GLOBAL EXISTENCE FOR SYSTEMS OF TRIPLED REACTION DIFFUSION EQUATIONS MODELLING SOME REVERSIBLE CHEMICAL REACTIONS

Said Kouachi

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Said Kouachi. GLOBAL EXISTENCE FOR SYSTEMS OF TRIPLED REACTION DIFFUSION EQUATIONS MODELLING SOME REVERSIBLE CHEMICAL REACTIONS. 2022. hal-03878666

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GLOBAL EXISTENCE FOR SYSTEMS OF TRIPLED REACTION DIFFUSION EQUATIONS MODELLING SOME REVERSIBLE CHEMICAL REACTIONS.

SAID KOUACHI

ABSTRACT. The purpose of this paper which is a continuation of S. Kouachi [13], is to prove global existence of solutions for systems of tripled reaction diffusion equations describing some reversible chemical reactions. In this case the nonlinearities present some difficulties since they change sign and then neither of the components of the solution is a priori bounded or at least bounded in some Lp-space for p large to permit the application of the well known regularizing effect principle (see beginning of the second section) or some functional methods to deduce the global existence in time for the systems in question. In this paper our techniques are based on Lyapunov functional method.

**Introduction.** We consider the reaction-diffusion system

$$ (1) \qquad \partial_t u - d_1 \Delta u = f(u, v, w), $$

$$ (2) \qquad \partial_t v - d_2 \Delta v = g(u, v, w) \quad \text{in } \mathbb{R}^+ \times \Omega, $$

$$ (3) \qquad \partial_t w - d_3 \Delta w = h(u, v, w), $$

with the boundary conditions

$$ (4) \qquad \begin{cases} \lambda_1 u + (1 - \lambda_1) \partial_\eta u = \beta_1, \\ \lambda_2 v + (1 - \lambda_2) \partial_\eta v = \beta_2, \\ \lambda_3 w + (1 - \lambda_3) \partial_\eta w = \beta_3, \end{cases} \quad \text{on } \mathbb{R}^+ \times \partial\Omega, $$

where, for $i = 1, 2$ and $3$, different type of boundary conditions are imposed: Nonhomogeneous Robin type ($0 < \lambda_i < 1$, $\beta_i \ge 0$) or homogeneous Neumann type ($\lambda_i = \beta_i = 0$) or homogeneous Dirichlet type ($1 - \lambda_i = \beta_i = 0, i = 1$ or $2$ or $3$ and $0 < \lambda_j < 1$, $\beta_j \ge 0, j = 1, 2, 3$, with $i \ne j$) can be assumed. $\Omega$ is an open bounded domain in $\mathbb{R}^N$ with smooth boundary $\partial\Omega$, $\partial_\eta$ denotes the outward normal derivative on $\partial\Omega$, $d_1, d_2$ and $d_3$ are positive constants called the diffusion terms, $0 \le \lambda_1, \lambda_2, \lambda_3 \le 1$ and $\beta_1, \beta_2$ and $\beta_3 \ge 0$ are in $C^1(\partial\Omega; \mathbb{R})$.

The initial data

$$ (5) \qquad u(0,x) = u_0(x), \quad v(0,x) = v_0(x), \quad w(0,x) = w_0(x) \quad \text{in } \Omega $$

<footnote>Date: November 2022.</footnote>

<footnote n="1991">1991 Mathematics Subject Classification. Primary 35K45, 35K57; Secondary 35K45.</footnote>

<footnote>Key words and phrases. Reaction-diffusion systems, Global existence, Lyapunov functionals.</footnote>

<footnote>This work was supported by the College of Science and Technology, University of Abbe Laghrour Khenchela, project code: C00L03UN400120220001.</footnote>
are assumed to be nonnegative and uniformly bounded on $\Omega$. The reaction terms are given by:

$$ (6) \qquad \begin{cases} f(u, v, w) = (-1)^{i_1} [a_1 u^{p_1} v^{q_1} w^{r_1} - a_2 u^{p_2} v^{q_2} w^{r_2}], \\ g(u, v, w) = (-1)^{i_2} [b_1 u^{p_1} v^{q_1} w^{r_1} - b_2 u^{p_2} v^{q_2} w^{r_2}], \\ h(u, v, w) = (-1)^{i_3} [c_1 u^{p_1} v^{q_1} w^{r_1} - c_2 u^{p_2} v^{q_2} w^{r_2}], \end{cases} $$

where $a_i, b_i, c_i, p_i, q_i, r_i$, $i = 1, 2$ are positive constants and $i_1, i_2, i_3$ are positive integers not odd or even at the same time to don't get the same sign in the nonlinearities 6. System 1-3 may be a model for the chemical reaction

