-\varepsilon , T)\times \mathbb {R}^d$ , we have $h(t,x)=0$ for all $(t,x)\in (T-\varepsilon , T)\times \mathbb {R}^d$ . Moreover, by $|f|\leq |N_*|$ , (4.4) and the finite speed of propagation, we have

$$\begin{align*}\operatorname{supp}_xh(t,\cdot)\subset \{x\in\mathbb{R}^d: |x|\leq 4(T-t)/3\},\quad\forall\ t\in[T-T_+, T).\end{align*}$$

Let $c_2\in (0, c_0)$ be such that $C_{0,2\beta }\cdot c_2<1/2$ , where $C_{0,2\beta }>0$ is given by (4.37). Note that $c_2$ is independent of f and $T_*$ . We claim that $T_+\geq c_2$ . We assume in contrary that $\varepsilon <T_+<c_2$ . Let

(4.39) $$ \begin{align} \mathscr{E}:=\left\{T_0\in(0, T_+): \|h(t,\cdot)\|_{L^\infty(\mathbb{R}^d)}\leq (T-t)^{2\beta-1} \text{ for all }t\in[T-T_0, T)\right\}. \end{align} $$

Then $(0,\varepsilon )\subset {\mathscr {E}}$ . Let $T_s:=\sup {\mathscr {E}}\in [\varepsilon , T_+]$ . By (4.38), we have $T_s<T_+$ , hence $T_s\in \mathscr {E}$ and $T_s<T_+<c_2$ . By (4.37), we have

$$ \begin{align*} |h(t,x)|\leq C_{0,2\beta}(T-t)^{2\beta}=C_{0,2\beta}(T-t)(T-t)^{2\beta-1}\leq C_{0,2\beta}c_2(T-t)^{2\beta-1}<\frac12(T-t)^{2\beta-1} \end{align*} $$

for all $t\in [T-T_s, T)\subset [T-c_2, T)$ . Thus, by the continuity we have $T_s+\delta \in \mathscr {E}$ for some $\delta>0$ . This contradicts with $T_s=\sup \mathscr {E}$ . Therefore, $T_+\geq c_2$ and $\|h(t,\cdot )\|_{L^\infty (\mathbb {R}^d)}\leq (T-t)^{2\beta -1}$ for all $t\in [T-c_2, T)$ . Now Lemma 4.5 (letting $T_*=c_2$ ) implies Lemma 4.4.

Let’s complete the proof of Lemma 4.5.

Proof of Lemma 4.5.

Assume that h solves (4.32). Then ( $\mathcal L$ is defined in (4.19))

$$\begin{align*}\mathcal Lh=\frac1{p-1}V_*\varphi_1(h)+N_*h+f.\end{align*}$$

We claim that for each $j\in \mathbb {Z}_{\geq 0}$ , there exists a constant $\widetilde C_j>0$ such that

(4.40) $$ \begin{align} \sqrt{E_j[h](t)}\leq \widetilde C_j\int_t^T\left(\frac{T-t}{T-s}\right)^{M_1}\sum_{i=0}^j\frac{\left\|D_x^i f(s)\right\|_{L_x^2}}{(T-s)^{(j-i)\beta}}\mathrm ds,\quad\forall\ t\in[T-T_*, T). \end{align} $$

By the definition of $\varphi _1$ , we know that $\varphi _1$ is a polynomial on $(h, \overline {h})$ of the form $\varphi _1=\sum _{2\leq i+j\leq p}c_{i,j}h^i\overline {h}^j$ , with $c_{i,j}\in \mathbb {R}$ , thus

(4.41) $$ \begin{align} |\varphi_1(h)|\lesssim |h|^{2}+|h|^p,\quad\forall\ h\in\mathbb{C}. \end{align} $$

Hence, $\operatorname {supp}_x\varphi _1(h)(t,\cdot )\subset \{x\in \mathbb {R}^d: |x|\leq 4(T-t)/3\}$ for all $t\in [T-T_*, T)$ . For $j\in \mathbb {Z}_{\geq 0}$ and $t\in [T-T_*, T)$ , by (4.10), (4.7) and Poincaré’s inequality, we have

$$ \begin{align*} \left\|D_x^j(V_*\varphi_1(h))(t)\right\|_{L_x^2}&\lesssim_j \sum_{i=0}^j\|D_x^{j-i}V_*(t)\|_{L_x^\infty}\|D_x^i\varphi_1(h)(t)\|_{L_x^2}\\ &\lesssim_j \sum_{i=0}^j(T-t)^{-(2\beta+j-i)}(T-t)^{j-i}\|D_x^j\varphi_1(h)(t)\|_{L_x^2}\\&\lesssim_j (T-t)^{-2\beta}\|D_x^j\varphi_1(h)(t)\|_{L_x^2}\end{align*} $$

Using the classical product estimate,

(4.42) $$ \begin{align} \|D_x^n(fg)\|_{L_x^2}\lesssim_n \|f\|_{L_x^\infty}\|D_x^ng\|_{L_x^2}+\|g\|_{L_x^\infty}\|D_x^nf\|_{L_x^2},\quad\forall\ n\in\mathbb{Z}_{\geq0}, \end{align} $$

and (4.35), we infer

$$ \begin{align*} &\|D_x^n(h^i\overline{h}^j)\|_{L_x^2}\lesssim_{n,i,j} \|h\|_{L_x^\infty}^{i+j-1}\|D_x^nh\|_{L_x^2},\quad\forall\ n,i,j\in\mathbb{Z}_{\geq0},\ i+j\geq2,\\ &\|D_x^j\varphi_1(h)(t)\|_{L_x^2}\lesssim_j\left(\|h(t)\|_{L_x^\infty}+\|h(t)\|_{L_x^\infty}^{p-1}\right)\|D_x^jh(t)\|_{L_x^2}\lesssim_j (T-t)^{2\beta-1}\|D_x^jh(t)\|_{L_x^2}, \\ &\left\|D_x^j(V_*\varphi_1(h))(t)\right\|_{L_x^2}\lesssim_j(T-t)^{-2\beta} (T-t)^{2\beta-1}\|D_x^jh(t)\|_{L_x^2}=(T-t)^{-1}\|D_x^jh(t)\|_{L_x^2}; \end{align*} $$

Similarly, by (4.7) and Poincaré’s inequality, we have

$$ \begin{align*} \left\|D_x^j(N_*h)(t)\right\|_{L_x^2}\lesssim_j(T-t)^{-1}\|D_x^jh(t)\|_{L_x^2}. \end{align*} $$

Therefore, for each $j\in \mathbb {Z}_{\geq 0}$ , there holds

(4.43) $$ \begin{align} \left\|D_x^j\mathcal Lh(t)\right\|_{L_x^2}\lesssim_j (T-t)^{-1}\|D_x^jh(t)\|_{L_x^2}+\|D_x^jf(t)\|_{L_x^2}. \end{align} $$

By (4.43), (4.22), for any $j\in \mathbb {Z}_{\geq 0}$ and $t\in [T-T_*, T)$ we have

(4.44) $$ \begin{align} \sqrt{E_j[h](t)}&\lesssim_j\int_t^T\left(\frac{T-t}{T-s}\right)^{M_1}\sum_{i=0}^j\frac{(T-s)^{-1}\|D_x^ih(s)\|_{L_x^2}+\|D_x^if(s)\|_{L_x^2}}{(T-s)^{(j-i)\beta}}\mathrm ds. \end{align} $$

It follows from Poincaré’s inequality and $\operatorname {supp}_xh(t,\cdot )\subset \{x\in \mathbb {R}^d: |x|\leq 4(T-t)/3\}$ that

(4.45) $$ \begin{align}(T-t)^{-1}\|D_x^jh(t)\|_{L_x^2}\lesssim\|D_x^{j+1}h(t)\|_{L_x^2}\leq\sqrt{2E_j[h](t)},\quad \forall\ t\in [T-T_*, T),\ j\in\mathbb{Z}_{\geq0}.\end{align} $$

Here we also used the definitions of $E_0$ and $E_j$ in (4.20) and (4.21). Next we use the induction argument to prove (4.40).

For $j=0$ , by (4.44) and (4.45), there exists a constant $C_0'>0$ satisfying

$$ \begin{align*} \sqrt{E_0[h](t)}\leq C_0'\int_t^T\left(\frac{T-t}{T-s}\right)^{M_1}\left(\sqrt{E_0[h](s)}+\|f(s)\|_{L_x^2}\right)\,\mathrm ds,\quad\forall\ t\in[T-T_*, T). \end{align*} $$

By Grönwall’s lemma, we get

$$ \begin{align*} (T-t)^{-M_1}\sqrt{E_0[h](t)}&\leq C_0'\int_t^T(T-s)^{-M_1}\text{e}^{C_0'(s-t)}\|f(s)\|_{L_x^2}\,\mathrm ds\\ &\leq C_0'\text{e}^{C_0'T}\int_t^T(T-s)^{-M_1}\|f(s)\|_{L_x^2}\,\mathrm ds \end{align*} $$

for all $t\in [T-T_*, T)$ . This proves (4.40) for $j=0$ . Let $n\in \mathbb {Z}_{+}$ , assume that (4.40) holds for $j\in \mathbb {Z}\cap [0, n-1]$ . By (4.20), (4.21) and $ \beta>1$ , we have

$$ \begin{align*} (T-s)^{-1}\|D_x^nh(s)\|_{L_x^2}\leq (T-s)^{-1}\sqrt{2E_{n-1}[h](s)}\leq (T-s)^{-\beta}\sqrt{2E_{n-1}[h](s)}, \end{align*} $$

for $ s\in [T-T_*, T)$ . Then by (4.44) for $j=n$ , (4.45) for $j=i<n$ , and the induction assumption, we have (as $0<T-t\leq T_*<c_0<1$ )

$$ \begin{align*} \sqrt{E_n[h](t)}&\lesssim_n\int_t^T\left(\frac{T-t}{T-s}\right)^{M_1}\left(\|D_x^nf(s)\|_{L_x^2}+\sum_{j=0}^{n-1}\frac{\sqrt{E_j[h](s)}+\|D_x^jf(s)\|_{L_x^2}}{(T-s)^{(n-j)\beta}}\right)\,\mathrm ds\\ &\lesssim_n \int_t^T\left(\frac{T-t}{T-s}\right)^{M_1}\sum_{j=0}^n\frac{\|D_x^jf(s)\|_{L_x^2}}{(T-s)^{(n-j)\beta}}\,\mathrm ds+I_n(t), \end{align*} $$

where

$$ \begin{align*} I_n(t)&:=\sum_{j=0}^{n-1}\int_t^T\frac{\big((T-t)/(T-s)\big)^{M_1}}{(T-s)^{(n-j)\beta}}\int_s^T\left(\frac{T-s}{T-\tau}\right)^{M_1}\sum_{i=0}^j\frac{\|D_x^if(\tau)\|_{L_x^2}}{(T-\tau)^{(j-i)\beta}}\,\mathrm d\tau\,\mathrm ds\\ &=\sum_{j=0}^{n-1}\sum_{i=0}^j\int_t^T\left(\frac{T-t}{T-\tau}\right)^{M_1}\frac{\|D_x^if(\tau)\|_{L_x^2}}{(T-\tau)^{(j-i)\beta}}\int_t^{\tau}\frac{\mathrm ds}{(T-s)^{(n-j)\beta}}\,\mathrm d\tau\\ &\leq \sum_{j=0}^{n-1}\sum_{i=0}^j\int_t^T\left(\frac{T-t}{T-\tau}\right)^{M_1}\frac{\|D_x^if(\tau)\|_{L_x^2}}{(T-\tau)^{(j-i)\beta}}\frac{1}{(T-\tau)^{(n-j)\beta}}\,\mathrm d\tau\\ &\leq n\sum_{i=0}^{n-1}\int_t^T\left(\frac{T-t}{T-\tau}\right)^{M_1}\frac{\|D_x^if(\tau)\|_{L_x^2}}{(T-\tau)^{(n-i)\beta}}\,\mathrm d\tau.\end{align*} $$

Thus, (4.40) holds for $j=n$ . Therefore, by the the induction, (4.40) holds for all $j\in \mathbb {Z}_{\geq 0}$ .

As $|D_x^jf|\leq |D_x^jN_*|$ for all $j\in \mathbb {Z}_{\geq 0}$ , by (4.4) and (4.8), for all $j\in \mathbb {Z}_{\geq 0}$ and $\lambda>0$ there exists a constant $\widetilde M_{j,\lambda }>0$ which is independent of f and $T_*$ such that $\|D_x^jf(t)\|_{L^2}\leq \widetilde M_{j,\lambda }(T-t)^\lambda $ for all $t\in [T-T_*, T)$ . Using (4.40), (4.21) and (4.45), for all $j\in \mathbb {Z}_{\geq 0}$ and $\lambda>0$ there exists a constant $M_{j,\lambda }'>0$ independent of f and $T_*$ (depending on $\widetilde M_{j,\lambda '} $ for some $\lambda '>\lambda $ ) such that

$$ \begin{align*} \|h(t)\|_{H_x^j}+\|\partial_th(t)\|_{H_x^j}\leq M_{j,\lambda}'(T-t)^\lambda,\quad\forall\ t\in[T-T_*, T). \end{align*} $$

By Sobolev’s embedding theorem ( $H_x^d(\mathbb {R}^d)\hookrightarrow L_x^\infty (\mathbb {R}^d)$ ), we have (4.37). It remains to prove (4.36). By (4.41), (4.35), (4.32), (4.37), (4.7) ( $j=0$ ) and (4.8) ( $j=0, \lambda =1$ ), we know that there there exist constants $ C_2'>0, C_3'>0$ such that

$$ \begin{align*} |\Box h|&\leq C_2'(T-t)^{-2\beta}|D^{\leq 1}h|+|f|\leq C_2'(T-t)^{-2\beta}(C_{0, 2\beta}+C_{1,2\beta})(T-t)^{2\beta}+|N_*| \leq C_3' \end{align*} $$

on $[T-T_*, T)\times \mathbb {R}^d$ , which implies (4.36).

4.4 Solving nonlinear wave equation

Proof of Proposition 2.8.

Let $\xi _1=1-\xi $ , then $\xi _1|_{[0,4/5]}=0, \xi _1|_{[1,+\infty )}=1$ . Let

$$\begin{align*}\varepsilon_n:=c_2/2^n,\quad f_n(t,x):=N_*(t,x)\xi_1((T-t)/\varepsilon_n), \quad \forall\ [T-c_0, T)\times\mathbb{R}^d,\ \forall\ n\in\mathbb{Z}_{+}.\end{align*}$$

Then for each $n\in \mathbb {Z}_{+}$ , we have $f_n\in C^\infty ([T-c_0, T)\times \mathbb {R}^d;\mathbb {C})$ and $\operatorname {supp}f_n\subset \{(t,x)\in [T-c_0, T-4\varepsilon _n/5]\times \mathbb {R}^d: |x|\leq 4(T-t)/3\}$ , hence $f_n\in C_c^\infty ([T-c_0, T)\times \mathbb {R}^d;\mathbb {C})$ ; as $f_n$ equals to $N_*$ multiplied by a function in t that takes values in $[0, 1]$ , we have $|D_x^jf_n|\leq |D_x^jN_*|$ for all $j\in \mathbb {Z}_{+}$ . By Lemma 4.4, for each $n\in \mathbb {Z}_{+}$ , there exists $h_n\in C_c^\infty ([T-c_2, T)\times \mathbb {R}^d;\mathbb {C})$ satisfying

(4.46) $$ \begin{align} \Box h_n+2\text{i} Xh_n+2Yh_n-V_*(h_n)_{\text{r}}-\frac1{p-1}V_*\varphi_1(h_n)-N_*h_n=f_n=N_*\xi_1\left(\frac{T-t}{\varepsilon_n}\right) \end{align} $$

on $[T-c_2, T)\times \mathbb {R}^d$ . Moreover, for $n\in \mathbb {Z}_{+}$ , we have

(4.47) $$ \begin{align} \operatorname{supp}_xh_n(t,\cdot)\subset\{x\in\mathbb{R}^d: |x|\leq4(T-t)/3\},\quad\forall\ t\in [T-c_2, T), \end{align} $$

and for $j\in \mathbb {Z}_{\geq 0},\lambda>0$ we have

(4.48) $$ \begin{align} |\Box h_n(t,x)|\leq C_\Box, \quad\forall\ t\in [T-c_2, T),\ \forall\ x\in\mathbb{R}^d, \end{align} $$
(4.49) $$ \begin{align} |D_x^jh_n(t,x)|+|\partial_tD_x^jh_n(t,x)|\leq C_{j,\lambda}(T-t)^\lambda,\quad \forall\ t\in [T-c_2, T),\ \forall\ x\in\mathbb{R}^d, \end{align} $$

where $C_\Box $ and $C_{j,\lambda }$ are given by Lemma 4.4. By $h_n\in C_c^\infty ([T-c_2, T)\times \mathbb {R}^d;\mathbb {C})$ , (4.47), (4.48), (4.49) and the Arzelà–Ascoli theorem, there exists a subsequence of $\{h_n\}_{n=1}^\infty $ , which is still denoted by $\{h_n\}_{n=1}^\infty $ Footnote 9, such that $h_n\to h$ in $C^1([T-c_0, T)\times \mathbb {R}^d)$ for some $h\in C^1([T-c_0, T)\times \mathbb {R}^d;\mathbb {C})$ and (here $D^{\leq 1}f:=(f,Df)$ )

(4.50) $$ \begin{align} \left\|D^{\leq 1}(h_n-h_{n+1})\right\|_{L^\infty([T-c_0, T)\times \mathbb{R}^d)}\leq 2^{-n},\quad \forall\ n\in\mathbb{Z}_{+}. \end{align} $$

Letting $j=2$ , $\lambda =1$ in (4.49), by (4.50) and the Gagliardo–Nirenberg inequality, we have

(4.51) $$ \begin{align} \left\|D_xD(h_n-h_{n+1})\right\|_{L^\infty([T-c_0, T)\times \mathbb{R}^d)}\leq \widetilde C_0 2^{-n/2},\quad \forall\ n\in\mathbb{Z}_{+} \end{align} $$

for some constant $\widetilde C_0>0$ which is independent of $n\in \mathbb {Z}_{+}$ . Letting $j=0$ and $\lambda =1$ in (4.49), by the definition of $\varphi _1$ , there exists a constant $\widetilde C_1>0$ such that for all $n\in \mathbb {Z}_+$ and $(t,x)\in [T-c_2, T)\times \mathbb {R}^d$ , we have

$$\begin{align*}|\varphi_1(h_n)-\varphi_1(h_{n+1})|\leq \widetilde C_1|h_n-h_{n+1}|.\end{align*}$$

Combining this with (4.46), (4.49), (4.7) ( $j=0$ ), and (4.8) ( $j=0, \lambda =1$ ), we know that there exist constants $\widetilde C_2>0, \widetilde C_3>0$ such that for all $n\in \mathbb {Z}_{+}$ , we haveFootnote 10

$$ \begin{align*} |\Box h_{n}-\Box h_{n+1}|&\leq \widetilde C_2(T-t)^{-2\beta}\left|D^{\leq 1}(h_n-h_{n+1})\right|+\widetilde C_2\varepsilon_n\\ &\overset{(4.49)}{\underset{(4.50)}{\leq}}\widetilde C_2(T-t)^{-2\beta}\min\left(2^{-n}, 2(C_{0, 4\beta}+C_{1,4\beta})(T-t)^{4\beta}\right)+\widetilde C_2\varepsilon_n\\ &\leq \widetilde C_3(2^{-n/2}+\varepsilon_n) \end{align*} $$

on $[T-c_2, T)\times \mathbb {R}^d$ . Hence, by (4.50) and (4.51) we know that $\{h_n\}_{n=1}^\infty $ is Cauchy in $C^2([T-c_2, T)\times \mathbb {R}^d;\mathbb {C})$ , hence $h\in C^2([T-c_2, T)\times \mathbb {R}^d;\mathbb {C})$ and $h_n\to h$ in $C^2$ . Moreover, by (4.49), (4.50) and the Gagliardo–Nirenberg inequality we know that $\{D_x^jD^{\leq 1}h_n\}$ is Cauchy in $L^\infty ([T-c_2, T)\times \mathbb {R}^d)$ for all $j\in \mathbb {Z}_{\geq 0}$ . Hence (also using (4.47)) $h(t,\cdot ), \partial _th(t,\cdot )\in C_c^\infty (\mathbb {R}^d)$ with $\operatorname {supp}_xh(t,\cdot )\subset \{x\in \mathbb {R}^d: |x|\leq 4(T-t)/3\}$ . Moreover, h solves the equation (as $f_n\to N_*$ )

(4.52) $$ \begin{align} \Box h+2\text{i} Xh+2Yh-V_*h_{\text{r}}-\frac1{p-1}V_*\varphi_1(h)-N_*h=N_*\quad\text{on}\quad [T-c_2, T)\times\mathbb{R}^d. \end{align} $$

By (4.49), we have

$$\begin{align*}|h(t,x)|\leq C_{0,1}(T-t),\quad \forall\ (t,x)\in[T-c_2, T)\times\mathbb{R}^d.\end{align*}$$

Let $c_1\in (0,c_2)\subset (0, T)$ be such that $C_{0, 1}c_1<1/2$ , hence $|h(t,x)|<1/2$ for all $(t,x)\in [T-c_1, T)\times \mathbb {R}^d$ . Let

$$ \begin{align*}u(t,x):=(1+h(t,x))w_*(t,x)\xi\left(\frac{3|x|}{5(T-t)}\right)\exp\left(\text{i}\Phi_*(t,x)\xi\left(\frac{3|x|}{5(T-t)}\right)\right)\end{align*} $$

for $(t,x)\in [T-c_1, T)\times \mathbb {R}^d$ . Then $u\in C^2([T-c_1, T)\times \mathbb {R}^d;\mathbb {C})$ with $u(t,\cdot ), \partial _tu(t,\cdot )\in C_c^\infty (\mathbb {R}^d)$ and $\operatorname {supp}_xu(t,\cdot ), \operatorname {supp}_x\partial _tu(t,\cdot )\subset \{x\in \mathbb {R}^d: |x|\leq 5(T-t)/3\}$ for $t\in [T-c_1, T)$ . Moreover, on $\mathcal C_0:=\{(t,x)\in (T-c_1, T)\times \mathbb {R}^d: |x|\leq T-t\}$ we have $u=(1+h)w_*\text {e}^{\text {i}\Phi _*}$ , and by (4.2), (4.3) and (4.52), we know that h satisfies (4.1) on $\mathcal C_0$ , hence by Lemma 4.1 we know that $\Box u=|u|^{p-1}u$ on $\mathcal C_0$ . Finally, by $|h|<1/2$ , (2.28) (for $j=0$ ) and (2.29) we have (2.31) on $\mathcal C_0$ .

This completes the proof of Proposition 2.8.

5 The linearized operator $\mathscr {L}$

5.1 Functional spaces

In this subsection, we define some functional spaces consisting of smooth functions. Let $I\subset [0,+\infty )$ be an interval. We denote $I^2:=\{x^2: x\in I\}$ andFootnote 11

(5.1) $$ \begin{align} C_{\text{e}}^\infty(I)&:=\left\{f\in C^\infty(I;\mathbb{C}): \exists \widetilde f\in C^\infty(I^2) \text{ s.t. }f(Z)=\widetilde f(Z^2),\quad \forall\ Z\in I\right\}, \end{align} $$
(5.2) $$ \begin{align} C_{\text{o}}^\infty(I)&:=\left\{f\in C^\infty(I;\mathbb{C}): \exists \widetilde f\in C^\infty(I^2) \text{ s.t. }f(Z)=Z\widetilde f(Z^2),\quad \forall\ Z\in I\right\}. \end{align} $$

Then $ C_{\text {e}}^\infty (I)$ is a ring, and $ C_{\text {o}}^\infty (I)$ is a linear vector space. Note that when $I=[0,+\infty )$ , the definitions in (5.1), (5.2) are the same as in (2.10), (2.11). For example, we have $f(Z)=Z\in C_{\text {o}}^\infty ([0,+\infty ))\setminus C_{\text {e}}^\infty ([0,+\infty ))$ and $f(Z)=Z^2\in C_{\text {e}}^\infty ([0,+\infty ))\setminus C_{\text {o}}^\infty ([0,+\infty ))$ .

Lemma 5.1. Let $f\in C_{\operatorname {e}}^\infty ([0,+\infty ))$ . Define $F(x)=f(|x|)$ for $x\in \mathbb {R}^d$ , then $F\in C^\infty (\mathbb {R}^d)$ .

Proof. As $f\in C_{\text {e}}^\infty ([0,+\infty ))$ , there exists a function $\widetilde f\in C^\infty ([0,+\infty ))$ such that $f(Z)=\widetilde f(Z^2)$ for all $Z\in [0,+\infty )$ , hence $F(x)=\widetilde f(|x|^2)$ for all $x\in \mathbb {R}^d$ . The smoothness of F follows from the smoothness of $\widetilde f$ and $x\mapsto |x|^2$ .

