1. Introduction
Some systems of reaction–diffusion equations have attracted much interest as a prototype model for oscillation and pattern formation in the book by Murray [Reference Murray1] and the references therein. The main purpose of this paper is to present mathematical tools for studying the positivity of solutions of reaction–diffusion systems. So, we deal with the following reaction–diffusion equations in the whole space ${\mathbb R}^n$ for $n \in{\mathbb N}$ .
This is a system of Lotka–Volterra type equations with diffusions. More precisely, this is a prey–predator model with dormancy of predators in [Reference Kuwamura2, Reference Kuwamura, Nakazawa and Ogawa3]. Here, $u \,:\!=\, u (x, t)$ , $v \,:\!=\, v (x, t)$ and $w \,:\!=\, w (x, t)$ stand for the density of prey, the density of active predator and the density of dormant predator, respectively, as the unknown scalar positive (or, nonnegative) functions at $x \in{\mathbb R}^n$ and $t \gt 0$ . To avoid effects from boundaries, the Cauchy problem is considered, in what follows. We have denoted the nonnegative constants by
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$\delta$ the diffusion coefficient of prey
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$h$ the constant of foraging efficiency and handling time
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$d$ the diffusion coefficient of active predator
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$\alpha$ the rate of awakening
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$r$ the growth rate of prey
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$\beta$ the mortality rate by competitions of active predators
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$k$ the capacity of prey
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$\tilde \iota$ the mortality rate of dormant predator
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$\gamma$ the mortality rate of prey
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$\iota$ the mortality rate of active predator
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$\theta$ the rate of sleeping
Also, $\mu \,:\!=\, \mu (u)$ and $\nu \,:\!=\, \nu (u)$ are smooth positive functions of $u$ denoting growth rates of active and dormant predators, respectively. In some mathematical research, $\mu (u)$ is given as a sigmoid function as
with some constants $\xi$ and $\eta$ , besides, $\nu (u) \,:\!=\, \gamma - \mu (u)$ ; see e.g. [Reference Kuwamura2]. In addition, we have used the notations of differentiation
By change of variables and constants, we may replace by $\delta = 1$ , $k = 1$ , $r = 1$ and $\beta = 1$ . For the simplicity of notations, we put $m \,:\!=\, \theta + \iota$ , $\rho \,:\!=\, \alpha + \tilde \iota$ , in addition, assume that $\mu$ and $\nu$ are positive constants independent of $u$ . So, we consider the initial value problem:
The bifurcation between stability and instability of stationary solutions to (LV) was concerned with some specific parameters, associated with numerical investigation [Reference Kuwamura, Nakazawa and Ogawa3]. Furthermore, a numerical study of Turing instability on (LV) was done [Reference Kuwamura2]. Besides, in this paper, we focus on the mathematical theory for the existence of time-global nonnegative unique classical solutions to (P) with nonnegative initial data, invariant regions and asymptotic behaviours. To do so, we estimate a priori bounds for solutions to (P) by enclosing and renormalising arguments of solutions to the corresponding ordinary differential equations.
This paper is organised as follows. In Section 2, we will present the main results of this paper and related works. In Section 3, we define function spaces and recall some properties of the heat semigroup and time-evolution operators. Section 4 will be devoted to the proof of the time-local existence of nonnegative unique classical solutions with nonnegative initial data. We will discuss the time-global solvability in Section 5, deriving a priori estimates of solutions and their derivatives, due to renormalisation arguments. In Section 6, some invariant regions and asymptotic behaviours of solutions to (p) will be argued.
Throughout this paper, we denote positive constants by $C$ the value of which may differ from one occasion to another.
2. Main results
We will state the main results in this paper. For the definition and properties of the set of all bounded and uniformly continuous functions $\text{BUC}$ , see Section 3, as well as $\text{BUC}^1$ .
Theorem 1. Let $n \in{\mathbb N}$ , $d$ , $h \gt 0$ , and let $m$ , $\theta$ , $\rho$ , $\alpha$ , $\gamma$ , $\mu$ , $\nu \geq 0$ . If $u_0$ , $v_0 \in \text{BUC} ({\mathbb R}^n)$ and $w_0 \in \text{BUC}^1 ({\mathbb R}^n)$ are nonnegative, then there exists a nonnegative time-global unique classical solutions to (P).
Remark 1. (i) We can find at most five stationary constant states, including the trivial solution $(0, 0, 0)$ . The trivial solution is always unstable, if $u_0 \gt 0$ . Besides, the stabilities of non-trivial constant states depend on parameters; see Theorem 2 and Remark 4 in below. (ii) Even if $\mu$ and $\nu$ are positive smooth functions of $u$ , the same time-global solvability can be proved. Here, we may relax the condition $\gamma = \mu + \nu$ , at least mathematically. (iii) In the case of $d = 0$ , we may obtain the same assertion, whenever $v_0 \in \text{BUC}^1$ . (iv) When $u_0$ , $v_0 \in L^\infty$ , we may also get the similar assertion, although there is a lack of continuity of solutions in $t$ at $t = 0$ .
