# Global existence for reaction diffusion systems with reactions changing sign.

Said Kouachi

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# Global existence for reaction diffusion systems with reactions changing sign.

Said Kouachi

Department of Mathematics and Informatics,
College of Science and Technology, University
Abbes Laghrour Khenchela. Algeria.

November 14, 2023

## Abstract

This study is a continuation of [10] where we aimed to prove global existence of solutions for coupled systems of reaction-diffusion equations with nonlinearities of non-constant sign. We aim here to correct an error in the proof of Theorem 1 (see Remark 2). We note that there aren't many results in this field, and the question of the global existence of strong solutions, even weak solutions, is still unsolved, even when the mass control and the quasi-positivity proprieties on the nonlinearities are satisfied. Based on refined Lyapunov techniques, proofs are provided.

# 1 Introduction

We consider the following reaction-diffusion system

$$ egin{cases} 
\frac{\partial u}{\partial t} - a\Delta u = f(u, v) = -c(t, x)\varphi(u, v) & \text{in } \mathbb{R}^+ \times \Omega, \\ 
\frac{\partial v}{\partial t} - b\Delta v = g(u, v) = c(t, x)\varphi(u, v) & \text{in } \mathbb{R}^+ \times \Omega, 
\end{cases} 
\quad (1) $$

with the boundary conditions

$$ \frac{\partial u}{\partial \eta} = \frac{\partial v}{\partial \eta} = 0 \quad \text{on } \mathbb{R}^+ \times \partial \Omega, \quad (2) $$

and the initial data

$$ u(0, x) = u_0(x), \quad v(0, x) = v_0(x) \quad \text{in } \Omega, \quad (3) $$

where $\Omega$ is an open bounded domain of class $C^1$ in $\mathbb{R}^N$, with boundary $\partial\Omega$, $\frac{\partial}{\partial\eta}$ denotes the outward normal derivative on $\partial\Omega$ and $a$ and $b$ are positive constants.
The initial data are assumed to be bounded on $\Omega$ and nonnegative. The function $\varphi \in C^1(\mathbb{R}^+ \times \mathbb{R}^+, \mathbb{R})$ with at most polynomial growth

$$|\varphi(u, v)| \leq C(u^s + v^s), \text{ for a all } u \geq 0, v \geq 0, \quad (4)$$

for some Positive constants $C$ and $s$ and satisfy the quasi-positivity property

$$\varphi(0, v) = \varphi(u, 0) = 0, \text{ for a all } u \geq 0, v \geq 0, \quad (5)$$

which assures the positivity of the solution on $\Omega$ at all time. The function $c(t, x) \in C(\mathbb{R}_+ \times \Omega, \mathbb{R})$ is not of constant sign. We suppose it possesses a finite number of zeros independent of the time, that is there exist $x_1, ..., x_k \in \Omega$ such that

$$c(t, x_1) = \dots = c(t, x_k) = 0, \text{ for a all } t \geq 0, \quad (6)$$

which is for example the case when $c(t, x) = h(t)c_1(x_1) \cdot c_2(x_2) \dots c_k(x_k) = 0$, for all $t \geq 0$. We denote by $\Omega_-$ and $\Omega_+$ the following subsets of $\Omega$

$$\left\{ \begin{array}{l} \Omega_- = \{x \in \Omega \text{ such that } c(t, x) > 0, \text{ for a all } t \geq 0\}, \\ \\ \Omega_+ = \{x \in \Omega \text{ such that } c(t, x) < 0, \text{ for a all } t \geq 0\}, \end{array} \right. \quad (7)$$

which are not time-dependent. The existence of a positive regular solution locally in time is classical (see for example [4]). The global existence in time of a regular solution is not so obvious. If $c(t, x)$ is a constant or is of constant sign, then global existence is immediate by application of the maximum principle (see for example [13]). The issue becomes problematic, though, if $c$ changes sign. In [19], the following case was investigated:

$$c(t, x) > 0 \text{ in } (-1, 0), \quad c(0) = 0, \quad c(t, x) < 0 \text{ in } (0, 1),$$

where it is shown that the solutions are locally uniformly bounded in $L_\infty([0, \infty) \times (0, 1)$ and $L_\infty([0, \infty) \times (-1, 0)$. The question with general $c(t, x)$ is still an open question in all space dimensions (see [16]). In general, $L_\infty(\Omega)$-blow-up may occur in finite time for quadratic systems with nonlinearities satisfying the quasi-positivity

$$f(0, v) \geq 0, \text{ for all } v \geq 0 \text{ and } g(u, 0) \geq 0, \text{ for all } u \geq 0, \quad (8)$$