We also have the following fundamental properties. Let $I\subset [0,+\infty )$ be an interval, then

(5.3) $$ \begin{align} f\in C_{\text{e}}^\infty(I)&\Longrightarrow f'\in C_{\text{o}}^\infty(I); \end{align} $$
(5.4) $$ \begin{align} f\in C_{\text{o}}^\infty(I)&\Longrightarrow f'\in C_{\text{e}}^\infty(I); \end{align} $$
(5.5) $$ \begin{align} f_1\in C_{\text{e}}^\infty(I), f_2\in C_{\text{e}}^\infty(I)&\Longrightarrow f_1f_2\in C_{\text{e}}^\infty(I); \end{align} $$
(5.6) $$ \begin{align} f_1\in C_{\text{o}}^\infty(I), f_2\in C_{\text{o}}^\infty(I)&\Longrightarrow f_1f_2\in C_{\text{e}}^\infty(I); \end{align} $$
(5.7) $$ \begin{align} f_1\in C_{\text{e}}^\infty(I), f_2\in C_{\text{o}}^\infty(I)&\Longrightarrow f_1f_2\in C_{\text{o}}^\infty(I). \end{align} $$

Moreover, if $\Omega \subset \mathbb {C}$ is open, $\varphi \in C^\infty (\Omega ;\mathbb {C})$ (not necessary to be holomorphic), and $f\in C_{\text {e}}^\infty (I)$ with $f(Z)\in \Omega $ for all $Z\in I$ , then the composition $\varphi \circ f\in C_{\text {e}}^\infty (I)$ . In particular,

(5.8) $$ \begin{align} f\in C_{\text{e}}^\infty(I)&\text{ with }f(Z)\neq 0 \ \forall\ Z\in I \Longrightarrow 1/f\in C_{\text{e}}^\infty(I), \end{align} $$
(5.9) $$ \begin{align} &f\in C_{\text{e}}^\infty(I)\Longrightarrow \exp f\in C_{\text{e}}^\infty(I), \end{align} $$
(5.10) $$ \begin{align} a\in\mathbb{R}, f\in C_{\text{e}}^\infty(I)&\text{ with }f(Z)>0 \text{ for all }Z\in I\Longrightarrow f^a\in C_{\text{e}}^\infty(I). \end{align} $$

Lemma 5.2. If $ f\in {\mathscr X}_0$ , then $\partial _\tau f, Z\partial _Z f, \partial _Z^2 f, \partial _Z f/Z\in {\mathscr X}_0$ .

Proof. By the definition of ${\mathscr X}_0$ in (2.20), it suffices to prove the result for $f=\widehat f(Z)\tau ^n$ for some $\widehat f\in C^\infty _{\text {e}}([0,+\infty ))$ and some $n\in \mathbb {Z}_{\geq 0}$ .

As $\partial _\tau f=n\widehat f(Z)\tau ^{n-1}$ , we have $\partial _\tau f\in {\mathscr X}_0$ ( $\partial _\tau f=0$ for $n=0$ ).

Note that $Z\partial _Z f=Z\widehat f'(Z)\tau ^n, \partial _Z^2 f=\widehat f"(Z)\tau ^n, \partial _Z f/Z=\frac {1}{Z}\widehat f'(Z)\tau ^n$ . As $\widehat f\in C^\infty _{\text {e}}([0,+\infty ))$ , by (5.3) we have $\widehat f'\in C^\infty _{\text {o}}([0,+\infty ))$ , then by the definitions of $C^\infty _{\text {e}}([0,+\infty ))$ and $C^\infty _{\text {o}}([0,+\infty ))$ we have $Z\widehat f'(Z),\frac {1}{Z}\widehat f'(Z)\in C^\infty _{\text {e}}([0,+\infty ))$ , by (5.4) we have $\widehat f"\in C^\infty _{\text {e}}([0,+\infty ))$ . Thus, $Z\partial _Z f$ , $\partial _Z^2 f$ , $\partial _Z f/Z\in {\mathscr X}_0$ .

Now we prove Lemma 2.3.

Proof.

  1. (i) Let $f(t,x)=(T-t)^\lambda \widehat f(\tau , Z)$ for some $\widehat f\in {\mathscr X}_0$ and $g(t,x)=(T-t)^\mu \widehat g(\tau , Z)$ for some $\widehat g\in {\mathscr X}_0$ . Then $(fg)(t,x)=(T-t)^{\lambda +\mu }(\widehat f\widehat g)(\tau , Z)$ . As ${\mathscr X}_0 $ is a ring, we have $\widehat f\widehat g\in {\mathscr X}_0$ and then $fg\in {\mathscr X}_{\lambda +\mu }$ . It is direct to check that

    $$ \begin{align*} \partial_t f=(T-t)^{\lambda-1}\left(\partial_\tau+Z\partial_Z-\lambda\right)\widehat f,\quad \Delta f=(T-t)^{\lambda-2}\left(\partial_Z^2+(k/Z)\partial_Z\right)\widehat f. \end{align*} $$
    Since $\widehat f\in {\mathscr X}_0$ , by Lemma 5.2 we have $\partial _\tau \widehat f, Z\partial _Z\widehat f, \partial _Z^2\widehat f, \partial _Z\widehat f/Z\in {\mathscr X}_0$ , which gives
    $$\begin{align*}\left(\partial_\tau+Z\partial_Z-\lambda\right)\widehat f\in{\mathscr X}_0,\quad \left(\partial_Z^2+(k/Z)\partial_Z\right)\widehat f\in{\mathscr X}_0.\end{align*}$$
    Thus, $\partial _t f\in {\mathscr X}_{\lambda -1}$ , $\Delta f\in {\mathscr X}_{\lambda -2}$ . As a consequence, we have $\partial _t^2 f\in {\mathscr X}_{\lambda -2}$ , $\Box f=-\partial _t^2 f+\Delta f\in {\mathscr X}_{\lambda -2}$ and (also using $fg\in {\mathscr X}_{\lambda +\mu } $ for $ f\in {\mathscr X}_{\lambda }$ , $ g\in {\mathscr X}_{\mu }$ )
    $$\begin{align*}\Box(fg)\in{\mathscr X}_{(\lambda+\mu)-2}, \quad (\Box f)g\in{\mathscr X}_{(\lambda-2)+\mu}, \quad f\Box g\in{\mathscr X}_{\lambda+(\mu-2)},\end{align*}$$
    hence,
    $$ \begin{align*} &\partial^\alpha f\partial_\alpha g=[\Box(fg)-(\Box f)g-f\Box g]/2\in{\mathscr X}_{\lambda+\mu-2},\\ & \partial^\alpha(f\partial_\alpha g)=\partial^\alpha f\partial_\alpha g+f\Box g\in{\mathscr X}_{\lambda+\mu-2}. \end{align*} $$
  2. (iii) Let $\lambda ,\mu \in \mathbb {R}$ and $j\in \mathbb {Z}_{\geq 0}$ be such that $\lambda \geq j+\mu $ . Let $f(t,x)=(T-t)^\lambda \widehat f(Z)$ for some $\widehat f\in C^\infty _{\text {e}}([0,+\infty ))$ . Assume that $\alpha _0,\alpha _1,\cdots ,\alpha _d\in \mathbb {Z}_{\geq 0}$ are such that $\alpha _0+\cdots +\alpha _d=j$ . We only need to prove that

    (5.11) $$ \begin{align} (T-t)^{-\mu}\partial_t^{\alpha_0}\partial_{x_1}^{\alpha_1}\cdots\partial_{x_d}^{\alpha_d}f\in L^\infty(\mathcal C). \end{align} $$
    Let $\widetilde f(x):=\widehat f(|x|)$ for $x\in \mathbb {R}^d$ , then by Lemma 5.1 we have $\widetilde f\in C^\infty (\mathbb {R}^d)$ . Let $j'=\alpha _1+\cdots +\alpha _d\in \mathbb {Z}\cap [0,j]$ , and we let
    $$\begin{align*}f_\alpha:=(-(\lambda-j')+x\cdot\nabla_x)(-(\lambda-j'-1)+x\cdot\nabla_x)\cdots(-(\lambda-j+1)+x\cdot\nabla_x)\partial_{x_1}^{\alpha_1}\cdots\partial_{x_d}^{\alpha_d}\widetilde f.\end{align*}$$
    Then $f_\alpha \in C^\infty (\mathbb {R}^d)$ and one can check by direct computation that
    $$\begin{align*}\partial_t^{\alpha_0}\partial_{x_1}^{\alpha_1}\cdots\partial_{x_d}^{\alpha_d}f(t,x)=(T-t)^{\lambda-j}f_\alpha(x/(T-t)),\quad \forall\ (t,x)\in[0, T)\times\mathbb{R}^d.\end{align*}$$
    As $|x/(T-t)|<2$ for $(t,x)\in \mathcal C$ and $\lambda -j-\mu \geq 0$ , we have (5.11).
  3. (ii) Let $\lambda ,\mu \in \mathbb {R}$ and $j\in \mathbb {Z}_{\geq 0}$ be such that $\lambda> j+\mu $ . By the definitions of ${\mathscr X}_0$ and ${\mathscr X}_\lambda $ , it suffices to prove $(T-t)^{-\mu }D^jf\in L^\infty (\mathcal C)$ for $f(t,x)=(T-t)^\lambda \widehat f(Z)\tau ^n$ for some $\widehat f\in C^\infty _{\text {e}}([0,+\infty ))$ and some $n\in \mathbb {Z}_{\geq 0}$ . Let $P(\tau ):=\tau ^n$ and $\widetilde P(t):=P(\tau )=P(-\ln (T-t))$ . Then by the induction, for any $i\in \mathbb {Z}_{\geq 0}$ , there is a polynomial $P_i(\tau )$ such that $\widetilde P^{(i)}(t)=(T-t)^{-i}P_i(\tau )$ . Hence,

    (5.12) $$ \begin{align} (T-t)^{i+\varepsilon}\widetilde P^{(i)}(t)\in L^\infty([0,T)),\quad\forall\ i\in\mathbb{Z}_{\geq 0},\ \forall\ \varepsilon>0. \end{align} $$
    Let $\widetilde f(t,x):=(T-t)^\lambda \widehat f(Z)$ for $(t,x)\in [0,T)\times \mathbb {R}^d$ , then $f(t,x)=\widetilde f(t,x)\widetilde P(t)$ for $(t,x)\in [0,T)\times \mathbb {R}^d$ , and by (iii) we have
    (5.13) $$ \begin{align} (T-t)^{i-\lambda}D^i\widetilde f\in L^\infty(\mathcal C),\quad \forall\ i\in\mathbb{Z}_{\geq 0}. \end{align} $$
    Assume that $\alpha _0,\alpha _1,\cdots ,\alpha _d\in \mathbb {Z}_{\geq 0}$ are such that $\alpha _0+\cdots +\alpha _d=j$ . It suffices to prove
    (5.14) $$ \begin{align} (T-t)^{-\mu}\partial_t^{\alpha_0}\partial_{x_1}^{\alpha_1}\cdots\partial_{x_d}^{\alpha_d}(\widetilde f(t,x)\widetilde P(t))\in L^\infty(\mathcal C). \end{align} $$
    By Leibnitz’s product rule, we have
    $$ \begin{align*} &(T-t)^{-\mu}\partial_t^{\alpha_0}\partial_{x_1}^{\alpha_1}\cdots\partial_{x_d}^{\alpha_d}(\widetilde f(t,x)\widetilde P(t))\\&\quad =(T-t)^{-\mu}\sum_{i=0}^{\alpha_0}\binom{\alpha_0}{i}\widetilde P^{(i)}(t)\partial_t^{\alpha_0-i}\partial_{x_1}^{\alpha_1}\cdots\partial_{x_d}^{\alpha_d}\widetilde f(t,x)\\&\quad =\sum_{i=0}^{\alpha_0}\binom{\alpha_0}{i}(T-t)^{\lambda-j-\mu+i}\widetilde P^{(i)}(t)\cdot(T-t)^{j-i-\lambda}\partial_t^{\alpha_0-i}\partial_{x_1}^{\alpha_1}\cdots\partial_{x_d}^{\alpha_d}\widetilde f(t,x). \end{align*} $$
    Then (5.14) follows from $\lambda -j-\mu>0$ , (5.12) and (5.13).

This completes the proof of Lemma 2.3.

5.2 $\mathscr {L}$ acting on ${\mathscr X}_\lambda $

Let’s first compute the linear operator $\mathscr {L}_\lambda $ induced by $\mathscr {L}$ acting on ${\mathscr X}_\lambda $ . The following lemma relies highly on the properties of the leading order profile $(\rho _0,\phi _0)$ . For readers’ convenience, we recall some notations. The linear operator $\mathscr L$ is defined in (2.19):

$$\begin{align*}\mathscr{L}(\phi):=\partial^\alpha\left(\rho_0^2\partial_\alpha\phi-\frac4{p-1}\rho_0^{3-p}\partial_\alpha\phi_0\partial^{\widetilde\alpha}\phi_0\partial_{\widetilde\alpha}\phi\right),\quad\phi=\phi(t,x)=\phi(t,r),\end{align*}$$

where according to (2.7),

$$\begin{align*}\phi_0(t, r)=(T-t)^{1-\beta}\widehat \phi_0(Z),\quad \rho_0(t,x)=(T-t)^{-\frac{2\beta}{p-1}}\widehat\rho_0(Z),\quad Z=\frac{r}{T-t}, \quad r=|x|.\end{align*}$$

By (3.13), $\widehat \phi _0$ satisfies

$$\begin{align*}\widehat\phi_0'(Z)=\frac{(\beta-1)\widehat\phi_0(Z)v(Z)}{1-Zv(Z)},\quad Z\widehat\phi_0'(Z)+(\beta-1)\widehat\phi_0(Z)=\frac{(\beta-1)\widehat\phi_0(Z)}{1-Zv(Z)},\end{align*}$$

and $\widehat \rho _0$ is defined by (2.12):

$$ \begin{align*} \widehat\rho_0(Z):=\frac{(\beta-1)^{\frac{2}{p-1}}\widehat\phi_0(Z)^{\frac{2}{p-1}}(1-v(Z)^2)^{\frac{1}{p-1}}} {(1-Zv(Z))^{\frac{2}{p-1}}}. \end{align*} $$

Lemma 5.3. There exist real-valued $A_0,\widetilde B_0, D_1, D_2\in C_{\operatorname {e}}^\infty ([0,+\infty ))$ and $\widehat B_0\in C_{\operatorname {o}}^\infty ([0,+\infty ))$ with

(5.15) $$ \begin{align} A_0(Z)=\widehat\rho_0(Z)^2\frac{(1-Zv(Z))^2-\ell(v(Z)-Z)^2}{1-v(Z)^2}=\widehat\rho_0(Z)^2\frac{\Delta_Z(Z, v(Z))}{Z(1-v(Z)^2)}, \end{align} $$

such that if we define

$$ \begin{align*} B_0(Z;\lambda):=Z^{-1}\widetilde B_0(Z)+\lambda \widehat B_0(Z), \quad D_0(Z;\lambda):=\lambda D_1(Z)+\lambda^2D_2(Z) \end{align*} $$

and

(5.16) $$ \begin{align} (\mathscr{L}_\lambda f)(Z):=A_0(Z)f"(Z)+B_0(Z;\lambda)f'(Z)+D_0(Z;\lambda)f(Z), \end{align} $$

then there hold (here $\mathscr {L}$ is defined in (2.19) and $\gamma :=4\beta /(p-1)+2=\beta (\ell -1)+2$ )

$$ \begin{align*} \mathscr{L}((T-t)^{\lambda}f(Z))=(T-t)^{\lambda-\gamma}(\mathscr{L}_\lambda f)(Z)\quad \text{for}\,\, f\in C_{\operatorname{e}}^\infty([0,+\infty)), \quad \lambda\in\mathbb{C}, \end{align*} $$

and

(5.17) $$ \begin{align} &\qquad A_0(0)=1,\quad A_0(Z_1)=0, \quad A_0'(Z_1)<0, \end{align} $$
(5.18) $$ \begin{align} &\quad A_0(Z)>0 \ \forall\ Z\in[0, Z_1),\quad A_0(Z)<0\ \forall\ Z\in(Z_1,+\infty), \end{align} $$
(5.19) $$ \begin{align} &\qquad\widehat B_0(Z)>0 \text{ for all }Z>0, \quad \widetilde B_0(0)=k\in \mathbb{Z}_+. \end{align} $$

Proof. We first consider the functions in the form of $(T-t)^{\lambda } f(\tau , Z)$ , where f is a smooth function and

(5.20) $$ \begin{align} \tau=\ln\frac1{T-t},\qquad Z=\frac r{T-t},\qquad r=|x|. \end{align} $$

Let $\gamma :=4\beta /(p-1)+2=\beta (\ell -1)+2$ , and let $f=f(\tau , Z)$ and $\lambda \in \mathbb {C}$ . Now we compute $\mathscr {L}\big ((T-t)^\lambda f\big )$ .

We will use the following identities: for $\lambda ,\mu \in \mathbb {C}, f=f(\tau , Z), g=g(\tau , Z)$ ,

(5.21) $$ \begin{align} &\partial^\alpha\left((T-t)^\lambda f\right)\partial_\alpha\left((T-t)^\mu g\right)\nonumber\\&\quad=(T-t)^{\lambda+\mu-2}\Big[-(\partial_\tau f+Z\partial_Z f-\lambda f) (\partial_\tau g+Z\partial_Zg-\mu g)+\partial_Z f\partial_Zg\Big], \end{align} $$
(5.22) $$ \begin{align}&\partial^\alpha\left((T-t)^\lambda f\partial_\alpha\left((T-t)^\mu g\right)\right)=(T-t)^{\lambda+\mu-2}\Big[-\big(\partial_\tau+Z\partial_Z-(\lambda+\mu-1)\big)\big(f(\partial_\tau g\nonumber \\&\qquad\qquad\qquad\qquad\qquad\qquad\quad\qquad\qquad\qquad + Z\partial_Zg-\mu g)\big)+(\partial_Z+k/Z)(f\partial_Zg)\Big]. \end{align} $$

Readers can check (5.21) and (5.22) by using direct computation.

By (2.7), (5.22) and $\gamma =4\beta /(p-1)+2$ , we have

$$ \begin{align*} \partial^\alpha\left(\rho_0^2\partial_\alpha\left((T-t)^\lambda f\right)\right)&=(T-t)^{\lambda-\gamma}\big\{-\big(\partial_\tau+Z\partial_Z-(\lambda-\gamma+1)\big)\left[\widehat\rho_0(Z)^2(\partial_\tau+Z\partial_Z-\lambda)f\right]\\& \quad +\left(\partial_Z+k/Z\right)\left(\widehat\rho_0(Z)^2\partial_Zf\right)\big\}. \end{align*} $$

By (2.7) and (5.21), we have

$$ \begin{align*} \partial^{\widetilde\alpha}\phi_0\partial_{\widetilde\alpha}\left((T-t)^\lambda f\right)=(T-t)^{\lambda-\beta-1}\left[-(Z\partial_Z\widehat\phi_0-(1-\beta)\widehat\phi_0)(\partial_\tau+Z\partial_Z-\lambda)f+\partial_Z\widehat\phi_0\partial_Zf\right]. \end{align*} $$

It follows from (3.13) that

$$ \begin{align*} -(Z\partial_Z\widehat\phi_0-(1-\beta)\widehat\phi_0)(\partial_\tau+Z\partial_Z-\lambda)f+\partial_Z\widehat\phi_0\partial_Zf&=\frac{(\beta-1)\widehat\phi_0}{1-Zv(Z)}\big(-\partial_\tau+(v-Z)\partial_Z+\lambda\big)f\\&=:g(\tau, Z), \end{align*} $$

and then $\partial ^{\widetilde \alpha }\phi _0\partial _{\widetilde \alpha }\left ((T-t)^\lambda f\right )=(T-t)^{\lambda -\beta -1}g(\tau , Z)$ . By (2.7) and (5.22), we have

$$ \begin{align*} \partial^\alpha&\left(\rho_0^{3-p}\partial_\alpha\phi_0\partial^{\widetilde\alpha}\phi_0\partial_{\widetilde\alpha}\left((T-t)^\lambda f\right)\right)=\partial^\alpha\left((T-t)^{\mu}\widehat\rho_0^{3-p}g\partial_\alpha\left((T-t)^{1-\beta}\widehat\phi_0\right)\right)\\&=(T-t)^{\lambda-\gamma}\Big\{-\big(\partial_\tau+Z\partial_Z-(\lambda-\gamma+1)\big)\left[\widehat\rho_0^{3-p}g\big(Z\partial_Z\widehat\phi_0-(1-\beta)\widehat\phi_0\big)\right]\\&\qquad\qquad\qquad+(\partial_Z+k/Z)\left(\widehat\rho_0^{3-p}g\partial_Z\widehat\phi_0\right)\Big\}, \end{align*} $$

where $\mu =-\frac {2\beta }{p-1}(3-p)+\lambda -\beta -1$ satisfies $\mu +(1-\beta )-2=\lambda -\gamma $ . By (2.12) and (3.13),

$$ \begin{align*} \widehat\rho_0(Z)^{3-p}\frac{(\beta-1)\widehat\phi_0(Z)}{1-Zv(Z)}\big(Z\partial_Z\widehat\phi_0-(1-\beta)\widehat\phi_0\big)&=\widehat\rho_0(Z)^2\widehat\rho_0(Z)^{1-p}\frac{(\beta-1)^2\widehat\phi_0(Z)^2}{(1-Zv(Z))^2}=\frac{\widehat\rho_0(Z)^2}{1-v(Z)^2},\\ \widehat\rho_0(Z)^{3-p}\frac{(\beta-1)\widehat\phi_0(Z)}{1-Zv(Z)}\partial_Z\widehat\phi_0&=\frac{\widehat\rho_0(Z)^2v(Z)}{1-v(Z)^2}, \end{align*} $$

thus

$$ \begin{align*} \widehat\rho_0^{3-p}g\big(Z\partial_Z\widehat\phi_0-(1-\beta)\widehat\phi_0\big)&=\frac{\widehat\rho_0(Z)^2}{1-v(Z)^2}\big(-\partial_\tau+(v(Z)-Z)\partial_Z+\lambda\big)f,\\ \widehat\rho_0^{3-p}g\partial_Z\widehat\phi_0&=\frac{\widehat\rho_0(Z)^2v(Z)}{1-v(Z)^2}\big(-\partial_\tau+(v(Z)-Z)\partial_Z+\lambda\big)f. \end{align*} $$

Therefore,

$$ \begin{align*} &\partial^\alpha\left(\rho_0^{3-p}\partial_\alpha\phi_0\partial^{\widetilde\alpha}\phi_0\partial_{\widetilde\alpha}\left((T-t)^\lambda f\right)\right)\\& \quad =(T-t)^{\lambda-\gamma}\Bigg\{-\big(\partial_\tau+Z\partial_Z-(\lambda-\gamma+1)\big)\left[\frac{\widehat\rho_0(Z)^2}{1-v(Z)^2}\big(-\partial_\tau+(v(Z)-Z)\partial_Z+\lambda\big)f\right]\\&\qquad\qquad\qquad+\left(\partial_Z+\frac kZ\right)\left[\frac{\widehat\rho_0(Z)^2v(Z)}{1-v(Z)^2}\big(-\partial_\tau+(v(Z)-Z)\partial_Z+\lambda\big)f\right]\Bigg\}. \end{align*} $$

Finally, recall that $ \mathscr {L}(\phi )=\partial ^\alpha \left (\rho _0^2\partial _\alpha \phi -\frac 4{p-1}\rho _0^{3-p}\partial _\alpha \phi _0\partial ^{\widetilde \alpha }\phi _0\partial _{\widetilde \alpha }\phi \right )$ , $\frac 4{p-1}=\ell -1$ , we obtain

$$ \begin{align*} \mathscr{L}\big((T-t)^\lambda f\big)=(T-t)^{\lambda-\gamma}\Bigg\{&-\big(\partial_\tau+Z\partial_Z-(\lambda-\gamma+1)\big)\left[\widehat\rho_0(Z)^2(\partial_\tau+Z\partial_Z-\lambda)f\right]\\&+\left(\partial_Z+k/Z\right)\left(\widehat\rho_0(Z)^2\partial_Zf\right)\\&+(\ell-1)\big(\partial_\tau+Z\partial_Z-(\lambda-\gamma+1)\big)\left[\frac{\widehat\rho_0(Z)^2}{1-v(Z)^2}\big(-\partial_\tau+(v(Z)-Z)\partial_Z+\lambda\big)f\right]\\&-(\ell-1)\left(\partial_Z+\frac kZ\right)\left[\frac{\widehat\rho_0(Z)^2v(Z)}{1-v(Z)^2}\big(-\partial_\tau+(v(Z)-Z)\partial_Z+\lambda\big)f\right]\Bigg\}. \end{align*} $$

For any $\lambda \in \mathbb {C}$ , we define a linear operator $\mathscr {L}_\lambda $ by

(5.23) $$ \begin{align} (\mathscr{L}_\lambda f)(Z):=&-\big(Z\partial_Z-(\lambda-\gamma+1)\big)\left[\widehat\rho_0(Z)^2(Z\partial_Z-\lambda)f\right]+\left(\partial_Z+k/Z\right)\left(\widehat\rho_0(Z)^2\partial_Zf\right)\nonumber\\&+(\ell-1)\big(Z\partial_Z-(\lambda-\gamma+1)\big)\left[\frac{\widehat\rho_0(Z)^2}{1-v(Z)^2}\big((v(Z)-Z)\partial_Z+\lambda\big)f\right]\nonumber\\&-(\ell-1)\left(\partial_Z+\frac kZ\right)\left[\frac{\widehat\rho_0(Z)^2v(Z)}{1-v(Z)^2}\big((v(Z)-Z)\partial_Z+\lambda\big)f\right], \end{align} $$

where $f=f(Z)$ depends only on $Z\in [0,+\infty )$ (not on $ \tau $ ). Assume that $f=f(Z)=f(Z; \lambda )$ satisfies $(\mathscr {L}_\lambda f)(Z)=g(Z)=g(Z; \lambda )$ , then (here $\mathscr {L}$ and $\mathscr {L}_\lambda $ do not act on $ \lambda $ )

(5.24) $$ \begin{align} \mathscr{L}\big((T-t)^\lambda f\big)=(T-t)^{\lambda-\gamma}g(Z). \end{align} $$

Now it is enough to prove that $\mathscr {L}_\lambda $ defined in (5.23) can be written in the form of (5.16) with $B_0(Z;\lambda )=Z^{-1}\widetilde B_0(Z)+\lambda \widehat B_0(Z)$ , $D_0(Z;\lambda )=\lambda D_1(Z)+\lambda ^2D_2(Z)$ and $A_0(Z),\widetilde B_0(Z),D_1(Z), D_2(Z)\in C_{\operatorname {e}}^\infty ([0,+\infty ))$ , $\widehat B_0\in C_{\operatorname {o}}^\infty ([0,+\infty ))$ satisfying (5.15), (5.17), (5.18), (5.19).