We will explain the strategy of the proof of Theorem 1, briefly. Using the heat semigroups, (P) is written as the forms of integral equations:
Once we obtain the existence of solutions to (1)–(3), the uniqueness and the regularity of solutions follow from these forms. However, it is not easy to get the existence of solutions, at least directly. Because, the nonnegativity of solutions or its approximation seems to be not ensured, in general. In fact, the following standard iteration scheme is often employed:
See, for example, the book by Smoller [Reference Smoller7]. With this approximation, it is not clear how to show $\bar u_\ell \gt - h$ for $\ell \geq 2$ , unfortunately. Thus, we have to look for the another approximation or integral forms for proving the existence of nonnegative solutions.
There are many mathematical articles to prove the existence of partial differential equation (PDE) with the Holling type $\rm I\!I$ nonlinear terms. As far as the authors know, the exiting techniques are as follows.
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Under the a priori assumption of the positivity of solutions or approximation, the existence of solutions is easily proved. However, it is not clear how to get the positivity.
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Apply the fixed point theorem of the mapping in the set of positive functions $K_+ \,:\!=\, \{ u, v, w \gt 0 \}$ . However, we have to verify its domain and range in $K_+$ .
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Consider the modified system taking the absolute value to $u$ or $\bar u_\ell$ in the denominator. To do so, we obtain weak solutions. However, it is not clear how to show that the weak solutions satisfy (P) in the classical sense, that is, $u \in C ((0, T);\, C^2)$ .
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By standard arguments, the corresponding ordinary differential equation (ODE) admit time-local solutions. So, the solution to (P) is estimated from below by that to ODE. However, the comparison principle can be applicable, after showing the existence of classical solutions.
Each approach in above has a flaw. Hence, we employ the same sprits in [Reference Kondo, Novrianti and Tsuge6]. To overcome difficulties, we will construct the solutions as the limits of the following successive approximations of abstract forms:
for $\ell \in{\mathbb N}$ . Our idea is to involve the coefficients of negative terms into the generators. We can rewrite them as
for $\ell \in{\mathbb N}$ . Here, we have used the time-evolution operators $ \{ U_\ell (t, s) \}$ and $ \{ V_\ell (t, s) \}$ associated with
for regarding $u_\ell$ , $v_\ell$ and $w_\ell$ as given nonnegative functions, respectively, starting at
The precise definition and estimates of time-evolution operators are given in Section 3. These approximations enable us to show the nonnegativities of $u_\ell$ , $v_\ell$ and $w_\ell$ for all $\ell \in{\mathbb N}$ , as well as its limit $u$ , $v$ and $w$ . We will derive the estimates $\| u_\ell, v_\ell, w_\ell \|_\infty$ by (4)–(6), inductively, in the fixed point arguments. Besides, for estimates $\| \partial _i u_\ell, \partial _i v_\ell, \partial _i w_\ell \|_\infty$ , we apply the heat semigroup representation of solutions. Once we derive uniform bounds of $u_\ell$ , $v_\ell$ , $w_\ell$ , $\partial _i u_\ell$ , $\partial _i v_\ell$ and $\partial _i w_\ell$ , we can easily see that the limit $(u, v, w)$ becomes a classical solution to (P).
On the other hand, it is rather standard to extend the obtained solutions time-globally, deriving a priori estimates of solutions. The key idea is to apply the maximum principle to the classical solutions. We can also investigate asymptotic behaviours of solutions, more precisely. Via analysis of solutions to the system of corresponding ODE, we obtain invariant regions.
Before stating results, we provide two stationary solutions. It is easy to verify that $(1, \overline v, \overline w)$ is a stationary solution to (P), where
Furthermore, $(\underline u, \underline v, \underline w)$ is also a stationary solution to (P), where
Theorem 2. Assume that $(u, v, w)$ is a solution to (P). (i) If $\overline v \leq 0$ , and if $u_0 \not \equiv 0$ , then $(u, v, w) \to (1, 0, 0)$ as $t \to \infty$ . Besides, if $u_0 \equiv 0$ , then $(u, v, w) \to (0, 0, 0)$ as $t \to \infty$ . (ii) If $\overline v \gt 0$ , then for any $0 \lt \varepsilon \lt \!\!\lt 1$ , there exists a $T_\varepsilon \geq 0$ such that
for $x \in{\mathbb R}^n$ and $t \geq T_\varepsilon$ . Moreover,
for $x \in{\mathbb R}^n$ and $t \gt 0$ . (iii) Let $\overline v \gt 0$ , $\underline u \gt 0$ , $\underline v \gt 0$ , and let $\underline w \gt 0$ . If $u$ , $v$ , $w \geq c_\star$ for $x \in{\mathbb R}^n$ at $t = t_\star \geq 0$ with some $c_\star \gt 0$ , then for $0 \lt \varepsilon \lt \!\!\lt 1$ , there exists a $T_\varepsilon^{\prime} \geq t_\star$ such that
for $x \in{\mathbb R}^n$ and $t \geq T_\varepsilon^{\prime}$ . Moreover,
for $x \in{\mathbb R}^n$ and $t \gt 0$ .
The sets $R_\ast$ and $R_\natural$ are invariant regions. The reader may find another (narrower) invariant regions for each individual parameter. Theorem 2 implies that an absorbing set always exists in $R_\ast$ or $R_\natural$ . Our conjecture is that we can also obtain the similar results in several domains with suitable boundary conditions.
