Global Existence of Solutions to a Class of Reaction–Diffusion Systems on ${\mathbb{R}}^n$

Abstract

We prove in this work the existence of a unique global nonnegative classical solution to the class of reaction–diffusion systemsuttx=aΔutx−guvm,vttx=dΔvtx+λtxguvm,

1. Introduction

In the sequel, we use the notations.

${\mathbb{R}}_{+}=\left[0,\infty \right[,$${\mathbb{R}}_{+}^{\ast }=\left]0,\infty \right[.$

$\mathbb{N}=\left\{0,1,\dots \right\}$ the set of natural numbers and ${\mathbb{N}}^{\ast }=\mathbb{N}\backslash \left\{0\right\}.$

For $p\in \mathbb{R}:\left[p\right]$ the integer part of $p$.

For $n\in {\mathbb{N}}^{\ast }$ and $x=\left(x_1,\dots ,x_n\right)\in {\mathbb{R}}^n:{\left|x\right|}^2={\sum }_{j=1}^n{x}_j^2.$

$\mathbb{Z}=\left\{\cdots ,-\mathrm{1,0,1},\cdots \right\}$ the set of integers.

For $x^{\left(0\right)}\in {\mathbb{R}}^n$ and $\rho \in {\mathbb{R}}_{+}^{\ast },$:

$B^{\prime }\left(x^{\left(0\right)},\rho \right)=\left\{x\in {\mathbb{R}}^n:\left|x-x^{\left(0\right)}\right|\le \rho \right\}$ the closed ball of center $x^{\left(0\right)}$ and radius $\rho .$

$S\left(x^{\left(0\right)},\rho \right)=\left\{x\in {\mathbb{R}}^n:\left|x-x^{\left(0\right)}\right|=\rho \right\}$ the boundary of $B^{\prime }\left(x^{\left(0\right)},\rho \right).$

Let $Q\subset {\mathbb{R}}^n$$\left(n\in {\mathbb{N}}^{\ast }\right)$ a subset. $\partial Q$ denote the boundary of $Q.$

$ln$: the natural logaritm function.

${\omega }_n\left(\rho \right)=\frac{2{\pi }^{n/2}{\rho }^{n-1}}{\mathrm{\Gamma }\left(n/2\right)}$ the surface area of $S\left(0,\rho \right)$, where $\mathrm{\Gamma }\left(x\right)={\int }_0^{\infty }{e}^{-t}{t}^{-x}\mathit{dt}$$\left(x\in {\mathbb{R}}_{+}^{\ast }\right)$ is the Gamma function.

$\mathit{BUC}\left({\mathbb{R}}^n\right)$ the Banach space of bounded and uniformly continuous functions on ${\mathbb{R}}^n$$$ with the supremum norm $\parallel u{\parallel }_{\infty }={sup}_{x\in {\mathbb{R}}^n}\left|u\left(x\right)\right|.$

$X=\mathit{BUC}\left({\mathbb{R}}^n\right)\times \mathit{BUC}\left({\mathbb{R}}^n\right)$ which is a Banach space endowed with the norm ${‖\left(u,v\right)‖}_X=\parallel u{\parallel }_{\infty }+\parallel v{\parallel }_{\infty }.$

For $u\in L^p\left({\mathbb{R}}^n\right)$$(p\in \left[1,\infty \right[$), we denote by $\parallel u{\parallel }_p^p={\int }_{{\mathbb{R}}^n}{\left|u\right|}^p\mathit{dx}.$

For $u,v:{\mathbb{R}}^n\to \mathbb{R}$ two regular functions, $\nabla u=\left(\frac{\partial u}{\partial x_1},\dots ,\frac{\partial u}{\partial x_n}\right)$ and $\nabla u.\nabla v={\sum }_{j=1}^n\frac{\partial u}{\partial x_j}.\frac{\partial v}{\partial x_j}.$

Reaction-Diffuison equations are nonlinear parabolic partial differential equations arises in many fields of sciences like chemistry, physics, biology, ecology and even medicine. It appears usually as coupled systems.

The somewhat general form of these systems of two equations is

$$\left\{\begin{array}{c}{u}_t\left(t,x\right)=a\mathrm{\Delta }u\left(t,x\right)+f_1\left(t,x,u,v\right),\\ v_t\left(t,x\right)=d\mathrm{\Delta }v\left(t,x\right)+f_2\left(t,x,u,v\right),\end{array}\right.$$

where $t>0,$$x\in \mathrm{\Omega }$ with $\mathrm{\Omega }\subset {\mathbb{R}}^n\left(n\in {\mathbb{N}}^{\ast }\right)$ is an open set, $\mathrm{\Delta }$ is the Laplacian operator, $a$, $d$ are two real positive constants called the coefficients of the diffusion. For a chemical reaction where two substances $S_1$ and $S_2$, $u$ and $v$ represent their concentrations at time $t$ and position $x$ respectively, and $f_1$ and $f_2$ represent the rate of production of these substances in the given order. For more details see [1, 2].

In this chapter, we are concerned with the existence of global solutions to the reaction–diffusion system

$$u_t\left(t,x\right)=a\mathrm{\Delta }u\left(t,x\right)-g\left(u\right)v^m,\left(t,x\right)\in {\mathbb{R}}_{+}^{\ast }\times {\mathbb{R}}^n,$$E1

$$v_t\left(t,x\right)=d\mathrm{\Delta }v\left(t,x\right)+\lambda \left(t,x\right)g\left(u\right)v^m,\left(t,x\right)\in {\mathbb{R}}_{+}^{\ast }\times {\mathbb{R}}^n,$$E2

with initial data

$$u\left(0,x\right)=u_0\left(x\right),v\left(0,x\right)=v_0\left(x\right),x\in {\mathbb{R}}^n.$$E3

Whe assume that.

(H1) The constants $a$, $d$ are such that $a,$$d\in {\mathbb{R}}_{+}^{\ast }.$

(H2)$\lambda :{\mathbb{R}}^{+}\times {\mathbb{R}}^n\to \mathbb{R}$ is a non-null, nonnegative and bounded function on ${\mathbb{R}}^{+}\times {\mathbb{R}}^n$ such that $\lambda \left(t,.\right)\in \mathit{BUC}\left({\mathbb{R}}^n\right)$ for all $t\in {\mathbb{R}}_{+}$. We denote ${\lambda }_{\infty }={sup}_{t\ge 0}\left({‖\lambda \left(t\right)‖}_{\infty }\right).$

(H3)$n$ and $m$ are positive integers, i.e. $n,m\in {\mathbb{N}}^{\ast }.$

(H4)$g$$:\mathit{BUC}\left({\mathbb{R}}^n\right)\to \mathit{BUC}\left({\mathbb{R}}^n\right)$ is a function defined on $\mathit{BUC}\left({\mathbb{R}}^n\right)$ such that:

i. $g\left(0\right)=0$ and $g\left(u\right)\ge 0$ pour $u\ge 0.$

ii. $g$ is of class $C^1$$$ and $\frac{\mathit{dg}\left(u\right)}{\mathit{du}}$ is bounded on $\mathbb{R}.$

(H5) The initial data $u_0$, $v_0$ are nonnegative and are in $\mathit{BUC}\left({\mathbb{R}}^n\right)$.

