This paper focuses on the Cauchy problem for a one-dimensional quasilinear hyperbolic–parabolic coupled system with initial data given on a line of parabolicity. The coupled system is derived from the Poiseuille flow of full Ericksen–Leslie model in the theory of nematic liquid crystals, which incorporates the crystal and liquid properties of the materials. The main difficulty comes from the degeneracy of the hyperbolic equation, which makes that the system is not continuously differentiable and then the classical methods for the strictly hyperbolic–parabolic coupled systems are invalid. With a choice of a suitable space for the unknown variable of the parabolic equation, we first solve the degenerate hyperbolic problem in a partial hodograph plane and express the smooth solution in terms of the original variables. Based on the smooth solution of the hyperbolic equation, we then construct an iterative sequence for the unknown variable of the parabolic equation by the fundamental solution of the heat equation. Finally, we verify the uniform convergence of the iterative sequence in the selected function space and establish the local existence and uniqueness of classical solutions to the degenerate coupled problem.

We investigate the global Cauchy problem for a two–phase flow model consisting of the pressureless Euler equations coupled with the isentropic compressible Navier–Stokes equations through a drag forcing term. This model was first derived by Choi–Kwon [J. Differential Equations, 261(1) (2016), pp. 654–711] by taking the hydrodynamic limit of the Vlasov/compressible Navier–Stokes equations. Under the assumption that the initial perturbation is sufficiently small, Choi–Kwon [J. Differential Equations, 261(1) (2016), pp. 654–711] established the global well–posedness and large time behaviour for the three dimensional periodic domain $\mathbb {T}^3$. However, up to now, the global well–posedness and large time behaviour for the three dimensional Cauchy problem still remain unsolved. In this paper, we resolve this problem by proving the global existence and optimal decay rates of classic solutions for the three dimensional Cauchy problem when the initial data is near its equilibrium. One of key observations here is that to overcome the difficulties arising from the absence of pressure in the Euler equations, we make full use of the drag forcing term and the dissipative structure of the Navier–Stokes equations to closure the energy estimates of the variables for the pressureless Euler equations.

In this study, we continue our study of the Cauchy problem associated with the Brinkman equations [see (1.1) and (1.2) below] which model fluid flow in certain types of porous media. Here, we will consider the flow in the upper half-space

\[ \mathbb{R}_{+}^{3}=\left\{\left(x,y,z\right) \in\mathbb{R}^{3}\left\vert z\geqslant 0\right.\right\}, \]

$z=0$

We consider the Cauchy problem for a general class of parabolic partial differential equations in the Euclidean space ℝN. We show that given a weighted Lp-space $L_w^p({\mathbb {R}}^N)$ with 1 ⩽ p < ∞ and a fast growing weight w, there is a Schauder basis $(e_n)_{n=1}^\infty$ in $L_w^p({\mathbb {R}}^N)$ with the following property: given an arbitrary positive integer m there exists nm > 0 such that, if the initial data f belongs to the closed linear span of en with n ⩾ nm, then the decay rate of the solution of the problem is at least t−m for large times t.

The result generalizes the recent study of the authors concerning the classical linear heat equation. We present variants of the result having different methods of proofs and also consider finite polynomial decay rates instead of unlimited m.

The Kudryashov–Sinelshchikov–Olver equation describes pressure waves in liquids with gas bubbles taking into account heat transfer and viscosity. In this paper, we prove the existence of solutions of the Cauchy problem associated with this equation.

The present paper concerns the system ut + [ϕ(u)]x = 0, vt + [ψ(u)v]x = 0 having distributions as initial conditions. Under certain conditions, and supposing ϕ, ψ: ℝ → ℝ functions, we explicitly solve this Cauchy problem within a convenient space of distributions u,v. For this purpose, a consistent extension of the classical solution concept defined in the setting of a distributional product (not constructed by approximation processes) is used. Shock waves, δ-shock waves, and also waves defined by distributions that are not measures are presented explicitly as examples. This study is carried out without assuming classical results about conservation laws. For reader's convenience, a brief survey of the distributional product is also included.