and the control mass guarantied by the following condition

$$f(u, v) + g(u, v) \leq 0, \text{ for all } u, v \geq 0.$$

Indeed, the authors in [18] showed $L_\infty(\Omega)$-blow-up for system (1) with appropriate boundary conditions. Recently they ameliorate their results to more
nonlinearities and homogeneous Neumann boundary conditions. For this type of reaction diffusion systems only two properties hold: the positivity of the solutions is preserved for all time and the total mass of the components is uniformly controlled in time

$$ \int_{\Omega} (u+v) dx \leq \int_{\Omega} (u_0 + v_0) dx, \quad t \geq 0. \qquad (9) $$

The uniform control on the mass (or, more mathematically, the solution's L1-norm) suggests that no blow-up should occur in finite time. It turns out that the situation is not so simple. This explains why there are so many incomplete findings in different directions are found in the literature on this topic, and why also the general question of global existence of strong solutions even weak solutions is still open, while numerous systems with these two natural features appear in applications. Here, we review the primary benefits and drawbacks for the global existence together with many references, a description of the still open problems and a few new results as well.

To show how the situation is difficult, the authors in [1] (see also [22]) and forced these reactions to satisfy a Lyapunov structure of LogL entropy type

$$ (\log u) f + (\log v) g \leq 0, \text{ for a all } u > 0, v > 0, \qquad (10) $$

to demonstrate that a class of quadratic reaction-diffusion systems has weak solutions that are global in time for whom the reaction terms change sign. But this type of structure is satisfied only for nonlinearities at most quadratic. The reaction terms have not a constant sign means that none of the equations is good in the sense that neither u nor v is a priori bounded or at least bounded in some Lp-space, allowing us to apply the well known regularizing effect and deduce the global existence of strong solutions in time for problem (1)-(3). Or to get uniform integrability of the right-hand side of (1) to obtain a good approximation of system (1) and then apply the Dominated Convergence Theorem of integrals which with compactness properties of the heat operator give together global existence of a weak solution. In the case when the nonlinearities have a constant sign many results have been obtained. S. L. Hollis, R. H. Martin and M. Pierre [6] established global existence of positive solutions for system (1) with appropriate boundary conditions, under the conditions of the uniform boundedness of u on $[0, T_{\max}] \times \Omega$ and

$$ f(r, s) + g(r, s) \leq C(r, s)(r + s + 1), \text{ for all } r \geq 0 \text{ and } s \geq 0, \qquad (11) $$

where $C(r, s)$ is positive and uniformly bounded function defined on $\mathbb{R}^+ \times \mathbb{R}^+$. One notices that, to prove global existence for solutions to system (1), authors impose, in addition to (6), to one of the components of the reaction term the
same condition.

In the case when the nonlinearities have not a constant sign or for which one of the components $u$ or $v$ is a priori uniformly bounded, there is not many results: A. J. Morgan [14] Generalized the results of S. L. Hollis, R. H. Martin and M. Pierre [6] to show that solutions of the m-components reaction diffusion systems exist globally ($m \ge 2$) where also, he imposed to $f$ and $f+g$ conditions (6). In S. Kouachi [12], we generalized the above results for two components reaction diffusion systems under the unique condition

$$f+Dg \le C(1+u+v), \text{ for a all } u \ge 0, v \ge 0, \quad (12)$$

for all positive constant $D$ sufficiently large, where $C$ is positive constant and we showed the global existence without imposing the boundedness of one of the components of the solution.

## 2 Notations and preliminary observations

It is well-known that to prove global existence of solutions to (1)-(3) (see A. Friedman [2], D. Henry [4], A. Pazy [15] and F. Rothe [20]), it suffices to derive a uniform estimate on $[0, T_{\max}]$ of the reactions in the space $L^p(\Omega)$ for some $p > N/2$ where $T_{\max}$ denotes the eventual bowing-up time in $L^\infty(\Omega)$. Our aim is to apply polynomial Lyapunov functional method (see M. Kirane and S. Kouachi [7], [9] and [8], S. Kouachi and A. Youkana [13] and S. Kouachi [11] and [12]) according to the solutions $(u, v)$ of system (1), to carry out their $L^p$-bounds and deduct their global existence. The nonnegativity of the solutions is preserved by application of classical results on invariant regions (see J. Smoller [21]), since the reaction is quasi-positive. The usual norms in the spaces $L^p(\Omega)$, $L^\infty(\Omega)$ and $C(\bar{\Omega})$ are respectively denoted by