Comparing the coefficients of $ \partial _Z^j$ ( $j=0,1,2$ ) in (5.23) and (5.16), we find

$$ \begin{align*} A_0(Z)&=\widehat\rho_0(Z)^2(1-Z^2)+(\ell-1)\frac{\widehat\rho_0(Z)^2}{1-v(Z)^2}Z(v(Z)-Z)-(\ell-1)\frac{\widehat\rho_0(Z)^2v(Z)}{1-v(Z)^2}(v(Z)-Z)\\&=\widehat\rho_0(Z)^2\left((1-Z^2)-(\ell-1)\frac{(v(Z)-Z)^2}{1-v(Z)^2}\right)=\widehat\rho_0(Z)^2\frac{(1-Zv(Z))^2-\ell(v(Z)-Z)^2}{1-v(Z)^2},\\B_0(Z;\lambda)&=-Z^2\partial_Z(\widehat\rho_0^2)-Z\widehat\rho_0^2+\lambda Z\widehat\rho_0^2+(\lambda-\gamma+1)Z\widehat\rho_0^2+\partial_Z(\widehat\rho_0^2)+(k/Z)\widehat\rho_0^2\\&\quad+(\ell-1)Z\partial_Z\left(\frac{\widehat\rho_0^2}{1-v^2}\right)(v-Z)+\frac{(\ell-1)\widehat\rho_0^2}{1-v^2}\big(Zv'-Z-(\lambda-\gamma+1)(v-Z)+\lambda Z\big)\\&\quad-(\ell-1)\partial_Z\left(\frac{\widehat\rho_0^2v}{1-v^2}\right)(v-Z)-(\ell-1)\frac{\widehat\rho_0^2v}{1-v^2}\big(v'-1+(k/Z)(v-Z)+\lambda\big),\\D_0(Z;\lambda)&=-\big(Z\partial_Z-(\lambda-\gamma+1)\big)\left(-\lambda\widehat\rho_0(Z)^2\right)+(\ell-1)\big(Z\partial_Z-(\lambda-\gamma+1)\big)\left(\lambda\frac{\widehat\rho_0(Z)^2}{1-v(Z)^2}\right)\\&\quad-(\ell-1)\left(\partial_Z+\frac kZ\right)\left(\lambda\frac{\widehat\rho_0(Z)^2v(Z)}{1-v(Z)^2}\right). \end{align*} $$

Then (5.23) and (5.16) are equivalent and $A_0$ satisfies (5.15).

By the expression of $ B_0(Z;\lambda ) $ , we have $B_0(Z;\lambda )=Z^{-1}\widetilde B_0(Z)+\lambda \widehat B_0(Z)$ with

$$ \begin{align*} \widehat B_0(Z)&=2Z\widehat\rho_0^2+\frac{(\ell-1)\widehat\rho_0^2}{1-v^2}(2Z-v)-\frac{(\ell-1)\widehat\rho_0^2v}{1-v^2} =\frac{2\widehat\rho_0^2}{1-v^2}\big(Z(1-v^2)+(\ell-1)(Z-v)\big).\\\widetilde{B}_0(Z)&= k\widehat \rho_0^2+(1-Z^2)Z\partial_Z(\widehat\rho_0^2)-\gamma Z^2\widehat\rho_0^2+(\ell-1)(v-Z)\bigg[Z^2\partial_Z\left(\frac{\widehat\rho_0^2}{1-v^2}\right)-Z\partial_Z\left(\frac{\widehat\rho_0^2v}{1-v^2}\right)\bigg]\\&\quad+(\ell-1)\frac{\widehat\rho_0^2}{1-v^2}\left(Z(Z-v)v'+(k+\gamma)Zv-\gamma Z^2-kv^2\right). \end{align*} $$

By the expression of $ D_0(Z;\lambda )$ , we have $D_0(Z;\lambda )=\lambda D_1(Z)+\lambda ^2D_2(Z)$ with

$$ \begin{align*} D_1(Z)=&\big(Z\partial_Z+\gamma-1\big)\left(\widehat\rho_0(Z)^2\right)+(\ell-1)\big(Z\partial_Z+\gamma-1\big)\left(\frac{\widehat\rho_0(Z)^2}{1-v(Z)^2}\right)\\ &-(\ell-1)\left(\partial_Z+\frac kZ\right)\left(\frac{\widehat\rho_0(Z)^2v(Z)}{1-v(Z)^2}\right),\\ D_2(Z)=&-\widehat\rho_0(Z)^2-(\ell-1)\frac{\widehat\rho_0(Z)^2}{1-v(Z)^2}=-\widehat\rho_0(Z)^2\frac{\ell-v(Z)^2}{1-v(Z)^2}. \end{align*} $$

It remains to prove that $A_0(Z),\widetilde B_0(Z),D_1(Z), D_2(Z)\in C_{\operatorname {e}}^\infty ([0,+\infty ))$ , $\widehat B_0\in C_{\operatorname {o}}^\infty ([0,+\infty ))$ and (5.17), (5.18), (5.19).

By Lemma A.6 we have $\widehat \rho _0\in C_{\text {e}}^\infty ([0,+\infty )), v\in C_{\text {o}}^\infty ([0,+\infty ))$ ; by (5.5) and (5.6) we have $\widehat \rho _0^2\in C_{\text {e}}^\infty ([0,+\infty ))$ , $(v(Z)-Z)^2\in C_{\text {e}}^\infty ([0,+\infty ))$ and $1-v(Z)^2\in C_{\text {e}}^\infty ([0,+\infty ))$ ; by (5.6) we have $Zv(Z)\in C_{\text {e}}^\infty ([0,+\infty ))$ , hence $1-Zv(Z)\in C_{\text {e}}^\infty ([0,+\infty ))$ , then using (5.5) we get $(1-Zv(Z))^2\in C_{\text {e}}^\infty ([0,+\infty ))$ . Therefore, by $v\in (-1, 1)$ (see Assumption 1), (5.8) and (5.15) we have $A_0(Z)\in C_{\text {e}}^\infty ([0,+\infty ))$ .

Similarly, by Lemma A.6 and (5.3)–(5.8) we have $\widehat B_0(Z)\in C_{\text {o}}^\infty ([0,+\infty ))$ and $\widetilde B_0(Z)$ , $D_1(Z)$ , $D_2(Z)\in C_{\operatorname {e}}^\infty ([0,+\infty ))$ .

It follows from $\widehat \rho _0(0)=1$ and $v(0)=0$ that $A_0(0)=1$ . By Remark 2.2 and $\widehat \rho _0(Z)>0$ for all $Z\in [0,+\infty )$ , we have $A_0(Z_1)=0$ and (5.18). Let $\Delta _0(Z)=\Delta _Z(Z, v(Z))$ , then by $\Delta _0(Z_1)=0$ and Remark 2.2, $A_0'(Z_1)=\widehat \rho _0(Z_1)^2\Delta _0'(Z_1)/(Z_1(1-v(Z_1)^2))\neq 0$ . This along with $A_0(Z_1)<A_0(Z)$ for all $Z\in [0, Z_1)$ implies $A_0'(Z_1)< 0$ . So we have (5.17).

As $v(Z)\in (0, 1)$ , $v(Z)<Z$ for all $Z>0$ (see Remark 2.2) and $\widehat \phi _0(Z)>0$ for all $Z\in [0,+\infty )$ , we have $\widehat B_0(Z)>0$ for all $Z>0$ ; as $\widehat \rho _0(0)=1$ and $v(0)=0$ we have $\widetilde B_0(0)=k\in \mathbb {Z}_{+} $ . This proves (5.19).

Next we compute the dual operator of $\mathscr {L}_\lambda $ . For any $\lambda \in \mathbb {C}$ , we define an operator $\mathscr {L}_\lambda ^*$ , called the dual operator of $\mathscr {L}_\lambda $ , by

(5.25) $$ \begin{align} \int_0^\infty (\mathscr{L}_\lambda f)(Z)g(Z)Z^k\,\mathrm dZ=\int_0^\infty f(Z)(\mathscr{L}_\lambda^*g)(Z)Z^k\,\mathrm dZ,\quad \forall\ f,g\in C_c^\infty((0,+\infty)). \end{align} $$

Lemma 5.4. For any $\lambda \in \mathbb {C}$ , we have $\mathscr {L}_\lambda ^*=\mathscr {L}_{-\lambda +\gamma -k-2}$ .

Proof. By the definition (5.25), it is enough to prove that

(5.26) $$ \begin{align}\int_0^\infty(\mathscr{L}_\lambda f)(Z)g(Z)Z^k\,\mathrm{d}Z=\int_0^\infty f(Z)(\mathscr{L}_{-\lambda+\gamma-k-2} g)(Z)Z^k\,\mathrm{d}Z\end{align} $$

for all $\lambda \in \mathbb {C}$ and $f,g\in C_c^\infty ((0,+\infty ))$ . We fix $\lambda \in \mathbb {C}$ and $f,g\in C_c^\infty ((0,+\infty ))$ . Let

$$ \begin{align*}\widetilde{f}(t,x):=(T-t)^{\lambda}f(Z),\quad \widetilde{g}(t,x):=(T-t)^{-\lambda+\gamma-k-2}g(Z), \quad \forall\ (t,x)\in [0, T)\times \mathbb{R}^d,\end{align*} $$

recalling $Z=|x|/(T-t)$ . Then by Lemma 5.3, we have $\mathscr {L}\widetilde {f}(t,x)=(T-t)^{\lambda -\gamma }(\mathscr {L}_\lambda f)(Z) $ and $\mathscr {L}\widetilde {g}(t,x)=(T-t)^{-\lambda -k-2}(\mathscr {L}_{-\lambda +\gamma -k-2} g)(Z) $ , thus $(\mathscr {L}\widetilde {f}\cdot \widetilde {g})(t,x)=(T-t)^{-k-2}(\mathscr {L}_\lambda f)(Z)g(Z) $ , and $(\widetilde {f}\cdot \mathscr {L}\widetilde {g})(t,x)=(T-t)^{-k-2}f(Z)(\mathscr {L}_{-\lambda +\gamma -k-2} g)(Z)$ . Recall that $d=k+1$ , $ Z=|x|/(T-t)$ , then we have (here $|S^k| $ is the area of the unit sphere $S^k$ in $\mathbb {R}^d=\mathbb {R}^{k+1}$ )

$$ \begin{align*} &\int_{\mathbb{R}^d}(\mathscr{L}\widetilde{f}\cdot\widetilde{g})(t,x)dx=(T-t)^{-1}|S^k|\int_0^\infty(\mathscr{L}_\lambda f)(Z)g(Z)Z^k\,\mathrm{d}Z,\\ &\int_{\mathbb{R}^d}(\widetilde{f}\cdot\mathscr{L}\widetilde{g})(t,x)dx=(T-t)^{-1}|S^k|\int_0^\infty f(Z)(\mathscr{L}_{-\lambda+\gamma-k-2} g)(Z)Z^k\,\mathrm{d}Z, \end{align*} $$

for all $t\in [0, T)$ . Thus, it is enough to prove that

(5.27) $$ \begin{align}\int_{\mathbb{R}^d}(\mathscr{L}\widetilde{f}\cdot\widetilde{g})(t,x)\,\mathrm dx=\int_{\mathbb{R}^d}(\widetilde{f}\cdot\mathscr{L}\widetilde{g})(t,x)\,\mathrm dx, \quad\ \forall\ t\in[0,T).\end{align} $$

Let $ \mathcal {J}:=\mathscr {L}\widetilde {f}\cdot \widetilde {g}-\widetilde {f}\cdot \mathscr {L}\widetilde {g}$ . Then (5.27) is further reduced to

(5.28) $$ \begin{align}\int_{\mathbb{R}^d}\mathcal{J}(t,x)\,\mathrm dx=0, \quad\ \forall\ t\in[0,T). \end{align} $$

By the definition of $\mathscr {L} $ in (2.19), we can write $ \mathcal {J}$ in the divergence form $ \mathcal {J}=\partial ^\alpha P_{\alpha }$ with

$$ \begin{align*} &P_{\alpha}:=\rho_0^2(\partial_\alpha\widetilde{f}\widetilde{g}-\widetilde{f}\partial_\alpha\widetilde{g})- \frac4{p-1}\rho_0^{3-p}\partial_\alpha\phi_0(\partial^{\widetilde\alpha}\phi_0\partial_{\widetilde\alpha}\widetilde{f}\widetilde{g}- \widetilde{f}\partial^{\widetilde\alpha}\phi_0\partial_{\widetilde\alpha}\widetilde{g}),\quad \forall\ \alpha\in\mathbb{Z}\cap[0, d]. \end{align*} $$

Let $E(t):=\int _{\mathbb {R}^d}P_0(t,x)\,\mathrm dx$ for $t\in [0, T)$ . By the divergence theorem (recalling $\partial ^0=-\partial _0=-\partial _t$ and the fact that $\operatorname {supp}_xP_\alpha (t,\cdot )$ is compact for each t and $\alpha $ ), we have

(5.29) $$ \begin{align} -\frac{\mathrm d}{\mathrm dt}E(t)=\int_{\mathbb{R}^d}\partial^0P_0(t,x)\,\mathrm dx=\int_{\mathbb{R}^d}\partial^\alpha P_\alpha(t,x)\,\mathrm dx=\int_{\mathbb{R}^d}\mathcal{J}(t,x)\,\mathrm dx,\quad\forall\ t\in[0, T). \end{align} $$

Thus, it is enough to prove that $E(t)$ is constant in t. We can write $P_0=P_{0,1}-\frac 4{p-1}P_{0,2}$ with

$$ \begin{align*} &P_{0,1}:=\rho_0^2(\partial_t\widetilde{f}\widetilde{g}-\widetilde{f}\partial_t\widetilde{g}),\quad P_{0,2}:=\rho_0^{3-p}\partial_t\phi_0(\partial^{\widetilde\alpha}\phi_0\partial_{\widetilde\alpha}\widetilde{f}\widetilde{g}- \widetilde{f}\partial^{\widetilde\alpha}\phi_0\partial_{\widetilde\alpha}\widetilde{g}). \end{align*} $$

As $\widetilde {f}(t,x)=(T-t)^{\lambda }f(Z) $ , $\widetilde {g}(t,x)=(T-t)^{-\lambda +\gamma -k-2}g(Z) $ , we have

$$ \begin{align*} &\partial_t\widetilde{f}(t,x)=(T-t)^{\lambda-1}f_1(Z)\text{ with } f_1(Z):=-\lambda f(Z)+Zf'(Z),\\& \partial_t\widetilde{g}(t,x)=(T-t)^{-\lambda+\gamma-k-3}g_1(Z)\text{ with } g_1(Z):=-(-\lambda+\gamma-k-2)g(Z)+Zg'(Z). \end{align*} $$

Then by (2.7), $\gamma =\frac {4\beta }{p-1}+2 $ and $d=k+1$ , we get

$$ \begin{align*} P_{0,1}(t,x)&=(T-t)^{-\frac{4\beta}{p-1}+\gamma-k-3}\widehat\rho_0(Z)^2[f_1(Z)g(Z)-f(Z)g_1(Z)]\\&=(T-t)^{-d}\widehat\rho_0(Z)^2[f_1(Z)g(Z)-f(Z)g_1(Z)]. \end{align*} $$

As $\widetilde {f}(t,x)=(T-t)^{\lambda }f(Z) $ , $\widetilde {g}(t,x)=(T-t)^{-\lambda +\gamma -k-2}g(Z) $ , we get by (2.7) and (5.21) that

$$ \begin{align*} &\partial^{\widetilde\alpha}\phi_0\partial_{\widetilde\alpha}\widetilde{f}(t,x)= (T-t)^{\lambda-\beta-1}f_2(Z),\quad \partial^{\widetilde\alpha}\phi_0\partial_{\widetilde\alpha}\widetilde{g}(t,x)= (T-t)^{-\lambda+\gamma-k-3-\beta}g_2(Z), \end{align*} $$

where

$$ \begin{align*} &f_2:=-(Z\partial_Z\widehat\phi_0-(1-\beta)\widehat\phi_0)(Z\partial_Z-\lambda)f+\partial_Z\widehat\phi_0\partial_Zf,\\ &g_2:=-(Z\partial_Z\widehat\phi_0-(1-\beta)\widehat\phi_0)(Z\partial_Z+\lambda-\gamma+k+2)g+\partial_Z\widehat\phi_0\partial_Zg. \end{align*} $$

Then by (2.7), (A.2) and $\gamma =\frac {4\beta }{p-1}+2=\frac {2(3-p)\beta }{p-1}+2\beta +2 $ , $d=k+1$ , we have

$$ \begin{align*} &P_{0,2}(t,x)\\& \quad =(T-t)^{-\frac{2(3-p)\beta}{p-1}-2\beta+\gamma-k-3}\widehat\rho_0(Z)^{3-p}[(\beta-1)\widehat\phi_0(Z)+Z\widehat\phi_0'(Z)] [f_2(Z)g(Z)-f(Z)g_2(Z)] \\& \quad =(T-t)^{-d}\widehat\rho_0(Z)^{3-p}[(\beta-1)\widehat\phi_0(Z)+Z\widehat\phi_0'(Z)] [f_2(Z)g(Z)-f(Z)g_2(Z)]. \end{align*} $$

As $P_0=P_{0,1}-\frac 4{p-1}P_{0,2}$ , we have $P_{0}(t,x)=(T-t)^{-d}H(Z) $ with

$$ \begin{align*} H(Z)&:=\widehat\rho_0(Z)^2[f_1(Z)g(Z)-f(Z)g_1(Z)]\\&\quad-\frac4{p-1}\widehat\rho_0(Z)^{3-p}[(\beta-1)\widehat\phi_0(Z)+Z\widehat\phi_0'(Z)] [f_2(Z)g(Z)-f(Z)g_2(Z)]. \end{align*} $$

Then by $d=k+1$ , $ Z=|x|/(T-t)$ , we have $E(t)=\int _{\mathbb {R}^d}P_0(t,x)\,\mathrm dx=|S^k|\int _0^\infty H(Z)Z^k\,\mathrm {d}Z$ , which is constant in t. By (5.29), we have (5.28), thus (5.27) and (5.26).

5.3 Surjection of $\mathscr {L}$

This subsection is devoted to the proof of Proposition 2.4, i.e., $\mathscr {L}:{\mathscr X}_\lambda \to {\mathscr X}_{\lambda -\gamma }$ is surjective for all $\lambda \in \mathbb {C}$ . For this, it suffices to show that

Lemma 5.5. If $R\in (k,+\infty )$ , then the linear operator $\mathscr {L}:{\mathscr X}_\lambda \to {\mathscr X}_{\lambda -\gamma }$ is surjective for all $\lambda \in B_R:=\{\lambda \in \mathbb {C}:|\lambda |<R\}$ , where $\gamma :=4\beta /(p-1)+2=\beta (\ell -1)+2$ .

From here until the end of this section, we fix an $R\in (k,+\infty )$ .

We consider functions depending analytically on a complex number $\lambda $ . Let $I\subset [0,+\infty )$ be an interval and let $\Omega \subset \mathbb {C}$ be an open set. We define

$$ \begin{align*} \operatorname{Hol}(\Omega)&:=\{\text{all holomorphic function on }\Omega\},\\ \mathcal{H}_I(\Omega)&:=\left\{f=f(Z;\lambda)\in C^\infty(I\times\Omega;\mathbb{C}): f(Z;\cdot)\in\operatorname{Hol}(\Omega)\text{ for all }Z\in I\right\},\\ \mathcal H_I^{\text{e}}(\Omega)&:=\{f\in C^\infty(I\times\Omega):\exists\ \widetilde{f}\in\mathcal H_{I^2}(\Omega),\text{ s.t. }f(Z;\lambda)=\widetilde f(Z^2;\lambda) \quad \forall\ Z\in I,\ \lambda\in \Omega\}. \end{align*} $$

Then $ \operatorname {Hol}(\Omega )$ , $ \mathcal H_I(\Omega )$ , $ \mathcal H_I^{\text {e}}(\Omega )$ are rings. Moreover, we have

$$\begin{align*}\mathcal H_{[0,+\infty)}^{\text{e}}(\Omega)=\mathcal H_{[0,a_2)}^{\text{e}}(\Omega)\cap\mathcal H_{(a_1,+\infty)}(\Omega), \quad \forall\ 0<a_1<a_2<+\infty.\end{align*}$$

The proof of Lemma 5.5 is based on the following result, which will be proved in next subsection.

Lemma 5.6. There exists $ \varphi \in \operatorname {Hol}(B_R)\setminus \{0\}$ such that if $g\in C_{\operatorname {e}}^\infty ([0,+\infty ))$ , then there exists $f=f(Z;\lambda )\in \mathcal H_{[0,+\infty )}^{\operatorname {e}}(B_R)$ such that $\mathscr {L}_\lambda f(\cdot ;\lambda )=\varphi (\lambda )\cdot g$ on $(0,+\infty )$ (for all $ \lambda \in B_R$ ).

Proof of Lemma 5.5.

We first prove that $\mathscr {L}$ maps ${\mathscr X}_\lambda $ to ${\mathscr X}_{\lambda -\gamma }$ .

Recall that $\mathscr {L}(\phi )=\partial ^\alpha \left (\rho _0^2\partial _\alpha \phi -\frac 4{p-1}\rho _0^{3-p}\partial _\alpha \phi _0\partial ^{\widetilde \alpha }\phi _0\partial _{\widetilde \alpha }\phi \right )$ , $\rho _0^2\in {\mathscr X}_{2\mu _0}$ , $\rho _0^{3-p}\in {\mathscr X}_{(3-p)\mu _0}$ , $\phi _0\in {\mathscr X}_{\lambda _0}$ , $\lambda _0=1-\beta $ , $\mu _0=-\frac {2\beta }{p-1}$ (see (2.22)) and $\gamma =4\beta /(p-1)+2$ . If $ \phi \in {\mathscr X}_\lambda $ , by Lemma 2.3 (i), we have $\partial ^\alpha (\rho _0^2\partial _\alpha \phi )\in {\mathscr X}_{\lambda -\gamma }$ , $\partial ^\alpha (\rho _0^{3-p}\partial _\alpha \phi _0\partial ^{\widetilde \alpha }\phi _0\partial _{\widetilde \alpha }\phi )\in {\mathscr X}_{\lambda -\gamma }$ , where we have used that $\lambda +2\mu _{0}-2=\lambda -\gamma =\lambda +\lambda _0-2+(3-p)\mu _0+\lambda _0-2$ , thus $ \mathscr {L}(\phi )\in {\mathscr X}_{\lambda -\gamma } $ .

Now we prove that $\mathscr {L}$ is surjective. By the definitions of ${\mathscr X}_0$ and ${\mathscr X}_\lambda $ , it suffices to prove that for every $g\in C^\infty _{\text {e}}([0,+\infty ))$ , $n\in \mathbb {Z}_{\geq 0}$ and $\lambda _*\in B_R$ , there exists $F_n\in {\mathscr X}_{\lambda _*}$ such that $\mathscr {L} F_n(t,x)=(T-t)^{\lambda _*-\gamma } g(Z)\tau ^n/n!$ . Now we fix $g\in C_{\text {e}}^\infty ([0,+\infty ))$ and $\lambda _*\in B_R$ .

By Lemma 5.6, there exist $ \varphi \in \operatorname {Hol}(B_R)\setminus \{0\}$ and a function $f=f(Z;\lambda )\in \mathcal H_{[0,+\infty )}^{\operatorname {e}}(B_R)$ such that $\mathscr {L}_\lambda f(Z;\lambda )=\varphi (\lambda )g(Z)$ for $Z\in (0,+\infty )$ , $ \lambda \in B_R$ . As $f\in \mathcal H_{[0,+\infty )}^{\operatorname {e}}(B_R)$ there exists $\widetilde {f}\in \mathcal H_{[0,+\infty )}(B_R)$ such that $f(Z;\lambda )=\widetilde {f}(Z^2;\lambda )$ for $Z\in [0,+\infty )$ , $ \lambda \in B_R$ . As $\lambda _*\in B_R$ , there exist $\delta _*>0$ and $m_*\in \mathbb {Z}_{\geq 0}$ such that $B_{2\delta _*}(\lambda _*)\subset B_R$ and

$$\begin{align*}\varphi(\lambda)=(\lambda-\lambda_*)^{m_*}\widetilde\varphi(\lambda)\text{ with }\widetilde \varphi(\lambda)\neq0,\quad\forall\ \lambda\in \Omega_*:=B_{2\delta_*}(\lambda_*),\text{ where }\widetilde \varphi\in \operatorname{Hol}(\Omega_*).\end{align*}$$

Here $B_{2\delta _*}(\lambda _*):=\{\lambda \in \mathbb {C}:|\lambda -\lambda _*|<2\delta _*\}$ and we have used the fact that if $\varphi \in \operatorname {Hol}(\Omega )\setminus \{0\}$ , then the zero set $\mathcal Z(\varphi ):=\{\lambda \in \Omega :\varphi (\lambda )=0\}$ is discrete.