One of the essential questions for (1)–(3) is the existence of global solutions and possibly bounds uniform in time. Recently, Collet and Xin in their paper [3] published in 1996 have studied the system (1)–(3) but with $a=\lambda =1$, $d>1$ and $\phi \left(u\right)=u.$ In this particular case, this system describes the evolution of $u$ the mass fraction of reactant $A$ and that $v$ of the product $B$ for the autocatalytic chemical reaction of the form $A+\mathit{mB}\to \left(m+1\right)B.$ They proved the existence of global solutions and showed that the $L^{\infty }$ norm of $v$ can not grow faster than $O\left(lnlnt\right)$ for any space dimension.

If we replace $g\left(u\right)v^m$ by $uexp\left\{-E/v\right\}$ where $E>0$ is a constant and take $\lambda =1$, there are many works on global solutions, see Avrin [4], Larrouturou [5] for results in one space dimension, among others.

It is worth mentioning here the result of S. Badraoui [6] who studied the system

$$u_t=a\mathrm{\Delta }u-{\mathit{uv}}^m,$$

$$v_t=b\mathrm{\Delta }u+d\mathrm{\Delta }v+{\mathit{uv}}^m,$$

where $a>0$, $d>0$, $b\ne 0$, $x\in {\mathbb{R}}^n$, $n\in {\mathbb{N}}^{\ast },$$m\in 2{\mathbb{N}}^{\ast }$ is an even positive integer. He has proved that if $u_0$, $v_0$ are nonnegative and are in $\mathit{BUC}\left({\mathbb{R}}^n\right)$ that:

If $a>d,$$b>0,$$v_0\ge \frac{b}{a-d}{u}_0$ on ${\mathbb{R}}^n$, then the solution is global and uniformly bounded.

If $a<d,$$b<0,$$v_0\ge \frac{b}{a-d}{u}_0$ on ${\mathbb{R}}^n,$ then the solution is global.

Our work here is a continuation of the work of Collet and Xin [3]. We treat the same question in a slightly general case. Inspired by the same ideas in [3] we prove that the system (1)–(3) under the assumptions (H1) to (H5) has a unique global nonnegative classical solution.

The chapter is organized as follows: In section 2, we treat the existence of local solution and reveal its positivity using the maximum principle.

In section 3, firstly, we prove by a simple comparison argument that if $a\ge d$, the solution is uniformly bounded and we give an upper bound of it. Afterwards, we attack the hard case in which $a<d$ where we used the Lyapunov functional $L\left(u,v\right)=\left[\alpha +2u-ln\left(1+u\right)\right]e^{\mathit{\epsilon v}}$$(\alpha ,$$\epsilon >0)$ and the cut-off function $\phi \left(x\right)={\left(1+{\left|x\right|}^2\right)}^{-n}.$ We show that for $\alpha$ sufficiently large and $\epsilon$ small enough we can control the $L^p$-norms of $v$$\left(p>max\left\{1,n/2\right\}\right)$ on every unit spacial cub in ${\mathbb{R}}^n$ from which we deduce the $L^{\infty }$-norm of $v$ at any time $t>0.$

We emphazise here that I have engaged to calculate the constants encountered in all equations and inequalities exactly.

2. Existence of a local solution and its positivity

We convert the system (1)–(3) to an abstract first order system in the Banach space $X≔\mathit{BUC}\left({\mathbb{R}}^n\right)\times \mathit{BUC}\left({\mathbb{R}}^n\right)$ of the form

$$\left\{\begin{array}{c}{w}^{\prime }\left(t\right)=\mathit{Aw}\left(t\right)+F\left(w\left(t\right)\right),t>0,\\ w\left(0\right)=w_0\in X.\end{array}\right.$$E4

Here $w\left(t\right)=\left(u\left(t\right),v\left(t\right)\right)$; the operator $A$ is defined as

$$\mathit{Aw}≔\left(\begin{array}{cc}a\mathrm{\Delta }& 0\\ 0& d\mathrm{\Delta }\end{array}\right)w=\left(a\mathrm{\Delta }u,d\mathrm{\Delta }v\right),$$

where $D\left(A\right)≔\left\{w=\left(u,v\right)\in X:\left(\mathrm{\Delta }u,\mathrm{\Delta }v\right)\in X\right\}$. The function $F$ is defined as $F\left(w\left(t\right)\right)=\left(-\phi \left(u\left(t\right)\right)v^m\left(t\right),\lambda \left(t\right)\phi \left(u\left(t\right)\right)v^m\left(t\right)\right)$.

It is known that for $c>0$ the operator $c\mathrm{\Delta }$ generates an analytic semigroup $G\left(t\right)$ in the space $\mathit{BUC}\left({\mathbb{R}}^n\right)$:

$$G\left(t\right)u={\left(4\mathit{\pi ct}\right)}^{-n/2}{\int }_{{\mathbb{R}}^n}exp\left\{-\frac{{\left|x-y\right|}^2}{4\mathit{ct}}\right\}u\left(y\right)\mathit{dy}.$$E5

Hence, the operator $A$ generates an analytic semigroup defined by

$$S\left(t\right)=\left(\begin{array}{cc}{S}_1\left(t\right)& 0\\ 0& S_2\left(t\right)\end{array}\right),$$E6

where $S_1\left(t\right)$ is the semigroup generated by the operator $a\mathrm{\Delta }$, and $S_2\left(t\right)$ is the semigroup generated by the operator $d\mathrm{\Delta }$.

Since the map $F$ is locally Lipschitz in $w$ in the space $X$, then proving the existence of a loacl classical solution on $\left[0,t_1\right]$ where $t_1\in {\mathbb{R}}_{+}^{\ast }$ is standard [7, 8].

For the positivity, let $w\left(t\right)=\left(u\left(t\right),v\left(t\right)\right)$ is a local solution of the problem (1)–(3) under the assumptions ${\left\{\mathit{Hj}\right\}}_{j=1}^5$ on the interval $\left[0,t_1\right]$.

We can write the first equation as

$$u_t-a\mathrm{\Delta }u+\left[v^m\frac{d}{du}g\left(\xi \right)\right]u=0,\left(t,x\right)\in \left]0,t_1\right]\times {\mathbb{R}}^n,$$E7

for some $\xi \in \mathbb{R}$. Thanks to the assumption (H4)-ii we deduce that $v^m\frac{\partial }{\partial u}g\left(\xi \right)$ is bounded on $\left[0,t_1\right]\times {\mathbb{R}}^n.$ Whence, by the theorem 9 on page 43 in [9], we obtain that

$$u\left(t,x\right)\ge 0,\text{for}\mathrm{all}\left(t,x\right)\in \left[0,t_1\right]\times {\mathbb{R}}^n,$$E8

The second equation can be written as

$$v_t-d\mathrm{\Delta }v+\left[-\mathit{\lambda g}\left(u\right)v^{m-1}\right]v,\left(t,x\right)\in \left]0,t_1\right]\times {\mathbb{R}}^n.$$E9

By the same theorem we get

$$v\left(t,x\right)\ge 0,\text{for}\mathrm{all}\left(t,x\right)\in \left[0,t_1\right]\times {\mathbb{R}}^n.$$E10

For the existence of a global solution, we use the contraposed of the characterization of the maximal existence time $t_{\max }$ ([8] on page 193) as follows

$$\left[\begin{array}{c}\text{there\hspace{0.17em}exists}\mathrm{a}\mathrm{map}C:{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}\text{such\hspace{0.17em}that}:\\ {‖u\left(t\right)‖}_{\infty }+{‖v\left(t\right)‖}_{\infty }\le C\left(t\right)\text{for}\mathrm{all}t\in {\mathbb{R}}_{+}\end{array}\right]\Rightarrow t_{\max }=+\infty .$$E11

3. Existence of a global solution

For this task we will use the fact that the solution is nonnegative.