$$\|u\|_p^p = \frac{1}{|\Omega|} \int_{\Omega} |u(x)|^p dx,$$

and

$$\|u\|_{\infty} = \max_{x \in \Omega} |u(x)|.$$

Since the nonlinear right hand side of (1) is continuously differentiable, then for any initial data in $C(\bar{\Omega})$ or $L^p(\Omega)$, $p \in (1, +\infty)$, it is easy to check directly its Lipschitz continuity on bounded subsets of the domain of a fractional power of the operator

$$\begin{pmatrix} -a\Delta & 0 \\ 0 & -b\Delta \end{pmatrix}.$$

Under these assumptions, the following local existence result is well-known (see A. Friedman [2], [4], [21] and [20]).
**Proposition 1** The system (1) admits a unique classical solution $(u; v)$ on $(0; T_{\max})$. If $T_{\max} < +\infty$, then

$$ \lim_{t \to \infty} T_{\max} (\|u(t, \cdot)\|_{\infty} + \|v(t, \cdot)\|_{\infty}) = +\infty. $$

Before the statement of the results, let us do the following

**Remark 2** We recall that this manuscript is a correction of an error which I committed when I submitted it to a journal. I defined for a fixed integer $p \ge 1$, the following polynomial functional which is of a great interest in the following

$$ t \mapsto F(t) = \int_{\Omega} G(u(t, x), v(t, x)) dx, \quad (13) $$

where

$$ G(u, v) = \sum_{i=0}^{p} C_{p}^{i} \theta_{i} u^{i} v^{p-i}, \quad (14) $$

and the coefficient $C_p^i$ is given by the formula

$$ C_p^i = \frac{p!}{i!(p-i)!}, $$

which is defined in terms of the factorial function $n!$, and

$$ \theta_i(x) = \begin{cases} \theta_i^-(x) = c^{i+1} K^{i^2}, & x \in \Omega_+, \\ \theta_i^+(x) = C^{i+1} K^{i^2}, & x \in \Omega_-, \end{cases} \quad (15) $$

where the constants $c$ and $C$ are chosen such that the finite sequence $\{\theta_i(x)\}$ is decreasing for $x \in \Omega_+$ and increasing for $x \in \Omega_-$. That is

$$ cK^{2p+1} < 1 < CK, \quad (16) $$

and where $K$ is any positive constant satisfying

$$ K^2 > \frac{(a+b)^2}{4ab}. \quad (17) $$

Clearly, from (15) we have

$$ \frac{\theta_i \theta_{i+2}}{\theta_{i+1}^2} = K^2, \quad \text{for all } x \in \Omega, \ i = 0, 1, \dots, p-2. \quad (18) $$

It was clear at first glance in order to apply below the Green's formula, I should solve the problem of discontinuity of the sequence $\{\theta_i\}$ at the origin. I multiplied it by a functions $\psi_n$ as follows

$$ \theta_i^n(x) = \psi_n(x) \theta_i(x), $$
where $\psi_n : ]-1, 1[ \mapsto [0, 1]$ are the functions defined as follows

$$ \psi_n(x) = \begin{cases} 0, & |x| < \frac{1}{2n}, \\ 1, & |x| > \frac{1}{n}. \end{cases} $$

I didn't take attention that the functions $\theta_i^n(x)$ wasn't $C^1$ but after a private communication with [23] who remarked to me that they become continues on $\Omega$ but the discontinuity persists for its first derivative and then we can't apply the well known "Green's formula". I asked myself what operation makes disappearing this discontinuity and the first idea coming to me was the course on the "Pseudo-Differential Operators" presented at Paris VII university in 1982 by my teacher the celebrated mathematician Laurent Schwartz who used forever the convolution to regularize functions. By this operation functions with only first $L^1$-derivatives can be regularized by themselves and become of second order $L^1$-derivatives, noteworthy if the regularization is used by functions more regular. This is because the convolution is a commutative operation, we have

$$ \frac{d^2 (f * g)}{dx^2} = f' * g'. $$

**Definition 3** The convolution of two functions $f$ and $g$ with compact supports in $\mathbb{R}$ is the function $f * g$ given by

$$ (f * g)(x) = \int_{\mathbb{R}} f(x - y)g(y)dx. $$

**Remark 4** $f * g$ is also called the generalized product of $f$ and $g$ and it also holds in the case that one of the function is a generalized function, like Dirac's delta.