$$ (7) \qquad p_1 U + q_1 V + r_1 W \rightleftharpoons p_2 U + q_2 V + r_2 W. $$

More precisely, we have the following reaction diffusion system

$$ (8) \qquad \left\{ \begin{array}{l} \frac{\partial u}{\partial t} - d_1 \Delta u = (p_2 - p_1) [k_1 u^{p_1} v^{q_1} w^{r_1} - k_2 u^{p_2} v^{q_2} w^{r_2}], \\ \frac{\partial v}{\partial t} - d_2 \Delta v = (q_2 - q_1) [k_1 u^{p_1} v^{q_1} w^{r_1} - k_2 u^{p_2} v^{q_2} w^{r_2}], \\ \frac{\partial w}{\partial t} - d_3 \Delta w = (r_2 - r_1) [k_1 u^{p_1} v^{q_1} w^{r_1} - k_2 u^{p_2} v^{q_2} w^{r_2}], \end{array} \right. \quad \text{in } \mathbb{R}^+ \times \Omega, $$

where $u, v$ and $w$ are the concentrations respectively of the reactants $U, V$ and $W$ and $k_1 > 0$ and $k_2 > 0$ are the reaction constants and the quantities $(p_2 - p_1), (q_2 - q_1)$ and $(r_2 - r_1)$ have not the same sign as in 6.

For example if we take $r_1 = p_2 = q_2 = 0$, we have the following chemical reaction

$$ (9) \qquad p_1 U + q_1 V \rightleftharpoons r_2 W, $$

and the system modelling this reaction is the following

$$ (10) \qquad \left\{ \begin{array}{l} \frac{\partial u}{\partial t} - d_1 \Delta u = -k_1 u^{p_1} v^{q_1} + k_2 w^{r_2}, \\ \frac{\partial v}{\partial t} - d_2 \Delta v = -k_1 u^{p_1} v^{q_1} + k_2 w^{r_2}, \\ \frac{\partial w}{\partial t} - d_3 \Delta w = k_1 u^{p_1} v^{q_1} - k_2 w^{r_2}. \end{array} \right. \quad \text{in } \mathbb{R}^+ \times \Omega, $$

As it was mentioned in S. Kouachi [17] for systems of coupled reaction diffusion equations, the difficulty for the type of this systems is that the reaction terms have not a constant sign and this means that none of the equations 1-3 is good is the sense that neither $u$ nor $v$ and $w$ is a priori bounded or at least bounded in some $L^p$-space for $p$ large to permits us the application of the well known regularizing effect principle (see beginning of the next section) and deduce the global existence in time for problem 1-5.

The special case of 10 where $p_1 = q_1 = r_2 = 1$ has been studied by Rothe [32] under homogeneous Neumann boundary conditions where he showed that $T_{\max} = \infty$ if $N \le 5$. J. Morgan [29] generalized the results of Rothe for every integer $N \ge 1$ and when all the components satisfy the same boundary conditions (Neumann or Dirichlet). S. L. Hollis [6] completed the work of Morgan and established global existence if $w$ satisfies the same type of boundary conditions as either $u$ or $v$. But if the boundary conditions of different types are imposed on $u$ and $v$, global existence follows regardless of the type of boundary condition that is imposed on $w$. In S. Kouachi [26] we have proved global existence of solutions to system 10 when $r_2 \le 1$ or $p_1 + q_1 \le 1$, under homogeneous Neumann boundary conditions, by studying the two coupled systems 1-3 and 2-3. However we have obtained
**Proposition 0.1.** Suppose that $r_2 \le 1$ or $p_1 + q_1 \le 1$, then all solutions of 10 with nonnegative uniformly bounded initial data and homogeneous Neumann boundary conditions are positive and exist globally for every positive constants $k_1$ and $k_2$.

Then, in [23], we generalized the above results to nonhomogeneous boundary conditions 4.

When the nonlinearities have not a constant sign, there is not many results and to prove global existence several conditions are imposed. Generally the following conditions are imposed

$$ (11) \qquad f + g + h \le C (1 + u + v + w), \quad \text{for all } u, v, w \ge 0, $$

and

$$ (12) \qquad f, g, h \le C'(1+u+v+w)^r, \quad \text{for all } u, v, w \ge 0, $$

for some $r > 1$ sufficiently large, where $C$ and $C'$ are positive constants. Condition 11 is known under the name "mass-control condition" and 12 reflects the polynomial growth of the nonlinearities. S. L. Hollis, R. H. Martin and M. Pierre [7] established global existence of positive solutions for systems of coupled equations

$$ (13) \qquad \partial_t u - d_1 \Delta u = f(u, v) \quad \text{in } \mathbb{R}^+ \times \Omega, $$

$$ (14) \qquad \partial_t v - d_2 \Delta v = g(u, v) \quad \text{in } \mathbb{R}^+ \times \Omega, $$

with the boundary conditions

$$ (15) \qquad \lambda_1 u + (1 - \lambda_1) \frac{\partial u}{\partial \eta} = \beta_1, \quad \lambda_2 v + (1 - \lambda_2) \frac{\partial v}{\partial \eta} = \beta_2 \qquad \text{on } \mathbb{R}^+ \times \partial \Omega, $$

where

$$ 0 < \lambda_1, \lambda_2 < 1, \lambda_1 = \lambda_2 = 1, \beta_1 \ge 0 \text{ and } \beta_2 \ge 0, \text{ or } \lambda_1 = \lambda_2 = \beta_1 = \beta_2 = 0, $$

under the condition of the uniform boundedness of $u$ on $[0, T_{\max}] \times \Omega$ and the following mass-control condition

$$ (16) \qquad f + g \le C(u, v)(u + v + 1), \quad \text{for all } u \ge 0 \text{ and } v \ge 0, $$

where $C(u, v)$ is positive and uniformly bounded function defined on $\mathbb{R}^+ \times \mathbb{R}^+$. A. J. Morgan [29] generalized the results of S. L. Hollis, R. H. Martin and M. Pierre [7] to show that solutions of the m-components reaction diffusion systems exist globally ($m \ge 2$) where also, in our case ($m = 3$), he imposed to $f, f+g$ and $f+g+h$ condition 11 under the boundary conditions 15. S. L. Hollis [6] extended these results, under the same conditions, to the boundary conditions 15 but he took

$$ 0 \le \lambda_1, \lambda_2 \le 1, \beta_1 \ge 0 \text{ and } \beta_2 \ge 0. $$