Let $\widetilde F({Z};\lambda ):=\widetilde f({Z};\lambda )/\widetilde \varphi (\lambda )$ , $F(Z;\lambda ):=\widetilde F(Z^2;\lambda )$ for $Z\in [0,+\infty )$ , $ \lambda \in B_R$ . Then $\widetilde F\in \mathcal H_{[0,+\infty )}(\Omega _*)$ , $F\in \mathcal H_{[0,+\infty )}^{\operatorname {e}}(\Omega _*)$ , $F(Z;\lambda )=f(Z;\lambda )/\widetilde \varphi (\lambda )$ , and

$$ \begin{align*}\mathscr{L}_\lambda F(Z;\lambda)=\varphi(\lambda)g(Z)/\widetilde \varphi(\lambda)=(\lambda-\lambda_*)^{m_*}g(Z),\quad\forall\ Z\in (0,+\infty), \lambda\in \Omega_*.\end{align*} $$

By Lemma 5.3, we have

$$ \begin{align*}\mathscr{L}((T-t)^{\lambda}F(Z;\lambda))=(T-t)^{\lambda-\gamma}\mathscr{L}_\lambda F(Z;\lambda)=(T-t)^{\lambda-\gamma}(\lambda-\lambda_*)^{m_*}g(Z)\end{align*} $$

for all $\lambda \in \Omega _*$ and $Z\in (0, +\infty )$ . Let

$$ \begin{align*}F_*(t,x;\lambda):=(T-t)^{\lambda}F(Z;\lambda)=(T-t)^{\lambda}\widetilde F(Z^2;\lambda),\quad G(t,x;\lambda):=(T-t)^{\lambda-\gamma}(\lambda-\lambda_*)^{m_*}g(Z).\end{align*} $$

Then $F_*,G\in C^{\infty }([0,T)\times \mathbb {R}^d\times \Omega _*) $ (as $Z^2=|x|^2/(T-t)^2$ is smooth on $[0,T)\times \mathbb {R}^d $ ) and $ \mathscr {L} F_*(t,x;\lambda )=G(t,x;\lambda )$ on $[0,T)\times \mathbb {R}^d\times \Omega _* $ (the case $Z=0$ follows by continuity).

Recall that $ \tau =\ln \frac {1}{T-t}$ and then

$$ \begin{align*} G(t,x;\lambda)=(T-t)^{\lambda_*-\gamma}\mathrm{e}^{-(\lambda-\lambda_*)\tau}(\lambda-\lambda_*)^{m_*}g(Z)=\sum_{n=0}^{\infty}(T-t)^{\lambda_*-\gamma}\frac{(-\tau)^n}{n!}(\lambda-\lambda_*)^{m_*+n}g(Z) \end{align*} $$

locally uniformly on $[0,T)\times \mathbb {R}^d\times B_{\delta _*}(\lambda _*)$ . By Cauchy’s integration formula (Theorem 4.4 in Chapter 2 of [Reference Stein and Shakarchi69]), we have (for $n\in \mathbb {Z}_{\geq 0}$ )

$$ \begin{align*} (T-t)^{\lambda_*-\gamma} g(Z)\frac{\tau^n}{n!}&=\frac{(-1)^n}{2\pi\text{i}}\oint_{|\lambda-\lambda_*|=\delta_*}\frac{G(t,x;\lambda)}{(\lambda-\lambda_*)^{m_*+n+1}}\,\mathrm d\lambda\\&= \frac{\delta_*^{-m_*-n}}{2\pi(-1)^n}\int_0^{2\pi}G(t,x;\lambda_*+\delta_*\mathrm{e}^{\text{i} \theta})\mathrm{e}^{-\text{i} (m_*+n)\theta}\,\mathrm d\theta. \end{align*} $$

Now let (for $n\in \mathbb {Z}_{\geq 0}$ )

$$ \begin{align*} F_n(t,x)&:=\frac{(-1)^n}{2\pi\text{i}}\oint_{|\lambda-\lambda_*|=\delta_*}\frac{F_*(t,x;\lambda)}{(\lambda-\lambda_*)^{m_*+n+1}}\,\mathrm d\lambda\\&= \frac{\delta_*^{-m_*-n}}{2\pi(-1)^n}\int_0^{2\pi}F_*(t,x;\lambda_*+\delta_*\mathrm{e}^{\text{i} \theta})\mathrm{e}^{-\text{i} (m_*+n)\theta}\,\mathrm d\theta. \end{align*} $$

Then $F_n\in C^{\infty }([0,T)\times \mathbb {R}^d) $ and $\mathscr {L} F_n(t,x)=(T-t)^{\lambda _*-\gamma } g(Z)\tau ^n/n!$ . It remains to prove that $F_n\in {\mathscr X}_{\lambda _*}$ .

As $ \tau =\ln \frac {1}{T-t}$ , $ F_*(t,x;\lambda )=(T-t)^{\lambda }\widetilde F(Z^2;\lambda )$ then

$$ \begin{align*} F_*(t,x;\lambda)=(T-t)^{\lambda_*}\mathrm{e}^{-(\lambda-\lambda_*)\tau}\widetilde F(Z^2;\lambda)=\sum_{j=0}^{\infty}(T-t)^{\lambda_*}\frac{(-\tau)^j}{j!}(\lambda-\lambda_*)^{j}\widetilde F(Z^2;\lambda), \end{align*} $$

locally uniformly on $[0,T)\times \mathbb {R}^d\times B_{\delta _*}(\lambda _*)$ , so we have

$$ \begin{align*} F_n(t,x)&=\sum_{j=0}^{\infty}(T-t)^{\lambda_*}\frac{(-\tau)^j}{j!}F_{n,j}(Z^2),\qquad\text{where}\\ F_{n,j}(\widetilde{Z})&:= \frac{(-1)^n}{2\pi\text{i}}\oint_{|\lambda-\lambda_*|=\delta_*}\frac{(\lambda-\lambda_*)^{j}\widetilde F(\widetilde{Z};\lambda)}{(\lambda-\lambda_*)^{m_*+n+1}}\,\mathrm d\lambda\\ &=\frac{\delta_*^{j-m_*-n}}{2\pi(-1)^n}\int_0^{2\pi}\widetilde F(\widetilde{Z};\lambda_*+\delta_*\mathrm{e}^{\text{i} \theta})\mathrm{e}^{\text{i} (j-m_*-n)\theta}\,\mathrm d\theta. \end{align*} $$

As $\widetilde F\in \mathcal H_{[0,+\infty )}(\Omega _*)\subset C^{\infty }([0,+\infty )\times \Omega _*)$ , we have $F_{n,j}\in C^\infty ([0,+\infty ))$ , $Z\mapsto F_{n,j}(Z^2)\in C_{\operatorname {e}}^\infty ([0,+\infty ))$ for every $n,j\in \mathbb {Z}_{\geq 0}$ ; moreover by Cauchy’s theorem (Corollary 2.3 in Chapter 2 of [Reference Stein and Shakarchi69]), we have $F_{n,j}=0 $ for $j>m_*+n$ , $n,j\in \mathbb {Z}_{\geq 0}$ . Thus,

$$ \begin{align*}F_n(t,x)=\sum_{j=0}^{m_*+n}(T-t)^{\lambda_*}\frac{(-\tau)^j}{j!}F_{n,j}(Z^2) \in {\mathscr X}_{\lambda_*}.\end{align*} $$

This completes the proof of Lemma 5.5.

5.4 Solvability of $\mathscr {L}_\lambda $

In this subsection, we prove Lemma 5.6.

Lemma 5.7. For $g\in C_{\operatorname {e}}^{\infty }([0, Z_1))$ , there exists $F=F(Z;\lambda )\in \mathcal H_{[0, Z_1)}^{\operatorname {e}}(B_R)$ satisfying $\mathscr {L}_\lambda F=g$ on $(0,Z_1)$ and $F(0;\lambda )=1$ for all $\lambda \in B_R$ .

Proof. By Lemma 5.3, we have $\widehat B_0\in C_{\operatorname {o}}^\infty ([0,+\infty ))$ and $A_0$ , $\widetilde B_0$ , $D_1$ , $D_2\in C_{\operatorname {e}}^\infty ([0,+\infty ))$ . Thus, there exist $\widetilde {A}_0,B_1,B_2,\widetilde {D}_1,\widetilde {D}_2\in C^\infty ([0,+\infty ))$ such that $A_0(Z)=\widetilde {A}_0(Z^2), \widetilde B_0(Z)=B_1(Z^2), \widehat B_0(Z)=ZB_2(Z^2), D_1(Z)=\widetilde {D}_1(Z^2), D_2(Z)=\widetilde {D}_2(Z^2)$ . Then $B_0(Z;\lambda )=Z^{-1}\widetilde B_0(Z)+\lambda \widehat B_0(Z)=Z^{-1} B_1(Z^2)+\lambda Z B_2(Z^2)$ , $D_0(Z;\lambda )=\lambda D_1(Z)+\lambda ^2D_2(Z)=\lambda \widetilde {D}_1(Z^2)+\lambda ^2\widetilde {D}_2(Z^2)$ . Let $\widetilde {Z}:=Z^2$ . Then for $ f(Z)=\widetilde {f}(Z^2)=\widetilde {f}(\widetilde {Z})$ , we have $ f'(Z)=2Z\widetilde {f}'(\widetilde {Z})$ , $ f"(Z)=2\widetilde {f}'(\widetilde {Z})+4Z^2\widetilde {f}"(\widetilde {Z})$ , and by (5.16),

$$ \begin{align*} &(\mathscr{L}_\lambda f)(Z)=A_0(Z)f"(Z)+B_0(Z;\lambda)f'(Z)+D_0(Z;\lambda)f(Z)\\& \quad =\widetilde{A}_0(Z^2)[2\widetilde{f}'(\widetilde{Z})+4Z^2\widetilde{f}"(\widetilde{Z})]+[Z^{-1} B_1(Z^2)+\lambda Z B_2(Z^2)]\cdot2Z\widetilde{f}'(\widetilde{Z})\\& \qquad + [\lambda \widetilde{D}_1(Z^2)+\lambda^2\widetilde{D}_2(Z^2)]\widetilde{f}(\widetilde{Z})\\& \quad =4\widetilde{Z}\widetilde{A}_0(\widetilde{Z})\widetilde{f}"(\widetilde{Z})+2[\widetilde{A}_0(\widetilde{Z})+B_1(\widetilde{Z})+\lambda \widetilde{Z} B_2(\widetilde{Z})]\widetilde{f}'(\widetilde{Z})+ [\lambda \widetilde{D}_1(\widetilde{Z})+\lambda^2\widetilde{D}_2(\widetilde{Z})]\widetilde{f}(\widetilde{Z}). \end{align*} $$

Let

$$ \begin{align*} &\widetilde{A}(\widetilde{Z})=4\widetilde{Z}\widetilde{A}_0(\widetilde{Z}),\ \widetilde{B}(\widetilde{Z};\lambda)=2[\widetilde{A}_0(\widetilde{Z})+B_1(\widetilde{Z})+\lambda \widetilde{Z} B_2(\widetilde{Z})],\ \widetilde{D}(\widetilde{Z};\lambda)=\lambda \widetilde{D}_1(\widetilde{Z})+\lambda^2\widetilde{D}_2(\widetilde{Z}). \end{align*} $$

Then we get

(5.30) $$ \begin{align} &(\mathscr{L}_\lambda f)(Z)=\widetilde{A}(\widetilde{Z})\widetilde{f}"(\widetilde{Z})+\widetilde{B}(\widetilde{Z};\lambda)\widetilde{f}'(\widetilde{Z})+ \widetilde{D}(\widetilde{Z};\lambda)\widetilde{f}(\widetilde{Z}),\ \text{for}\ f(Z)=\widetilde{f}(\widetilde{Z}),\ \widetilde{Z}=Z^2. \end{align} $$

Let $I_1=[0,Z_1^2)$ . As $\widetilde {A}_0,B_1,B_2,\widetilde {D}_1,\widetilde {D}_2\in C^\infty ([0,+\infty ))$ , we have $\widetilde {A}\in C^\infty (I_1)$ , $\widetilde {B}, \widetilde {D}\in \mathcal H_{I_1}(\mathbb {C})$ , and $\widetilde {A}'(0)=4\widetilde {A}_0(0)=4{A}_0(0)=4\neq 0$ (using (5.17)). By (5.18), we have $\widetilde {A}_0(Z^2)=A_0(Z)>0$ for $Z\in [0,Z_1)$ . Thus, $\widetilde {A}(\widetilde {Z})=4\widetilde {Z}\widetilde {A}_0(\widetilde {Z})=0$ has a unique solution $\widetilde {Z}=0 $ in $I_1=[0,Z_1^2)$ .

Moreover, we have $\widetilde {B}(\widetilde {Z};\lambda )=\widetilde B_1(\widetilde {Z})+\lambda \widetilde {B}_2(\widetilde {Z})$ , where $\widetilde {B}_1(\widetilde {Z}):=2[\widetilde {A}_0(\widetilde {Z})+B_1(\widetilde {Z})], \widetilde {B}_2(\widetilde {Z}):=2\widetilde {Z}B_2(\widetilde {Z})$ , then $\widetilde B_1(0)=2[\widetilde {A}_0(0)+B_1(0)]=2[{A}_0(0)+\widetilde B_0(0)] =2(1+k)>0$ (using (5.17) and (5.19)) and $\widetilde {B}_2(0)=0$ . As a consequence, for any $\lambda \in \mathbb {C}$ and $n\in \mathbb {Z}_{\geq 0}$ we have $n\widetilde {A}'(0)+\widetilde {B}(0;\lambda )=4n+2(1+k)\neq 0$ . As $g\in C_{\operatorname {e}}^{\infty }([0, Z_1))$ , there exists $\widetilde {g}\in C^{\infty }([0,Z_1^2))$ such that $g(Z)=\widetilde {g}(Z^2)$ . By Proposition B.4, there exists $\widetilde F=\widetilde F(\widetilde {Z};\lambda )\in \mathcal H_{I_1}(B_R)$ satisfying

$$ \begin{align*} &\widetilde{A}(\widetilde{Z})\widetilde F"(\widetilde{Z};\lambda)+\widetilde{B}(\widetilde{Z};\lambda)\widetilde F'(\widetilde{Z};\lambda)+ \widetilde{D}(\widetilde{Z};\lambda)\widetilde F(\widetilde{Z};\lambda)=\widetilde{g}(\widetilde{Z}),\quad \widetilde F(0;\lambda)=1, \end{align*} $$

where the prime $'$ denotes the derivative with respect to $\widetilde {Z}$ . Now we define

$$\begin{align*}F(Z;\lambda):=\widetilde F(Z^2;\lambda),\qquad\forall\ Z\in[0,Z_1), \forall\ \lambda\in B_R,\end{align*}$$

then $F\in \mathcal H_{[0, Z_1)}^{\operatorname {e}}(B_R)$ , $F(0;\lambda )=\widetilde F(0;\lambda )=1$ and $\mathscr {L}_\lambda F=g$ in $(0, Z_1)$ by recalling (5.30).

In view of Lemma 5.3 and Proposition B.4, we let

(5.31) $$ \begin{align} \Lambda_*:=\{\lambda\in\mathbb{C}: nA_0'(Z_1)+B_0(Z_1;\lambda)=0\text{ for some }n\in\mathbb{Z}_{\geq 0}\}. \end{align} $$

By $B_0(Z_1;\lambda )=Z_1^{-1}\widetilde B_0(Z_1)+\lambda \widehat B_0(Z_1)$ and $\widehat B_0(Z_1)>0$ , we know that $\Lambda _*\subset \mathbb {C}$ is a non-empty (countable) discrete set.

Lemma 5.8. There exists a nonzero polynomial $ \psi _1(\lambda )$ satisfying $\{\lambda \in B_R: \psi _1(\lambda )=0\}=\Lambda _*\cap B_R$ such that for $g\in C^{\infty }((0, +\infty ))$ , there exists a function $F=F(Z;\lambda )\in \mathcal H_{(0,+\infty )}(B_R)$ satisfying $\mathscr {L}_\lambda F=\psi _1(\lambda )\cdot g$ on $(0, +\infty )$ and $F(Z_1;\lambda )=\psi _1(\lambda )$ for all $\lambda \in B_R$ .

Proof. By Lemma 5.3, we have $A_0(Z_1)=0$ , $A_0'(Z_1)\neq 0$ , $\widehat B_0(Z_1)>0$ and $Z_1$ is the unique solution of $A_0(Z)=0$ in $(0, +\infty ) $ . Hence Lemma 5.8 follows from Proposition B.4.

Taking $g=0$ in Lemma 5.7 we know that there exists $\Psi _1=\Psi _1(Z;\lambda )\in \mathcal H_{[0, Z_1)}^{\operatorname {e}}(B_R)$ satisfying $\mathscr {L}_\lambda \Psi _1=0$ on $(0,Z_1)$ and $\Psi _1(0;\lambda )=1$ for all $\lambda \in B_R$ . Taking $g=0$ in Lemma 5.8 we know that there exists $\Psi _2=\Psi _2(Z;\lambda )\in \mathcal H_{(0,+\infty )}(B_R)$ satisfying $\mathscr {L}_\lambda \Psi _2=0$ on $(0, +\infty )$ and $\Psi _2(Z_1;\lambda )=\psi _1(\lambda )$ for all $\lambda \in B_R$ . We define the Wronski

(5.32) $$ \begin{align}W(Z;\lambda):=\Psi_1(Z;\lambda)\Psi_2'(Z;\lambda)-\Psi_1'(Z;\lambda)\Psi_2(Z;\lambda),\quad \forall\ Z\in(0, Z_1),\ \forall\ \lambda\in B_R,\end{align} $$

where the prime $'$ denotes the derivative with respect to Z. Then we have

(5.33) $$ \begin{align} A_0(Z)W'(Z;\lambda)+B_0(Z;\lambda)W(Z;\lambda)=0,\qquad\forall\ Z\in(0, Z_1),\quad \lambda\in B_R. \end{align} $$

Lemma 5.9. Fix $Z_0\in (0,Z_1)$ . Let $\psi _2(\lambda ):=W(Z_0;\lambda )$ for all $\lambda \in B_R$ and $\lambda _0^*:=\gamma -k-2$ . Then $\psi _2\in \operatorname {Hol}(B_R)$ , $0<-\lambda _0^*<k<R$ and $\psi _2(\lambda _0^*)\neq 0$ .

Proof. As $\Psi _1(Z;\lambda )\in \mathcal H_{[0, Z_1)}^{\operatorname {e}}(B_R)$ , $\Psi _2(Z;\lambda )\in \mathcal H_{(0,+\infty )}(B_R)$ , by (5.32) we have $W(Z;\lambda )\in \mathcal H_{(0, Z_1)}(B_R)$ , then by $Z_0\in (0,Z_1)$ we have $\psi _2(\lambda )=W(Z_0;\lambda )\in \operatorname {Hol}(B_R)$ . As $\beta>0$ , $\ell>1$ , $\gamma =\beta (\ell -1)+2$ , $\lambda _0^*=\gamma -k-2$ , we get by (2.9) that

(5.34) $$ \begin{align} \lambda_0^*=\gamma-k-2=\beta(\ell-1)+2-k-2=\beta(\ell-1)-k<\beta(\ell-1)-\beta(\ell+\sqrt\ell)<0, \end{align} $$

and $R>k>k-\beta (\ell -1)=-\lambda _0^*>0$ , then $\lambda _0^*\in B_R$ . It remains to prove that $\psi _2(\lambda _0^*)\neq 0$ .

We consider the dual $\mathscr {L}_0^*$ of $\mathscr {L}_0$ , defined by (5.25). On one hand, we get by Lemma 5.4 that

(5.35) $$ \begin{align} \mathscr{L}_0^*=\mathscr{L}_{\lambda_0^*}=A_0\partial_Z^2+B_0(\cdot;\lambda_0^*)\partial_Z+D_0(\cdot;\lambda_0^*). \end{align} $$

On the other hand, by (recalling that $D_0(\cdot ;0)=0$ )

$$ \begin{align*}\mathscr{L}_0f=A_0\partial_Z^2+B_0(\cdot;0)\partial_Z=A_0\partial_Z^2+Z^{-1}\widetilde B_0\partial_Z\end{align*} $$

and (5.25), we compute that

(5.36) $$ \begin{align} (\mathscr{L}_0^*f)(Z)=\frac{1}{Z^k}\left(\partial_Z^2(Z^kA_0f)(Z)-\partial_Z(Z^{k-1}\widetilde B_0f)(Z)\right),\quad\forall\ Z\in(0,+\infty). \end{align} $$

Comparing the coefficients of $\partial _Z$ in (5.35) and (5.36), we obtain

$$ \begin{align*} Z^{-1}\widetilde B_0(Z)+\lambda_0^*\widehat B_0(Z)&=B_0(Z;\lambda_0^*)=[2\partial_Z(Z^kA_0)(Z)-Z^{k-1}\widetilde B_0(Z)]/{Z^k}\\ &={2k}{Z}^{-1}A_0(Z)+2A_0'(Z)-Z^{-1}\widetilde B_0(Z) \end{align*} $$

for all $Z\in (0,+\infty )$ . Letting $Z=Z_1$ , we get(as $A_0(Z_1)=0$ , see (5.17))

(5.37) $$ \begin{align} Z_1^{-1}\widetilde B_0(Z_1)+{\lambda_0^*}\widehat B_0(Z_1)/{2}=A_0'(Z_1). \end{align} $$

For any $n\in \mathbb {Z}_{\geq 0}$ , by (5.37), (5.17), (5.19) and $\lambda _0^*<0$ (i.e. (5.34)), we have

(5.38) $$ \begin{align} \begin{aligned} nA_0'(Z_1)+B_0(Z_1;\lambda_0^*)&=nA_0'(Z_1)+Z_1^{-1}\widetilde B_0(Z_1)+\lambda_0^*\widehat B_0(Z_1)\\ &=(n+1)A_0'(Z_1)+{\lambda_0^*}\widehat B_0(Z_1)/{2}<0, \end{aligned} \end{align} $$

It follows from (5.31) and (5.38) that $\lambda _0^*\notin \Lambda _*$ . Then by $\lambda _0^*\in B_R$ and Lemma 5.8, we have $\psi _1(\lambda _0^*)\neq 0$ . Let $f_1=\Psi _1(\cdot ;\lambda _0^*)$ and $f_2=\Psi _2(\cdot ;\lambda _0^*)$ , then $f_1\in C_{\text {e}}^\infty ([0,Z_1))$ , $f_2\in C^\infty ((0,+\infty ))$ and $(\mathscr {L}_{\lambda _0^*}f_j)(Z)=0$ for $Z\in (0, Z_1)$ , $j\in \{1, 2\}$ . By (5.35) and (5.36), we get

$$\begin{align*}\partial_Z^2(Z^kA_0f_j)(Z)-\partial_Z(Z^{k-1}\widetilde B_0f_j)(Z)=0,\quad\forall\ Z\in(0, Z_1), j\in\{1,2\}.\end{align*}$$

By $f_1,A_0,\widetilde B_0\in C_{\text {e}}^\infty ([0,Z_1))$ , $k\geq 3$ we have $[\partial _Z(Z^kA_0f_1)(Z)-Z^{k-1}\widetilde B_0(Z)f_1(Z)]|_{Z=0}=0$ , so

(5.39) $$ \begin{align} \partial_Z(Z^kA_0f_1)(Z)-Z^{k-1}\widetilde B_0(Z)f_1(Z)=0,\quad\forall\ Z\in(0, Z_1). \end{align} $$

For $f_2$ , since $A_0(Z_1)=0$ (see (5.17)), we have

(5.40) $$ \begin{align} &\partial_Z(Z^kA_0f_2)(Z)-Z^{k-1}\widetilde B_0(Z)f_2(Z)=[\partial_Z(Z^kA_0f_2)(Z)-Z^{k-1}\widetilde B_0(Z)f_2(Z)]|_{Z=Z_1}\notag\\& \quad =Z_1^kA_0'(Z_1)f_2(Z_1)-Z_1^{k-1}\widetilde B_0(Z_1)f_2(Z_1)=Z_1^k(A_0'(Z_1)-Z_1^{-1}\widetilde B_0(Z_1))\psi_1(\lambda_0^*)=:C' \end{align} $$

for all $Z\in (0, Z_1)$ , where we have used $f_2(Z_1)=\Psi _2(Z_1;\lambda _0^*)=\psi _1(\lambda _0^*)$ (recalling Lemma 5.8). Moreover, by (5.37), $\lambda _0^*<0$ (in (5.34)) and $\widehat B_0(Z_1)>0$ (in (5.19)), we have $A_0'(Z_1)-Z_1^{-1}\widetilde B_0(Z_1)={\lambda _0^*}\widehat B_0(Z_1)/{2}<0$ , then by $\psi _1(\lambda _0^*)\neq 0$ we have $C'\neq 0$ . We claim that

(5.41) $$ \begin{align} f_1(Z)\neq 0,\quad\forall\ Z\in(0, Z_1). \end{align} $$

Indeed, if $f_1(Z^*)=0$ for some $Z^*\in (0, Z_1)$ , by the uniqueness of solutions to (5.39) in $(0, Z_1)$ with $f_1(Z^*)=0$ , we have $f_1(Z)=0$ for all $Z\in (0, Z_1)$ , which contradicts with $1=f_1(0)=\lim _{Z\to 0+}f_1(Z)$ . This proves (5.41). As $f_1=\Psi _1(\cdot ;\lambda _0^*)$ , $f_2=\Psi _2(\cdot ;\lambda _0^*)$ , by (5.32), (5.39), (5.40), (5.41) and $C'\neq 0$ , we have

$$ \begin{align*} &Z^kA_0(Z)W(Z;\lambda_0^*)=Z^kA_0(Z)[f_1(Z)f_2'(Z)-f_1'(Z)f_2(Z)]\\&\quad=f_1(Z)\partial_Z(Z^kA_0f_2)(Z)-\partial_Z(Z^kA_0f_1)(Z)f_2(Z)\\&\quad=f_1(Z)[Z^{k-1}\widetilde B_0(Z)f_2(Z)+C']-Z^{k-1}\widetilde B_0(Z)f_1(Z)f_2(Z)=C'f_1(Z)\neq0, \end{align*} $$

for all $Z\in (0, Z_1)$ . Thus, $W(Z;\lambda _0^*)\neq 0 $ for all $Z\in (0, Z_1)$ , and $\psi _2(\lambda _0^*)=W(Z_0;\lambda _0^*)\neq 0$ .