**Remark 5** The convolution operation for $L^1$-functions is commutative, associative, distributive with respect to the addition and if $f$ is $C^n(\mathbb{R})$ for some positive integer $n$ with compact support in $\mathbb{R}$ and $g$ is $L^1(\mathbb{R})$, then $f * g$ is $C^n(\mathbb{R})$ and we have $(f * g)^{(n)} = f^{(n)} * g$.

**Proposition 6** For all positive $\epsilon$, there exists a nonnegative function $\rho \in C^\infty(\mathbb{R})$ with compact support in the interval $(-1, 1)$ satisfying

$$ \int_{\mathbb{R}} \rho(x) dx = \int_{|x| \le \epsilon} \rho_{\epsilon}(x) dx = 1. \qquad (20) $$

**Proof.** We use the following function

$$ \sigma(x) = \begin{cases} \exp\left(-\frac{1}{1-x}\right), & |x| < 1, \\ 0, & |x| \ge 1, \end{cases} $$
then we dilate to obtain

$$ \rho(x) = \frac{1}{\int_{\mathbb{R}} \sigma(x) dx} \sigma(x) $$

and a simple dilation-renormalization gives

$$ \rho_{\epsilon}(x) = \frac{1}{\epsilon} \rho\left(\frac{x}{\epsilon}\right). $$

<figure><img src="image_2.png" /></figure>

# 3 Statement and proof of the main results

The main result of the paper is the following

**Theorem 7** Suppose that conditions (4) and (5) are satisfied, then for any solution $(u(t, \cdot), v(t, \cdot))$ of the problem (1)-(3) with positive initial data, the functional defined by (21a)-(24) below is uniformly bounded on the interval $(0; T_{\max})$.

**Proof.** We treat the special case when $\Omega = ]-1, 1[$ and the function $c(t, x) = c(x)$ independent of the time and possess a unique zero, for example $c(0) = 0$. Suppose $c(t, x) > 0$ for $x \in ]-1, 0[$ and $c(t, x) < 0$ for $x \in ]0, 1[$. The general case can be deduced easily. The $\theta_i$'s given by (15)-(18) can be regularized as follows

$$ \bar{\theta}_i(x) = (\rho * \theta_i)(x), \quad x \in \mathbb{R}, \quad i = 0, 1, \dots, p, \qquad (21a) $$

where

$$ \rho = \frac{1}{2} (\rho^{-} + \rho^{+}), \qquad (22) $$

and $\rho^{-}$ and $\rho^{+}$ are two $C^{\infty}(\mathbb{R})$ nonnegative functions given by proposition 6 with $\text{Supp}(\rho^{-}) \subset (-1, 0)$ and $\text{Supp}(\rho^{+}) \subset (0, 1)$. Then the function $\rho$ is $C^{\infty}(\mathbb{R})$ with $\text{Supp}(\rho) \subset (-1, 1)$ and satisfies (20). Then, we easily check that the functions $\bar{\theta}_i(x)$ are $C^{\infty}(\mathbb{R})$ with $\text{Supp}(\bar{\theta}_i) \subset (-2, 2)$. Let

$$ t \mapsto L(t) = \int_{\Omega} H(u(t, x), v(t, x)) dx, \qquad (23) $$

where

$$ H(u, v) = \sum_{i=0}^{p} C_{p}^{i} \bar{\theta}_{i} u^{i} v^{p-i}. \qquad (24) $$
By differentiating $L$ with respect to $t$, we get

$$L'(t) = \sum_{i=1}^{p} \int_{\Omega} (i C_p^i \bar{\theta}_i u^{i-1} v^{p-i}) \frac{\partial u}{\partial t} dx \\ + \sum_{i=1}^{p} \int_{\Omega} ((p-i) C_p^i \bar{\theta}_i u^{i} v^{p-i-1}) \frac{\partial v}{\partial t} dx. \quad (25)$$

Using the fact that

$$iC_p^i = pC_{p-1}^{i-1}, \text{ for all } i = 1, \dots, p, \quad (26)$$

and interchanging the index, we get

$$L'(t) = p \sum_{i=0}^{p-1} C_{p-1}^{i} \int_{\Omega} u^{i} v^{p-1-i} \left[ (a \bar{\theta}_{i+1} \Delta u + b \bar{\theta}_{i} \Delta v) + (\bar{\theta}_{i+1} f + \bar{\theta}_{i} g) \right] dx \\ = I + J,$$

where

$$I = p \sum_{i=0}^{p-1} \int_{\Omega} C_{p-1}^{i} u^{i} v^{p-1-i} (a \bar{\theta}_{i+1} \Delta u + b \bar{\theta}_{i} \Delta v) dx, \quad (27)$$

and

$$J = p \sum_{i=0}^{p-1} C_{p-1}^{i} \int_{\Omega} [(-\bar{\theta}_{i+1} + \bar{\theta}_{i}) c(t, x) \varphi(u, v) u^{i} v^{p-1-i}] dx. \quad (28)$$