In S. Kouachi [26], respectively in S. Kouachi [23] and finally in S. Abdelmalek and S. Kouachi [1], we generalized the above results respectively for two, three and finally m components systems:

$$ (17) \qquad \frac{\partial u_i}{\partial t} - d_i \Delta u_i = f_i(u_1, \dots, u_m) \quad \text{in } \mathbb{R}^+ \times \Omega; \quad i = 1, \dots, m, $$

under the unique condition

$$ (18) \qquad \sum_{i=1}^{m} D_i f_i(u_1, \dots, u_m) \le C'' \left[ 1 + \sum_{i=1}^{m} u_i \right] $$
for positive constants $D_i$ sufficiently large ($D_1 = 1$), where $C''$ is a positive constant and we showed the global existence without imposing the boundedness of one of the components of the solution.

In the case when the nonlinearities have a constant sign (for example $a_i = b_i = c_i = 0$ for $i = 1$ or $i = 2$) many results have been obtained for coupled reaction diffusion systems (which can be generalized easily to tripled systems): When $f(u, v) = -uv^\beta$ (witch implies the uniform boundedness of $u$) and $g(u, v) = uv^\beta$, N. Alikakos [2] established global existence and $L^\infty$-bounds of positive solutions for 1-2 when $1 < \beta < \frac{(n+2)}{n}$. K. Masuda [28] showed that the solutions exist globally for every $\beta > 1$. Haraux and A. Youkana [4] simplified the demonstration of K. Masuda while using techniques based on a Lyapunov functional and while taking handle nonlinearities $f(u, v) = -g(u, v) = -ue^{v^\gamma}$ with $0 < \gamma < 1$ sufficiently small. S. Kouachi and A. Youkana [27] generalized the results of A. Haraux and A. Youkana [4] to the triangular case while adding $-c\Delta u$, with $c > 0$, to the right-hand side of equation 14 under the condition

$$ \lim_{s \to +\infty} \left[ \frac{\log(1 + f(r, s))}{s} \right] < \alpha^*, \text{ for any } r \ge 0, $$

with

$$ \alpha^* = \frac{2d_1d_2}{N(d_1 - d_2)^2 \|u_0\|_{\infty}}, $$

condition reflecting the weak exponential growth of the reaction term $f$.

**Preliminaries.** It is well known under the name "the regularizing effect principle" that to prove global existence of solutions to 1-3 (see Henry [5]), it suffices to derive a uniform estimate of $\|f(u, v, w)\|_p$ (which equal to $\|g(u, v, w)\|_p$ and $\|h(u, v, w)\|_p$) on $[0, T_{\max}]$ for some $p > N/2$, where $T_{\max}$ denotes the eventual blowing-up time in $L^\infty(\Omega)$. Our aim is to apply polynomial Lyapunov functional method (see M. Kirane and S. Kouachi [13], [11] and [8], S. Kouachi and A. Youkana [27] and S. Kouachi, [17], [26] and [23] and S. Abdelmalek and S. Kouachi [1] according to the solution $(u, v, w)$ of system 1-3, 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 [33]), since the reaction 6 is quasi-positive, i.e.:

$$ (19) \qquad f(0, v, w) \ge 0, \quad g(u, 0, w) \ge 0, \quad h(u, v, 0) \ge 0, \quad \text{for all } u, v, w \ge 0. $$

Since the nonlinearities 6 are continuously differentiable on $\mathbb{R}^{+3}$, then for any initial data in $C(\bar{\Omega})$ or $L^p(\Omega)$, $p \in (1, +\infty)$, it is easy to check directly their Lipschitz continuity on bounded subsets of the domain of a fractional power of the operator $I_3(d_1, d_2, d_3)^t \Delta$, where $I_3$ the three dimensional identity matrix, $\Delta$ is the Laplacian operator and $()^t$ denotes the transposition. Under these assumptions, the following local existence result is well known (see A. Friedman [3], D. Henry [5], A. Pazy [30], J. Smoller [33] and F. Rothe [32]).

**Proposition 0.2.** The system 1-3 admits a unique, classical solution $(u, v, w)$ on $[0, T_{\max}[ \times \Omega$. If $T_{\max} < \infty$ then

$$ (20) \qquad \lim_{t \to T_{\max}} \{ \|u(t, \cdot)\|_{\infty} + \|v(t, \cdot)\|_{\infty} + \|w(t, \cdot)\|_{\infty} \} = \infty. $$
We obtained in S. Kouachi [23], in particular for tripled reaction diffusion systems the following result

**Proposition 0.3.** Suppose that the functions f, g and h are polynomial growth and satisfy conditions 18 for m = 3. Then all positive solutions of problem 1-5 with initial data in $L^\infty(\Omega)$ are global.