Now we fix $Z_0\in (0,Z_1)$ , $\psi _2(\lambda )=W(Z_0;\lambda )$ , $\lambda _0^*:=\gamma -k-2$ . Let $ \psi _1(\lambda )$ be given by Lemma 5.8 and $\varphi (\lambda ):=\psi _1(\lambda )\psi _2(\lambda )$ for all $\lambda \in B_R$ . Let $g\in C_{\text {e}}^\infty ([0,+\infty ))$ , we need to prove that there exists a function $f=f(Z;\lambda )\in \mathcal H_{[0,+\infty )}^{\operatorname {e}}(B_R)$ such that $\mathscr {L}_\lambda f(\cdot ;\lambda )=\varphi (\lambda )\cdot g$ on $(0,+\infty )$ .

We first consider the case when g is supported near $Z=0$ .

Lemma 5.10. Assume that $g\in C_{\operatorname {e}}^\infty ([0,+\infty ))$ satisfies $\operatorname {supp}g\subset [0, Z_1)$ , then there exists a function $f=f(Z;\lambda )\in \mathcal H_{[0,+\infty )}^{\operatorname {e}}(B_R)$ such that $\mathscr {L}_\lambda f(\cdot ;\lambda )=\varphi (\lambda )\cdot g$ on $(0,+\infty )$ .

Proof. By Lemma 5.7, there exists $f_0\in \mathcal H_{[0, Z_1)}^{\text e}(B_R)$ such that $\mathscr {L}_\lambda f_0=g$ on $(0, Z_1)$ with $f_0(0;\lambda )=1$ for all $\lambda \in B_R$ . We assume that $\operatorname {supp}g\subset [0,\delta )$ for some $\delta \in (0, Z_1)$ , then $(\mathscr {L}_\lambda f_0)(Z)=0$ for $Z\in [\delta , Z_1)$ . For $\lambda \in B_R$ , let

(5.42) $$ \begin{align} C_1(\lambda)&:={f_0(\delta;\lambda)\Psi_2'(\delta;\lambda)-f_0'(\delta;\lambda)\Psi_2(\delta;\lambda)}\in\mathbb{C}, \end{align} $$
(5.43) $$ \begin{align} C_2(\lambda)&:={f_0'(\delta;\lambda)\Psi_1(\delta;\lambda)-f_0(\delta;\lambda)\Psi_1'(\delta;\lambda)}\in\mathbb{C}. \end{align} $$

Then $C_1, C_2$ are holomorphic functions on $B_R$ and for all $\lambda \in B_R$ there holds

$$ \begin{align*}&W(\delta;\lambda)f_0(\delta;\lambda)=C_1(\lambda)\Psi_1(\delta;\lambda)+C_2(\lambda)\Psi_2(\delta;\lambda),\\& W(\delta;\lambda)f_0'(\delta;\lambda)=C_1(\lambda)\Psi_1'(\delta;\lambda)+C_2(\lambda)\Psi_2'(\delta;\lambda).\end{align*} $$

By the uniqueness of the solution on $[\delta , Z_1)$ , we have

$$ \begin{align*}W(\delta;\lambda)f_0(Z;\lambda)=C_1(\lambda)\Psi_1(Z;\lambda)+C_2(\lambda)\Psi_2(Z;\lambda),\qquad\forall\ Z\in[\delta, Z_1),\ \forall\ \lambda\in B_R.\end{align*} $$

For $\lambda \in B_R$ , let

$$\begin{align*}f_*(Z;\lambda):=\begin{cases} W(\delta;\lambda)f_0(Z;\lambda)-C_1(\lambda)\Psi_1(Z;\lambda) & \text{if }Z\in [0, Z_1),\\ C_2(\lambda)\Psi_2(Z;\lambda) & \text{if }Z\in[\delta, +\infty). \end{cases}\end{align*}$$

Then $f_*\in \mathcal H_{[0,+\infty )}^{\text e}(B_R)$ and $\mathscr {L}_\lambda f_*=W(\delta ;\lambda )\cdot g$ on $(0,+\infty )$ . By (5.33), we have $W(Z;\lambda )=W(Z_0;\lambda )\mathrm {e}^{- A_*(Z;\lambda )}=\psi _2(\lambda )\mathrm {e}^{- A_*(Z;\lambda )}$ with $A_*(Z;\lambda ):=\int _{Z_0}^Z\frac {B_0(Z;\lambda )}{A_0(Z)}\mathrm {d}Z\in \mathcal H_{(0,Z_1)}(B_R) $ (using Lemma 5.3). Recall that $\varphi =\psi _1\psi _2$ , $ \psi _1$ is a polynomial, then $\varphi (\lambda )=\psi _1(\lambda )W(\delta ;\lambda )\mathrm {e}^{A_*(\delta ;\lambda )} $ , and the result follows by taking $ f(Z;\lambda ):=\psi _1(\lambda )\mathrm {e}^{A_*(\delta ;\lambda )}f_*(Z;\lambda )$ for $Z\in [0, +\infty ),\lambda \in B_R$ .

Now we consider the case when g is supported away from $Z=0$ .

Lemma 5.11. Assume that $g\in C_{\operatorname {e}}^\infty ([0,+\infty ))$ satisfies $\operatorname {supp}g\subset (0, +\infty )$ , then there exists a function $f=f(Z;\lambda )\in \mathcal H_{[0,+\infty )}^{\operatorname {e}}(B_R)$ such that $\mathscr {L}_\lambda f(\cdot ;\lambda )=\varphi (\lambda )\cdot g$ on $(0,+\infty )$ .

Proof. By Lemma 5.8, there exists $f_0=f_0(Z;\lambda )\in \mathcal H_{(0,+\infty )}(B_R)$ such that $\mathscr {L}_\lambda f_0=\psi _1(\lambda )\cdot g$ on $(0, +\infty )$ . We assume that $\operatorname {supp} g\subset (\delta , +\infty )$ for some $\delta \in (0, Z_1)$ , then $(\mathscr {L}_\lambda f_0)(Z)=0$ for $Z\in (0,\delta ]$ . For $\lambda \in B_R$ , let $C_1(\lambda ), C_2(\lambda )$ be defined by (5.42) and (5.43) respectively. For the same reason as in the proof of Lemma 5.10, we have

$$ \begin{align*}W(\delta;\lambda)f_0(Z;\lambda)=C_1(\lambda)\Psi_1(Z;\lambda)+C_2(\lambda)\Psi_2(Z;\lambda),\qquad\forall\ Z\in(0, \delta],\ \forall\ \lambda\in B_R.\end{align*} $$

For $\lambda \in B_R$ , let

$$\begin{align*}f_*(Z;\lambda):=\begin{cases} W(\delta;\lambda)f_0(Z;\lambda)-C_2(\lambda)\Psi_2(Z;\lambda) & \text{if }Z\in (0, +\infty),\\ C_1(\lambda)\Psi_1(Z;\lambda) & \text{if }Z\in[0,\delta]. \end{cases}\end{align*}$$

Then $f_*\in \mathcal H_{[0,+\infty )}^{\operatorname {e}}(B_R)$ and $\mathscr {L}_\lambda f_*=W(\delta ;\lambda )\psi _1(\lambda )\cdot g$ on $(0,+\infty )$ . As in the proof of Lemma 5.10, we have $\varphi (\lambda )=\psi _1(\lambda )W(\delta ;\lambda )\mathrm {e}^{A_*(\delta ;\lambda )} $ and $A_*(Z;\lambda )\in \mathcal H_{(0,Z_1)}(B_R) $ , then the result follows by taking $ f(Z;\lambda ):=\mathrm {e}^{A_*(\delta ;\lambda )}f_*(Z;\lambda )$ for $Z\in [0, +\infty ),\lambda \in B_R$ .

Now we are in a position to prove Lemma 5.6.

Proof of Lemma 5.6.

We fix $Z_0\in (0,Z_1)$ , $\psi _2(\lambda )=W(Z_0;\lambda )$ , $\lambda _0^*:=\gamma -k-2$ . Let $ \psi _1(\lambda )$ be given by Lemma 5.8 and $\varphi (\lambda ):=\psi _1(\lambda )\psi _2(\lambda )$ for all $\lambda \in B_R$ . By Lemma 5.9, we have $\lambda _0^*\in B_R$ and $\psi _2\in \operatorname {Hol}(B_R)\setminus \{0\}$ . By Lemma 5.8 we have $\psi _1\in \operatorname {Hol}(B_R)\setminus \{0\}$ . Thus, $\varphi =\psi _1\psi _2\in \operatorname {Hol}(B_R)\setminus \{0\}$ .

Let $\zeta \in C^\infty (\mathbb {R};[0,1])$ satisfy $\operatorname {supp}\zeta \subset (Z_1/2, +\infty )$ and $\zeta (Z)=1$ for $Z\in [3Z_1/4, +\infty )$ . Let $g_1(Z)=g(Z)(1-\zeta (Z)), g_2(Z)=g(Z){\zeta }(Z)$ for all $Z\in [0,+\infty )$ . Then

$$\begin{align*}\operatorname{supp}g_1\subset[0, 3Z_1/4],\quad \operatorname{supp}g_2\subset[Z_1/2,+\infty),\quad g_1, g_2\in C_{\text{e}}^\infty([0,+\infty)),\quad g=g_1+g_2.\end{align*}$$

By Lemma 5.10, there exists $f_1=f_1(Z;\lambda )\in \mathcal H_{[0,+\infty )}^{\operatorname {e}}(B_R)$ such that $\mathscr {L}_\lambda f_1=\varphi (\lambda )g_1$ on $(0,+\infty )$ . By Lemma 5.11, there exists $f_2=f_2(Z;\lambda )\in \mathcal H_{[0,+\infty )}^{\operatorname {e}}(B_R)$ such that $\mathscr {L}_\lambda f_2=\varphi (\lambda )g_2$ on $(0,+\infty )$ . Let $f=f_1+f_2$ , then $f\in \mathcal H_{[0,+\infty )}^{\operatorname {e}}(B_R)$ satisfies $\mathscr {L}_\lambda f=\varphi (\lambda )g$ on $(0,+\infty )$ .

A The derivation and properties of ODE (2.8)

A.1 The derivation of ODE (2.8)

Lemma A.1. Let $\beta>1$ and $v=v(Z)\in C^\infty ([0, +\infty );(-1, 1))$ be given by Assumption 1. We define $\widehat \phi _0(Z), \widehat \rho _0(Z)$ according to (2.12) and we define $\phi _0(t,x),\rho _0(t,x)$ by (2.7). Then $(\phi _0,\rho _0)$ solves the leading order equation (2.6).

Proof. Recall that $Z=r/(T-t)$ with $r=|x|$ , we know that $\phi _0=\phi _0(t,r)$ and $\rho _0=\rho _0(t,r)$ are radially symmetric. Hence, (2.6) is equivalent to

(A.1) $$ \begin{align} \rho_0^{p-1}-|\partial_t\phi_0|^2+|\partial_r\phi_0|^2=0,\qquad -\partial_t(\rho_0^2\partial_t\phi_0)+\partial_r(\rho_0^2\partial_r\phi_0)+\frac{k}{r}\rho_0^2\partial_r\phi_0=0, \end{align} $$

where $k=d-1\in \mathbb {Z}_{\geq 1}$ . It follows from (2.7) that

(A.2) $$ \begin{align}\partial_t\phi_0(t,x)=(T-t)^{-\beta}\big((\beta-1)\widehat\phi_0(Z)+Z\widehat\phi_0'(Z)\big),\quad \partial_r\phi_0(t,x)=(T-t)^{-\beta}\widehat\phi_0'(Z),\end{align} $$

where the prime $'$ stands for the derivative with respect to Z. By (2.12), we have

(A.3) $$ \begin{align} \widehat\phi_0'(Z)=\frac{(\beta-1)\widehat\phi_0(Z)v(Z)}{1-Zv(Z)}\Longleftrightarrow\big((\beta-1)\widehat\phi_0(Z)+Z\widehat\phi_0'(Z)\big)v(Z)=\widehat\phi_0'(Z), \end{align} $$

thus

$$\begin{align*}\partial_t\phi_0(t,x)=\frac{(T-t)^{-\beta}(\beta-1)\widehat\phi_0(Z)}{1-Zv(Z)},\qquad \partial_r\phi_0(t,x)=\frac{(T-t)^{-\beta}(\beta-1)\widehat\phi_0(Z)v(Z)}{1-Zv(Z)},\end{align*}$$

and then we have

$$\begin{align*}|\partial_t\phi_0|^2-|\partial_r\phi_0|^2=\frac{(T-t)^{-2\beta}(\beta-1)^2\widehat\phi_0(Z)^2(1-v(Z)^2)}{(1-Zv(Z))^2}.\end{align*}$$

Using (2.7) and (2.12) for $\rho _0$ and $\widehat \rho _0$ , we obtain the first equation in (A.1).

Now we define

(A.4) $$ \begin{align} \ell=\frac4{p-1}+1>1,\quad \widetilde\phi_0(Z):=\frac{\widehat\phi_0(Z)^\ell(1-v(Z)^2)^{\frac{2}{p-1}}}{(1-Zv(Z))^\ell}=\frac{\widehat\phi_0(Z)^\ell(1-v(Z)^2)^{\frac{\ell-1}{2}}}{(1-Zv(Z))^\ell}>0. \end{align} $$

Then we compute that

$$ \begin{align*} \rho_0^2\partial_t\phi_0(t,x)&=(T-t)^{-\beta\ell}(\beta-1)^\ell\widetilde\phi_0(Z),\\ \rho_0^2\partial_r\phi_0(t,x)&=(T-t)^{-\beta\ell}(\beta-1)^\ell\widetilde\phi_0(Z)v(Z),\\ \partial_t(\rho_0^2\partial_t\phi_0)(t,x)&= (T-t)^{-\beta\ell-1}(\beta-1)^\ell\big(\beta\ell\widetilde\phi_0(Z)+Z\widetilde\phi_0'(Z)\big),\\ \partial_r(\rho_0^2\partial_r\phi_0)(t,x)&=(T-t)^{-\beta\ell-1}(\beta-1)^\ell(\widetilde\phi_0 v)'(Z),\\ \frac{k}{r}\rho_0^2\partial_r\phi_0(t,x)&=(T-t)^{-\beta\ell-1}(\beta-1)^\ell\frac{k}{Z}(\widetilde\phi_0 v)(Z). \end{align*} $$

Therefore, the second equation in (A.1) is equivalent to

(A.5) $$ \begin{align} \beta\ell\widetilde\phi_0+Z\widetilde\phi_0'=(\widetilde\phi_0 v)'+\frac{k}{Z}(\widetilde\phi_0 v)\Longleftrightarrow (\beta\ell-v'-kv/Z)\widetilde\phi_0=(v-Z)\widetilde\phi_0'. \end{align} $$

Recall from (A.3) and (A.4) that

$$ \begin{align*} \frac{\widetilde\phi_0'}{\widetilde\phi_0} &=\ell\frac{\widehat\phi_0'}{\widehat\phi_0}-(\ell-1)\frac{vv'}{1-v^2}+\ell\frac{v+Zv'}{1-Zv}\\ &=\ell\frac{(\beta-1)v}{1-Zv}+\ell\frac{v}{1-Zv}-(\ell-1)\frac{vv'}{1-v^2}+\ell\frac{Zv'}{1-Zv}\\ &=\frac{\beta\ell v}{1-Zv}+\frac{vv'}{1-v^2}+\frac{\ell(Z-v)v'}{(1-v^2)(1-Zv)}, \end{align*} $$

hence (A.5) is equivalent to

$$ \begin{align*} \beta\ell-v'-\frac{kv}{Z}=(v-Z)\frac{\widetilde\phi_0'}{\widetilde\phi_0} =\frac{\beta\ell v(v-Z)}{1-Zv}+\frac{v(v-Z)v'}{1-v^2}-\frac{\ell(v-Z)^2v'}{(1-v^2)(1-Zv)}, \end{align*} $$

or equivalently,

$$ \begin{align*} \beta\ell-\frac{\beta\ell v(v-Z)}{1-Zv}-\frac{kv}{Z} &=v'+\frac{v(v-Z)v'}{1-v^2}-\frac{\ell(v-Z)^2v'}{(1-v^2)(1-Zv)},\\\frac{\beta\ell(1-v^2)}{1-Zv}-\frac{kv}{Z} &=\frac{(1-Zv)v'}{1-v^2}-\frac{\ell(v-Z)^2v'}{(1-v^2)(1-Zv)}, \end{align*} $$

which is a direct consequence of (2.8).

A.2 Properties of solutions to ODE (2.8)

In this part, we prove Remark 2.2.

Lemma A.2. Under Assumption 1, we have $v(Z_1)=v_1$ , where

(A.6) $$ \begin{align} Z_1=\frac{k}{\sqrt{\ell}(k-\beta(\ell-1))},\qquad v_1=\frac{\beta\sqrt{\ell}}{k-\beta\ell}. \end{align} $$

Proof. We define a function $F_0\in C^\infty ([0, +\infty ))$ by

(A.7) $$ \begin{align} F_0(Z):=1-Zv(Z)+\sqrt\ell(v(Z)-Z),\quad\forall\ Z\in[0, +\infty). \end{align} $$

Then $F_0(0)=1$ and $F_0(1)=(\sqrt \ell -1)(v(1)-1)<0$ , where we have used that $v(1)\in (-1, 1)$ , recalling Assumption 1. By the intermediate value theorem, there exists $Z_0\in (0,1)$ such that $F_0(Z_0)=0$ . Thus, $\Delta _Z(Z_0, v(Z_0))=Z_0F_0(Z_0)\big (1-Z_0v(Z_0)-\sqrt \ell (v(Z_0)-Z_0)\big )=0$ . Then we have $\Delta _v(Z_0, v(Z_0))=\Delta _Z(Z_0, v(Z_0))v'(Z_0)=0$ , i.e., $\Delta _v(Z_0, v(Z_0))=\Delta _Z(Z_0, v(Z_0))=0$ . On the other hand, it is direct to check that

(A.8) $$ \begin{align} \left\{(Z,v)\in (0,+\infty)\times(-1,1): \Delta_v(Z,v)=\Delta_Z(Z,v)=0\right\}=\{(Z_1, v_1)\}, \end{align} $$

where $Z_1, v_1$ are given by (A.6). Moreover, it follows from (2.9) that $0<v_1<Z_1<1$ . Hence, we must have $(Z_0, v(Z_0))=(Z_1, v_1)$ , which implies that $v(Z_1)=v_1$ .

Lemma A.3. Under Assumption 1, let $\Delta _0(Z):=\Delta _Z(Z, v(Z))$ for $Z\in [0, +\infty )$ , then we have $\Delta _0(Z)>0$ for $Z\in (0, Z_1)$ , $\Delta _0(Z)<0$ for $Z\in (Z_1, +\infty )$ and $\Delta _0'(Z_1)\neq 0$ .

Proof. By the definition of $\Delta _0(Z)$ , we have $\Delta _0\in C^\infty ([0, +\infty ))$ and (see (2.8))

(A.9) $$ \begin{align} \Delta_0(Z)=ZF_0(Z)\widetilde F_0(Z),\quad \forall\ Z\in[0, +\infty), \end{align} $$

where $F_0\in C^\infty ([0, +\infty ))$ is defined by (A.7) and $\widetilde F_0\in C^\infty ([0, +\infty ))$ is defined by

(A.10) $$ \begin{align} \widetilde F_0(Z):=1-Zv(Z)-\sqrt\ell(v(Z)-Z),\quad\forall\ Z\in[0, +\infty). \end{align} $$

If $\Delta _0(Z_*)=0$ for some $Z_*\in (0, +\infty )$ , then $\Delta _v(Z_*, v(Z_*))=\Delta _0(Z_*)v'(Z_*)=0$ , and by (A.8) we obtain $Z_*=Z_1$ , hence (using Lemma A.2, (A.8) and (A.9))

(A.11) $$ \begin{align} \{Z\in[0, +\infty):\Delta_0(Z)=0\}=\{0, Z_1\}. \end{align} $$

Now we prove that $\Delta _0'(Z_1)\neq 0$ . By the proof of Lemma A.2, we have $F_0(Z_1)=0$ . Then by (A.7) and (A.10), we have $\widetilde F_0(Z_1)=2(1-Z_1v(Z_1))>0$ . Thus (using (A.9)), $\Delta _0'(Z_1)=Z_1F_0'(Z_1)\widetilde F_0(Z_1) $ . Assume on the contrary that $\Delta _0'(Z_1)=0$ , then $F_0'(Z_1)=0$ .

Let $\Delta _1(Z):=\Delta _v(Z, v(Z))$ . Then (2.8) becomes $\Delta _0(Z)v'(Z)=\Delta _1(Z)$ . Taking derivative with respect to Z at $Z=Z_1$ on both sides, we obtain(using $\Delta _0(Z_1)=0$ ) $\Delta _1'(Z_1)=\Delta _0'(Z_1)v'(Z_1)=0$ and $\Delta _1(Z_1)=0$ . By (2.8), we have $\Delta _1(Z)=F_1(Z)\widetilde F_1(Z)$ with $F_1(Z):=Z-g_1(v(Z)) $ , $\widetilde F_1(Z):=(1-v(Z)^2)(\beta \ell +(k-\beta \ell )v(Z)^2) $ , $g_1(v):=kv/(\beta \ell +(k-\beta \ell )v^2) $ (note that $0<\beta \ell <k$ using (2.9)). As $v(Z_1)\in (-1,1)$ , we have $\widetilde F_1(Z_1)>0$ , then by $\Delta _1'(Z_1)=0$ and $\Delta _1(Z_1)=0$ we have $F_1(Z_1)=0$ , $\Delta _1'(Z_1)=F_1'(Z_1)\widetilde F_1(Z_1)=0$ and $F_1'(Z_1)=0$ . Thus, $0=1-g_1'(v(Z_1))v'(Z_1)=1-g_1'(v_1)v'(Z_1) $ (using Lemma A.2).

Similarly, by (A.7), we have $F_0(Z)=F_2(Z)\widetilde F_2(Z)$ with $F_2(Z):=g_2(v(Z))-Z $ , $\widetilde F_2(Z):=v(Z)+\ell $ , $g_2(v):=(1+\sqrt \ell v)/(v+\sqrt \ell ) $ , and $\widetilde F_2(Z_1)>0$ . Thus, $F_2(Z_1)=0$ , $F_2'(Z_1)=0$ (using $F_0(Z_1)= F_0'(Z_1)=0$ ) and $0=g_2'(v_1)v'(Z_1)-1$ .

Now we have $1=g_1'(v_1)v'(Z_1)=g_2'(v_1)v'(Z_1)$ and $g_1'(v_1)=g_2'(v_1) $ . On the other hand,

$$ \begin{align*} g_1(v)-g_2(v)&=\frac{kv}{\beta \ell+(k-\beta \ell)v^2}-\frac{1+\sqrt\ell v}{v+\sqrt\ell}= \frac{(1-v^2)(kv\sqrt\ell-\beta \ell(1+\sqrt\ell v)}{(\beta \ell+(k-\beta \ell)v^2)(v+\sqrt\ell)}\\ &=\frac{(1-v^2)(k-\beta \ell)\sqrt\ell(v-v_1)}{(\beta \ell+(k-\beta \ell)v^2)(v+\sqrt\ell)}, \end{align*} $$

here we used (A.6), thus

$$ \begin{align*} g_1'(v_1)-g_2'(v_1)&=\frac{(1-v_1^2)(k-\beta \ell)\sqrt\ell}{(\beta \ell+(k-\beta \ell)v_1^2)(v_1+\sqrt\ell)}>0, \end{align*} $$

which is a contradiction. Therefore, $\Delta _0'(Z_1)\neq 0$ .

By $F_0(0)=\widetilde F_0(0)=1>0$ , (A.9), and (A.11), we have $\Delta _0(Z)>0$ for $Z\in (0, Z_1)$ . Finally, using $\Delta _0'(Z_1)\neq 0$ and (A.11), we have $\Delta _0'(Z_1)<0$ and $\Delta _0(Z)<0$ for all $Z\in (Z_1, +\infty )$ .