3. Semigroups and time-evolution operators
In this section, we recall the definitions of function spaces and properties of the heat semigroup, as well as time-evolution operators.
Let $n \in{\mathbb N}$ , $1 \leq p \lt \infty$ , and let $L^p \,:\!=\, L^p ({\mathbb R}^n)$ be the space of all $p$ th integrable functions in ${\mathbb R}^n$ with the norm $\displaystyle \| f \|_p \,:\!=\, \left( \int _{{\mathbb R}^n} |f(x)|^p dx \right)^{1/p}$ . We often omit the notation of the domain $({\mathbb R}^n)$ , if no confusion occurs. We do not distinguish scalar-valued functions and vector, as well as function spaces. Let $L^\infty$ be the space of all bounded functions with the norm $\| f \| \,:\!=\, \| f \|_\infty \,:\!=\, \textrm{ess}.\textrm{sup}_{x \in{\mathbb R}^n} | f(x) |$ ; $\text{BUC}$ as the space of all bounded uniformly continuous functions. Note that $L^p$ , $L^\infty$ and $\text{BUC}$ are Banach spaces. For $k \in{\mathbb N}$ , let $W^{k, \infty }$ be a set of all bounded functions whose $k$ -th derivatives are also bounded. Furthermore, define
In the whole space ${\mathbb R}^n$ , for $\vartheta _0 \in L^\infty ({\mathbb R}^n)$ , the heat equation
admits a time-global unique smooth solution
in $C_w ((0, \infty );\, L^\infty ({\mathbb R}^n))$ , that is, $\vartheta \in C ([\tau, \infty );\, L^\infty ({\mathbb R}^n))$ for any small $\tau \gt 0$ . Here, $H_t \,:\!=\, H_t (x) \,:\!=\, (4 \pi t)^{-n/2} \exp ({-}|x|^2/4t)$ is the heat kernel. Since $\| H_t \|_1 = 1$ for $t \gt 0$ , by Young’s inequality we have $\| \vartheta (t) \| \leq \| H_t \|_1 \| \vartheta _0 \| = \| \vartheta _0 \|$ for $t \gt 0$ . In particular, if $\vartheta _0 (x) \geq 0$ for $x \in{\mathbb R}^n$ , then $\vartheta (x, t) \geq 0$ holds true for $x \in{\mathbb R}^n$ and $t \gt 0$ , so-called the maximum principle. Furthermore, if additionally $\vartheta _0 \in \text{BUC} ({\mathbb R}^n)$ and $\vartheta _0 \not \equiv 0$ , then $\vartheta (x, t) \gt 0$ for $x \in{\mathbb R}^n$ and $t \gt 0$ , so-called the strong maximum principle. For $\vartheta _0 \in L^\infty ({\mathbb R}^n)$ , there is a lack of the continuity of solutions to (H) in time at $t = 0$ , in general. Note that $e^{t \Delta } \vartheta _0 \to \vartheta _0$ in $L^\infty$ as $t \to 0$ , if and only if $\vartheta _0 \in \text{BUC} ({\mathbb R}^n)$ . The reader may find its proof in [Reference Giga, Inui and Matsui5]. Indeed, if $\vartheta _0 \in \text{BUC} ({\mathbb R}^n)$ , then $\vartheta \in C ([0, \infty );\, \text{BUC} ({\mathbb R}^n))$ .
We can easily see that for $j \in{\mathbb N}$ , splitting the heat semigroup into $j$ -th parts, there exists a positive constant $C_\sharp \,:\!=\, \pi ^{-1/2} \lt 1$ such that
for $t \gt 0$ and $1 \leq i \leq n$ . So, $\vartheta (t) \in \text{BUC}^j ({\mathbb R}^n)$ for any $j \in{\mathbb N}$ and $t \gt 0$ , which implies that $\vartheta (t) \in C^\infty ({\mathbb R}^n)$ for $t \gt 0$ . Moreover, $\vartheta \in C^\infty ({\mathbb R}^n \times (0, \infty ))$ by using (H).
In what follows, we recall some properties and estimates for time-evolution operators. Let us consider the following autonomous Cauchy problem with non-constant coefficients.
Here, $d \gt 0$ is a constant, and $\psi (x, t)$ is a given bounded function. We establish the time-local solvability of $\rm (P_A)$ with upper bounds of $\varphi$ .
Lemma 1 ([Reference Kondo, Novrianti and Tsuge6]). Let $n \in{\mathbb N}$ , $d$ , $T \gt 0$ and $\psi \in L^\infty ([0, T];\, W^{1, \infty } ({\mathbb R}^n))$ . If $\varphi _0 \in \text{BUC} ({\mathbb R}^n)$ , then there exist a $T_\ast \in (0, T]$ and a time-local unique classical solution to $\rm (P_A)$ , having $\| \varphi (t) \| \leq \dfrac 43 \| \varphi _0 \|$ for $t \in [0, T_\ast ]$ . Moreover, if $\varphi _0 \geq 0$ , then $\varphi \geq 0$ .