**The main ingredient of the proof:** We used, for a given positive integer $p$, the following polynomial functional

$$ (21) \qquad L(t) = \int_{\Omega} H_p(u(t, x), v(t, x), w(t, x)) dx, $$

khj??hgk where

$$ (22) \qquad H_p(u, v, w) = \sum_{q=0}^{p} \sum_{r=0}^{q} C_p^q C_q^r \theta_r \sigma_q u^r v^{q-r} w^{p-q}, $$

By differentiating $L$ with respect to $t$ and then by a simple use of Green's formula, we got

$$ (23) \qquad L'(t) = I + J, $$

where

$$ (24) \qquad I = I_1 + I_2. $$

$I_1$ is uniformly bounded by using the boundary conditions 4 and $I_2$ is the integral of a quadratic form with respect to $\nabla u, \nabla v$ and $\nabla w$. We proved that $I_2 \le 0$ by choosing the sequences $\{\theta_r\}$ and $\{\sigma_q\}$ satisfying the conditions

$$ A = \begin{pmatrix} \theta_2 \sigma_2 & \theta_1 \sigma_2 & \theta_1 \sigma_1 \\ \theta_1 \sigma_2 & \theta_0 \sigma_2 & \theta_0 \sigma_1 \\ \theta_1 \sigma_1 & \theta_0 \sigma_1 & \theta_0 \sigma_0 \end{pmatrix} \to |A| = \theta_0 \sigma_2 (\theta_0 \theta_2 - \theta_1^2) (\sigma_0 \sigma_2 - \sigma_1^2) $$

$$ (25) \qquad \frac{\theta_r \theta_{r+2}}{\theta_{r+1}^2} \ge \overline{d_3}^2, \quad r = 1, \dots, q, $$

and

$$ (26) \qquad \left( \frac{\sigma_q \sigma_{q+2}}{\sigma_{q+1}^2} - \overline{d_1}^2 \right) \left( \frac{\theta_r \theta_{r+2}}{\theta_{r+1}^2} - \overline{d_3}^2 \right) \ge \left( \overline{d_2} - \overline{d_1 d_3} \right)^2, \quad r = 1, \dots, q, \quad q = 1, \dots, p, $$

$$ \left( \begin{array}{ccc} a\theta_{q+2}\sigma_{p+2} & \left(\frac{a+b}{2}\right)\theta_{q+1}\sigma_{p+2} & \left(\frac{a+c}{2}\right)\theta_{q+1}\sigma_{p+1} \\ \left(\frac{a+b}{2}\right)\theta_{q+1}\sigma_{p+2} & b\theta_{q}\sigma_{(p+2)} & \left(\frac{b+c}{2}\right)\theta_{q}\sigma_{(p+1)} \\ \left(\frac{a+c}{2}\right)\theta_{q+1}\sigma_{p+1} & \left(\frac{b+c}{2}\right)\theta_{q}\sigma_{(p+1)} & c\theta_{q}\sigma_{p} \end{array} \right) $$

where

$$ \overline{d_k} = \frac{d_i + d_j}{2\sqrt{d_i d_j}}, \quad i \neq j \neq k, \quad i, j, k = 1, 2, 3. $$

Concerning the integral $J$

$$ (\theta_2 \sigma_2 f + \theta_1 \sigma_2 g) + \theta_1 \sigma_1 h \le 0, \quad (\theta_1 \sigma_2 f + \theta_0 \sigma_2 g) + \theta_0 \sigma_1 h \le 0, \quad (\sigma_1 \theta_1 f + \theta_0 \sigma_1 g) + \sigma_0 \theta_0 h \le 0 $$

$$ \Delta = \begin{vmatrix} \theta_2 \sigma_2 & \theta_1 \sigma_2 & \theta_1 \sigma_1 \\ \theta_1 \sigma_2 & \theta_0 \sigma_2 & \theta_0 \sigma_1 \\ \theta_1 \sigma_1 & \theta_0 \sigma_1 & \theta_0 \sigma_0 \end{vmatrix} = \theta_0 \sigma_2 (\sigma_1^2 - \sigma_0 \sigma_2) (\theta_1^2 - \theta_0 \theta_2) \to $$
we obtained