For the integrals $I$ and since the functions $\bar{\theta}_i(x)$ are $C^1(\mathbb{R})$, we can apply Green's formula to obtain

$$I = -p \sum_{i=0}^{p-1} C_{p-1}^{i} \int_{\Omega} [a \nabla (\bar{\theta}_{i+1} u^{i} v^{p-1-i}) \nabla u + b \nabla (\bar{\theta}_{i} u^{i} v^{p-1-i}) \nabla v] dx,$$

which can be written as follows

$$I = I_1 + I_2,$$

where

$$I_1 = -p \sum_{i=0}^{p-1} C_{p-1}^{i} \int_{\Omega} [a \bar{\theta}_{i+1} \nabla (u^{i} v^{p-1-i}) \cdot \nabla u + b \bar{\theta}_{i} \nabla (u^{i} v^{p-1-i}) \cdot \nabla v] dx,$$
and

$$I_2 = -p \sum_{i=0}^{p-1} C_{p-1}^i \int_{\Omega} (u^i v^{p-1-i}) [a \nabla \bar{\theta}_{i+1} \cdot \nabla u + b \nabla \bar{\theta}_i \cdot \nabla v] dx.$$

Using (26) another time and interchanging the indexes, the integral $I_1$ becomes

$$I_1 = -p(p-1) \sum_{i=0}^{p-2} C_{p-2}^i \int_{\Omega} T_i (\nabla u, \nabla v) u^i v^{p-2-i} dx, \quad (29)$$

where

$$T_i(\nabla u, \nabla v) = (a\bar{\theta}_{i+2} |\nabla u|^2 + (a+b)\bar{\theta}_{i+1} \nabla u \nabla v + b\bar{\theta}_i |\nabla v|^2) dx. \quad (30)$$

Then from the expression of $\rho$, we have

$$\bar{\theta}_i := \frac{1}{2} (\rho^- + \rho^+) * \theta_i = \frac{1}{2} \rho_i^- + \frac{1}{2} \rho_i^+,$$

where

$$\bar{\rho}_i^- = \rho^- * \theta_i \text{ and } \bar{\rho}_i^+ = \rho^+ * \theta_i,$$

and since $\text{Supp}(\rho^-) \subset (-1, 0)$, $\text{Supp}(\rho^+) \subset (0, 1)$ and $\text{Supp}(\theta_i) = (-1, 1)$ then

$$\text{Supp}\bar{\rho}_i^- = [-2, 1] \text{ and } \text{Supp}\bar{\rho}_i^+ = [-1, 2].$$

We have

$$\bar{\rho}_i^- (x) = \rho^- * \theta_i = \int_{D_1 \cap D_2^+} \rho^- (x-y) \theta_i (y) dy,$$

where

$$D_1: -1 < y < 1 \text{ and } D_2^+: -1 < x - y < 0.$$

This gives

$$\rho_i^- (x) = \begin{cases} \theta_i^- \int_{-1}^{x+1} \rho^- (z) dz, & \text{for } -2 < x < -1, \\ \theta_i^+ \int_{0}^{x+1} \rho^- (z) dz, & \text{for } -1 < x < 0. \end{cases}$$

Since we integrate on the interval $(-1, 0)$, then we have

$$\bar{\rho}_i^- (x) = \theta_i^+ \int_{0}^{x+1} \rho^- (z) dz, \text{ for } -1 < x < 0. \quad (31)$$
The same reasoning gives

$$ \rho_i^+(x) = \rho^+ * \theta_i = \int_{D_1 \cap D_2^+} \rho^+(x-y) \theta_i(y) dy, $$

where

$$ D_1: -1 < y < 1 \text{ and } D_2^+: 0 < x - y < 1. $$

The calculations give

$$ \rho_i(x) = \begin{cases} \theta_i^- \int_{x-1}^0 \rho^-(z) dz, & \text{for } 0 < x < 1, \\ \theta_i^+ \int_{x-1}^1 \rho^-(z) dz, & \text{for } 1 < x < 2. \end{cases} $$

Since we integrate on the interval $(0, 1)$, we take

$$ \rho_i^+(x) = \theta_i^- \int_{x-1}^0 \rho^-(z) dz, \quad \text{for } 0 < x < 1. \qquad (32) $$