To finish the proof of Remark 2.2, it remains to show that $v(Z)<Z$ and $Zv(Z)<1$ for all $Z\in (0, +\infty )$ . We use the barrier function method. For any $V\in C^1((0, +\infty ))$ , we define

(A.12) $$ \begin{align} (\mathscr AV)(Z):=-\Delta_Z(Z, V(Z))V'(Z)+\Delta_v(Z, V(Z)),\quad\forall\ Z\in(0, +\infty). \end{align} $$

Then $\mathscr Av=0$ if v is given by Assumption 1.

Lemma A.4. Under Assumption 1, we have $v(Z)<Z$ for all $Z\in (0, +\infty )$ .

Proof. Since $v(Z)\in (-1, 1)$ for all $Z\in (0, +\infty )$ by Assumption 1, it suffices to prove $v(Z)<Z$ for all $Z\in (0, 1)$ . We first show that $v(Z)<Z$ for all $Z\in (Z_1, 1)$ . By (A.9), $F_0(0)=1, F_0(Z_1)=0$ and (A.11), we have

$$\begin{align*}\{Z\in[0, +\infty): F_0(Z)=0\}=\{Z_1\}.\end{align*}$$

As $F_0(1)<0$ and $Z_1\in (0,1)$ , we have $F_0(Z)<0$ for all $Z\in (Z_1, 1)$ , hence

$$\begin{align*}v(Z)<\frac{\sqrt\ell Z-1}{\sqrt{\ell}-Z}<Z,\quad\forall\ Z\in(Z_1, 1).\end{align*}$$

Next we prove that $v(Z)<Z$ for all $Z\in (0, Z_1)$ . Let $V_1(Z):=Z$ for all $Z\in [0, +\infty )$ , then we have

(A.13) $$ \begin{align} (\mathscr AV_1)(Z)=(\beta\ell-k-1)Z(1-Z^2)^2<0,\quad \forall\ Z\in(0, 1), \end{align} $$

where we have used $\beta \ell -k-1<0$ , which follows from (2.9). On the other hand, letting $Z\to 0+$ in $\mathrm dv/\mathrm dZ=\Delta _v(Z, v(Z))/\Delta _Z(Z,v(Z))$ , by L’Hôpital’s rule, we have

$$\begin{align*}v'(0)=\frac{\partial_Z\Delta_v(0, 0)+\partial_v\Delta_v(0,0)v'(0)}{\partial_Z\Delta_Z(0, 0)+\partial_v\Delta_Z(0,0)v'(0)}=\beta\ell-kv'(0),\end{align*}$$

hence $v'(0)=\beta \ell /(k+1)<1=V_1'(0)$ . As $v(0)=0=V_1(0)$ , there exists $\delta \in (0, Z_1)$ such that $v(Z)<V_1(Z)$ for all $Z\in (0, \delta )$ . Assume for contradiction that $Z_*\in (0, Z_1)\subset (0, 1)$ satisfies $v(Z)<V_1(Z)$ for all $Z\in (0, Z_*)$ and $v(Z_*)=V_1(Z_*)=Z_*$ , then $v'(Z_*)\geq V_1'(Z_*)$ . Thus, by $\Delta _Z(Z_*, V_1(Z_*))=\Delta _Z(Z_*, v(Z_*))=Z_*(1-Z_*^2)^2>0$ , we have

$$ \begin{align*} (\mathscr AV_1)(Z_*)&=-\Delta_Z(Z_*, V_1(Z_*))V_1'(Z_*)+\Delta_v(Z_*, V_1(Z_*))\\ &\geq -\Delta_Z(Z_*, v(Z_*))v'(Z_*)+\Delta_v(Z_*, v(Z_*))=(\mathscr Av)(Z_*)=0, \end{align*} $$

which contradicts with (A.13). Therefore, we have $v(Z)<V_1(Z)=Z$ for all $Z\in (0, Z_1)$ .

Finally, by $v(Z_1)=v_1<Z_1<1$ , we obtain $v(Z)<Z$ for all $Z\in (0, 1)$ .

Lemma A.5. Under Assumption 1, we have $v(Z)<1/Z$ for all $Z\in (0, +\infty )$ .

Proof. Since $v(Z)\in (-1, 1)$ for all $Z\in (0, +\infty )$ by Assumption 1, it suffices to prove $v(Z)<1/Z$ for all $Z\in (1, +\infty )$ . Let $V_2(Z):=1/Z$ for $Z\in (0, +\infty )$ , then we have

(A.14) $$ \begin{align} (\mathscr AV_2)(Z)=(\beta-1)\ell Z\left(1-1/{Z^2}\right)^2>0,\quad\forall\ Z\in(1,+\infty), \end{align} $$

where we have used $\beta>1$ by (2.9). As $v(Z_1)=v_1<1=V_2(Z_1)$ , there exists $\delta>0$ such that $v(Z)<V_2(Z)$ for $Z\in [Z_1, Z_1+\delta )$ . Assume for contradiction that $Z^*\in (1, +\infty )$ satisfies $v(Z)<V_2(Z)$ for all $Z\in (Z_1, Z^*)$ and $v(Z^*)=V_2(Z^*)$ , then $v'(Z^*)\geq V_2'(Z^*)$ . Thus, by $\Delta _Z(Z^*, V_2(Z^*))=\Delta _Z(Z^*, v(Z^*))=-\ell (1-Z_*^2)^2/Z_*<0$ , we have

$$ \begin{align*} (\mathscr AV_2)(Z^*)&=-\Delta_Z(Z^*, V_2(Z^*))V_2'(Z^*)+\Delta_v(Z^*, V_2(Z^*))\\ &\leq-\Delta_Z(Z^*, v(Z^*))v'(Z^*)+\Delta_v(Z^*, v(Z^*))=(\mathscr Av)(Z^*)=0, \end{align*} $$

which contradicts with (A.14). Therefore, $v(Z)<V_2(Z)=1/Z$ for all $Z\in (1,+\infty )$ .

The proof of Remark 2.2 is completed now. To conclude this appendix, we prove that $\widehat \phi _0, \widehat \rho _0\in C_{\operatorname {e}}^\infty ([0,+\infty ))$ , where $\widehat \phi _0$ and $\widehat \rho _0$ are defined by (2.12).

Lemma A.6. Let $v\in C_{\operatorname {o}}^\infty ([0, +\infty ))$ be given by Assumption 1, and define $\widehat \phi _0, \widehat \rho _0$ by (2.12). Then we have $\widehat \phi _0, \widehat \rho _0\in C_{\operatorname {e}}^\infty ([0,+\infty ))$ .

Proof. We first claim that

(A.15) $$ \begin{align} f\in C_{\text{o}}^\infty([0, +\infty))\Longrightarrow F(Z):=\int_0^Zf(s)\,\mathrm ds\in C_{\text{e}}^\infty([0, +\infty)). \end{align} $$

Now we prove that $\widehat \phi _0\in C_{\text {e}}^\infty ([0,+\infty ))$ . By $v\in C_{\text {o}}^\infty ([0, +\infty ))$ , we have $1-Zv(Z)\in C_{\text {e}}^\infty ([0,+\infty ))$ . Since $Zv(Z)<1$ for all $Z\in [0,+\infty )$ by Lemma A.5, it follows from (5.8) that $0<\frac 1{1-Zv(Z)}\in C_{\text {e}}^\infty ([0,+\infty ))$ , hence by $v\in C_{\text {o}}^\infty ([0, +\infty ))$ and (5.7) we have $\frac {v(Z)}{1-Zv(Z)}\in C_{\text {o}}^\infty ([0, +\infty ))$ , then by (A.15) we obtain

$$\begin{align*}(\beta-1)\int_0^Z \frac{v(s)}{1-sv(s)}\,\mathrm ds\in C_{\text{e}}^\infty([0, +\infty)).\end{align*}$$

Thus, by (5.9) and (2.12) we have $\widehat \phi _0\in C_{\text {e}}^\infty ([0,+\infty ))$ .

As for $\widehat \rho _0$ , by $0<\frac 1{1-Zv(Z)}\in C_{\text {e}}^\infty ([0,+\infty ))$ , and (5.10), we have $1/{(1-Zv(Z))^{\frac 2{p-1}}}\in C_{\text {e}}^\infty ([0,+\infty )).$ Similarly, using $\widehat \phi _0\in C_{\text {e}}^\infty ([0,+\infty ))$ and $\widehat \phi _0(Z)>0$ for all $Z\in [0, +\infty )$ , we get $\widehat \phi _0(Z)^{2/(p-1)}\in C_{\text {e}}^\infty ([0,+\infty ))$ . It follows from $1-v(Z)^2\in C_{\text {e}}^\infty ([0,+\infty ))$ , $v(Z)\in (-1, 1)$ for all $Z\in [0, +\infty )$ and (5.10) that $(1-v(Z)^2)^{\frac 1{p-1}}\in C_{\text {e}}^\infty ([0,+\infty )).$ Therefore, by (2.12) and (5.5), we have $\widehat \rho _0\in C_{\text {e}}^\infty ([0,+\infty ))$ .

Finally, it suffices to show the claim (A.15). By (2.11), there exists $\widetilde f\in C^\infty ([0, +\infty ))$ such that $f(Z)=Z\widetilde f(Z^2)$ for all $Z\in [0, +\infty )$ . Let

$$\begin{align*}\widetilde F(Z):=\frac12\int_0^Z\widetilde f(s)\,\mathrm ds,\quad\forall\ Z\in[0, +\infty),\end{align*}$$

then $\widetilde F\in C^\infty ([0, +\infty ))$ . Moreover, we have

$$\begin{align*}F(Z)=\int_0^Zf(s)\,\mathrm ds=\int_0^Z s\widetilde f(s^2)\,\mathrm ds=\frac12\int_0^{Z^2}\widetilde f(s)\,\mathrm ds=\widetilde F(Z^2),\quad\forall\ Z\in[0, +\infty).\end{align*}$$

Hence by (2.10), we have $F\in C_{\text {e}}^\infty ([0,+\infty ))$ .

B Linear ODEs with singular points

In this appendix, we establish the well-posedness theory for a class of second order linear ODEs with singular points.

First of all, we introduce a preliminary lemma, which ensures that the functions we are considering are smooth in the sense of multi-variable functions. Let $I\subset \mathbb {R}$ be an interval and let $\Omega \subset \mathbb {C}$ be an open subset. We define

(B.1) $$ \begin{align} \operatorname{Hol}(\Omega)&:=\{x=x(\lambda) \text{ is holomorphic (or equivalently, analytic) on }\Omega\}, \end{align} $$
(B.2) $$ \begin{align} \mathcal H_I^{0}(\Omega)&:=\big\{x=x(t;\lambda)\in C(I\times\Omega;\mathbb{C}): x(\cdot;\lambda)\in C^\infty(I) \text{ for all }\lambda\in\Omega,\nonumber \\&\qquad x(t;\cdot)\in\operatorname{Hol}(\Omega) \text{ for all }t\in I \text{ and }\partial_t^jx\in L^\infty(I\times\Omega)\text{ for all }j\in\mathbb{Z}_{\geq 0}\big\}, \end{align} $$
(B.3) $$ \begin{align} \mathcal{H}_I(\Omega)&:=\left\{x=x(t;\lambda)\in C^\infty(I\times\Omega;\mathbb{C}): x(t;\cdot)\in\operatorname{Hol}(\Omega)\text{ for all }t\in I\right\}. \end{align} $$

Then $ \operatorname {Hol}(\Omega )$ , $ \mathcal H_I^{0}(\Omega )$ , $ \mathcal H_I(\Omega )$ are rings and the definitions in (B.1), (B.3) are the same as in section 5.3. This appendix is only used in the proof of Lemma 5.6, which does not require the definition of Z in $Z=|x|/(T-t)$ .

So, with abuse of notation, we replace Z by t and use x to denote a general function of $(t;\lambda )$ . We stress that here $(t,x)$ has nothing to do with the coordinates in $\mathbb {R}^{1+d}$ .

Lemma B.1. Let $I\subset \mathbb {R}$ be an interval and $\Omega \subset \mathbb {C}$ be an open subset. Then $\mathcal H_I^0(\Omega )\subset \mathcal H_I(\Omega )$ .

Proof. Let $x=x(t;\lambda )\in \mathcal H_I^0(\Omega )$ . Pick $\lambda _0\in \Omega $ and let $r\in (0,1)$ be such that $B_r(\lambda _0):=\{\lambda \in \mathbb {C}:|\lambda -\lambda _0|<r\}\subset \Omega $ . By Cauchy’s integration formula (Theorem 4.4 in Chapter 2 of [Reference Stein and Shakarchi69]), for any $t\in I, \lambda \in \Omega $ we have

(B.4) $$ \begin{align} x(t;\lambda)=\sum_{k=0}^\infty x_k(t)(\lambda-\lambda_0)^k, \end{align} $$

where

(B.5) $$ \begin{align} x_k(t)=\frac1{2\pi\text{i}}\int_{|\lambda-\lambda_0|=r}\frac{x(t;\lambda)}{(\lambda-\lambda_0)^{k+1}}\,\mathrm d\lambda=\frac{r^{-k}}{2\pi}\int_0^{2\pi}x\left(t;\lambda_0+re^{\text{i}\theta}\right)e^{-\text{i} k\theta}\,\mathrm d\theta \end{align} $$

for all $t\in I, k\in \mathbb {Z}_{\geq 0}$ . Since $x(\cdot ;\lambda )\in C^\infty (I)$ for all $\lambda \in \Omega $ and $\partial _t^jx\in L^\infty (I\times \Omega )$ , by (B.5) and the dominated convergence theorem, we have $x_k\in C^\infty (I)$ and

(B.6) $$ \begin{align} \left\|x_k^{(j)}\right\|_{L^\infty(I)}\leq \|\partial_t^jx\|_{L^\infty(I\times\Omega)}r^{-k},\quad \forall\ j\in\mathbb{Z}_{\geq 0},\ \forall\ k\in\mathbb{Z}_{\geq 0}. \end{align} $$

Using (B.6), we know that $\sum _{k=0}^\infty x_k^{(j)}(t)\partial _\lambda ^\alpha ((\lambda -\lambda _0)^k)$ is uniformly absolutely convergent on $I\times B_{r/2}(\lambda _0)$ for all $j\in \mathbb {Z}_{\geq 0}$ and $\alpha \in (\mathbb {Z}_{\geq 0})^2$ , hence (B.4) implies that $x\in C^\infty (I\times B_{r/2}(\lambda _0))$ . Since $\lambda _0\in \Omega $ is arbitrary, we have $x\in C^\infty (I\times \Omega )$ . Hence $x\in \mathcal {H}_I(\Omega )$ .

Remark B.2. As smoothness is a local property, we have $\mathcal {H}_{I,\text {loc}}^0(\Omega )\subset \mathcal H_I(\Omega )$ , where

$$ \begin{align*} \mathcal{H}_{I,\text{loc}}^0(\Omega):=\left\{x=x(t;\lambda):I\times\Omega\to\mathbb{C}\Big|x\in \mathcal H_{J}^0(\Omega) \text{ for any compact sub-interval }J\subset I\right\}. \end{align*} $$

Moreover, we have $\mathcal H_I(\Omega )\subset \mathcal H_{I,\text {loc}}^0(\Omega ')$ for any open subset $\Omega '\subset \subset \Omega $ (i.e. there exists a compact set K such that $\Omega '\subset K\subset \Omega $ ).

Lemma B.3. Let $I\subset \mathbb {R}$ be an interval and let $A(t)\in C^\infty (I; \mathbb {C})$ be such that $A(t)=0$ has a unique solution $t=t_0$ in I Footnote 12 with $A'(t_0)\neq 0$ . Let $\Omega \subset \mathbb {C}$ be an open subset and let $B(t;\lambda ), D(t;\lambda )\in C^\infty (I\times \Omega ;\mathbb {C})$ be such that $B, D\in \mathcal H_{I,\operatorname {loc}}^0(\Omega )$ , which implies

$$ \begin{align*} N_0^*:=\sup_{\lambda\in\Omega}\left(\max\left\{3, -\operatorname{Re}\left(\frac{B(t_0;\lambda)}{A'(t_0)}\right)+1\right\}\right)<+\infty. \end{align*} $$

Then there exists $N_0>N_0^*$ such that for all $N\in \mathbb {Z}\cap (N_0, +\infty )$ , if $f\in \mathcal {H}_{I,\operatorname {loc}}^0(\Omega )$ is such that

(B.7) $$ \begin{align} \frac{|f(t;\lambda)|}{|t-t_0|^N}\in L^\infty(J\times\Omega) \quad\text{ for any compact sub-interval }\quad J\subset I, \end{align} $$

then the linear ODE (here the prime $'$ refers to the derivative with respect to t)

(B.8) $$ \begin{align} A(t)x"(t;\lambda)+B(t;\lambda)x'(t;\lambda)+D(t;\lambda)x(t;\lambda)=f(t;\lambda) \end{align} $$

has a (complex-valued) smooth solution $x(t;\lambda )$ on $I\times \Omega $ such that $x\in \mathcal H_{I}(\Omega )$ .

Proof. Without loss of generality, we assume that $t_0=0\in {I}$ , and there exists $\delta _0\in (0, 1)$ such that $I_0=[-\delta _0, \delta _0]\subset I$ or $I_0=[0, \delta _0]= I\cap [-\delta _0, \delta _0]$ .

Step 1. Existence of a $C^2$ local solution. We define the Banach space

$$ \begin{align*} Y_N:=\left\{y\in C(I_0\times\Omega;\mathbb{C}): {y(t;\lambda)}/{|t|^N}\in L^\infty(I_0\times\Omega) \text{ and }y(t;\cdot)\in\operatorname{Hol}(\Omega)\ \forall\ t\in I_0\right\}, \end{align*} $$

where $N\geq 3$ is an integer, with the norm $\|y\|_{Y_N}:=\left \|{y(t;\lambda )}/{|t|^N}\right \|_{L^\infty (I_0\times \Omega )}\kern-1pt.$

We define a linear operator $\mathcal T_N: Y_N\to Y_N$ by

$$\begin{align*}(\mathcal T_Ny)(t;\lambda):=\int_0^t\left(\frac{B(s;\lambda)}{A(s)}y(s;\lambda)+\frac{D(s;\lambda)}{A(s)}\int_0^sy(\tau;\lambda)\,\mathrm d\tau\right)\,\mathrm ds, \qquad \forall\ t\in I_0,\ \forall\ \lambda\in\Omega.\end{align*}$$

By the hypotheses on the coefficients $A, B, D$ , we have

(B.9) $$ \begin{align} M:=\sup_{s\in I_0}\left|\frac s{A(s)}\right|+\sup_{s\in I_0,\lambda\in\Omega}\left|\frac{s B(s;\lambda)}{A(s)}\right|+\sup_{s\in I_0,\lambda\in\Omega}\left|\frac{s D(s;\lambda)}{A(s)}\right|\in (0, +\infty). \end{align} $$

Hence, for all $t\in I_0\subset [-1,1]$ and for all $\lambda \in \mathbb {C}$ we have

$$ \begin{align*} \left|(\mathcal T_Ny)(t;\lambda)\right|&=\left|\int_0^t\left(\frac{s B(s;\lambda)}{A(s)}\frac{y(s;\lambda)}{s^N}s^{N-1}+\frac{s D(s;\lambda)}{A(s)}\frac1s\int_0^s\frac{y(\tau;\lambda)}{\tau^N}\tau^N\,\mathrm d\tau\right)\,\mathrm ds\right|\\ &\leq \frac{M}{N}|t|^N\|y\|_{Y_N}, \end{align*} $$

which gives

(B.10) $$ \begin{align} \left\|\mathcal T_N\right\|_{Y_N\to Y_N}\leq \frac{M}{N},\qquad\forall\ N\in\mathbb{Z}\cap[3,+\infty). \end{align} $$

We also define an operator $\mathcal F_N:Y_N\to Y_N$ by

$$\begin{align*}(\mathcal F_Nf)(t;\lambda):=\int_0^t\frac{f(s;\lambda)}{A(s)}\,\mathrm ds, \qquad \forall\ t\in I_0,\ \forall\ \lambda\in\Omega.\end{align*}$$

Using (B.9), we know that $\mathcal F_N:Y_N\to Y_N$ is a bounded linear operator with $\left \|\mathcal F_Nf\right \|_{Y_N}\leq \frac {M}{N}\|f\|_{Y_N}$ for all $f\in Y_N$ . Now we take $N_0\in \mathbb {Z}$ such that $N_0>N_0^*+2M$ . For any $N\in \mathbb {Z}\cap (N_0, +\infty )$ , by (B.10) we know that $\|\mathcal T_N\|_{Y_N\to Y_N}\leq 1/2$ , hence $\operatorname {id}+\mathcal {T}_N:Y_N\to Y_N$ is invertible, then $(\operatorname {id}+\mathcal T_N)^{-1}\mathcal F_N: Y_N\to Y_N$ is a bounded linear operator with

$$ \begin{align*} \left\|(\operatorname{id}+\mathcal T_N)^{-1}\mathcal F_N\right\|_{Y_N\to Y_N}\leq \left\|(\operatorname{id}+\mathcal T_N)^{-1}\right\|_{Y_N\to Y_N}\left\|\mathcal F_N\right\|_{Y_N\to Y_N}\leq \frac{2M}{N}. \end{align*} $$

For any $N\in \mathbb {Z}\cap (N_0, +\infty )$ , given $f\in \mathcal {H}_{I,\text {loc}}^0(\Omega )$ satisfying (B.7) (then $f\in Y_N$ ), we define

$$ \begin{align*}y=(\operatorname{id}+\mathcal T_N)^{-1}\mathcal F_Nf\in Y_N,\quad x(t;\lambda)=\int_0^ty(s;\lambda)\,\mathrm ds,\ \forall\ t\in I_0,\ \forall\ \lambda\in\Omega,\end{align*} $$

then $x\in C(I_0\times \Omega )$ , $x(\cdot ;\lambda )\in C^1(I_0)$ for all $\lambda \in \Omega $ , $x(t;\cdot )\in \operatorname {Hol}(\Omega )$ for all $t\in I_0$ ,

$$\begin{align*}x'(t;\lambda)=\int_0^t\left(-\frac{B(\tau;\lambda)}{A(\tau)}x'(\tau;\lambda)-\frac{D(\tau;\lambda)}{A(\tau)}x(\tau;\lambda)+ \frac{f(\tau;\lambda)}{A(\tau)}\right)\,\mathrm d\tau, \quad \forall\ t\in I_0,\ \forall\ \lambda\in\Omega,\end{align*}$$

and $x'(t;\lambda )=y(t;\lambda )$ , $x(t;\lambda )=\int _0^tx'(s;\lambda )\,\mathrm ds$ for $t\in I_0$ and $\lambda \in \Omega $ . Moreover, we have (recalling $t_0=0$ )

(B.11) $$ \begin{align} \frac{x(t;\lambda)}{|t|^{N+1}}\in L^\infty(I_0\times\Omega),\qquad \frac{x'(t;\lambda)}{|t|^{N}}\in L^\infty(I_0\times\Omega). \end{align} $$

On the other hand, since $x'(t;\lambda )=\int _0^t X(s;\lambda )\,\mathrm ds$ for $t\in I_0, \lambda \in \Omega $ , where

$$ \begin{align*} X(s;\lambda)&:=-\frac{B(s;\lambda)}{A(s)}x'(s;\lambda)-\frac{D(s;\lambda)}{A(s)}x(s;\lambda)+\frac{f(s;\lambda)}{A(s)}\\ &=-\frac{sB(s;\lambda)}{A(s)}\frac{x'(s;\lambda)}{s^N}s^{N-1}-\frac{sD(s;\lambda)}{A(s)}\frac{x(s;\lambda)}{s^{N+1}}s^N+ \frac{s}{A(s)}\frac{f(s;\lambda)}{s^N}s^{N-1} \end{align*} $$

for $s\in I_0\setminus \{0\}$ and $\lambda \in \Omega $ , thus there exists a constant $C>0$ such that we have $|X(s;\lambda )|\leq C|s|^{N-1}$ for $s\in I_0\setminus \{0\}, \lambda \in \Omega $ . As $N>1$ , we know that $x'(\cdot ;\lambda )\in C^1(I_0)$ (thus $x(\cdot ;\lambda )\in C^2(I_0)$ ) and $x"(0;\lambda )=0$ for $\lambda \in \Omega $ . Hence, x solves (B.8) on $(t,\lambda )\in I_0\times \mathbb {C}$ . We also have

(B.12) $$ \begin{align} \frac{x"(t;\lambda)}{|t|^{N-1}}\in L^\infty(I_0\times\Omega). \end{align} $$

Step 2. Smoothness of the $C^2$ local solution. In this step, we show that $x(\cdot ;\lambda )\in C^\infty (I_0)$ for any $\lambda \in \Omega $ . By standard ODE theory, we have $x(\cdot ;\lambda )\in C^\infty (I_0\setminus \{0\})$ for all $\lambda \in \Omega $ . We claim that for any $k\in \mathbb {Z}\cap [0, N]$ , there exists a constant $C_k>0$ such that

(B.13) $$ \begin{align} x^{(k)}(0;\lambda)=0\qquad\text{and}\qquad |x^{(k)}(t;\lambda)|\leq C_k|t|^{N+1-k},\qquad\forall \ t\in I_0,\ \forall\ \lambda\in\Omega. \end{align} $$