$$
\begin{align*}
(2\mathcal{L}) &= p \sum_{q=0}^{p-1} \sum_{r=0}^{q} \int_{\Omega} C_{p-1}^{q} C_{q}^{r} (\theta_{r+1} \sigma_{q+1} f + \theta_{r} \sigma_{q+1} g + \theta_{r} \sigma_{q} h) u^{r} v^{q-r} w^{p-1-q} dx \\
&= p \sum_{q=0}^{p-1} \sum_{r=0}^{q} \int_{\Omega} C_{p-1}^{q} C_{q}^{r} \left( \frac{\theta_{r+1}}{\theta_{r}} \frac{\sigma_{q+1}}{\sigma_{q}} f + \frac{\sigma_{q+1}}{\sigma_{q}} g + h \right) \theta_{r} \sigma_{q} u^{r} v^{q-r} w^{p-1-q} dx \\
&= p \sum_{q=0}^{p-1} \sum_{r=0}^{q} \int_{\Omega} C_{p-1}^{q} C_{q}^{r} \left[ \frac{\sigma_{q+1}}{\sigma_{q}} \left( \frac{\theta_{r+1}}{\theta_{r}} f + g \right) + h \right] \theta_{r} \sigma_{q} u^{r} v^{q-r} w^{p-1-q} dx \\
&\qquad \left( \frac{\theta_{1}}{\theta_{0}} f + g \right) + h \le 0, \quad \frac{\sigma_{2}}{\sigma_{1}} \left( \frac{\theta_{1}}{\theta_{0}} f + g \right) + h \le 0, \quad \frac{\sigma_{2}}{\sigma_{1}} \left( \frac{\theta_{2}}{\theta_{1}} f + g \right) + h \le 0. \\
&\qquad \left( \frac{\sigma_{1} \theta_{1}}{\sigma_{0} \theta_{0}} f + \frac{\sigma_{1}}{\sigma_{0}} g \right) + h \le 0, \quad \left( \frac{\sigma_{2} \theta_{1}}{\sigma_{1} \theta_{0}} f + \frac{\sigma_{2}}{\sigma_{1}} g \right) + h \le 0, \quad \left( \frac{\sigma_{2} \theta_{2}}{\sigma_{1} \theta_{1}} f + \frac{\sigma_{2}}{\sigma_{1}} g \right) + h \le 0 \\
&\qquad (\sigma_{1} \theta_{1} f + \theta_{0} \sigma_{1} g) + \sigma_{0} \theta_{0} h \le 0, \quad (\theta_{1} \sigma_{2} f + \theta_{0} \sigma_{2} g) + \theta_{0} \sigma_{1} h \le 0, \quad (\theta_{2} \sigma_{2} f + \theta_{1} \sigma_{2} g) + \theta_{1} \sigma_{1} h \le 0 \\
(28) \qquad \Delta &= \begin{vmatrix} \theta_{1} \sigma_{1} & \theta_{0} \sigma_{1} & \theta_{0} \sigma_{0} \\ \theta_{1} \sigma_{2} & \theta_{0} \sigma_{2} & \theta_{0} \sigma_{1} \\ \theta_{2} \sigma_{2} & \theta_{1} \sigma_{2} & \theta_{1} \sigma_{1} \end{vmatrix} = -\theta_{0} \sigma_{2} (\sigma_{1}^{2} - \sigma_{0} \sigma_{2}) (\theta_{1}^{2} - \theta_{0} \theta_{2}) \rightarrow \Delta = \begin{vmatrix} \theta_{1} \sigma_{1} & \theta_{0} \sigma_{1} & \theta_{0} \sigma_{0} \\ \theta_{1} \sigma_{2} & \theta_{0} \sigma_{2} & \theta_{0} \sigma_{1} \\ \theta_{2} \sigma_{2} & \theta_{1} \sigma_{2} & \theta_{1} \sigma_{1} \end{vmatrix}
\end{align*}
$$

which gave, under condition 7 for $m=3$

$$ (29) \qquad J \le C_1 L(t), \quad t \in [0, T_{\max}[. $$

Then, the functional $L$ satisfied the differential inequality

$$ (30) \qquad L'(t) \le C_2 L(t), $$

where $C_1$ and $C_2$ are positive constants. The resolution of this linear differential inequality gave the uniform boundedness of the functional $L$ on the interval $[0, T_{\max}]$, what finished, by using the preliminary observations of this section, the proof.

In S. Kouachi [17], by direct application of a proposition analogous to proposition 0.3 to the coupled reaction diffusion system 13-14 with the following reactions

$$ f(u, v) = -a_1 u^{p_1} v^{q_1} + a_2 u^{p_2} v^{q_2} \quad \text{in } \mathbb{R}^+ \times \Omega, $$

$$ g(u, v) = b_1 u^{p_1} v^{q_1} - b_2 u^{p_2} v^{q_2} \quad \text{in } \mathbb{R}^+ \times \Omega, $$

we obtained, in the case $a_1 = b_1$ and $a_2 = b_2$, Corollary ??

**Corollary 0.4.** Suppose that

$$ (31) \qquad p_i + q_i \le 1, $$

for $i=1$ or $i=2$, or

$$ (32) \qquad p_j + q_j > p_i + q_i > 1 \quad \text{and} \quad p_i - p_j < p_i q_j - q_i p_j < q_j - q_i, $$

for $i=1, j=2$ or $i=2, j=1$, then solutions of 13-14 with the boundary conditions 15 and positive uniformly bounded initial data, exist for all $t > 0$.