Although the proof is written in [Reference Kondo, Novrianti and Tsuge6], we give it here. The idea is to use the standard iteration. Let $\varphi _1 (t) \,:\!=\, e^{d t \Delta } \varphi _0$ , and let
for each $\ell \in{\mathbb N}$ , successively. It is easy to see that for $\ell \in{\mathbb N}$ , $\| \varphi _\ell (t) \| \leq \dfrac 43 \| \varphi _0 \|$ for $t \in [0, T_\ast ]$ with some $T_\ast \gt 0$ independent of $\ell$ . We can easily show that $ \{ \varphi _\ell \}_{\ell = 1}^\infty$ is a Cauchy sequence in $C ([0, T_\ast ];\, \text{BUC} ({\mathbb R}^n))$ . So, the limit $\varphi \,:\!=\, \lim _{\ell \to \infty } \varphi _\ell$ exists and satisfies $\rm (P_A)$ , having the estimate $\| \varphi (t) \| \leq \dfrac 43 \| \varphi _0 \|$ for $t \in [0, T_\ast ]$ . It is rather straightforward to obtain the uniqueness and regularity of $\varphi$ . Moreover, the nonnegativity of $\varphi$ easily follows from the maximum principle.
Note that if $\| \varphi _0 \| \leq L$ and $\textrm{sup}_{0 \leq t \leq T} \| \psi (t) \| \leq L$ with some $L \gt 0$ , then we may derive the estimate $T_\ast \geq C/L$ with $C \gt 0$ . The solution to $({\textrm{P}}_{\textrm{A}})$ can be rewritten as $\varphi (t) = U (t, 0) \varphi _0$ , using time-evolution operators $ \{ U (t, s) \}_{t \geq s \geq 0}$ associated with $A \,:\!=\, A(x, t) \,:\!=\, d \Delta - \psi (x, t)$ , see the book by Tanabe [Reference Tanabe8]. The boundedness of solutions $\varphi$ implies that $\| U (t, 0) \|_{L^\infty \to L^\infty } \leq 4/3$ for $t \in [0, T_\ast ]$ , and then $\| U (t, s) \|_{L^\infty \to L^\infty } \leq 4/3$ for $0 \leq s \leq t \leq T_\ast$ . Here, we have used the notation of an operator-norm $\|{\mathcal O} \|_{X \to Y} \,:\!=\, \textrm{sup}_{x \in X} \|{\mathcal O} x \|_Y/ \| x \|_X$ .
4. Time-local solvability
We give a proof of the time-local solvability on (P) in this section. Recall $\| \cdot \| \,:\!=\, \| \cdot \|_\infty$ , and put $M \,:\!=\, \max \{ \| u_0 \|, \| v_0 \|, \| w_0 \|, \| \partial _i w_0 \| \}$ .
Proposition 1. Let $n \in{\mathbb N}$ , $d \gt 0$ , and let those other parameters be nonnegative. Assume that $u_0$ , $v_0 \in \text{BUC} ({\mathbb R}^n)$ and $w_0 \in \text{BUC}^1 ({\mathbb R}^n)$ are nonnegative, then there exist a $T_0 \gt 0$ and a time-local unique classical solutions to (P), having $0 \leq u, v, w \leq 2 M$ for $x \in{\mathbb R}^n$ and $t \in [0, T_0]$ . Furthermore, $T_0 \geq C_\ast/(M^4+1)$ holds with some constant $C_\ast \gt 0$ independent of $M$ .
Proof. For the sake of simplicity, we assume that all parameter is positive. Making the approximation sequences, we begin with (7). For $\ell \in{\mathbb N}$ , we successively define $u_{\ell + 1}$ , $v_{\ell + 1}$ and $w_{\ell + 1}$ by (4)–(6). So, $u_{\ell + 1}$ , $v_{\ell + 1}$ and $w_{\ell + 1}$ also satisfy their abstract equations for $x \in{\mathbb R}^n$ and $t \gt 0$ with nonnegative functions $u_0$ , $v_0$ , $w_0$ , $u_\ell$ , $v_\ell$ and $w_\ell$ , formally.
In what follows, we estimate $u_\ell$ , $v_\ell$ , $w_\ell$ , $\partial _i u_\ell$ , $\partial _i v_\ell$ and $\partial _i w_\ell$ . Put
for $T \gt 0$ , $\ell \in{\mathbb N}$ and $1 \leq i \leq n$ . To derive uniform estimates, we argue the induction of $\ell$ , taking $T$ small.