Theorem 3.1. Let $\left(u,v\right)$ be the solution of the problem (1)–(3) under the assumptions ${\left\{\mathit{Hj}\right\}}_{j=1}^5$ and such that

$$a\ge d.$$E12

Then, the solution is global and uniformly bounded on ${\mathbb{R}}^{+}\times {\mathbb{R}}^n$. More precisely, we have the estimates

$${‖u\left(t\right)‖}_{\infty }\le {‖u_0‖}_{\infty },\text{for}\mathrm{all}t\in {\mathbb{R}}_{+},$$E13

$${‖v\left(t\right)‖}_{\infty }\le ‖v_0‖+{\lambda }_{\infty }{\left(\frac{a}{d}\right)}^{n/2}{‖u_0‖}_{\infty },\text{for}\mathrm{all}t\in {\mathbb{R}}_{+}.$$E14

Proof. By the comparison principle we get (13).

The solution $\left(u,v\right)$ satisfies the integral equations

$$u\left(t,x\right)=S_1\left(t\right)u_0-{\int }_0^t{S}_1\left(t-\tau \right)g\left(u\left(\tau \right)\right)v^m\left(\tau \right)\mathit{d\tau },$$E15

$$v\left(t,x\right)=S_2\left(t\right)v_0+{\int }_0^t{S}_2\left(t-\tau \right)\lambda \left(\tau \right)g\left(u\left(\tau \right)\right)v^m\left(\tau \right)\mathit{d\tau }.$$E16

Here $S_1\left(t\right)$ and $S_2\left(t\right)$ are the semigroups generated by the operators $a\mathrm{\Delta }$ and $d\mathrm{\Delta }$ in the space $\mathit{BUC}\left({\mathbb{R}}^n\right)$ respectively. As $u$ is nonnegative, then from (15) we get

$${\int }_0^t{S}_1\left(t-\tau \right)g\left(u\left(\tau \right)\right)v^m\left(\tau \right)\mathit{d\tau }\le S_1\left(t\right)u_0.$$E17

Since $a\ge d$, using the explicit expression of $S_1\left(t-\tau \right)g\left(u\left(\tau \right)\right)v^m\left(\tau \right)$ and $S_2\left(t-\tau \right)g\left(u\left(\tau \right)\right)v^m\left(\tau \right)$, one can observe that (see [10])

$$\begin{array}{l}{\int }_0^t{S}_2\left(t-\tau \right)\lambda \left(\tau \right)g\left(u\left(\tau \right)\right)v^m\left(\tau \right)\mathit{d\tau }\le {\left(\frac{a}{d}\right)}^{n/2}{\int }_0^t{S}_1\left(t-\tau \right)\lambda \left(\tau \right)g\left(u\left(\tau \right)\right)v^m\left(\tau \right)\mathit{d\tau }\\ \le {\lambda }_{\infty }{\left(\frac{a}{d}\right)}^{n/2}{\int }_0^t{S}_1\left(t-\tau \right)g\left(u\left(\tau \right)\right)v^m\left(\tau \right)\mathit{d\tau }.\end{array}$$E18

From (17) and (18) into (16) we get

$$v\left(t\right)\le S_2\left(t\right)v_0+{\lambda }_{\infty }{\left(\frac{a}{d}\right)}^{n/2}{S}_1\left(t\right)u_0.$$E19

This last inequality leads to the veracity of (14).

Thus, from (13) and (14), we deduce that the solution $\left(u,v\right)$ is global and uniformly bounded on ${\mathbb{R}}_{+}\times {\mathbb{R}}^n$. $♦$

In the case where $d>a$, it seems that the idea of comparison cannot be applied. Nevertheless, we can prove the existence of global classical solutions; but it appears that their boundedness is not assured.

Theorem 3.2. Let $\left(u,v\right)$ be the solution of the problem (1)–(3) with the assumptions ${\left\{\mathit{Hj}\right\}}_{j=1}^5$. If

$$a<d,$$E20

the solution $\left(u,v\right)$ is global. More precisely we have the estimates (13) and (83).

Proof. In this case, it is not easy to prove global existence. But can derive estimates of solutions independent of $t_1$ by using the same method used in [3] and the same form of the functional used in [6] but with different coefficients.

We need some lemmas.

Lemma 3.3. Let $\left(u,v\right)$ be the solution of the problem (1)–(3) under the assumptions ${\left\{\mathit{Hj}\right\}}_{j=1}^5$ on the local interval time $\left[0,t_1\right]$. Define the functional

$$L\left(u,v\right)=\left[\alpha +2u-ln\left(1+u\right)\right]e^{\mathit{\epsilon v}}\text{with}\alpha ,\epsilon \in {\mathbb{R}}_{+}^{\ast }.$$E21

Then for any $\phi =\phi \left(x\right)$$\left(x\in {\mathbb{R}}^n\right)$ a smooth nonnegative function with exponential spacial decay at infinity, we have

$$\begin{array}{l}\frac{d}{\mathit{dt}}{\int }_{{\mathbb{R}}^n}\mathit{\text{φLdx}}=d{\int }_{{\mathbb{R}}^n}\mathrm{\Delta }\mathit{\text{φLdx}}+\left(d-a\right){\int }_{{\mathbb{R}}^n}{L}_1\nabla \phi .\nabla \mathit{udx}\\ -{\int }_{{\mathbb{R}}^n}\phi \left[{\mathit{aL}}_{11}{\left|\nabla u\right|}^2+\left(a+d\right)L_{12}\nabla u\nabla v+{\mathit{dL}}_{22}{\left|\nabla v\right|}^2\right]\mathit{dx}\\ +{\int }_{{\mathbb{R}}^n}\phi \left(\lambda L_2-L_1\right)g\left(u\right)v^m\mathit{dx},\end{array}$$E22

where

$$\begin{array}{ll}{L}_1& \equiv \frac{\partial L}{\partial u}=\left(2-\frac{1}{1+u}\right)e^{\mathit{\epsilon v}},L_2\equiv \frac{\partial L}{\partial v}=\epsilon \left[\alpha +2u-ln\left(1+u\right)\right]e^{\mathit{\epsilon v}},\\ L_{11}& \equiv \frac{{\partial }^{2}L}{\partial u^2}=\frac{1}{{\left(1+u\right)}^2}{e}^{\mathit{\epsilon v}},L_{12}\equiv \frac{{\partial }^{2}L}{\partial u\partial v}=\epsilon \left(2-\frac{1}{1+u}\right)e^{\mathit{\epsilon v}},\\ L_{22}& \equiv \frac{{\partial }^{2}L}{\partial v^2}={\epsilon }^2\left[\alpha +2u-ln\left(1+u\right)\right]e^{\mathit{\epsilon v}}.\end{array}$$E23