M. Pierre [17] generalized our results to the case where

$$ (33) \qquad a_2 b_1 \le a_1 b_2, $$
and proved global existence of solutions if

$$ (34) \quad \begin{cases} q_2 > q_1 \text{ and } q_2p_1 - q_1p_2 \le q_2 - q_1 \text{ or } q_2 = q_1 \text{ and } p_1 < p_2, \\ \text{or } p_1 > p_2 \text{ and } q_2p_1 - q_1p_2 \le p_1 - p_2 \text{ or } p_1 = p_2 \text{ and } q_2 < q_1, \end{cases} $$

and global weak solutions (solutions that are not in $L^\infty(\Omega)$ but continue to live in $L^1(\Omega)$) for all $p_i, q_i \ge 1, i = 1, 2$.

In S. Kouachi [17] we presented some generalizations of the above results obtained by M. Pierre [31] without condition 33 or 34 to get

**Proposition 0.5.** *Under conditions ??, the solutions of problem 13-14 exist globally in time for all $a_i, b_i > 0, i = 1, 2$.*

**Proposition 0.6.** *Under conditions 33 and for $a_1$ or $b_2$ sufficiently large or $a_2$ or $b_1$ sufficiently small, the solutions of problem 13-14 with boundary conditions 15 and positive uniformly bounded conditions exist globally in time for all $p_i, q_i \ge 1, i = 1, 2$.*

For the case when $p_1 < p_2$ and $q_2 < q_1$ which remained an open problem, we obtained

**Proposition 0.7.** *Suppose that $p_1 < p_2$, $q_2 > q_1$ and $a_2b_1 = a_1b_2$, then all solutions of system 13-14 with boundary conditions 15 and positive uniformly bounded conditions exist globally in time.*

**The main results and their proofs.** We suppose that, for $i = 1$ or $i = 2$, we have

$$ (35) \qquad p_i + q_i + r_i \le 1, $$

or for $i = 1, j = 2$ or $i = 2, j = 1$, we have

$$ (36) \qquad p_j + q_j + r_j > p_i + q_i + r_i > 1, $$

and

$$ (37) \quad \begin{cases} p_i - p_j < (p_i q_j - p_j q_i) + (p_i r_j - p_j r_i), \\ q_i - q_j < (q_i p_j - q_j p_i) + (q_i r_j - q_j r_i), \\ r_i - r_j < (r_i p_j - r_j p_i) + (r_i q_j - r_j q_i). \end{cases} $$

**Proposition 0.8.** *Under conditions 35 or 36-37 the solutions of problem 1-5 exist globally in time for all $a_i, b_i > 0, i = 1, 2, 3$.*

**Proposition 0.9.** *For the reaction constants $(-1)^{i_k} a_i$ or $(-1)^{j_k} b_j$ which are negative and sufficiently large in absolute value or which are positive and sufficiently small in absolute value, the solutions of problem 1-5 exist globally in time for all $p_i, q_i, r_i \ge 0, i = 1, 2$.*

We begin by proving Proposition 0.8

*Proof.* Without loss of generality, take in $6$ $i_1 = 1, i_2 = i_3 = 2$ and consider homogenous Neumann boundary conditions, then by differentiating $L$ given by 22 with respect to $t$, we get 23, where the integral $I$ given by 24 is negative under conditions 25 and 26 on the sequences $\{\theta_r\}$ and $\{\sigma_q\}$. For the second integral $J$ given by 27, we have

$$ (38) \quad \begin{cases} \theta_{r+1}\sigma_{q+1}f + \theta_r\sigma_{q+1}g + \theta_r\sigma_qh = \\ \quad \quad [(-\theta_{r+1}a_1 + \theta_r b_1)\sigma_{q+1} + \theta_r\sigma_q c_1] u^{p_1}v^{q_1}w^{r_1} + \\ \quad \quad [(\theta_{r+1}a_2 - \theta_r b_2)\sigma_{q+1} - \theta_r\sigma_q c_2] u^{p_2}v^{q_2}w^{r_2}. \end{cases} $$
The case $p_1 + q_1 + r_1 \le 1$ is trivial while applying young inequality to the term $u^{p_1}v^{q_1}w^{r_1}$ and choosing the first two terms of the sequence $\{\theta_r\}$ sufficiently small such that

$$ (39) \qquad \theta_{r+1}a_2 - \theta_r b_2 \le 0, r = 0, 1, \dots, q, $$

for $q$ sufficiently large, then we get 29 which gives a differential inequality analogous to 30 and then the uniform boundedness of the functional $L$ on the interval $[0, T_{\max}]$. We treat by the same way the case $p_2 + q_2 + r_2 \le 1$ while exchanging the roles of $p_1, q_1, r_1$ with $p_2, q_2, r_2$ and choosing the first two terms of the sequences $\{\theta_r\}$ and $\{\sigma_q\}$ sufficiently large such that

$$ (40) \qquad (-\theta_{r+1}a_1 + \theta_r b_1) \sigma_{q+1} + \theta_r \sigma_q c_1 \le 0, r = 0, 1, \dots, q, \quad q = 0, 1, \dots, p-1, $$

for $p$ sufficiently large.