Let's summarize, we have

$$ \bar{\theta}_i = (\rho * \theta_i) = \begin{cases} \bar{\rho}_-(x) \theta_i^+, & \text{for } -1 < x < 0, \\ \bar{\rho}_+(x) \theta_i^-, & \text{for } 0 < x < 1, \end{cases} \qquad (33) $$

where

$$ \left\{ \begin{array}{l} \bar{\rho}_-(x) = \displaystyle\int_0^{x+1} \rho^-(z) dz, \quad \text{for } -1 < x < 0, \\ \text{and} \\ \bar{\rho}_+(x) = \displaystyle\int_{x-1}^0 \rho^-(z) dz, \quad \text{for } 0 < x < 1. \end{array} \right. \qquad (34) $$

Consequently we get

$$ T_i(\nabla u, \nabla v) = T_i^-(\nabla u, \nabla v) + T_i^+(\nabla u, \nabla v), $$

where

$$ T_i^-(\nabla u, \nabla v) = (a\theta_{i+2}^+ |\nabla u|^2 + (a+b)\theta_{i+1}^+ \nabla u \nabla v + b\theta_i^+ |\nabla v|^2) \bar{\rho}_-(x), \quad \text{on } (0, T_{\max}) \times \Omega_-, $$

and

$$ T_i^+(\nabla u, \nabla v) = (a\theta_{i+2}^- |\nabla u|^2 + (a+b)\theta_{i+1}^- \nabla u \nabla v + b\theta_i^- |\nabla v|^2) \bar{\rho}_+(x) dx, \quad \text{on } (0, T_{\max}) \times \Omega_+. $$
The above quadratic forms of the two variables $\nabla u$ and $\nabla v$ under the relations (15)-(18) together with the positivity of $\bar{\rho}_-$ and $\bar{\rho}_+$ are positive and satisfy the following inequalities

$$ T_i^-(\nabla u, \nabla v) \geq \bar{\rho}_-(x) \Delta_i^+ \cdot \left( \frac{|\nabla u|^2}{b\theta_i^+} + \frac{|\nabla v|^2}{a\theta_{i+2}^+} \right), \quad (35) $$

and

$$ T_i^+(\nabla u, \nabla v) \geq \bar{\rho}_+(x) \Delta_i^- \cdot \left( \frac{|\nabla u|^2}{b\theta_i^-} + \frac{|\nabla v|^2}{a\theta_{i+2}^-} \right), \quad (36) $$

for all $i = 0, 1, \dots, p-2$ and all $t \in (0, T_{\max})$ where

$$ \Delta_i^{\mp} = \frac{4ab\theta_i^{\mp}\theta_{i+2}^{\mp} - (a+b)^2 (\theta_{i+1}^{\mp})^2}{8}, $$

is positive, for all $i = 0, 1, \dots, p-2$. In conclusion the integrals $I_1^-$ and $I_1^+$ are non positive on $(0, T_{\max})$ and then we have

$$ I_1 =: I_1^- + I_1^+ \leq 0, \text{ for all } t \in (0, T_{\max}), $$

where

$$ I_1^- = -p(p-1) \sum_{i=0}^{p-2} C_{p-2}^i T_i^-(\nabla u, \nabla v), $$

and

$$ I_1^+ = -p(p-1) \sum_{i=0}^{p-2} C_{p-2}^i T_i^+(\nabla u, \nabla v). $$

In the following our strategy is to subtract the integrals $I^2$ which are summation until $p-1$ from the integrals $I^1$ which are summation until $p$. To do this, we use the following formula

$$ C_{p-1}^i = C_{p-2}^{i-1} + C_{p-2}^i, \quad i = 1, \dots, p-2. $$

Then the integrals $I^2$ can be written as follows

$$ I^2 = I^{21} + I^{21}, $$

where

$$ \begin{aligned} I^{21} &= -p \int_{\Omega} [av\nabla\bar{\theta}_1\nabla u + bv\nabla\bar{\theta}_0\nabla v] \cdot v^{p-2}dx \\ &= -p \sum_{i=1}^{p-2} C_{p-2}^i \int_{\Omega} [av\nabla\bar{\theta}_{i+1}\nabla u + bv\nabla\bar{\theta}_i\nabla v] \cdot u^i v^{p-2-i}dx \\ &= -p \sum_{i=0}^{p-2} C_{p-2}^i [av\nabla\bar{\theta}_{i+1}\nabla u + bv\nabla\bar{\theta}_i\nabla v] \cdot u^i v^{p-2-i}dx, \end{aligned} $$
and