$\ell = 1$ For $0 \leq u_0(x), v_0(x), w_0(x) \leq M$ , by the maximum principle and the fact that $e^{t(d \Delta -m)} = e^{-mt} e^{dt \Delta }$ , we can easily see that
for $x \in{\mathbb R}^n$ and $t \gt 0$ by $m, \rho \geq 0$ . In addition, it is easy to obtain that
for $t \gt 0$ and $1 \leq i \leq n$ by the estimate of the heat kernel. Here and hereafter, we replace the constant $C_\sharp \,:\!=\, \pi ^{-1/2} \lt 1$ by $1$ , for the sake of simplicity. Thus, we have
$\ell = 2$ Before estimating $u_2$ and $v_2$ , we will confirm bounds for time-evolution operators $U_1$ and $V_1$ . By $u_1 \geq 0$ and (8), it holds that
for $t \gt 0$ . By Lemma 1, for $ \{ U_1 (t, s) \}_{t \geq s \geq 0}$ with $A_1 (x, t) \,:\!=\, \Delta - \eta _1 (x, t)$ , we thus see that $\displaystyle 0 \leq U_1 (t, s) u_0 \leq \frac{4}{3} \| u_0 \|$ for $x \in{\mathbb R}^n$ and $0 \leq s \leq t \leq T^{\prime}_2$ with some $T^{\prime}_2 \gt 0$ depending only on $\overline \eta _1$ . By (4) with $\ell = 1$ , we have
with $\zeta _1 (x, t) \,:\!=\, u_1 (x, t)$ and $0 \leq \zeta _1 (x, s) \leq \overline \zeta _1 \,:\!=\, M$ , provided $0 \leq s \leq t \leq T_2^\dagger$ with $T_2^\dagger \,:\!=\, \min \{ T^{\prime}_2, 1/2 \}$ . Similarly, since
for $x \in{\mathbb R}^n$ and $t \gt 0$ , we may define the time-evolution operator $ \{ V_1 (t, s) \}_{t \geq s \geq 0}$ associated with $B_1 (x, t) \,:\!=\, d \Delta - \xi _1 (x, t)$ , having a uniform bound. Applying Lemma 1, we see that $\displaystyle 0 \leq V_1 (t, s) v_0 \leq \frac{4}{3} \| v_0 \|$ for $0 \leq s \leq t \leq T_2^\sharp$ with some $T_2^\sharp \gt 0$ depending only on $\overline \xi _1$ . By (5)
hold with $\chi _1 (x, t) \,:\!=\, \mu u_1 (x, t) v_1 (x, t)/ \{ u_1 (x, t) + h \} + \alpha w_1 (x, t)$ and $0 \leq \chi _1 (x, s) \leq \overline \chi _1 \,:\!=\, (\mu M/h + \alpha ) M$ , provided if $0 \leq s \leq t \leq T_2^\flat$ with $T_2^\flat \,:\!=\, \min \{ T_2^\dagger, T_2^\sharp, h/(2 \mu M + 2 \alpha h) \}$ . For the estimate of $w_2$ , we obtain
for $0 \leq s \leq t \leq T_2^\natural$ with $T_2^\natural \,:\!=\, \min \{ T_2^\flat, h/(\nu M + h \theta ) \}$ . To derive the estimate for $\partial _i u_2$ , we use the heat semigroup expression:
by rewriting (4). Hence, it holds that
for $t \in (0, T_2^\heartsuit ]$ with $T_2^\heartsuit \,:\!=\, \min \left\{ T_2^\natural, h/ (2 h + 4 h M + 4 \gamma M) \right\}$ . As the similar way, for $\partial _i v_2$ , we appeal to the heat semigroup expression again:
for $t \in \big(0, T_2^\diamondsuit \big]$ with $T_2^\diamondsuit \,:\!=\, \min \left\{ T_2^\heartsuit, h/ (2 \mu M + 2 \alpha h + 4 hm + 4 h M) \right\}$ . Furthermore,
holds true, and this implies that
for $t \in [0, T_2]$ with
Therefore, it is shown that $u_2, v_2, w_2 \geq 0$ and
$\ell = 3$ We stand for the time-evolution operator $ \{ U_2 (t, s) \}_{t \geq s \geq 0}$ associated with $A_2 (x, t) \,:\!=\, \Delta - \eta _2 (x, t)$ and
By Lemma 1, $U_2 (t, s) u_0 \geq 0$ holds and $\| U_2 (t, s) \|_{L^\infty \to L^\infty } \leq 4/3$ for $0 \leq s \leq t \leq T^{\prime}_3$ with some $T^{\prime}_3 \gt 0$ , since $0 \leq \eta _2 (x, t) \leq \overline \eta \,:\!=\, 2 M + 2 \gamma M/h$ by (9). So, we get
for $x \in{\mathbb R}^n$ and $t \in [0, T_3^\dagger ]$ with $T_3^\dagger \,:\!=\, \min \{ T^{\prime}_3, 1/4 \}$ . Here, we used that
Similarly, we denote the time-evolution operator by $ \{ V_2 (t, s) \}_{t \geq s \geq 0}$ associated with $B_2 (x, t) \,:\!=\, d \Delta - \xi _2 (x, t)$ , where
We seek that $V_2(t, s) v_0 \geq 0$ and $\| V_2 (t, s) \|_{L^\infty \to L^\infty } \leq 4/3$ for $0 \leq s \leq t \leq T_3^\sharp$ with some $T_3^\sharp \gt 0$ by Lemma 1. Hence, we can see that
for $x \in{\mathbb R}^n$ and $t \in [0, T_3^\flat ]$ with $T_3^\flat \,:\!=\, \min \{ T_3^\dagger, T_3^\sharp, h/(8 \mu M + 4 \alpha h) \}$ . Here, we have used
by (9). It is also easy to show that
for $x \in{\mathbb R}^n$ and $t \in [0, T_3^\natural ]$ with $T_3^\natural \,:\!=\, \min \{ T_3^\flat, h/(4 \nu M + 2 h \theta ) \}$ . By the heat semigroup expression, we obtain that
for $t \in (0, T_3^\heartsuit ]$ with $T_3^\heartsuit \,:\!=\, \min \{ T_3^\natural, h/ (4 h + 8 h M + 8 \gamma M) \}$ . As the similar way, we derive
for $t \in (0, T_3^\diamondsuit ]$ with $T_3^\diamondsuit \,:\!=\, \min \{ T_3^\heartsuit, h/ (4 h m + 8 h M + 8 \mu M + 4 \alpha h) \}$ . For $\partial _i w_3$ , see
for $t \in (0, T_0]$ with
Note that the estimate $T_0 \geq C/(M^4+1)$ is yielded with some $C \gt 0$ .