We use the induction. By (B.11) and (B.12), we know that (B.13) holds for $k\in \{0,1,2\}$ . Assume that for some $K\in \mathbb {Z}\cap [1, N-1]$ , (B.13) holds for all $k\in \mathbb {Z}\cap [0, K]$ . Now we prove that (B.13) holds for $k=K+1$ . By our induction hypotheses, $x(\cdot ;\lambda )\in C^{(K)}(I_0)$ and $x^{(K+1)}(0;\lambda )=\lim _{t\to 0}(x^{(K)}(t;\lambda )/t)=0$ for all $\lambda \in \Omega $ . For $t\in I_0\setminus \{0\}$ , taking derivative $K-1$ times on both sides of (B.8) with respect to t, we obtain

(B.14) $$ \begin{align} A(t)x^{(K+1)}(t;\lambda)+\sum_{j=0}^{K}A_{j, K}(t;\lambda)x^{(j)}(t;\lambda)=f^{(K-1)}(t;\lambda),\quad \forall\ t\in I_0\setminus\{0\},\ \forall\ \lambda\in\Omega, \end{align} $$

where $A_{j,K}$ ’s are linear combinations of $A, B, D$ and their derivatives, hence $A_{j,K}(t;\lambda )\in C^\infty (I\times \Omega )\cap L^\infty (I_0\times \Omega )$ for all $j\in \mathbb {Z}\cap [0, K]$ .Footnote 13 As $f\in \mathcal H_{I,\text {loc}}^0(\Omega )$ satisfies (B.7), we haveFootnote 14 $f^{(k)}(t;\lambda )/|t|^{N-k}\in L^\infty (I_0\times \Omega )$ for all $k\in \mathbb {Z}\cap [0, N]$ . Therefore,

$$\begin{align*}\frac{\left|x^{(K+1)}(t;\lambda)\right|}{|t|^{N-K}}=\frac{|t|}{|A(t)|}\frac{\left|f^{(K-1)}(t;\lambda)-\sum_{j=0}^{K}A_{j, K}(t;\lambda)x^{(j)}(t;\lambda)\right|}{|t|^{N-(K-1)}}\in L^\infty(I_0\times\Omega).\end{align*}$$

This proves (B.13) for $k=K+1$ . Hence, (B.13) holds by the induction and thus $x(\cdot ;\lambda )\in C^N(I_0)$ for all $\lambda \in \Omega $ and

(B.15) $$ \begin{align} \sup_{t\in I_0,\lambda\in\Omega}|x^{(k)}(t;\lambda)|<+\infty,\quad\forall\ k\in\mathbb{Z}\cap[0, N]. \end{align} $$

Next we claim that for $k\in \mathbb {Z}\cap [N,+\infty )$ we have

(B.16) $$ \begin{align} \sup_{t\in I_0\setminus\{0\},\lambda\in\Omega}|x^{(k)}(t;\lambda)|<+\infty. \end{align} $$

By (B.13), we know that (B.16) holds for $k=N$ . Assume that for some $k\in \mathbb {Z}_{\geq N}$ we have

(B.17) $$ \begin{align} \sup_{t\in I_0\setminus\{0\},\lambda\in\Omega}|x^{(N)}(t;\lambda)|<+\infty,\cdots, \sup_{t\in I_0\setminus\{0\},\lambda\in\Omega}|x^{(k)}(t;\lambda)|<+\infty. \end{align} $$

For $t\in I_0\setminus \{0\}$ and $\lambda \in \Omega $ , by (B.14) for $K=k+1$ and footnote 13 we have

(B.18) $$ \begin{align} A(t)x^{(k+2)}&(t;\lambda)+(kA'(t)+B(t;\lambda))x^{(k+1)}(t;\lambda)=F_k(t;\lambda), \end{align} $$
(B.19) $$ \begin{align} &F_k(t;\lambda):=f^{(k)}(t;\lambda)-\sum_{j=0}^{k}A_{j, k+1}(t;\lambda)x^{(j)}(t;\lambda). \end{align} $$

Then by $f\in \mathcal H_{I_0}^0(\Omega )$ , (B.15) and (B.17) we have $\sup _{t\in I_0\setminus \{0\},\lambda \in \Omega }|F_k(t;\lambda )|<+\infty $ . Let $\widetilde A(t):=\int _0^1A'(ts)\,\mathrm ds$ and $\widetilde B(t;\lambda ):=B(t;\lambda )-\frac {B(0;\lambda )}{A'(0)}\widetilde A(t)$ for $t\in I_0, \lambda \in \Omega $ , then $\widetilde B(0;\lambda )=0$ , $\widetilde A\in C^{\infty }(I_0)$ , $\widetilde B\in C(I_0\times \Omega )$ for $\lambda \in \Omega $ , $\partial _t\widetilde B\in L^{\infty }(I_0\times \Omega ) $ and $\widetilde A(t)={A(t)}/{t}$ for $ t\in I_0\setminus \{0\}$ . Thus, $\widetilde B(t;\lambda )/t\in L^{\infty }(I_0\times \Omega )$ and $\widetilde B(t;\lambda )/A(t)\in L^{\infty }(I_0\times \Omega )$ (using (B.9)).

Let $\eta (t;\lambda ):=|t|^{\frac {B(0;\lambda )}{A'(0)}}\exp \left (\int _0^t\frac {\widetilde B(s;\lambda )}{A(s)}\,\mathrm ds\right ) $ then (here $\eta $ is different from the one in (2.23))

(B.20) $$ \begin{align} C_\eta^{-1}|t|^{\operatorname{Re}\frac{B(0;\lambda)}{A'(0)}}\leq|\eta(t;\lambda)|\leq C_\eta|t|^{\operatorname{Re}\frac{B(0;\lambda)}{A'(0)}},\qquad \forall\ t\in I_0\setminus\{0\},\ \forall\ \lambda\in\Omega \end{align} $$

for some constant $C_\eta>0$ . We also have $\eta \in C^\infty ((I_0\setminus \{0\})\times \Omega ;\mathbb {C}\setminus \{0\})$ and

$$ \begin{align*} \frac{\eta'(t;\lambda)}{\eta(t;\lambda)}=\frac{\widetilde B(t;\lambda)}{A(t)}+\frac{B(0;\lambda)}{A'(0)t}=\frac{ B(t;\lambda)}{A(t)}-\frac{B(0;\lambda)}{A'(0)}\frac{\widetilde A(t)}{A(t)}+\frac{B(0;\lambda)}{A'(0)t}=\frac{ B(t;\lambda)}{A(t)},\quad \forall\ t\in I_0\setminus\{0\}. \end{align*} $$

Here we used $\widetilde A(t)={A(t)}/{t}$ . It follows from (B.18) that

$$ \begin{align*} \left(A(t)^k\eta(t;\lambda)x^{(k+1)}(t;\lambda)\right)'=A(t)^{k-1}\eta(t;\lambda)F_k(t;\lambda),\qquad\forall\ t\in I_0\setminus\{0\},\ \forall\ \lambda\in\Omega. \end{align*} $$

By (B.14) for $K=k$ and (B.15), (B.17), we have $A(t)x^{(k+1)}(t;\lambda )\in L^\infty ((I_0\setminus \{0\})\times \Omega )$ ; using (B.20), $|A(t)|\sim |t|$ as $t\to 0$ and

(B.21) $$ \begin{align}k\geq N>N_0>-\inf_{\lambda\in\Omega}\operatorname{Re}\left(B(0;\lambda)/A'(0)\right)+1,\end{align} $$

we have

$$\begin{align*}\lim_{t\to0} A(t)^k\eta(t;\lambda)x^{(k+1)}(t;\lambda)=0,\quad \forall\ \lambda\in\Omega,\end{align*}$$

hence

$$\begin{align*}A(t)^k\eta(t;\lambda)x^{(k+1)}(t;\lambda)=\int_0^tA(s)^{k-1}\eta(s;\lambda)F_k(s;\lambda)\,\mathrm ds,\qquad\forall\ t\in I_0\setminus\{0\},\ \forall\ \lambda\in\Omega.\end{align*}$$

As a consequence, we have

$$ \begin{align*} \left|x^{(k+1)}(t;\lambda)\right|=\frac{\left|\int_0^tA(s)^{k-1}\eta(s;\lambda)F_k(s;\lambda)\,\mathrm ds\right|}{|A(t)|^k|\eta(t;\lambda)|}\leq C_{k+1}\frac{\int_0^{|t|}s^{k-1}s^{\operatorname{Re}(B(0;\lambda)/A'(0))}\,\mathrm ds}{|t|^k|t|^{\operatorname{Re}(B(0;\lambda)/A'(0))}}\leq \widetilde C_{k+1} \end{align*} $$

for all $t\in I_0\setminus \{0\}$ and $\lambda \in \Omega $ , where $C_{k+1}>0$ and $\widetilde C_{k+1}>0$ are constants. Here we have used $k+\inf _{\lambda \in \Omega }\operatorname {Re}(B(0;\lambda )/A'(0))>0$ , which follows from (B.21). This proves (B.16).

Next we use once again the induction to prove that

(B.22) $$ \begin{align} x^{(k)}(0;\lambda) \text{ exists and } \lim_{t\to0}x^{(k)}(t;\lambda)=x^{(k)}(0;\lambda),\qquad\forall\ \lambda\in\Omega,\ \forall\ k\in\mathbb{Z}\cap[0,+\infty). \end{align} $$

We know from (B.13) that (B.22) holds for $k\leq N$ . Now we assume that for some $k\in \mathbb {Z}_{\geq N}$ , (B.22) holds for $0, 1,\cdots , k$ . Then by (B.19), we have $F_k(\cdot ;\lambda )\in C(I_0)$ , by (B.16) with k replaced by $k+2$ and $A(0)=0$ we have $\lim _{t\to 0}A(t)x^{(k+2)}(t;\lambda )=0$ , and by (B.18) we have

$$\begin{align*}\lim_{t\to0} x^{(k+1)}(t;\lambda)=\frac {F_k(0;\lambda)}{kA'(0)+B(0;\lambda)}\in\mathbb{C},\qquad\forall\ \lambda\in\Omega,\end{align*}$$

where we have used $kA'(0)+B(0;\lambda )\neq 0$ , which follows from $k>-\inf _{\lambda \in \Omega }\operatorname {Re}(B(0;\lambda )/A'(0))$ (see (B.21)). Finally, we get by L’Hôpital’s rule that

$$\begin{align*}x^{(k+1)}(0;\lambda)=\lim_{t\to0}\frac{x^{(k)}(t;\lambda)-x^{(k)}(0)}{t}=\lim_{t\to0}x^{(k+1)}(t;\lambda),\qquad\forall\lambda\in\Omega.\end{align*}$$

This proves (B.22) for $k+1$ . Then (B.22) holds for all $k\in \mathbb {Z}_{\geq N}$ by the induction. Hence, $x(\cdot ;\lambda )\in C^\infty (I_0)$ for all $\lambda \in \Omega $ . Moreover, combining (B.15) and (B.16) gives that

(B.23) $$ \begin{align} \partial_t^jx\in L^\infty(I_0\times\Omega),\qquad\forall\ j\in\mathbb{Z}_{\geq 0}. \end{align} $$

Step 3. $x\in C^\infty (I_0\times \Omega )$ . Recall that $x\in C(I_0\times \Omega )$ satisfies $x(t;\cdot )\in \operatorname {Hol}(\Omega )$ for all $t\in I_0$ (in Step 1) and $x(\cdot ;\lambda )\in C^\infty (I_0)$ for all $\lambda \in \Omega $ (in Step 2). Using (B.23) we have $x\in \mathcal H_{I_0}^0(\Omega )$ . Then Lemma B.1 implies that $x\in C^\infty (I_0\times \Omega )$ .

Step 4. Extension of the smooth local solution. For any fixed $\lambda \in \Omega $ , we have constructed a local solution $x_L(\cdot ;\lambda )\in C^\infty (I_0)$ of (B.8) on $I_0\subset I$ . Moreover, we have $x_L\in C^\infty (I_0\times \Omega )$ and $x_L(t;\cdot )\in \operatorname {Hol}(\Omega )$ for all $t\in I_0$ . By standard ODE theory, the initial value problem

$$\begin{align*}\begin{cases} x"(t;\lambda)+\frac{B(t;\lambda)}{A(t)}x'(t;\lambda)+\frac{D(t;\lambda)}{A(t)}x(t;\lambda)=\frac{f(t;\lambda)}{A(t)},\\ x(\delta_0/2;\lambda)=x_L(\delta_0/2;\lambda), x'(\delta_0/2;\lambda)=x_L'(\delta_0/2;\lambda) \end{cases}\end{align*}$$

has a unique solution $x=x(t;\lambda )$ on $((0, +\infty )\cap I)\times \Omega $ and $x\in C^\infty (((0, +\infty )\cap I)\times \Omega )$ . Moreover, by the analytic dependence on parameters (Lemma B.5), we have $x(t;\cdot )\in \operatorname {Hol}(\Omega )$ for all $t\in (0, +\infty )\cap I$ . Hence, $x_L$ can be extended to be a smooth solution of (B.8) on $((0, +\infty )\cap I)\times \Omega $ ; Similarly we can extend $x_L$ on the negative direction (for the case $I_0=[-\delta _0,\delta _0]$ ). And for the extended solution x, we have $x\in \mathcal H_I(\Omega )$ .

Proposition B.4. Let $I\subset \mathbb {R}$ be an interval. Let $A(t)\in C^\infty (I;\mathbb {C})$ be such that $A(t)=0$ has a unique solution $t=t_0$ in I with $A'(t_0)\neq 0$ . Let $B(t;\lambda ), D(t;\lambda )\in \mathcal H_I(\mathbb {C})$ . Assume that $B(t;\lambda )=\widetilde B(t)+\lambda \widehat B(t)$ for $t\in I$ and $\lambda \in \mathbb {C}$ , where $\widetilde B, \widehat B\in C^\infty (I;\mathbb {C})$ . Suppose that

(B.24) $$ \begin{align} \text{either }\widehat B(t_0)\neq0\quad \text{or}\quad \widehat B(t_0)=0 \text{ and }-\widetilde B(t_0)/A'(t_0)\notin\mathbb{Z}_{\geq 0}. \end{align} $$

We define

(B.25) $$ \begin{align} \Lambda_*:=\left\{\lambda\in\mathbb{C}: nA'(t_0)+B(t_0;\lambda)=0\text{ for some }n\in\mathbb{Z}_{\geq 0}\right\}. \end{align} $$

Then $\Lambda _*\subset \mathbb {C}$ is a (probably empty) discrete set. Let $R\in (0,+\infty )$ . There exists a nonzero polynomial $ \psi _1(\lambda )$ satisfying $\{\lambda \in B_R: \psi _1(\lambda )=0\}=\Lambda _*\cap B_R$ such that for every $f(t;\lambda )\in \mathcal H_I(\mathbb {C}) $ , the inhomogeneous ODE

(B.26) $$ \begin{align} \begin{cases} A(t)x"(t;\lambda)+B(t;\lambda)x'(t;\lambda)+D(t;\lambda)x(t;\lambda)=\psi_1(\lambda)f(t;\lambda),\quad t\in I, \lambda\in B_R,\\ x(t_0;\lambda)=\psi_1(\lambda),\quad \lambda\in B_R, \end{cases} \end{align} $$

where the prime $'$ refers to the derivative with respect to $t\in I$ , has a solution $x=x(t;\lambda )\in \mathcal H_I(B_R)$ . Moreover, if $\widehat B(t_0)=0$ , then $ \psi _1(\lambda )=1$ .

Proof. We first show that $\Lambda _*$ is a discrete set. If $\widehat B(t_0)\neq 0$ , then $\Lambda _*=\{-nA'(t_0)/\widehat B(t_0)-\widetilde B(t_0)/\widehat B(t_0): n\in \mathbb {Z}_{\geq 0}\}$ , hence $\Lambda _*$ is a discrete set. If $\widehat B(t_0)= 0$ , then $\lambda _*\in \Lambda _*$ if and only if $0=nA'(t_0)+B(t_0;\lambda _*)=nA'(t_0)+\widetilde B(t_0)=0$ for some $n\in \mathbb {Z}_{\geq 0}$ , which implies that $-\widetilde B(t_0)/A'(t_0)\in \mathbb {Z}_{\geq 0}$ , and this is a contradiction with our assumption (B.24). As a consequence, if $\widehat B(t_0)=0$ (and $-\widetilde B(t_0)/A'(t_0)\notin \mathbb {Z}_{\geq 0}$ ), then $\Lambda _*=\emptyset $ .

Next, we construct $ \psi _1(\lambda )$ . Let $N_0$ be given by Lemma B.3 (for $\Omega =B_R$ ) and fix an integer $N>\max \{N_0+1,-\inf _{\lambda \in B_R}\operatorname {Re}\left (B(t_0;\lambda )/A'(t_0)\right )+1\}$ . Let $ \psi _1(\lambda ):=1$ for the case $\widehat B(t_0)=0$ and $ \psi _1(\lambda ):=\prod _{j=0}^{N-1}\big (jA'(t_0)+B(t_0,\lambda )\big )$ for the case $\widehat B(t_0)\neq 0$ .

Claim 1. $ \psi _1(\lambda )$ is a nonzero polynomial. If $\widehat B(t_0)=0$ , then $ \psi _1(\lambda )\equiv 1$ is a polynomial of degree $0$ ; if $\widehat B(t_0)\neq 0$ , as $ B(t_0,\lambda )=\widetilde B(t_0)+\lambda \widehat B(t_0)$ , then $ \psi _1(\lambda )$ is a polynomial of degree N.

Claim 2. $\{\lambda \in B_R: \psi _1(\lambda )=0\}=\Lambda _*\cap B_R$ . If $\widehat B(t_0)=0$ , then $\{\lambda \in B_R: \psi _1(\lambda )=0\}=\emptyset =\Lambda _*=\Lambda _*\cap B_R$ . For the case $\widehat B(t_0)\neq 0$ , if $\psi _1(\lambda )=0 $ then $jA'(t_0)+B(t_0,\lambda )=0$ for some $j\in \mathbb {Z}\cap [0,N-1]$ and $\lambda \in \Lambda _*$ , thus $\{\lambda \in B_R: \psi _1(\lambda )=0\}\subseteq \Lambda _*\cap B_R$ . On the other hand, if $\lambda _0\in \Lambda _*\cap B_R$ (and $\widehat B(t_0)\neq 0$ ), then $nA'(t_0)+B(t_0;\lambda _0)=0$ for some $n\in \mathbb {Z}_{\geq 0}$ , and $n=-B(t_0;\lambda _0)/A'(t_0)\leq -\inf _{\lambda \in B_R}\operatorname {Re}\left (B(t_0;\lambda )/A'(t_0)\right )<N-1$ , thus $n\in \mathbb {Z}\cap [0,N-1]$ and $\psi _1(\lambda _0)=0 $ . So $\Lambda _*\cap B_R\subseteq \{\lambda \in B_R: \psi _1(\lambda )=0\}$ .

It remains to construct $x(t;\lambda )$ . For any $n\in \mathbb {Z}\cap [0, N]$ and $\lambda \in \mathbb {C}$ , let $ \psi _{1,n}(\lambda ):=1$ for the case $\widehat B(t_0)=0$ and $ \psi _{1,n}(\lambda ):=\prod _{j=n}^{N-1}\big (jA'(t_0)+B(t_0,\lambda )\big )$ (here $\psi _{1, N}(\lambda ):=1$ ) for the case $\widehat B(t_0)\neq 0$ . Then $ \psi _{1}(\lambda )=\psi _{1,0}(\lambda )$ for all $\lambda \in \mathbb {C}$ .

Claim 3. If $n\in \mathbb {Z}\cap [0,N]$ , $g(t;\lambda )\in \mathcal H_I(\mathbb {C}) $ , $ \partial _t^ig(t_0;\lambda )=0$ for $i\in \mathbb {Z}$ , $0\leq i<n$ . Then

(B.27) $$ \begin{align} A(t)y"(t;\lambda)+B(t;\lambda)y'(t;\lambda)+D(t;\lambda)y(t;\lambda)=\psi_{1,n}(\lambda)g(t;\lambda),\ y(t_0;\lambda)=0,\ \lambda\in B_R, \end{align} $$

has a solution $y=y(t;\lambda )\in \mathcal H_I(B_R)$ .

Let $g(t;\lambda )=f(t;\lambda )-D(t;\lambda )$ , $n=0$ , then by Claim 3, (B.27) has a solution $y=y(t;\lambda )\in \mathcal H_I(B_R)$ with $n=0$ . $x(t;\lambda )=y(t;\lambda )+\psi _{1}(\lambda )\in \mathcal H_I(B_R)$ solves (B.26) (using $ \psi _{1}(\lambda )=\psi _{1,0}(\lambda )$ ).

It remains to prove Claim 3. We use the (backward) induction. We need to prove that:

  1. (i) Claim 3 holds for $n=N$ ;

  2. (ii) if $j\in \mathbb {Z}\cap [0,N-1]$ , Claim 3 holds for $n=j+1$ , then Claim 3 holds for $n=j$ .

Proof of (i). As $g\in \mathcal H_I(\mathbb {C})\subset \mathcal H_{I,\text {loc}}^0(B_R)$ , $n=N$ , by Taylor’s theorem with integral remainders, we have $g(t;\lambda )/|t-t_0|^{N}\in L^\infty _{\text {loc}}(I\times \mathbb {C})$ , and we also have $ \psi _{1,n}(\lambda )=\psi _{1,N}(\lambda )=1$ . Then the result follows from Lemma B.3.

Proof of (ii). We fix $j\in \mathbb {Z}\cap [0,N-1]$ and assume $g(t;\lambda )\in \mathcal H_I(\mathbb {C}) $ , $ \partial _t^ig(t_0;\lambda )=0$ for $i\in \mathbb {Z}$ , $0\leq i<j$ . For $t\in I, \lambda \in \mathbb {C}$ , let $x_j(t):=(t-t_0)^{j+1}$ , $y_j(t;\lambda ):=A(t)x_j"(t)+B(t;\lambda )x_j'(t)+D(t;\lambda )x_j(t)$ , then $x_j\in C^{\infty }(I)$ , $x_j(t_0)=0$ , $y_j(t;\lambda )\in \mathcal H_I(\mathbb {C})$ , and

$$ \begin{align*} &y_j(t;\lambda)=A(t)j(j+1)(t-t_0)^{j-1}+B(t;\lambda)(j+1)(t-t_0)^{j}+D(t;\lambda)(t-t_0)^{j+1}. \end{align*} $$

By Taylor’s formula, we have $\partial _t^iy_j(t_0;\lambda )=0$ for $i\in \mathbb {Z}$ , $0\leq i<j$ and

$$ \begin{align*} \partial_t^jy_j(t_0;\lambda)&=j!\lim_{t\to t_0}\frac{y_j(t;\lambda)}{(t-t_0)^j}=j!\lim_{t\to t_0}j(j+1)\frac{A(t)}{t-t_0}+j!B(t_0;\lambda)(j+1)\\&=j!j(j+1)A'(t_0)+(j+1)!B(t_0;\lambda)=(j+1)!(jA'(t_0)+B(t_0;\lambda)). \end{align*} $$

For the case of $\widehat B(t_0)\neq 0$ , let $a_j=(j+1)!$ , $b_j(\lambda )=jA'(t_0)+B(t_0;\lambda )$ then $a_j\neq 0$ , $b_j\in \operatorname {Hol}(\mathbb {C})$ . As $ \psi _{1,n}(\lambda )=\prod _{j=n}^{N-1}\big (jA'(t_0)+B(t_0;\lambda )\big )$ for $n\in \mathbb {Z}\cap [0,N]$ we have $ \psi _{1,j}(\lambda )=b_j(\lambda )\psi _{1,j+1}(\lambda )$ and $ \partial _t^jy_j(t_0;\lambda )=(j+1)!(jA'(t_0)+B(t_0;\lambda ))=a_jb_j(\lambda )$ for all $\lambda \in \mathbb {C}$ .

For the case of $\widehat B(t_0)=0$ , we have $jA'(t_0)+B(t_0,\lambda )=jA'(t_0)+\widetilde B(t_0)\neq 0$ (using (B.24)). Let $a_j=(j+1)!(jA'(t_0)+\widetilde B(t_0))$ , $b_j(\lambda )=1$ then $a_j\neq 0$ , $b_j\in \operatorname {Hol}(\mathbb {C})$ , $\partial _t^jy_j(t_0;\lambda )=a_j=a_jb_j $ . As $ \psi _{1,n}(\lambda )=1$ for $n\in \mathbb {Z}\cap [0,N]$ we have $\psi _{1,j}(\lambda )=b_j(\lambda )\psi _{1,j+1}(\lambda )$ for all $\lambda \in \mathbb {C}$ .

Thus, we always have $a_j\neq 0$ , $b_j\in \operatorname {Hol}(\mathbb {C})$ , $\psi _{1,j}(\lambda )=b_j(\lambda )\psi _{1,j+1}(\lambda )$ , $ \partial _t^jy_j(t_0;\lambda )=a_jb_j(\lambda )$ .

For $t\in I, \lambda \in \mathbb {C}$ , let $\widetilde g(t;\lambda ):=b_j(\lambda )g(t;\lambda )-\partial _t^jg(t_0;\lambda )\cdot y_j(t;\lambda )/a_j $ then $\widetilde g\in \mathcal H_I(\mathbb {C})$ ,Footnote 15 and $\partial _t^j\widetilde g(t_0;\lambda )=0$ . As $ \partial _t^ig(t_0;\lambda )=0$ , $\partial _t^iy_j(t_0;\lambda )=0$ for $i\in \mathbb {Z}$ , $0\leq i<j$ , we have $ \partial _t^i\widetilde g(t_0;\lambda )=0$ , for $i\in \mathbb {Z}$ , $0\leq i<j$ . Thus, $ \partial _t^i\widetilde g(t_0;\lambda )=0$ , for $i\in \mathbb {Z}$ , $0\leq i\leq j$ .