$$
\begin{align*}
I^{22} &= -p \sum_{i=1}^{p-2} C_{p-2}^{i-1} \int_{\Omega} [au \nabla \bar{\theta}_{i+1} \nabla u + bu \nabla \bar{\theta}_i \nabla v] \cdot \nabla \bar{\rho} u^{i-1} v^{p-1-i} dx \\
&\quad -p \int_{\Omega} [au \nabla \bar{\theta}_p \nabla u + bu \nabla \bar{\theta}_{p-1} \nabla v] \cdot \nabla \bar{\rho} u^{p-2} dx \\
&:= -p \sum_{i=0}^{p-2} \int_{\Omega} C_{p-2}^{i} [au \nabla \bar{\theta}_{i+2} \nabla u + bu \nabla \bar{\theta}_{i+1} \nabla v] u^i v^{p-2-i}.
\end{align*}
$$

Then

$$
I^2 = -p \sum_{i=0}^{p-2} \int_{\Omega} C_{p-2}^{i} [a\sigma_{i+1} \cdot \nabla u + b\sigma_{i} \cdot \nabla v] u^{i} v^{p-2-i} dx, \quad (37)
$$

were

$$
\sigma_i = (u \nabla \bar{\theta}_{i+1} + v \nabla \bar{\theta}_i), \quad i = 0, 1, \dots, p-1. \tag{38}
$$

Before the subtraction, we apply Young's inequality to the new form of the integral $I^2$ given by (37). That is for all $\epsilon > 0$ we can find a positive constant $C_\epsilon$ such that

$$
|\sigma_{i+1} \nabla u| \le \epsilon(p-1) \bar{\theta}_{i+2} |\nabla u|^2 + C_{\epsilon} \frac{|\sigma_{i+1}|^2}{\epsilon(p-1)\bar{\theta}_{i+2}},
$$

and

$$
|\sigma_i \nabla v| \le \epsilon(p-1) \bar{\theta}_i |\nabla v|^2 + C_\epsilon \frac{|\sigma_i|^2}{\epsilon(p-1)\bar{\theta}_i}.
$$

Put

$$
I^{2a} = \sum_{i=0}^{p-2} C_{p-2}^{i} \int_{\Omega} u^i v^{p-2-i} \left( a \bar{\theta}_{i+2} |\nabla u|^2 + b \bar{\theta}_i |\nabla v|^2 \right) dx,
$$

and

$$
I^{2b} = \int_{\Omega} \sum_{i=0}^{p-2} C_{p-2}^{i} \mu_{i}(x, u, v) u^{i} v^{p-2-i} dx, \quad (39)
$$

where

$$
\mu_i(x, u, v) = \frac{|\sigma_{i+1}|^2}{\epsilon(p-1)\bar{\theta}_{i+2}} + \frac{|\sigma_i|^2}{\epsilon(p-1)\bar{\theta}_i}, \quad (40)
$$

is a second degree polynomial of the variables *u* and *v*. Then we have

$$
I^2 \leq \epsilon p (p-1) I^{2a} + C_{\epsilon} p I^{2b}.
$$

In conclusion we subtract $I^{2a}$ from $I^1$ as follows

$$
I := I^1 + I^2 \le (I^1 + \epsilon p (p-1) I^{2a}) + C_{\epsilon} p I^{2b}.
$$
Since $\int_{\Omega} \rho dx = 1$ and

$$0 < \theta_i^- \le \theta_i \le \theta_i^+, \text{ on } \Omega, i = 0, 1, \dots, p, \quad (41)$$

then using the properties of the convolution, we have

$$\theta_i^- \le \bar{\theta}_i(x) \le \theta_i^+, \text{ on } \Omega.$$

As the $\bar{\theta}_i$'s are positive and uniformly bounded on $\bar{\Omega}$ and from (18), we can choose $K$ sufficiently large such that

$$\frac{\bar{\theta}_i \bar{\theta}_{i+2} (1 - \epsilon)^2}{(\bar{\theta}_{i+1})^2} > \frac{(a + b)^2}{4ab}, \quad i = 0, 1, \dots, p-2,$$

for some $\epsilon > 0$ sufficiently small. This gives

$$I_{\epsilon}^{1} := -p(p-1) \sum_{i=0}^{p-2} C_{p-2}^{i} \int_{\Omega} u^{i} v^{p-2-i} T_{i\epsilon} (\nabla u, \nabla v) dx < 0,$$

where

$$T_{i\epsilon}(\nabla u, \nabla v) = a\bar{\theta}_{i+2}(1-\epsilon)|\nabla u|^2 + (a+b)\bar{\theta}_{i+1}\nabla u\nabla v + b\bar{\theta}_i(1-\epsilon)|\nabla v|^2.$$