Therefore, we see that $u_3, v_3, w_3 \geq 0$ and
$\ell = 4, 5, \ldots$ Let $\ell \geq 4$ . We assume that $u_\ell$ , $v_\ell$ , $w_\ell \geq 0$ and
hold true. We will compute estimates for $u_{\ell + 1}$ , $v_{\ell + 1}$ and $w_{\ell + 1}$ . Note that $\eta _\ell \leq \overline \eta$ , $\zeta _\ell \leq \overline \zeta$ , $\xi _\ell \leq \overline \xi$ and $\chi _\ell \leq \overline \chi$ hold, independently of $\ell \geq 3$ . So, as the same discussion in the case $\ell = 3$ above, we can see that $u_{\ell +1}$ , $v_{\ell +1}$ , $w_{\ell +1} \geq 0$ and
The detail is omitted here. Hence, the nonnegativities of approximations and (10) hold true for all $\ell \in{\mathbb N}$ .
We can see that $u_\ell$ , $v_\ell$ and $w_\ell$ are continuous in $t \in [0, T_0]$ for $\ell \in{\mathbb N}$ . And also, it is easy to see that $ \left\{ u_\ell, v_\ell, w_\ell, t^{1/2} \partial _i u_\ell, t^{1/2} \partial _i v_\ell, \partial _i w_\ell \right\}_{\ell =1}^\infty$ are Cauchy sequences in $C ([0, T_0];\, \text{BUC})$ , choosing $T_0$ small again, if necessary. Let
in the topology of $C ([0, T_0];\, \text{BUC})$ . Obviously, the coincidences $\hat u = t^{1/2} \partial _i u$ , $\hat v = t^{1/2} \partial _i v$ and $\hat w = \partial _i w$ hold by construction. Furthermore, it is also ensured that
The uniqueness follows from (1)–(3) and Gronwall’s inequality, directly. If fact, let $(u, v, w)$ and $(u^\ast, v^\ast, w^\ast )$ be solutions to (P) in $[0, T_0]$ with the same initial data, then $u \equiv u^\ast$ , $v \equiv v^\ast$ and $w \equiv w^\ast$ simultaneously hold. Thanks to the boundedness of the first derivatives, it is easy to control the second derivatives in $x$ of $u$ and $v$ for $t \in (0, T_0]$ , as well as the first derivatives in $t$ of solutions. So, we see that $( u, v, w )$ is a time-local unique classical solution to (P). This completes the proof of Proposition 1.
Remark 2. (i) $( u, v, w )$ is smooth in $x$ and $t$ , if $w_0$ is smooth.(ii) The instability of the trivial solution $(0, 0, 0)$ is easily obtained with $u_0 \not \equiv 0$ . Moreover, by strong maximum principle for solutions to the heat equation, $u \gt 0$ for $x \in{\mathbb R}^n$ and $t \in (0, T_0]$ . This means that $\textrm{supp} \, u(t) ={\mathbb R}^n$ for any small $t \gt 0$ , even if $\textrm{supp} \, u_0$ is compact. That is, the propagation speed of solutions to (P) is infinite, as the same as the heat equation. In addition, $v \gt 0$ and $w \gt 0$ for $x \in{\mathbb R}^n$ and $t \gt 0$ , if either $v_0 \not \equiv 0$ or $w_0 \not \equiv 0$ .
5. Global well-posedness
In this section, we will derive a priori bounds of solutions and their derivatives. To do so, our first task is to obtain upper bounds of solutions to (P) with large initial data. For the case when $\| u_0 \| \leq 1$ , we will discuss in Remark 3 (ii) below and Section 6.
Proposition 2. Suppose the assumption of Proposition 1 . If $\| u_0 \| \gt 1$ , then $0 \lt u \lt \| u_0 \|$ , $0 \leq v \leq \widetilde v$ , and $0 \leq w \leq \widetilde w$ hold for $x \in{\mathbb R}^n$ and $t \gt 0$ with some positive constants $\widetilde v$ and $\widetilde w$ depending on $\| u_0 \|$ , $\| v_0 \|$ and $\| w_0 \|$ , as long as the classical solutions exist.
If $v_0 \equiv 0$ and $w_0 \equiv 0$ , then $v \equiv w \equiv 0$ for $t \gt 0$ . Assume either $v_0 \not \equiv 0$ or $w_0 \not \equiv 0$ . So, as seen in Remark 2 (iii), we have $u$ , $v$ , $w \gt 0$ . For observing the behaviour of $u$ , we consider the following logistic equation:
where $\kappa _0 = \| u_0 \|$ . By maximum principle, $u (x, t) \leq \kappa (t)$ holds for $x \in{\mathbb R}^n$ and $t \gt 0$ , as long as the classical solution $u$ exists. Since
for $t \gt 0$ , it is clear that $u \lt \kappa _0$ .