We prove the global logarithmic stability of the Cauchy problem for $H^{2}$-solutions of an anisotropic elliptic equation in a Lipschitz domain. The result is based on existing techniques used to establish stability estimates for the Cauchy problem combined with related tools used to study an inverse medium problem.

This paper is concerned with modelling the dynamics of social outbursts of activity, such as protests or riots. In this sequel to our work in Berestycki et al. (Networks and Heterogeneous Media, vol. 10, no. 3, 1–34), written in collaboration with J-P. Nadal, we model the effect of restriction of information and explore its impact on the existence of upheaval waves. The system involves the coupling of an explicit variable representing the intensity of rioting activity and an underlying (implicit) field of social tension. We prove the existence of global solutions to the Cauchy problem in ${\mathbb R}^d$ as well as the existence of traveling wave solutions in certain parameter regimes. We furthermore explore the effects of heterogeneities in the environment with the help of numerical simulations, which lead to pulsating waves in certain cases. We analyse the effects of periodic domains as well as the barrier problem with the help of numerical simulations. The barrier problem refers to the potential blockage of a wavefront due to a spatial heterogeneity in the system which leads to an area of low excitability (referred to as the barrier). We conclude with a variety of open problems.

In this paper, a Cauchy problem of two-dimensional heat conduction equation is investigated. This is a severely ill-posed problem. Based on the solution of Cauchy problem of two-dimensional heat conduction equation, we propose to solve this problem by modifying the kernel, which generates a well-posed problem. Error estimates between the exact solution and the regularized solution are given. We provide a numerical experiment to illustrate the main results.

In this paper, we consider the Cauchy problem

$$\left\{ \begin{align} & {{u}_{t}}=\Delta ({{u}^{m}}),\,\,\,\,\,x\in {{\mathbb{R}}^{N}},t>0,N\ge 3, \\ & u(x,0)={{u}_{0}}(x),\,\,\,\,\,x\in {{\mathbb{R}}^{N}}. \\ \end{align} \right.$$

We will prove that

(i) for ${{m}_{c}}\,<\,m,\,{{m}_{0}}\,<\,1,\,\left| u(x,\,t,m)-u(x,\,t,{{m}_{0}}) \right|\,\to \,0$ as $m\,\to \,{{m}_{0}}$ uniformly on every compact subset of ${{\mathbb{R}}^{N}}\,\times \,{{\mathbb{R}}^{+}}$, where ${{m}_{c}}\,=\,\frac{{{(N-2)}_{+}}}{N}$;

(ii) there is a ${{C}^{*}}$ that explicitly depends on $m$ such that

$${{\left\| u(\cdot ,\cdot ,m)-u(\cdot ,\cdot ,1) \right\|}_{{{L}^{2}}({{\mathbb{R}}^{N}}\times {{\mathbb{R}}^{+}})}}\le {{C}^{*}}\left| m-1 \right|.$$

In this paper, we give a general proof on convergence estimates for some regularization methods to solve a Cauchy problem for the Laplace equation in a rectangular domain. The regularization methods we considered are: a non-local boundary value problem method, a boundary Tikhonov regularization method and a generalized method. Based on the conditional stability estimates, the convergence estimates for various regularization methods are easily obtained under the simple verifications of some conditions. Numerical results for one example show that the proposed numerical methods are effective and stable.

We consider the fourth-order thin film equation, with a stable second-order diffusion term. For the first critical exponent, where N ≥ 1 is the space dimension, the Cauchy problem is shown to admit countable continuous branches of source-type self-similar very singular solutions of the form These solutions are inherently oscillatory in nature and will be shown in Part II to be the limit of appropriate free-boundary problem solutions. For p ≠ p0, the set of very singular solutions is shown to be finite and to be consisting of a countable family of branches (in the parameter p) of similarity profiles that originate at a sequence of critical exponents {pl, l ≥ 0}. At p = pl, these branches appear via a non-linear bifurcation mechanism from a countable set of similarity solutions of the second kind of the pure thin film equation Such solutions are detected by the ‘Hermitian spectral theory’, which allows an analytical n-branching approach. As such, a continuous path as n → 0+ can be constructed from the eigenfunctions of the linear rescaled operator for n = 0, i.e. for the bi-harmonic equation ut = −Δ2u. Numerics are used, wherever appropriate, to support the analysis.