By the induction assumption (for $n=j+1$ ), there exists $\widetilde y(t;\lambda )\in \mathcal H_I(B_R)$ such that

$$ \begin{align*} &A(t)\widetilde y"(t;\lambda)+B(t;\lambda)\widetilde y'(t;\lambda)+D(t;\lambda)\widetilde y(t;\lambda)=\psi_{1,j+1}(\lambda)\widetilde g(t;\lambda),\quad \widetilde y(t_0;\lambda)=0,\quad \lambda\in B_R. \end{align*} $$

For all $t\in I, \lambda \in B_R$ , let $ y(t;\lambda ):=\widetilde y(t;\lambda )+\psi _{1,j+1}(\lambda )\partial _t^jg(t_0;\lambda )\cdot x_j(t)/a_j$ , then $y\in \mathcal H_I(B_R)$ , $y(t_0;\lambda )=0 $ and

$$ \begin{align*} &A(t) y"(t;\lambda)+B(t;\lambda) y'(t;\lambda)+D(t;\lambda) y(t;\lambda)\\&\quad =\psi_{1,j+1}(\lambda)\widetilde g(t;\lambda)+\psi_{1,j+1}(\lambda)\partial_t^jg(t_0;\lambda)\cdot y_j(t;\lambda)/a_j=\psi_{1,j+1}(\lambda)b_j(\lambda)g(t;\lambda)=\psi_{1,j}(\lambda)g(t;\lambda), \end{align*} $$

where we have used $y_j(t;\lambda )=A(t)x_j"(t)+B(t;\lambda )x_j'(t)+D(t;\lambda )x_j(t)$ , $\widetilde g(t;\lambda )+\partial _t^jg(t_0;\lambda )\cdot y_j(t;\lambda )/a_j =b_j(\lambda )g(t;\lambda )$ and $\psi _{1,j}(\lambda )=b_j(\lambda )\psi _{1,j+1}(\lambda )$ . Thus, $y\in \mathcal H_I(B_R)$ solves (B.27) for $n=j$ . This completes the proof.

In the end of this appendix, we prove the analytic dependence on parameters of solutions to linear regular ODEs. The following lemma has been used in Step 4 of the proof of Lemma B.3, to show that the extended smooth solution is analytic with respect to the parameter $\lambda $ .

Lemma B.5. Let $\Omega \subset \mathbb {C}$ be an open set and $I\subset \mathbb {R}$ . Let $p(t;\lambda ), q(t;\lambda ), f(t;\lambda )\in C^\infty (I\times \Omega ;\mathbb {C})$ be such that $p(t;\cdot ), q(t;\cdot ), f(t;\cdot )$ are analytic on $\Omega $ for each $t\in I$ . Let $x_0(\lambda ), x_1(\lambda )$ be two analytic functions on $\Omega $ and let $t_0\in I$ . For each $\lambda \in \Omega $ , let $x(t;\lambda ) (t\in I)$ be the unique smooth solution to the initial value problem

$$ \begin{align*} x"(t;\lambda)+p(t;\lambda)x'(t;\lambda)+q(t;\lambda)x(t;\lambda)=f(t;\lambda),\quad x(t_0;\lambda)=x_0(\lambda), x'(t_0;\lambda)=x_1(\lambda), \end{align*} $$

where the prime $'$ refers to the derivative with respect to $t\in I$ . Then for each $t\in I$ , the function $\lambda \in \Omega \mapsto x(t;\lambda )$ is analytic.

Proof. By the standard ODE theory, we know that $x\in C^\infty (I\times \Omega )$ . For any complex function $\varphi =\varphi (\lambda ):\mathbb {C}\to \mathbb {C}$ of class $C^1$ seen as a function on $\mathbb {R}^2$ , we can define the Wirtinger derivatives

$$\begin{align*}\partial_{\bar\lambda}\varphi(\lambda)=\frac12\big(\partial_1\varphi(\lambda)+\text{i}\partial_2\varphi(\lambda)\big),\quad \partial_{\lambda}\varphi(\lambda)=\frac12\big(\partial_1\varphi(\lambda)-\text{i}\partial_2\varphi(\lambda)\big).\end{align*}$$

Now it suffices to show that $\partial _{\bar \lambda }x(t;\lambda )=0$ for all $(t, \lambda )\in I\times \Omega $ . Since $x\in C^\infty (I\times \Omega )$ , the derivative with respect to t and $\partial _{\bar \lambda }$ are commutable. By the analyticity of coefficients and the initial data, we know that $\partial _{\bar \lambda }x$ satisfies

$$\begin{align*}(\partial_{\bar\lambda}x)"(t;\lambda)+p(t;\lambda)(\partial_{\bar\lambda}x)'(t;\lambda)+q(t;\lambda)\partial_{\bar\lambda}x(t;\lambda)=0,\quad \partial_{\bar\lambda}x(t_0;\lambda)=(\partial_{\bar\lambda}x)'(t_0;\lambda)=0.\end{align*}$$

By the uniqueness, we have $\partial _{\bar \lambda }x(t;\lambda )=0$ for all $(t, \lambda )\in I\times \Omega $ .

Acknowledgments

We would like to warmly thank the anonymous referee who gave many insightful comments and helpful suggestions that substantially improved the manuscript.

Competing interest

The authors have no competing interests to declare.

Financial support

D. Wei is partially supported by the National Key R&D Program of China under the grant 2021YFA1001500. Z. Zhang is partially supported by NSF of China under Grant 12288101.

Footnotes

1 Here we use the Einstein’s summation convention.

2 In particular, we have $k>\ell $ , which is equivalent to $p>1+4/(d-2)$ . So we are in the supercritical case. Nevertheless, we can not cover the whole supercritical range using the method of current paper.

3 System (2.6) is exactly the same as (2.5) and (2.6) in [Reference Shao, Wei and Zhang65] as long as we let $\ell =1+4/(p-1)$ and $\varrho =\rho ^{p+1}$ .

4 Note that $\beta $ in this paper is not the same as $\beta $ in [Reference Shao, Wei and Zhang65]. In fact, $\beta _{\text {in this paper}}=\beta _{\text {in [40]}}/(\ell +1)$ . Hence, $\beta _{\text {in this paper}}>1$ is equivalent to $\beta _{\text {in [65]}}>\ell +1$ , see Lemma A.7 in [Reference Shao, Wei and Zhang65].

5 ODE (2.8) is exactly the same as (2.17) in [Reference Shao, Wei and Zhang65], as long as we let $m=\beta \ell $ .

6 For fixed $(t,x)\in [0,T)\times \mathbb {R}^d$ , the summations in (2.23) are both finite sums. Indeed, we have $T-t>0$ , then $\lim _{n\to \infty }(T-t)/T_n=+\infty $ , thus $(T-t)/T_n>2$ for all sufficiently large n and hence $\eta \big ((T-t)/T_n\big )=0$ for all sufficiently large n. As a consequence, we have $\rho _*, \phi _*\in C^\infty ([0,T)\times \mathbb {R}^d)$ .

7 Note that $1<\ell ^*(3)<\ell _1(4)<3/2$ , thus if (2.33) holds then $\ell +\sqrt \ell <2\ell <3\leq k$ .

8 Here we explain the notations to avoid ambiguities. For a smooth function $f(t,x)$ , we denote the action of the vector field X on f by $Xf$ , i.e., $Xf=X_\alpha \partial ^\alpha f=X^\alpha \partial _\alpha f$ , where $X_0=\partial _t\Phi _*\xi , X^0=-X_0$ and $X_j=X^j=\partial _{j}\Phi _*\xi $ for $j\in \mathbb {Z}\cap [1,d]$ . The same clarification holds also for Y. Moreover, in (4.2), although $D\Phi _*(t,x)$ is only defined for $(t,x)\in \mathcal C$ , we just simply let $X(t,x)=0$ for $(t,x)\in ([0, T)\times \mathbb {R}^d)\setminus \mathcal C$ , noting that $\xi (3|x|/(5(T-t)))=0$ near the boundary of $\mathcal C$ . The same clarification holds also for Y, and $V_*, N_*$ in (4.3).

9 Then $\{\varepsilon _n\}$ becomes its subsequence satisfying $\varepsilon _n\leq c_2/2^n$ .

10 Here, we need to estimate $|f_n-f_{n+1}|$ , which is achieved by combining (4.8) ( $j=0, \lambda =1$ ) and

$$ \begin{align*} \left|\xi_1\left(\frac{T-t}{\varepsilon_n}\right)-\xi_1\left(\frac{T-t}{\varepsilon_m}\right)\right|\leq \int_{\varepsilon_m}^{\varepsilon_n}\frac{T-t}{\varepsilon^2}\left|\xi_1'\left(\frac{T-t}{\varepsilon}\right)\right|\,\mathrm d\varepsilon\leq \|z^2\xi_1'(z)\|_{L^\infty}(T-t)^{-1}\varepsilon_n \end{align*} $$

for all positive integers $m>n$ , where we have used the fact that $\operatorname {supp}\xi _1'\subset [-1, 1]$ .

11 In (5.1), “e” stands for “even”; In (5.2), “o” stands for “odd”. Please don’t confuse $C_{\text {o}}^\infty (I)$ with “ $C_0^\infty (I)$ ”.

12 It means that $\{t\in I:A(t)=0\}=\{t_0\}$ , and also for Proposition B.4.

13 (B.14) and the properties of $A_{j,K}$ ’s holds for all $K\in \mathbb {Z}_+$ (not merely for $K\in \mathbb {Z}\cap [1, N-1]$ ), and we also have $A_{K, K}(t;\lambda )=(K-1)A'(t)+B(t;\lambda )$ for all $t\in I_0, \lambda \in \Omega $ .

14 Indeed, (B.7) implies that $f^{(k)}(0;\lambda )=0$ for any $k\in \mathbb {Z}\cap [0, N-1]$ and any $\lambda \in \Omega $ . As a consequence, we have $|f^{(N-1)}(t;\lambda )|=\left |\int _0^tf^{(N)}(s;\lambda )\,\mathrm ds\right |\leq \left (\sup _{s\in I_0,\lambda \in \Omega }|f^{(N)}(s;\lambda )|\right )|t|$ for all $t\in I_0, \lambda \in \Omega $ , where we have used $f\in \mathcal H_{I_0}^0(\Omega )$ . Similarly one shows that $f^{(k)}(t;\lambda )/|t|^{N-k}\in L^\infty (I_0\times \Omega )$ for all $k\in \mathbb {Z}\cap [0, N]$ .

15 Here we use the fact that if $x\in \mathcal H_I(\Omega )$ , then $x^{(n)}\in \mathcal H_I(\Omega )$ for any $n\in \mathbb {Z}_{\geq 0}$ .

References

Alinhac, S., Blowup for nonlinear hyperbolic equations (Progr. Nonlinear Differential Equations Appl.) vol. 17 (Birkhäuser Boston, Inc., Boston, MA, 1995).Google Scholar
Bourgain, J., ‘Problems in Hamitonian PDE’s’, Gemo. Funct. Anal. Special Volume, Part I (2000), 3256.Google Scholar
Brenner, P. and von Wahl, W., ‘Global classical solutions of nonlinear wave equations’, Math. Z. 176(1) (1981), 87121.CrossRefGoogle Scholar
Bulut, A., ‘Global well-posedness and scattering for the defocusing energy-supercritical cubic nonlinear wave equation’, J. Funct. Anal. 263(6) (2012), 16091660.Google Scholar
Bulut, A., ‘The defocusing energy-supercritical cubic nonlinear wave equation in dimension five’, Trans. Amer. Math. Soc. 367(9) (2015), 60176061.Google Scholar
Christ, M., Colliander, J. and Tao, T., ‘Ill-posedness for nonlinear Schrodinger and wave equations’, Preprint, 2003, arXiv:math/0311048.Google Scholar
Collot, C., Raphaël, P. and Szeftel, J., ‘On the stability of type I blow up for the energy super critical heat equation’, Mem. Amer. Math. Soc. 260(1255) (2019), v+97 pp.Google Scholar
Collot, C., Raphaël, P. and Szeftel, J., ‘Strongly anisotropic type II blow up at an isolated point’, J. Amer. Math. Soc. 33(2) (2020), 527607.Google Scholar
Cortázar, C., del Pino, M. and Musso, M., ‘Green’s function and infinite-time bubbling in the critical nonlinear heat equation’, J. Eur. Math. Soc. (JEMS) 22(1) (2020), 283344.Google Scholar
Dávila, J., del Pino, M. and Wei, J., ‘Singularity formation for the two-dimensional harmonic map flow into ${S}^2$ ’, Invent. Math. 219(2) (2020), 345466.Google Scholar
del Pino, M., Musso, M. and Wei, J., ‘Infinite-time blow-up for the 3-dimensional energy-critical heat equation’, Anal. PDE 13(1) (2020), 215274.CrossRefGoogle Scholar
Donninger, R., ‘Strichartz estimates in similarity coordinates and stable blowup for the critical wave equation’, Duke Math. J. 166(9) (2017), 16271683.CrossRefGoogle Scholar
Duyckaerts, T., Jia, H., Kenig, C. E. and Merle, F., ‘Soliton resolution along a sequence of times for the focusing energy critical wave equation’, Geom. Funct. Anal. 27(4) (2017), 798862.Google Scholar
Duyckaerts, T., Jia, H., Kenig, C. E. and Merle, F., ‘Universality of blow up profile for small blow up solutions to the energy critical wave map equation’, Int. Math. Res. Not. IMRN 2018 (22) (2018), 69617025.Google Scholar
Duyckaerts, T., Kenig, C. E. and Merle, F., ‘Profiles of bounded radial solutions of the focusing, energy-critical wave equation’, Geom. Funct. Anal. 22(3) (2012), 639698.Google Scholar
Duyckaerts, T., Kenig, C. E. and Merle, F., ‘Classification of radial solutions of the focusing, energy-critical wave equation’, Camb. J. Math. 1(1) (2013), 75144.Google Scholar
Duyckaerts, T., Kenig, C. E. and Merle, F., ‘Soliton resolution for the radial critical wave equation in all odd space dimensions’, Acta Math. 230(1) (2023), 192.Google Scholar
Duyckaerts, T. and Yang, J., ‘Blow-up of a critical Sobolev norm for energy-subcritical and energy-supercritical wave equations’, Anal. PDE 11(4) (2018), 9831028.CrossRefGoogle Scholar
Ginibre, J. and Velo, G., ‘The global Cauchy problem for the nonlinear Klein-Gordon equation’, Math. Z. 189(4) (1985), 487505.Google Scholar
Ginibre, J. and Velo, G., ‘The global Cauchy problem for the nonlinear Klein-Gordon equation. II’, Ann. Inst. H. Poincaré Anal. Non Linéaire 6(1) (1989), 1535.Google Scholar
Grillakis, M., ‘Regularity and asymptotic behaviour of the wave equation with a critical nonlinearity’, Ann. of Math. 132(3) (1990), 485509.Google Scholar
Grillakis, M., ‘Regularity for the wave equation with a critical nonlinearity’, Comm. Pure Appl. Math. 45(6) (1992), 749774.CrossRefGoogle Scholar
Harada, J., ‘A type II blowup for the six dimensional energy critical heat equation’, Ann. PDE 6(2) (2020), Paper No. 13, 63 pp.Google Scholar
Hardy, M., ‘Combinatorics of partial derivatives’, Electron. J. Combin. 13(1) (2006), Research Paper 1, 13 pp.Google Scholar
Hörmander, L., Lectures on nonlinear hyperbolic differential equations (Mathématiques & Applications (Berlin)) vol. 26 (Springer-Verlag, New York, 1997).Google Scholar
Ibrahim, S., Majdoub, M. and Masmoudi, N., ‘Well- and ill-posedness issues for energy supercritical waves’, Anal. PDE 4(2) (2011), 341367.Google Scholar
Jendrej, J., ‘Construction of type II blow-up solutions for the energy-critical wave equation in dimension 5’, J. Funct. Anal. 272(3) (2017), 866917.CrossRefGoogle Scholar
Jendrej, J. and Lawrie, A., ‘Bubble decomposition for the harmonic map heat flow in the equivariant case’, Calc. Var. Partial Differential Equations 62(9) (2023), Paper No. 264, 36 pp.Google Scholar
Jendrej, J. and Lawrie, A., ‘Soliton resolution for the energy-critical nonlinear wave equation in the radial case’, Ann. PDE 9(2) (2023), Paper No. 18, 117 pp.Google Scholar
John, F., ‘Blow-up of solutions of nonlinear wave equations in three space dimensions’, Manuscripta Math. 28(1-3) (1979), 235268.Google Scholar
Jörgens, K., ‘Das Anfangswertproblem im Grossen für eine Klasse nichtlinearer Wellengleichungen’, Math. Z. 77 (1961), 295308.Google Scholar
Kenig, C. E., Lectures on the energy critical nonlinear wave equation (CBMS Reg. Conf. Ser. Math.) vol. 122 (Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2015).CrossRefGoogle Scholar
Kenig, C. E. and Merle, F., ‘Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case’, Invent. Math. 166(3) (2006), 645675.Google Scholar
Kenig, C. E. and Merle, F., ‘Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation’, Acta Math. 201(2) (2008), 147212.Google Scholar
Killip, R. and Visan, M., ‘The defocusing energy-supercritical nonlinear wave equation in three space dimensions’, Trans. Amer. Math. Soc. 363(7) (2011), 38933934.Google Scholar
Killip, R. and Visan, M., ‘The radial defocusing energy-supercritical nonlinear wave equation in all space dimensions’, Proc. Amer. Math. Soc. 139(5) (2011), 18051817.Google Scholar
Kim, K. and Merle, F., ‘On classification of global dynamics for energy-critical equivariant harmonic map heat flows and radial nonlinear heat equation’, Preprint, 2024, arXiv:2404.04247.Google Scholar
Krieger, J. and Miao, S., ‘On the stability of blowup solutions for the critical corotational wave-map problem’, Duke Math. J. 169(3) (2020), 435532.Google Scholar
Krieger, J., Miao, S. and Schlag, W., ‘A stability theory beyond the co-rotational setting for critical Wave Maps blow up’, Preprint, 2024, arXiv:2009.08843v2.Google Scholar
Krieger, J. and Schlag, W., ‘Full range of blow up exponents for the quintic wave equation in three dimensions’, J. Math. Pures Appl. (9) 101(6) (2014), 873900.Google Scholar
Krieger, J., Schlag, W. and Tataru, D., ‘Renormalization and blow up for charge one equivariant critical wave maps’, Invent. Math. 171(3) (2008), 543615.Google Scholar
Krieger, J., Schlag, W. and Tataru, D., ‘Slow blow-up solutions for the ${H}^1({\mathbb{R}}^3)$ critical focusing semilinear wave equation’, Duke Math. J. 147(1) (2009), 153.Google Scholar
Levine, H. A., ‘Instability and nonexistence of global solutions to nonlinear wave equations of the form $P{u}_{tt}=- Au+(u)$ ’, Trans. Amer. Math. Soc. 192 (1974), 121.Google Scholar
Lindblad, H. and Sogge, C. D., ‘Long-time existence for small amplitude semilinear wave equations’, Amer. J. Math. 118(5) (1996), 10471135.Google Scholar
Luk, J., Introduction to nonlinear wave equations, https://web.stanford.edu/~jluk/NWnotes.pdf.Google Scholar
Martel, Y., Yvan, and Merle, F., ‘Inelasticity of soliton collisions for the 5D energy critical wave equation’, Invent. Math. 214(3) (2018), 12671363.CrossRefGoogle Scholar
Matano, H. and Merle, F., ‘On nonexistence of type II blowup for a supercritical nonlinear heat equation’, Comm. Pure Appl. Math. 57(11) (2004), 14941541.CrossRefGoogle Scholar
Merle, F. and Raphaël, P., ‘On a sharp lower bound on the blow-up rate for the ${L}^2$ critical nonlinear Schrödinger equation’, J. Amer. Math. Soc. 19(1) (2006), 3790.CrossRefGoogle Scholar
Merle, F., Raphaël, P. and Rodnianski, I., ‘Blowup dynamics for smooth data equivariant solutions to the critical Schrödinger map problem’, Invent. Math. 193(2) (2013), 249365.Google Scholar
Merle, F., Raphaël, P. and Rodnianski, I., ‘Type II blow up for the energy supercritical NLS’, Camb. J. Math. 3(4) (2015), 439617.Google Scholar
Merle, F., Raphaël, P., Rodnianski, I. and Szeftel, J., On blow up for the energy super critical defocusing nonlinear Schrödinger equations, Invent. Math. 227(1) (2022), 247413.Google Scholar
Merle, F., Raphaël, P., Rodnianski, I. and Szeftel, J., ‘On the implosion of a compressible fluid I: Smooth self-similar inviscid profiles’, Ann. of Math. (2) 196(2) (2022), 567778.CrossRefGoogle Scholar
Merle, F., Raphaël, P., Rodnianski, I. and Szeftel, J., ‘On the implosion of a compressible fluid II: Singularity formation’, Ann. of Math. (2) 196(2) (2022), 779889.Google Scholar
Merle, F., Raphaël, P. and Szeftel, J., ‘Stable self-similar blow-up dynamics for slightly ${L}^2$ super-critical NLS equations’, Geom. Funct. Anal. 20(4) (2010), 10281071.CrossRefGoogle Scholar
Merle, F., Raphaël, P. and Szeftel, J., ‘On collapsing ring blow-up solutions to the mass supercritical nonlinear Schrödinger equation’, Duke Math. J. 163(2) (2014), 369431.CrossRefGoogle Scholar
Merle, F., Raphaël, P. and Szeftel, J., ‘On strongly anisotropic type I blowup’, Int. Math. Res. Not. IMRN 2020 (2) (2020), 541606.Google Scholar
Merle, F. and Zaag, H., ‘Stability of the blow-up profile for equations of the type ${u}_t=\varDelta u+|{\left.u\right|}^{p-1}u$ ’, Duke Math. J. 86(1) (1997), 143195.Google Scholar
Merle, F. and Zaag, H., ‘Determination of the blow-up rate for the semilinear wave equation’, Amer. J. Math. 125(5) (2003), 11471164.Google Scholar
Perelman, G., ‘On the formation of singularities in solutions of the critical nonlinear Schrödinger equation’, Ann. Henri Poincaré 2(4) (2001), 605673.CrossRefGoogle Scholar
Perelman, G., ‘Blow up dynamics for equivariant critical Schrödinger maps’, Comm. Math. Phys. 330(1) (2014), 69105.Google Scholar
Perelman, G., ‘Formation of singularities in nonlinear dispersive PDEs’, in ICM—International Congress of Mathematicians. Vol. 5. Sections 9-11 (EMS Press, Berlin, 2023), 38543879.Google Scholar
Raphaël, P. and Rodnianski, I., ‘Stable blow up dynamics for the critical co-rotational wave maps and equivariant Yang-Mills problems’, Publ. Math. Inst. Hautes Études Sci. 115 (2012), 1122.CrossRefGoogle Scholar
Raphaël, P. and Schweyer, R., ‘Stable blowup dynamics for the 1-corotational energy critical harmonic heat flow’, Comm. Pure Appl. Math. 66(3) (2013), 414480.Google Scholar
Rodnianski, I. and Sterbenz, J., ‘On the formation of singularities in the critical $O(3)$ $\sigma$ -model’, Ann. of Math. (2) 172(1) (2010), 187242.CrossRefGoogle Scholar
Shao, F., Wei, D. and Zhang, Z., ‘Self-similar imploding solutions of the relativistic Euler equations’, Preprint, 2024, arXiv:2403.11471.Google Scholar
Shatah, J. and Struwe, M., ‘Regularity results for nonlinear wave equations’, Ann. of Math. (2) 138(3) (1993), 503518.CrossRefGoogle Scholar
Shatah, J. and Struwe, M., ‘Well-posedness in the energy space for semilinear wave equations with critical growth’, Int. Math. Res. Not. IMRN 1994 (7) (1994), 303309.CrossRefGoogle Scholar
Sogge, C., Lectures on Nonlinear Wave Equations (Monographs in Analysis, II. International Press, Boston, MA, 1995).Google Scholar
Stein, E. M. and Shakarchi, R., Complex analysis (Princeton Lectures in Analysis) vol. 2 (Princeton University Press, Princeton, NJ, 2003).Google Scholar
Strauss, W. A., Nonlinear wave equations (CBMS Regional Conference Series in Mathematics) vol. 73 (American Mathematical Society, Providence, RI, 1989).Google Scholar
Struwe, M., ‘Globally regular solutions to the ${u}^5$ Klein-Gordon equation’, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 15(3) (1988), 495513.Google Scholar
Tao, T., ‘Global regularity for a logarithmically supercritical defocusing nonlinear wave equation for spherically symmetric data’, J. Hyperbolic Differ. Equ. 4(2) (2007), 259265.Google Scholar
Tao, T., ‘Finite-time blowup for a supercritical defocusing nonlinear wave system’, Anal. PDE 9(8) (2016), 19992030.CrossRefGoogle Scholar
Tao, T., ‘Finite time blowup for a supercritical defocusing nonlinear Schrödinger system’, Anal. PDE 11(2) (2018), 383438.CrossRefGoogle Scholar
Yang, S., ‘Global behaviors of defocusing semilinear wave equations’, Ann. Sci. Éc. Norm. Supér. (4), 55(2) (2022), 405428.CrossRefGoogle Scholar