Proof. Note that $L>0$, $L_1>0$, $L_2>0$, $L_{11}>0$, $L_{12}>0$ and $L_{22}>0$. We can differentiate under the integral symbol

$$\frac{d}{\mathit{dt}}{\int }_{{\mathbb{R}}^n}\mathit{\text{φLdx}}=a{\int }_{{\mathbb{R}}^n}\phi L_1\mathit{udx}+d{\int }_{{\mathbb{R}}^n}\phi L_2\mathrm{\Delta }\mathit{vdx}+{\int }_{{\mathbb{R}}^n}\phi \left(\lambda L_2-L_1\right)g\left(u\right)v^m\mathit{dx}.$$E24

Using integration by parts, we get

$$\begin{array}{l}{\int }_{{\mathbb{R}}^n}\phi L_1\mathrm{\Delta }\mathit{udx}={\int }_{{\mathbb{R}}^n}\left(\phi L_1\right)\mathrm{\Delta }\mathit{udx}=-{\int }_{{\mathbb{R}}^n}\nabla \left(\phi L_1\right)\nabla \mathit{udx}=-{\int }_{{\mathbb{R}}^n}{L}_1\nabla \phi \nabla \mathit{udx}\\ -{\int }_{{\mathbb{R}}^n}\phi L_{11}{\left|\nabla u\right|}^2\mathit{dx}-{\int }_{{\mathbb{R}}^n}\phi L_{12}\nabla u\nabla \mathit{vdx},\end{array}$$E25

In fact, let $\rho \in {\mathbb{R}}_{+}^{\ast },$ then we have by the Geen theorem

$$\begin{array}{ll}{\int }_{B^{\prime }\left(0,\rho \right)}\phi L_1\mathrm{\Delta }\mathit{udx}& ={\int }_{B^{\prime }\left(0,\rho \right)}\left(\phi L_1\right)\mathrm{\Delta }\mathit{udx}\\ & =-{\int }_{B^{\prime }\left(0,\rho \right)}\nabla \left(\phi L_1\right).\nabla \mathit{udx}+{\int }_{S\left(0,\rho \right)}\left(\phi L_1\right)\frac{\partial u}{\partial \nu }\mathit{dx},\end{array}$$E26

where $\frac{\partial u}{\partial \nu }$ is the derivative of $u$ with respect to the unit outer normal $\nu$ to the boundary $S\left(0,\rho \right)$.

We have

$$\begin{array}{l}\left|{\int }_{S\left(0,\rho \right)}\left(\phi L_1\left(t\right)\right)\frac{\partial u\left(t\right)}{\partial \nu }\mathit{dx}\right|\le 2e^{\epsilon {‖v\left(t\right)‖}_{\infty }}{‖\frac{\partial u\left(t\right)}{\partial \nu }‖}_{\infty }{\int }_{S\left(0,\rho \right)}\mathit{\phi dx}\\ \le 2e^{\epsilon {‖v\left(t\right)‖}_{\infty }}{‖\frac{\partial u\left(t\right)}{\partial \nu }‖}_{\infty }\frac{1}{{\left(1+{\rho }^2\right)}^n}\frac{2{\pi }^{n/2}{\rho }^{n-1}}{\mathrm{\Gamma }\left(n/2\right)}.\end{array}$$E27

From (27) we obtain

$$\underset{\rho \to \infty }{lim}{\int }_{S\left(x_0,\rho \right)}\left[\phi L_1\left(t\right)\right]\frac{\partial u\left(t\right)}{\partial \nu }\mathit{dx}=0.$$E28

We pass to the limit for $\rho \to \infty$ in (26) taking into account (28) we obtain (25).

By the same way we get

$${\int }_{{\mathbb{R}}^n}\phi L_2\mathrm{\Delta }\mathit{vdx}=-{\int }_{{\mathbb{R}}^n}{L}_2\nabla \phi .\nabla \mathit{vdx}-{\int }_{{\mathbb{R}}^n}\phi L_{22}{\left|\nabla v\right|}^2\mathit{dx}-{\int }_{{\mathbb{R}}^n}\phi L_{12}\nabla u\nabla \mathit{vdx},$$E29

$${\int }_{{\mathbb{R}}^n}L\mathrm{\Delta }\mathit{\phi dx}=-{\int }_{{\mathbb{R}}^n}{L}_1\nabla \phi .\nabla \mathit{udx}-{\int }_{{\mathbb{R}}^n}{L}_2\nabla \phi .\nabla \mathit{vdx}.$$E30

From (30) we find that

$${\int }_{{\mathbb{R}}^n}{L}_2\nabla \phi .\nabla \mathit{vdx}=-{\int }_{{\mathbb{R}}^n}{L}_1\nabla \phi .\nabla \mathit{udx}-{\int }_{{\mathbb{R}}^n}L\mathrm{\Delta }\mathit{\phi dx}.$$E31

From (25), (29) and (31) into (24) we get our basic identity (22). $♦$

Lemma 3.4. There exist two positive real constants $\alpha =\alpha \left(a,d,{\gamma }_1,{‖u_0‖}_{\infty }\right)$ and $\epsilon =\epsilon \left(a,d,{\gamma }_1,{\gamma }_2,{\lambda }_{\infty },{‖u_0‖}_{\infty }\right)$ such that

$$\begin{array}{l}\frac{d}{\mathit{dt}}{\int }_{{\mathbb{R}}^n}\mathit{\text{φLdx}}\le d{\int }_{{\mathbb{R}}^n}\mathit{\text{LΔφdx}}+\left(d-a\right){\int }_{{\mathbb{R}}^n}{L}_1\nabla \phi .\nabla \mathit{udx}\\ -{\gamma }_1{\int }_{{\mathbb{R}}^n}\phi \left[{\mathit{aL}}_{11}{\left|\nabla u\right|}^2+{\mathit{dL}}_{22}{\left|\nabla v\right|}^2\right]\mathit{dx}-{\gamma }_2{\int }_{{\mathbb{R}}^n}\phi L_{1}g\left(u\right)v^m\mathit{dx},\end{array}$$E32

where ${\gamma }_1,{\gamma }_2\in \left]0,1\right[$ are two arbitrary constants.

Proof. We seek $L$ such that

$${\mathit{aL}}_{11}{\left|\nabla u\right|}^2+\left(a+d\right)L_{12}\nabla u\nabla v+{\mathit{dL}}_{22}{\left|\nabla v\right|}^2\ge {\gamma }_1\left[{\mathit{aL}}_{11}{\left|\nabla u\right|}^2+{\mathit{dL}}_{22}{\left|\nabla v\right|}^2\right]$$E33

and

$$\lambda L_2-L_1\le -{\gamma }_2{L}_1$$E34

for ${\gamma }_1,{\gamma }_2\in \left]0,1\right[.$

The inequality (33) is satisfied if

$$\frac{{\left(a+d\right)}^2{L}_{12}^2}{4\mathit{ad}{\left(1-{\gamma }_1\right)}^2{L}_{11}{L}_{12}}\le 1.$$E35

From (23); (35), then (33) is satisfied if

$$\alpha \ge \frac{{\left(a+d\right)}^2{\left[1+2{‖u_0‖}_{\infty }\right]}^2}{4\mathit{ad}{\left(1-{\gamma }_1\right)}^2}.$$E36