Suppose that 36 and 37 are satisfied and take $i = 1, j = 2$. We'll prove that the functional $L$ given by 22 satisfies the differential inequality 30 and deduce its uniform boundedness on the interval $[0, T_{\max}]$:

Put

$$ \nu_1 = \frac{p_2 + q_2 + r_2 - 1}{p_1 + q_1 + r_1 - 1} \quad \text{and} \quad \nu_2 = \frac{\nu_1}{\nu_1 - 1} = \frac{p_2 + q_2 + r_2 - 1}{(p_2 + q_2 + r_2) - (p_1 + q_1 + r_1)}, $$

then (3.2) implies

$$ \nu_1 > 1, \nu_2 > 1 \text{ and } \frac{1}{\nu_1} + \frac{1}{\nu_2} = 1. $$

We can write

$$ p_1 = p'_1 + p''_1, q_1 = q'_1 + q''_1 \text{ and } r_1 = r'_1 + r''_1 $$

where

$$ p'_1 = \frac{p_2}{\nu_1} = \frac{p_1 + q_1 + r_1 - 1}{p_2 + q_2 + r_2 - 1} p_2, p''_1 = \frac{p_1 (p_2 + q_2 + r_2 - 1) - p_2 (p_1 + q_1 + r_1 - 1)}{p_2 + q_2 + r_2 - 1}, $$

$$ q'_1 = \frac{q_2}{\nu_1} = \frac{p_1 + q_1 + r_1 - 1}{p_2 + q_2 + r_2 - 1} q_2, q''_1 = \frac{q_1 (p_2 + q_2 + r_2 - 1) - q_2 (p_1 + q_1 + r_1 - 1)}{p_2 + q_2 + r_2 - 1}, $$

$$ r'_1 = \frac{r_2}{\nu_1} = \frac{p_1 + q_1 + r_1 - 1}{p_2 + q_2 + r_2 - 1} r_2, r''_1 = \frac{r_1 (p_2 + q_2 + r_2 - 1) - r_2 (p_1 + q_1 + r_1 - 1)}{p_2 + q_2 + r_2 - 1}. $$

Then 37 implies $p''_1, q''_1, r''_1 > 0$. By choosing the first two terms of the sequences $\{\theta_r\}$ and $\{\sigma_q\}$ sufficiently large such that 39 is satisfied; for the terms for which 40 is satisfied, we have

$$ (41) \qquad [(-\theta_{r+1}a_1 + \theta_r b_1) \sigma_{q+1} + \theta_r \sigma_q c_1] u^{p_1} v^{q_1} w^{r_1} + [(-\theta_{r+1}a_2 - \theta_r b_2) \sigma_{q+1} - \theta_r \sigma_q c_2] u^{p_2} v^{q_2} w^{r_2} \le 0. $$

For the remained terms, we apply Young inequality to get

$$ u^{p_1} v^{q_1} w^{r_1} \le \frac{-(\theta_{r+1}a_2 - \theta_r b_2) \sigma_{q+1} + \theta_r \sigma_q c_2}{[(-\theta_{r+1}a_1 + \theta_r b_1) \sigma_{q+1} + \theta_r \sigma_q c_1]} (u^{p'_1} v^{q'_1} w^{r'_1})^{\nu_1} + C_i (u^{p''_1} v^{q''_1} w^{r''_1})^{\nu_2}, $$

$i = 0, 1, \dots, p-1$, where $C_i$ ($i = 0, 1, \dots, p-1$) are positive constants and

$$ (u^{p'_1} v^{q'_1} w^{r'_1})^{\nu_1} = u^{p_2} v^{q_2} w^{r_2} \text{ and } \nu_2(p''_1 + q''_1 + r''_1) = 1. $$

Finally, while applying Young inequality another time to the second term of the left hand side of inequality 41, one from there deducts 29, then 30 and finally the uniform boundedness of the functional $L$ on the interval $[0, T_{\max}]$.
For the case $i = 2, j = 1$, we follow the same reasoning while exchanging the roles of $p_1, q_1, r_1$ with $p_2, q_2, r_2$. □

*Proof of proposition 3.2.* Without lost of generality, take $i_1 = 1, i_2 = i_3 = 2$ and homogenous Neumann boundary conditions. We'll prove that the functional $L$ given by 22 is decreasing: By differentiating it with respect to $t$, we get 23, where $I$ given by 24 is negative under condition 27 on the sequences $\{\theta_r\}$ and $\{\sigma_q\}$. The integral $J$ is negative if 39 and 40 are satisfied, that is

$$ (42) \quad \left( -\frac{\theta_{r+1}}{\theta_r} a_1 + b_1 \right) \frac{\sigma_{q+1}}{\sigma_q} + c_1 \le 0 \quad \text{and} \quad \left( \frac{\theta_{r+1}}{\theta_r} a_2 - b_2 \right) \frac{\sigma_{q+1}}{\sigma_q} - c_2 \le 0, $$

for all $r = 0, 1, \dots, q$, $q = 0, 1, \dots, p-1$. Then, if $a_1$ and $b_2$ or $c_2$ are sufficiently large or $b_1$ and $c_1$ and $a_2$ are sufficiently small, we can choose the first $q$ elements of the sequence $\{\theta_r\}$ and the first $p$ elements of the sequence $\{\sigma_q\}$ such that $J$ is negative. This implies the uniform boundedness of $\|f(u, v, w)\|_p$ ($\|g(u, v, w)\|_p$ and $\|h(u, v, w)\|_p$) on $[0, T_{\max}]$ and global existence becomes from the regularizing effect principle. That completes the proof. □

**Remark 0.1.** For reaction diffusion coupled equations, corollary 2.3 is a particular case of proposition 0.8 by choosing $r_1 = r_2 = 0$ (or $p_1 = p_2 = 0$ or $q_1 = q_2 = 0$) and failing equation 3 (or 1 or 2 respectively) of our system.