Next, we investigate on upper bounds of $v$ and $w$ . We will use renormalising arguments to ODE. Let a pair $\sigma = \sigma (t)$ and $\omega = \omega (t)$ be solutions to
Here, $m_\star \,:\!=\, m - \mu \kappa _0/ ( \kappa _0 + h )$ and $\theta _\star \,:\!=\, \theta + \nu \kappa _0/ ( \kappa _0 + h )$ . Since $(e^{\rho t} \omega )^{\prime} = \theta _\star e^{\rho t} \sigma$ , we have
for $t \gt 0$ . Inserting it into the first equation of (12), it holds that
for $t \gt 0$ . Therefore, we can see that $\sigma (t) \lt \widetilde v \,:\!=\, \max \{ \sigma _0, \bar \sigma \} + 1$ for $t \gt 0$ , where
Note that $\bar \sigma$ satisfies $\alpha ( \omega _0 + (\theta _\star/ \rho ) \bar \sigma ) - m_\star \bar \sigma - \bar \sigma ^2 = 0$ . Indeed, if there exists some $t_\star \gt 0$ such that $\sigma (t_\star ) = \widetilde v \geq \bar \sigma + 1$ and $\sigma (t) \lt \widetilde v$ for $t \in [0, t_\star )$ , then $\sigma^{\prime} (t_\star ) \geq 0$ . This contradicts $\sigma^{\prime} (t_\star ) \lt 0$ . We can similarly deduce $\omega (t) \leq \widetilde w$ holds for $t \gt 0$ , where $\widetilde w \,:\!=\, \max \{ \omega _0, \theta _\star \widetilde v/ \rho \} + 1$ .
We will use enclosing arguments, that is, applying the comparison principle between solutions to PDE and those to ODE. Put $V \,:\!=\, \sigma - v$ and $W \,:\!=\, \omega - w$ . Hence, $V(0) \geq 0$ and $W(0) \geq 0$ . Also, we see
and
We thus find the fact that $V \geq 0$ and $W \geq 0$ for $t \gt 0$ , as the same discussion in the proof of Proposition 1. This implies that
for $x$ and $t$ . Therefore, we conclude that $0 \leq v \leq \widetilde v$ and $0 \leq w \leq \widetilde w$ .
Remark 3. (i) By definitions of $\overline v$ and $\overline w$ , it is clear that $\widetilde v \geq \overline v$ and $\widetilde w \geq \overline w$ , if $| u_0 \| \geq 1$ . Besides, $\widetilde v \leq \overline v$ and $\widetilde w \leq \overline w$ , if $| u_0 \| \leq 1$ , $\| v_0 \| \leq \overline v$ and $\| w_0 \| \leq \overline w$ ; see Section 6. (ii) Even if $\| u_0 \| \leq 1$ , then uniform bounds on $v$ and $w$ are obtained; $v \leq \widetilde v$ and $w \leq \widetilde w$ hold, replacing $m_\star$ by $m_1 \,:\!=\, m - \mu/ (1+h)$ and $\theta _\star$ by $\theta _1 \,:\!=\,\theta + \nu/ (1+h)$ . (iii) Although we take the maximum values of solutions to ODE (12) by the comparison method (finding $t$ as $\sigma^{\prime}(t) = 0$ or $\omega^{\prime}(t) = 0$ ), such critical points do not always give the maximum values of solutions to PDE, in general. Hence, we have to use enclosing and renormalising arguments in above.
In what follows, we give the a priori estimate for $\| \partial _i w (t) \|$ , which may grow in $t$ . As seen in Proposition 2, and by using definitions of $\overline v$ and $\overline w$ in Theorem 2, we prove that $0 \leq u, v, w \leq N$ as long as the classical solutions exist, if $N$ is chosen as
Proposition 3. Let $T$ , $N \gt 0$ . If $0 \leq u, v, w \leq N$ for $x \in{\mathbb R}^n$ and $t \in [0, T]$ , then there exists a $C \gt 0$ independent of $N$ and $T$ such that
We first derive the estimate for $\partial _i u$ . By (1), we have
for $t \in [0, T]$ and $1 \leq i \leq n$ with some $C$ . Similarly, by (2), we seek
with some $C$ . Finally, by (3) and estimates above, it turns out that
for $t \in [0, T]$ and $1 \leq i \leq n$ with some positive constant $C$ depending on parameters, however, independent of $N$ and $T$ .
Note that the proof of Theorem 1 is now complete. In fact, Theorem 1 follows from Propositions 1, 2, 3 and $T_0 \geq C_\ast/(M^4+1)$ in Proposition 1, since we can extend the obtained unique classical solutions time-globally, repeating the construction.
6. Invariant regions
This section will be devoted to observing invariant regions. The proof of Theorem 2 (i) is easy, since $(1, 0, 0)$ is only one stable constant state. So, we skip it here.
We are now in position to give a proof of Theorem 2 (ii). The key step is to deduce a priori bounds of solutions, due to the maximum principle and comparison with solutions to the system of corresponding ordinary differential equations of $\kappa$ , $\sigma$ and $\omega$ given by (11) and (12). Let us recall the assumptions:
and $R_\ast \,:\!=\, [0, 1] \times [0, \overline v] \times [0, \overline w]$ .