Also, (34) is satisfied if $\frac{{\mathit{\epsilon \lambda }}_{\infty }\left(\alpha +2{‖u_0‖}_{\infty }\right)}{1-{\gamma }_2}\le 1$, i.e. $\epsilon \le \frac{1-{\gamma }_2}{{\lambda }_{\infty }\left(\alpha +2{‖u_0‖}_{\infty }\right)},$ and from (36) we get

$$0<\epsilon \le \frac{1-{\gamma }_2}{{\lambda }_{\infty }}\frac{4\mathit{ad}{\left(1-{\gamma }_1\right)}^2}{{\left(a+d\right)}^2{\left[1+2{‖u_0‖}_{\infty }\right]}^2+8\mathit{ad}{\left(1-{\gamma }_1\right)}^2{‖u_0‖}_{\infty }}.$$E37

Whence, if $\alpha$ satisfies (36) and $\epsilon$ satisfies (37), we obtain (32). $♦$

As a consequence of (33) we have

$$\frac{d}{\mathit{dt}}{\int }_{{\mathbb{R}}^n}\mathit{\text{φLdx}}\le d{\int }_{{\mathbb{R}}^n}\mathit{\text{LΔφdx}}+\left(d-a\right){\int }_{{\mathbb{R}}^n}{L}_1\nabla \phi .\nabla \mathit{udx}-{\gamma }_{1}a{\int }_{{\mathbb{R}}^n}\phi L_{11}{\left|\nabla u\right|}^2\mathit{dx}.$$E38

Lemma 3.5. With the functional $L$ defined in (21) and $\alpha$, $\epsilon$ defined in (36) and (37) respectively and with the truncation function $\phi :{\mathbb{R}}^n\to \mathbb{R}$ defined by

$$\phi \left(x\right)=\frac{1}{{\left(1+{\left|x-x_0\right|}^2\right)}^n}.$$E39

We have

$$\frac{d}{\mathit{dt}}{\int }_{{\mathbb{R}}^n}\mathit{\text{φLdx}}\le {\mathit{dk}}_1\left(n\right){\int }_{{\mathbb{R}}^n}\mathit{\text{φLdx}}+\frac{1}{4{\gamma }_{1}a}{\left(d-a\right)}^2{k}_2^2\left(n\right){\int }_{{\mathbb{R}}^n}\phi \frac{L_1^2}{L_{11}}\mathit{dx},$$E40

where

$$k_1\left(n\right)=2n\left(3n+2\right),k_2\left(n\right)=2n.$$E41

Proof. Calulate $\mathrm{\Delta }\phi$ and estimate it

$$\mathrm{\Delta }\phi =-\frac{2n^2}{{\left(1+{\left|x-x_0\right|}^2\right)}^{n+1}}-\frac{4n\left(n+1\right){\left|x-x_0\right|}^2}{{\left(1+{\left|x-x_0\right|}^2\right)}^{n+2}};$$

whence

$$\left|\mathrm{\Delta }\phi \right|\le 2n\left(3n+2\right)\phi .$$E42

Calulate $\nabla \phi$ and estimate it

$${\left|\nabla \phi \right|}^2=4n^2\frac{{\left|x-x_0\right|}^2}{{\left(1+{\left|x-x_0\right|}^2\right)}^{2\left(n+2\right)}};$$

whence

$$\left|\nabla \phi \right|\le 2\mathit{n\phi }.$$E43

Using the Cauchy-Schwarz inequality $\nabla \phi .\nabla u\le \left|\nabla \phi \right|\left|\nabla u\right|$ and the inequalities (42) and (43) into (38) we get

$$\frac{d}{\mathit{dt}}{\int }_{{\mathbb{R}}^n}\mathit{\text{φLdx}}\le {\mathit{dk}}_1\left(n\right){\int }_{{\mathbb{R}}^n}\mathit{\text{φLdx}}+\left(d-a\right)k_2\left(n\right){\int }_{{\mathbb{R}}^n}\phi L_1\left|\nabla \phi \right|\mathit{dx}-{\gamma }_{1}a{\int }_{{\mathbb{R}}^n}\phi L_{11}{\left|\nabla u\right|}^2\mathit{dx}.$$E44

We pove that

$$\left(d-a\right)k_2\left(n\right)\phi L_1\left|\nabla \phi \right|-{\gamma }_1\mathit{a\phi }{L}_{11}{\left|\nabla u\right|}^2\le \frac{1}{4{\gamma }_1}\frac{{\left(d-a\right)}^2}{a}{k}_2^2\left(n\right)\phi \frac{L_1^2}{L_{11}}.$$E45

To do this, it sufficies to compute the discriminant of the trinoma in $\left|\nabla \phi \right|$

$$\mathrm{\Delta }=-{\gamma }_1\mathit{a\phi }{L}_{11}{\left|\nabla u\right|}^2+\left(d-a\right)k_2\left(n\right)\phi L_1\left|\nabla \phi \right|-\frac{1}{4{\gamma }_1}\frac{{\left(d-a\right)}^2}{a}{k}_2^2\left(n\right)\phi \frac{L_1^2}{L_{11}}.$$

From (45) into (42) we find the desired result (40). $♦$

Lemma 3.6. For $\alpha$ and $\epsilon$ defined in (36) and (37) respectively and for all real constant $\gamma$

$$\gamma \ge max\left\{\frac{1}{a},8{‖u_0‖}_{\infty }+4\right\},$$E46

we have

$${\int }_{{\mathbb{R}}^n}\mathit{\text{φLdx}}\le \beta e^{\mathit{\sigma t}},\text{for}\mathrm{all}t\in {\mathbb{R}}_{+};$$E47

where

$$\beta =\frac{2}{n}\left(\alpha +2{‖u_0‖}_{\infty }\right){\omega }_n{e}^{\epsilon {‖v_0‖}_{\infty }},$$E48

and

$$\sigma ={\mathit{dk}}_1\left(n\right)+\frac{\gamma }{4{\gamma }_{1}a}{\left(d-a\right)}^2{k}_2^2\left(n\right).$$E49

Proof. We seek a constant $\gamma \in {\mathbb{R}}_{+}^{\ast }$ such that

$$\frac{L_1^2}{L_{11}}\le \mathit{\gamma L},\text{for}\mathrm{all}u\in \left[0,{‖u_0‖}_{\infty }\right].$$E50

The inequality (50) is equivalent to ${\left(2u+1\right)}^2{e}^{\mathit{\epsilon v}}\le \gamma \left[\alpha +2u-ln\left(1+u\right)\right].$ We prove that if $\gamma$ satisfies (46) then (50) follows.

Whence, from (50) into (40) we obtain

$$\frac{d}{\mathit{dt}}{\int }_{{\mathbb{R}}^n}\mathit{\text{φLdx}}\le \left[{\mathit{dk}}_1\left(n\right)+\frac{\gamma }{4{\gamma }_{1}a}{\left(d-a\right)}^2{k}_2^2\left(n\right)\right]{\int }_{{\mathbb{R}}^n}\mathit{\text{φLdx}},\text{forall}t\in {\mathbb{R}}_{+}.$$E51

As

$${\int }_{{\mathbb{R}}^n}\mathit{\phi L}\left(t=0\right)\mathit{dx}={\int }_{{\mathbb{R}}^n}\phi \left[\alpha +2u_0-ln\left(1+u_0\right)\right]e^{\epsilon v_0}\mathit{dx};$$E52

then, from (51) and (52) we get

$${\int }_{{\mathbb{R}}^n}\mathit{\text{φLdx}}\le \left(\alpha +2{‖u_0‖}_{\infty }\right){‖\phi ‖}_1\left[exp\left(\epsilon {‖v_0‖}_{\infty }\right)\right]e^{\mathit{\sigma t}},\text{for}\mathrm{all}t\in {\mathbb{R}}_{+},$$E53

where $\sigma$ is defined by (49).