Then we can subtract the integral $\epsilon p(p-1)I^{2a}$ from $I_{\epsilon}^{1}$ to get

$$I^1 = I_{\epsilon}^1 + \epsilon p(p-1)I^{2a} \le 0.$$

Finally we have

$$I \le C_{\epsilon} p I^{2b}.$$

Using the fact that the functions $\{\bar{\theta}_i(x)\}$ are $C^{\infty}(\mathbb{R})$ with compact supports then the coefficients of $u$ and $v$ in the expression of $\sigma_i$ given by (38) are uniformly bounded and consequently the coefficients $\mu_i$ (given by (40)) of the polynomial under the integration sign in formula (39) intervening in the integrals $I^{2b}$ are uniformly bounded by a constant independent of the time. Consequently we have

$$I^{2b} \le C_p \int_{\Omega} \sum_{i=0}^{p-2} C_{p-2}^{i} u^{i} v^{p-2-i} (u^2 + v^2) dx.$$

For the remaining integral $J$, we have

$$J = p \sum_{i=0}^{p-1} C_{p-1}^{i} \int_{\Omega} [(-\bar{\theta}_{i+1} + \bar{\theta}_{i}) c(t, x) \varphi(u, v) u^{i} v^{p-1-i}] dx \\ = J^{-} + J^{+},$$
where

$$J^{-} = p \sum_{i=0}^{p-1} C_{p-1}^{i} \int_{\Omega_{-}} [(-\theta_{i+1}^{+} + \theta_{i}^{+}) \bar{\rho}_{-}(x) c(t, x) \varphi(u, v) u^{i} v^{p-1-i}] dx$$

and

$$J^{+} = p \sum_{i=0}^{p-1} C_{p-1}^{i} \int_{\Omega_{+}} [(-\theta_{i+1}^{-} + \theta_{i}^{-}) \bar{\rho}_{+}(x) c(t, x) \varphi(u, v) u^{i} v^{p-1-i}] dx.$$

Since the sequence {$\theta_i^+$} is increasing with $c(t, x) > 0$ on $\Omega_-$, and the sequence {$\theta_i^-$} is decreasing with $c(t, x) < 0$ on $\Omega$, then $J^-$ and $J^+$ are nonpositive on $(0, T_{\max})$. That is

$$J \le 0, \text{ for all } t \in (0, T_{\max}).$$

In conclusion, we have

$$L'(t) \le C'_{p} (u + v)^{p}, \quad 0 \le t < T_{\max}.$$

But from (41) we can find two positive constants $C_p^-$ and $C_p^+$ such that

$$C_p^- (u + v)^p \le L(t) \le C_p^+ (u + v)^p, \quad 0 \le t < T_{\max}, \qquad (42)$$

then we have for some positive constant $C_p''$ the following inequality

$$L'(t) \le C_p'' L(t), \quad 0 \le t < T_{\max},$$

which gives

$$L(t) \le L(0) e^{C_p'' t}, \quad 0 \le t < T_{\max}. \qquad (43)$$

This shows that the Functional $L(t)$ is uniformly bounded on $[0, T_{\max}]$ and ends the proof of the Theorem. ■

**Corollary 8** All solutions of problem (1)-(3) with uniformly bounded positive initial data in $\Omega$ are global in time.

**Proof.** Using (43) and (42), we get

$$\|u(t, \cdot) + v(t, \cdot)\|_p \le C e^{\frac{C_p''}{p} t} \|u_0 + v_0\|_p.$$

Since, the function $c(t, x)$ is uniformly bounded on $\Omega$ and the function $\varphi(u, v)$ is of polynomial growth, the reactions are $L_p(\Omega)$ for all $p > 1$ which gives, from the preliminary remarks the global existence of the solution. ■
# 4 Conclusion

With the exception of a few partial results obtained under very strict constraints the question of the global existence of solutions of systems of coupled reaction-diffusion equations with nonlinearities with non constant sign had been unsolved until just lately. The failure of traditional approaches, such as those based on the entropy inequality or those employing the duality argument, etc., is the main reason for the fascinating outcomes in such situations. This question was solved in this study using a relatively straightforward functional yet powerful approach.

**Acknowledgement 9** This work was supported by the College of Science and Technology, University of Abbe Laghrour Khenchela, project code: C00L03UN400120220001.

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