Let P(r, θ) be the two-dimensional Poisson kernel in the unit disc D. It is proved that there exists a special sequence {ak} of points of D which is non-tangentially dense for ∂D and such that any function on ∂D can be expanded in series of P(|ak|, (·)–arg ak) with coefficients depending continuously on f in various classes of functions. The result is used to solve a Cauchy-type problem for Δu = μ, where μ is a measure supported on {ak}.

This paper investigates the Cauchy problem for two classes of parabolic systems with localised sources. We first give the blowup criterion, and then deal with the possibilities of simultaneous blowup or non-simultaneous blowup under some suitable assumptions. Moreover, when simultaneous blowup occurs, we also establish precise blowup rate estimates. Finally, using similar ideas and methods, we shall consider several nonlocal problems with homogeneous Neumann boundary conditions.

The well-posedness of the Ostrovsky equation is considered. Local well-posedness for data in $\tilde{H}^s(\mathbb{R})$ $(s\geq-\frac{1}{8})$ and global well-posedness for data in $\tilde{L}^{2}(\mathbb{R})$ are obtained.

Let $1\leq p,q\leq\infty$, $s\in\mathbb{R}$ and let $X$ be a Banach space. We show that the analogue of Marcinkiewicz’s Fourier multiplier theorem on $L^p(\mathbb{T})$ holds for the Besov space $B_{p,q}^s(\mathbb{T};X)$ if and only if $1\ltp\lt\infty$ and $X$ is a UMD-space. Introducing stronger conditions we obtain a periodic Fourier multiplier theorem which is valid without restriction on the indices or the space (which is analogous to Amann’s result (Math. Nachr. 186 (1997), 5–56) on the real line). It is used to characterize maximal regularity of periodic Cauchy problems.

AMS 2000 Mathematics subject classification: Primary 47D06; 42A45

We discuss the problem of the uniqueness of the solution to the Cauchy problem for second-order, linear, uniformly parabolic differential equations. For most uniqueness theorems the solution must be uniformly bounded with respect to the time variable $t$, but some authors have shown an interest in relaxing the growth conditions in time.

In 1997, Chung proved that, in the case of the heat equation, uniqueness holds under the restriction: $|u(x,t)|\leq C\exp[(a/t)^{\alpha}+a|x|^2]$, for some constants $C,a>0$, $0\lt\alpha\lt1$. The proof of Chung’s theorem is based on ultradistribution theory, in particular it relies heavily on the fact that the coefficients are constants and that the solution is smooth. Therefore, his method does not work for parabolic operators with arbitrary coefficients. In this paper we prove a uniqueness theorem for uniformly parabolic equations imposing the same growth condition as Chung on the solution $u(x,t)$. At the centre of the proof are the maximum principle, Gaussian-type estimates for short cylinders and a boot-strapping argument.

AMS 2000 Mathematics subject classification: Primary 35K15

On an arbitrary reflexive Banach space, we build asymptotic observers for an abstract class of nonlinear control systems with possible compact outputs. An important part of this paper is devoted to various examples, where we discuss the existence of persistent inputs which make the system observable. These results make a wide generalization to a nonlinear framework of previous works on the observation problem in infinite dimension (see [11,18,22,26,27,38,40] and other references therein).

The Cauchy problem in the form of (1.11) with linear and constant coefficients is discussed. The solution (1.10) can be given in explicit form when the stochastic process is a multidimensional autoregression (AR) type, or Ornstein–Uhlenbeck process. Functionals of (1.10) form were studied by Kac in the Brownian motion case. The solutions are obtained with the help of the Radon–Nikodym transformation, proposed by Novikov [12].