Now, let us estimate ${‖\phi ‖}_1.$ We have ([11] on page 485)

$${‖\phi ‖}_1={\int }_{{\mathbb{R}}^n}\mathit{\phi dx}={\int }_{{\mathbb{R}}^n}\frac{1}{{\left(1+{\left|x\right|}^2\right)}^n}\mathit{dx}={\omega }_n{\int }_0^{\infty }{r}^{n-1}\frac{1}{{\left(1+r^2\right)}^n}\mathit{dr}.$$

As

$$\begin{array}{l}{\int }_0^{\infty }\frac{r^{n-1}}{{\left(1+r^2\right)}^n}\mathit{dr}={\int }_0^1\frac{r^{n-1}}{{\left(1+r^2\right)}^n}\mathit{dr}+{\int }_1^{\infty }\frac{r^{n-1}}{{\left(1+r^2\right)}^n}\mathit{dr}\\ \le {\int }_0^1{r}^{n-1}\mathit{dr}+{\int }_1^{\infty }\frac{1}{r^{n+1}}\mathit{dr}\le \frac{2}{n},\end{array}$$

then

$${‖\phi ‖}_1\le \frac{2}{n}{\omega }_n.$$E54

Thus, from (54) in (53) we get the estimate (47) with $\beta$ and $\sigma$ given by (48) and (49). $♦$

In the following step we trie to control the second component $v$ of the solution on any unit spacial cube in the $L^p-$ norms with $p\in \left[1,\infty \right[.$

Let $x^{\left(0\right)}=\left(x_1^{\left(0\right)},\dots ,x_n^{\left(0\right)}\right)\in {\mathbb{R}}^n$ be an arbitrary fixed point and

$$Q=\left\{x=\left(x_1,\dots ,x_n\right)\in {\mathbb{R}}^n:\left|x_k-x_k^{\left(0\right)}\right|\le \frac{1}{2},\text{for}\mathrm{all}k=1,\dots ,n\right\}.$$E55

Lemma 3.7. Let $\left(u,v\right)$ be the solution of the problem in consideration. For $\alpha$ and $\epsilon$ satisfying (36) above and (63) below respectively, then for any unit cube $Q$ of ${\mathbb{R}}^n$ of the form (55) we have

$${\int }_Q{v}^p\mathit{dx}\le \frac{\beta {\left(p+1\right)}^{p+1}}{{\mathit{\alpha \epsilon }}^{\left[p\right]+1}}{\left(\frac{4+n}{4}\right)}^n{e}^{\mathit{\sigma t}},\text{for}\mathrm{all}\left(p,t\right)\in \left[1,\infty \right[\times {\mathbb{R}}_{+}.$$E56

Proof. It’s obvious that

$$\phi \left(x\right)\ge {\left(\frac{4}{4+n}\right)}^n,\text{for}\mathrm{all}x\in {\mathbb{R}}^n,$$E57

and

$$e^{\mathit{\epsilon v}}\ge \frac{{\epsilon }^k}{k!}{v}^k,\text{for}\mathrm{all}k\in {\mathbb{N}}^{\ast }.$$E58

Then

$${\int }_{{\mathbb{R}}^n}\mathit{\text{φLdx}}\ge \frac{{\mathit{\alpha \epsilon }}^k}{k!}{\left(\frac{4}{4+n}\right)}^n{\int }_Q{v}^k\mathit{dx}.$$E59

Let us combine (47) and (59)

$${\int }_Q{v}^k\mathit{dx}\le \frac{\mathit{\beta k}!}{{\mathit{\alpha \epsilon }}^k}{\left(\frac{4+n}{4}\right)}^n{e}^{\mathit{\sigma t}},\text{for}\mathrm{all}\left(k,t\right)\in {\mathbb{N}}^{\ast }\times {\mathbb{R}}_{+}.$$E60

By induction we prove that

$$k!\le p^p,\text{for}\mathrm{all}k\in {\mathbb{N}}^{\ast }\text{and}p\ge k.$$E61

Let $p\ge 1$ and $k=\left[p\right]+1,$ then we have by the imbedding theorem for $L^p-$spaces

$${\int }_Q{v}^p\mathit{dx}\le {\left({\int }_Q{v}^k\mathit{dx}\right)}^{p/k}.$$E62

Taking $\epsilon$ enough small such that $\frac{\mathit{\beta k}!}{{\mathit{\alpha \epsilon }}^k}{\left(\frac{4+n}{4}\right)}^n\ge 1$. Combining this with (37)

$$0<\epsilon \le min\left\{\begin{array}{c}\frac{1-{\gamma }_2}{{\lambda }_{\infty }}\frac{4\mathit{ad}{\left(1-{\gamma }_1\right)}^2}{{\left(a+d\right)}^2{\left[1+2{‖u_0‖}_{\infty }\right]}^2+8\mathit{ad}{\left(1-{\gamma }_1\right)}^2{‖u_0‖}_{\infty }},\\ {\left[\frac{1}{\alpha }\mathit{\beta k}!{\left(\frac{4+n}{4}\right)}^n\right]}^{1/k}\end{array}\right\}.$$E63

From (60), (61) and (63) into (62) we get (56). $♦$

Lemma 3.8. Let $Q_i$ et $Q_j$ be two different unit cubes of center $x^{\left(i\right)}=\left(x_1^{\left(i\right)},\dots ,x_n^{\left(i\right)}\right)$ and $x^{\left(j\right)}=\left(x_1^{\left(j\right)},\dots ,x_n^{\left(j\right)}\right)$ respectively of the form

$$\begin{array}{l}{Q}_i=\left\{x=\left(x_1,\dots ,x_n\right)\in {\mathbb{R}}^n:\left|x_k-x_k^{\left(i\right)}\right|\le 1/2\right\},\text{for}\mathrm{all}k=1,\dots ,n,\\ Q_j=\left\{x=\left(x_1,\dots ,x_n\right)\in {\mathbb{R}}^n:\left|x_k-x_k^{\left(j\right)}\right|\le 1/2\right\},\text{for}\mathrm{all}k=1,\dots ,n,\end{array}$$E64

with $x^{\left(j\right)}=x^{\left(i\right)}+l,$ where $l=\left(l_1,\dots ,l_n\right)\in {\mathbb{Z}}^n\backslash 0_{{\mathbb{Z}}^n}$. Then, there exists a positive constant

$$\delta \left(n\right)={\left(2+\sqrt{n}\right)}^2,$$E65

such that

$$\mathit{\text{dist}}{\left(x^{\left(i\right)},Q_j\right)}^2\le {\left|x^{\left(i\right)}-y\right|}^2\le \delta \left(n\right)\mathit{\text{dist}}{\left(x^{\left(i\right)},Q_j\right)}^2,\text{for}\mathrm{all}y\in Q_j.$$E66

Proof. By Pythagorean theorem we have

$$\left|x^{\left(j\right)}-y\right|\le \frac{\sqrt{n}}{2}.$$E67

As $\left|x^{\left(i\right)}-x^{\left(j\right)}\right|\ge 1$, then from (67)

$$\left|x^{\left(j\right)}-y\right|\le \frac{\sqrt{n}}{2}\left|x^{\left(i\right)}-x^{\left(j\right)}\right|.$$E68