Proof. We first show that $R_\ast$ is an invariant region. Let $(u_0, v_0, w_0) \in R_\ast$ . By construction of time-local solutions in Proposition 1, the nonnegativity of solutions is clarified. Note that $(0, 0, 0)$ and $(1, 0, 0)$ are classical solutions in $R_\ast$ . If $u_0 \equiv 0$ , then $u \equiv 0$ , in addition, $v \in [0, \overline v]$ and $w \in [0, \overline w]$ , since $v^\flat \,:\!=\, \alpha \theta/ \rho - m \leq \overline v$ and $w^\flat \,:\!=\, \theta (\alpha \theta - m \rho )/ \rho ^2 \leq \overline w$ . Also, it is easy to see that $v \equiv 0$ and $w \equiv 0$ hold for $t \gt 0$ , provided if $v_0 \equiv 0$ and $w_0 \equiv 0$ .
Let $u_0 \not \equiv 0$ and either $v_0 \not \equiv 0$ or $w_0 \not \equiv 0$ . As seen in Remark 2 (iii), it is clear that the classical solutions $u$ , $v$ , $w$ never touch $0$ , as long as they exist. Moreover, with $u_0 \leq 1$ , we observe that $u(\tau ) \lt 1$ for small $\tau \gt 0$ by the strong maximum principle. Similarly, it turns out that $v(\tau ) \lt \overline v$ by $v_0 \leq \overline v$ , as well as $w(\tau ) \lt \overline w$ . So, regarding $\tau$ as the initial time, we can assume $(u_0, v_0, w_0) \in R_\ast ^\circ \,:\!=\, (0, 1) \times (0, \overline v) \times (0, \overline w) = R_\ast \setminus \partial R_\ast$ , without loss of generality.
Put $\hat t \in (0, T_0]$ is the first time when $u$ touches $1$ at $\hat x \in{\mathbb R}^n$ . We may assume $|\hat x| \lt \infty$ by Oleinik’s argument on the maximum principle; see e.g. [Reference Giga, Giga and Saal4]. Since $u (\hat x, \hat t) = 1$ is the local maximum, at $(\hat x, \hat t)$ we see that $\partial _t u \geq 0$ , $\Delta u \leq 0$ , $(1 - u) u = 0$ and $-\gamma uv/(u+h) \lt 0$ by $v \gt 0$ . This contradicts to that $u$ is a solution to (P). Hence, $u$ never touches $1$ .
The same argument works on $v$ and $w$ . Indeed, let $0 \lt u \lt 1$ , $0 \lt w \lt \overline w$ , and if there exists $(\check x, \check t) \in{\mathbb R}^n \times (0, T_0]$ such that $\check t$ is the first time when $v$ touches $\overline v$ at $\check x$ . So, at $(\check x, \check t)$ , we see that $\partial _t v \geq 0$ , $d \Delta v \leq 0$ and
So, $v$ never touches $\overline v$ . As the same as above, we can confirm that $w$ never touches $\overline w$ as long as classical solutions exist. This means that the solutions always remain in $R_\ast ^\circ$ .
Next, we show the asymptotic behaviour of solutions, briefly. Even if $\| u_0 \| \gt 1$ , by $u (x, t) \leq \kappa (t)$ , then there exists a $T_\varepsilon ^\ast \gt 0$ such that $\| u (t) \| \lt 1 + \varepsilon$ for $t \gt T_\varepsilon ^\ast$ . From this and the comparison $v(x, t) \leq \sigma (t)$ , there exists $T_\varepsilon ^\sharp \gt T_\varepsilon ^\ast$ such that $\| v (t) \| \lt \overline v + \varepsilon$ for $t \gt T_\varepsilon ^\sharp$ . Finally, we can also show that there exists $T_\varepsilon \gt T_\varepsilon ^\sharp$ such that $\| w (t) \| \lt \overline w + \varepsilon$ for $t \gt T_\varepsilon$ , by the similar way. This completes the proof of Theorem 2 (ii).
The proof of Theorem 2 (iii) is essentially similar to above. So, we omit it here.
Remark 4. The stability of non-trivial constant states to the system of corresponding ODE can be easily obtained. For example, if
are chosen, then the bifurcation occurs, that is, the stability of a constant state $(u, v, w) = (1/2, 1/2, 1/2)$ is changed in $h$ at $0$ . Indeed, the constant state $(1/2, 1/2, 1/2)$ is stable for any $h \gt 0$ , while this is unstable for any $-1/2 \lt h \lt 0$ . The authors believe that such stability is still valid for solutions to (P). For studying the Turing instability, we need to deal with more complicated situation, for example, when $\mu$ and $\nu$ are sigmoid functions of $u$ .
Acknowledgements
The authors would like to express their sincere gratitude to Professor Yoshio Yamada for his numerous valuable comments and suggestions on this manuscript. The authors would also like to express their sincere gratitude to Professor Shintaro Kondo for his many benefit comments. Okihiro Sawada and Naoki Tsuge contributed equally to this work.
Financial support
N. Tsuge’s research was partially supported by Grant-in-Aid for Scientific Research (C) 17K05315, Japan.
Conflicts of interest
None.