Also, it’s clear that $\mathit{\text{dist}}\left(x^{\left(i\right)},Q_j\right)=\mathit{\text{dist}}\left(x^{\left(i\right)},\partial Q_j\right)$, but every point $z=\left(z_1,\dots ,z_n\right)\in \partial Q_j$ is of the form

$$z=x^{\left(j\right)}+s,$$E69

where $s=\left(s_1,\dots ,s_n\right)\ne 0$ and $s_k\in \left[-\frac{1}{2},\frac{1}{2}\right]$$,$ for all $k=1,\dots ,n$ with at least one of the $s_k\in \left\{-\frac{1}{2},\frac{1}{2}\right\}.$

It’s easy to prove that

$$\left|x_k^{\left(j\right)}-x_k^{\left(i\right)}\right|\le 2\left|x_k^{\left(j\right)}-x_k^{\left(i\right)}+s_k\right|,\text{for}\mathrm{all}k=1,\dots ,n.$$E70

Then

$$\left|x^{\left(j\right)}-x^{\left(i\right)}\right|\le 2\mathit{\text{dist}}\left(x^{\left(i\right)},Q_j\right).$$E71

As $\left|x^{\left(i\right)}-y\right|\le \left|x^{\left(i\right)}-x^{\left(j\right)}\right|+\left|x^{\left(j\right)}-y\right|$ we get from (68) and (71) the estimate

$$\begin{array}{l}\left|x^{\left(i\right)}-y\right|\le 2\mathit{\text{dist}}\left(x^{\left(i\right)},Q_j\right)+\frac{\sqrt{n}}{2}\left|x^{\left(i\right)}-x^{\left(j\right)}\right|\\ \le \left(2+\sqrt{n}\right)\mathit{\text{dist}}\left(x^{\left(i\right)},Q_j\right).\end{array}$$E72

We have obviously

$$\left|x^{\left(i\right)}-y\right|\ge \mathit{\text{dist}}\left(x^{\left(i\right)},Q_j\right).$$E73

From (71) and (73) we get (66). $♦$

Proof of theorem 3.2.

Let $x\in {\mathbb{R}}^n$ an arbitrary point and ${\left\{Q_j\right\}}_{j\in \mathbb{N}}$ be the family of pairwise disjoint measurable cubes of the form (64) covering ${\mathbb{R}}^n$ such that the center of $Q_0$ is $x^{\left(0\right)}=x$.

Firstly, using the fact that ${\mathbb{R}}^n={\cup }_{j=0}^{\infty }{Q}_j$ and applying the left-hand inequality in (66)

$$\begin{array}{l}{\int }_{{\mathbb{R}}^n}{e}^{-\frac{{\left|x-y\right|}^2}{4d\left(t-s\right)}}\mathit{\lambda g}\left(u\right)v^m\mathit{dy}=\sum _{j=0}^{\infty }{\int }_{Q_j}{e}^{-\frac{{\left|x-y\right|}^2}{8d\left(t-s\right)}}{e}^{-\frac{{\left|x-y\right|}^2}{8d\left(t-s\right)}}\mathit{\lambda g}\left(u\right)v^m\mathit{dy}\\ \le \sum _{j=0}^{\infty }\left\{e^{-\frac{\mathit{\text{dist}}{\left(x,Q_j\right)}^2}{8d\left(t-s\right)}}{{\int }_{Q_j}}^{-\frac{{\left|x-y\right|}^2}{8d\left(t-s\right)}}\mathit{\lambda g}\left(u\right)v^m\mathit{dy}\right\}.\end{array}$$E74

By Hölder ineguality with $p>max\left\{1,\frac{n}{2}\right\}$ and $q=1-\frac{1}{p}$

$${{\int }_{Q_j}}^{-\frac{{\left|x-y\right|}^2}{8d\left(t-s\right)}}\mathit{\lambda g}\left(u\right)v^m\mathit{dy}\le {\left[{{\int }_{Q_j}}^{-\frac{q{\left|x-y\right|}^2}{8d\left(t-s\right)}}\mathit{dy}\right]}^{1/q}{\left[{\int }_{Q_j}{\lambda }^p{g}^p\left(u\right)v^{\mathit{pm}}\mathit{dy}\right]}^{1/p}.$$E75

As

$${\int }_{Q_j}{e}^{-\frac{q{\left|x-y\right|}^2}{8d\left(t-s\right)}}\mathit{dy}\le {{\int }_{{\mathbb{R}}^n}}^{-\frac{q{\left|x-y\right|}^2}{8d\left(t-s\right)}}\mathit{dy}={\left(\frac{8\mathit{\pi d}}{q}\right)}^{n/2}{\left(t-s\right)}^{n/2}$$E76

and by (56) we have

$${\int }_{Q_j}{\lambda }^p{g}^p\left(u\right)v^{\mathit{pm}}\mathit{dy}\le {\lambda }_{\infty }^p{g}_{\infty }^p\beta \frac{{\left(\mathit{pm}+1\right)}^{\mathit{pm}+1}}{{\mathit{\alpha \epsilon }}^{\left[\mathit{pm}\right]+1}}{\left(\frac{4+n}{4}\right)}^n{e}^{\mathit{\sigma t}},$$E77

where

$$g_{\infty }=\underset{u\in \left[0,{‖u_0‖}_{\infty }\right]}{sup}g\left(u\right).$$E78

Then, from (76) and (77) into (75)

$${{\int }_{Q_j}}^{-\frac{{\left|x-y\right|}^2}{8d\left(t-s\right)}}\mathit{\lambda g}\left(u\right)v^m\mathit{dy}\le K{\left(\frac{8\mathit{\pi d}}{q}\right)}^{\frac{n}{2}\left(1-\frac{1}{p}\right)}{\left(t-s\right)}^{\frac{n}{2}\left(1-\frac{1}{p}\right)}{\lambda }_{\infty }{g}_{\infty }{e}^{\left(\sigma /p\right)t},$$E79

where

$$K=K\left(p,m,n,\alpha ,\epsilon \right)={\left[\beta \frac{{\left(\mathit{pm}+1\right)}^{\mathit{pm}+1}}{{\mathit{\alpha \epsilon }}^{\left[\mathit{pm}\right]+1}}{\left(\frac{4+n}{4}\right)}^n\right]}^{1/p}.$$E80

On the other hand, we deduce from the right-hand inequality in (66) that

$${\int }_{Q_j}{e}^{-\frac{{\left|x-y\right|}^2}{8\mathit{d\delta }\left(n\right)\left(t-s\right)}}\mathit{dy}\ge e^{-\frac{\mathit{\text{dist}}{\left(x,Q_j\right)}^2}{8d\left(t-s\right)}},\text{for}\mathrm{all}j\in {\mathbb{N}}^{\ast }.$$E81

Then

$$\begin{array}{c}\sum _{j=0}^{\infty }{e}^{-\frac{\mathit{\text{dist}}{\left(x,Q_j\right)}^2}{8d\left(t-s\right)}}\le 1+\sum _{j=1}^{\infty }{e}^{-\frac{\mathit{\text{dist}}{\left(x,Q_j\right)}^2}{8d\left(t-s\right)}}\le 1+\sum _{j=1}^{\infty }{\int }_{Q_j}{e}^{-\frac{{\left|x-y\right|}^2}{8\mathit{d\delta }\left(n\right)\left(t-s\right)}}\mathit{dy}\\ \le 1+{\int }_{{\mathbb{R}}^n}{e}^{-\frac{{\left|x-y\right|}^2}{8\mathit{d\delta }\left(n\right)\left(t-s\right)}}\mathit{dy}\le 1+{\left[8\mathit{\pi d\delta }\left(n\right)\right]}^{n/2}{\left(t-s\right)}^{n/2}.\end{array}$$E82

We have from (79) and (82) into (74)

$$\begin{array}{l}\frac{1}{{\left[4\mathit{\pi d}\left(t-s\right)\right]}^{n/2}}{\int }_{{\mathbb{R}}^n}{e}^{-\frac{{\left|x-y\right|}^2}{4d\left(t-s\right)}}\mathit{\lambda g}\left(u\right)v^m\mathit{dy}\\ \le 2^{n/2}{\left(1-\frac{1}{p}\right)}^{\frac{n}{2}\left(1-\frac{1}{p}\right)}{\left(t-s\right)}^{-\frac{n}{2p}}K{\lambda }_{\infty }{g}_{\infty }{e}^{\left(\sigma /p\right)t}\left\{1+{\left[8\mathit{\pi d\delta }\left(n\right)\right]}^{n/2}{\left(t-s\right)}^{n/2}\right\}\\ \le 2^{n/2}{\left(1-\frac{1}{p}\right)}^{\frac{n}{2}\left(1-\frac{1}{p}\right)}K{\lambda }_{\infty }{g}_{\infty }{e}^{\left(\sigma /p\right)t}\left\{{\left(t-s\right)}^{-\frac{n}{2p}}+{\left[8\mathit{\pi d\delta }\left(n\right)\right]}^{n/2}{\left(t-s\right)}^{\frac{n}{2}\left(1-\frac{1}{p}\right)}\right\}.\end{array}$$

Whence

$$\begin{array}{l}{\int }_0^t{S}_2\left(t-s\right)\mathit{\lambda g}\left(u\right)v^m\mathit{ds}\\ \le 2^{n/2}{\left(1-\frac{1}{p}\right)}^{\frac{n}{2}\left(1-\frac{1}{p}\right)}K{\lambda }_{\infty }{g}_{\infty }{e}^{\left(\sigma /p\right)t}\left[\frac{2p}{2p-n}{t}^{1-\frac{n}{2p}}+{\left[8\mathit{\pi d\delta }\left(n\right)\right]}^{n/2}\frac{2p}{p\left(n+2\right)+2p}{t}^{\frac{n}{2}\left(1-\frac{1}{p}\right)+1}\right]\end{array}$$

and finally we have for all $t\in {\mathbb{R}}_{+}$

$${‖v\left(t\right)‖}_{\infty }\le {‖v_0‖}_{\infty }+2^{n/2}{\left(1-\frac{1}{p}\right)}^{\frac{n}{2}\left(1-\frac{1}{p}\right)}K{\lambda }_{\infty }{g}_{\infty }{e}^{\left(\sigma /p\right)t}\left[\begin{array}{c}\frac{2p}{2p-n}{t}^{1-\frac{n}{2p}}\\ +{\left[8\mathit{\pi d\delta }\left(n\right)\right]}^{n/2}\frac{2p}{p\left(n+2\right)-n}{t}^{\frac{n}{2}\left(1-\frac{1}{p}\right)+1}\end{array}\right]$$E83

As $p>max\left\{1,\frac{n}{2}\right\},$ the function in $t$ on the right-hand side of the estimate (83) is continuous on ${\mathbb{R}}_{+}$. As ${‖u\left(t\right)‖}_{\infty }\le {‖u_0‖}_{\infty }$ on $\left[0,t_{\max }\right[$ and $v$ satisfied (83), we conclude from (11) that $t_{\max }=+\infty$. Whence, the solution is global. $♦$

Remark. We can extend the system to the case where instead of $v^m$ we put $\mathit{vh}\left(v\right)$ provided that.

i. $h:\mathit{BUC}\left(\mathbb{R}\right)\to \mathit{BUC}\left(\mathbb{R}\right)$ is a locally continuous Lipschitz function, namely: for all constant $\rho \in {\mathbb{R}}_{+},$ there exists a constant $c\left(\rho \right)\in {\mathbb{R}}_{+}^{\ast }$ such that for all $u,v\in \mathit{BUC}\left({\mathbb{R}}^n\right)$ with ${‖u‖}_{\infty }\le \rho$ and ${‖v‖}_{\infty }\le \rho$ we have

$${‖h\left(u\right)-g\left(v\right)‖}_{\infty }\le c\left(\rho \right){‖u-v‖}_{\infty }.$$

ii. There exist two constants $M\in {\mathbb{R}}_{+}^{\ast }$ and $r\in \mathbb{N}$ such that:

$$0\le h\left(v\right)\le {\mathit{Mv}}^r,\text{for}\mathrm{all}v\in {\mathbb{R}}_{+}.$$

In this more general case, by examining the proof of the theorem 3.2; we see that under the same assumptions above, the system has also a global nonnegative classical solution. $♦$

4. Illustrative example

To illustrate the previous study about global existence, we give the following reaction–diffusion system

$$\left\{\begin{array}{c}{u}_t\left(t,x\right)=a\mathrm{\Delta }u\left(t,x\right)-\frac{c_1{u}^3}{c_2+c_3{u}^2}{v}^m,\left(t,x\right)\in {\mathbb{R}}_{+}^{\ast }\times {\mathbb{R}}^n,\\ v_t\left(t,x\right)=d\mathrm{\Delta }v\left(t,x\right)+c_4{e}^{-c_{5}t{\left|x\right|}^2}\frac{u^3}{c_2+c_3{u}^2}{v}^m,\left(t,x\right)\in {\mathbb{R}}_{+}^{\ast }\times {\mathbb{R}}^n,\\ u\left(0,x\right)=u_0\left(x\right),v\left(0,x\right)=v_0\left(x\right),x\in {\mathbb{R}}^n,\end{array}\right.$$E84

where $c_k,$$k=1,\dots ,4$ are real positive constants and $c_5$ is a real nonnegative constant. If $a,b\in {\mathbb{R}}_{+},$$n,m\in {\mathbb{N}}^{\ast },$$u_0,v_0$$\in \mathit{BUC}\left({\mathbb{R}}^n\right)$ and are nonnegative; the system (84) admits a unique global nonnegative classical solution $\left(u,v\right)\in C\left({\mathbb{R}}_{+};X\right)\cap C^1\left({\mathbb{R}}_{+}^{\ast };X\right).$$♦$

5. Conclusion and perspectives

We have prouved in the case where $a<d$ that the solution is global, but it remains an interesting question that if it is uniformly bounded or not.

As perspectives, we will replace the function $g=g\left(u\right)$ satisfying the hypothesis (H4) by the function $g\left(u\right)=u^r$ with $r\ge 1$ is a real constant and replace the term $v^m$ by $e^{\mathit{\alpha v}}$ with $\alpha >0$; namely that reaction term is of exponential growth. The system was studied on bounded domain by J. I. Kanel and M. Mokhtar in [12]. $♦$