The proposition 0.8 is not applicable to the cases

$$ (43) \quad \left\{ \begin{array}{l} p_2 \ge p_1, q_2 > q_1 \text{ and } r_1 < r_2, \\ \text{or } p_2 \ge p_1, q_2 < q_1 \text{ and } r_1 > r_2, \\ \text{or } p_2 < p_1, q_2 > q_1 \text{ and } r_1 < r_2, \\ \text{or } p_2 < p_1, q_2 < q_1 \text{ and } r_1 > r_2 \end{array} \right. $$

(or the analogous cases obtained by exchanging the roles the $p'_i$'s with those of $q'_i$'s or $r'_i$'s) and hypothesis of proposition 0.9 are not satisfied if we suppose

$$ (44) \quad \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}. $$

Namely, we have the following result

**Proposition 0.10.** Under condition 44 all solutions of system 1-3 with boundary conditions 4 and positive uniformly bounded conditions 5 exist globally in time, in the following cases:

1. $p_1 < p_2, q_2 > q_1$ and $r_1 = r_2 = r$ (or $p_1 < p_2, r_2 > r_1$ and $q_2 = q_1$ or $q_2 < q_1, r_2 > r_1$ and $p_1 = p_2$),

2. $p_2 + q_2 + r_2 < r_1$.

*Proof.* 1. Without lost of generality, take $i_1 = 2, i_2 = i_3 = 1$ and homogenous Neumann boundary conditions. We can write the reactions as follows

$$ \begin{aligned} f(u, v, w) &= (a_1 u^{p_1} v^{q_1} - a_2 u^{p_2} v^{q_2}) w^r, \\ g(u, v, w) &= k (-a_1 u^{p_1} v^{q_1} + a_2 u^{p_2} v^{q_2}) w^r, \\ h(u, v, w) &= l (-a_1 u^{p_1} v^{q_1} + a_2 u^{p_2} v^{q_2}) w^r, \end{aligned} $$
where $l$ and $k$ are positive constants. We have, for $p, p' > 1$ and $\gamma > 0$

$$ 
\begin{aligned}
\frac{d}{dt} \int_{\Omega} (u^p + \gamma v^{p'}) dx &= \int_{\Omega} (d_1 p u^{p-1} \Delta u + d_2 \gamma p' v^{p'-1} \Delta v) dx \\
&\quad + a_1 \int_{\Omega} (p u^{p-1} - \gamma p' k v^{p'-1}) (u^{p_1} v^{q_1} - h u^{p_2} v^{q_2}) w^r dx \\
&= I + J,
\end{aligned}
$$

where $h = \frac{a_2}{a_1}$. By application of Green formula to the first integral and taking $u^{p_1} v^{q_1}$ as factor in the second, we get

$$ I = - \int_{\Omega} \left( d_1 p (p-1) u^{p-2} |\nabla u|^2 + d_2 \gamma p' (p' - 1) v^{p'-2} |\nabla v|^2 \right) dx $$

and

$$ 
\begin{aligned}
J &= - \int_{\Omega} u^{p_1} v^{q_1} \left( p u^{p-1} - \gamma p' k v^{p'-1} \right) \left( h u^{p_2-p_1} - v^{q_2-q_1} \right) w^r dx \\
&= - p h \int_{\Omega} u^{p_1} v^{q_1} \left( u^{p-1} - \frac{\gamma p' k}{p} v^{p'-1} \right) \left( u^{p_2-p_1} - h^{-1} v^{q_2-q_1} \right) w^r dx.
\end{aligned}
$$

Then $I \le 0$. By choosing $p$ and $p'$ satisfying

$$ \frac{p-1}{p_2-p_1} = \frac{p'-1}{q_2-q_1} = \mu, $$

and putting

$$ U = u^{p_2-p_1} \quad \text{and} \quad V = v^{q_2-q_1}, $$

we get

$$ J = -p h \int_{\Omega} U^{\frac{p_1}{p_2-p_1}} V^{\frac{q_1}{q_2-q_1}} \left( U^{\mu} - \frac{\gamma p' k}{p} V^{\mu} \right) (U - h^{-1} V) w^r dx. $$

By choosing $\gamma$ satisfying

$$ \frac{\gamma p' k}{p} = h^{-\mu}, $$

we can deduce that

$$ U^{\mu} - \frac{\gamma p' k}{p} V^{\mu} = (U^{\mu} - (h^{-1}V)^{\mu}) = (U - h^{-1}V) \sum_{i=0}^{\mu-1} U^{\mu-1-i} (h^{-1}V)^i $$

and this gives $J \le 0$. As $p$ and $p'$ are arbitrary, we deduce the uniform boundedness of $u$ and $v$ on $[0, T_{\max}]$. Since

$$ (45) \qquad k (\partial_t u - d_1 \Delta u) + (\partial_t w - d_3 \Delta w) = 0, $$

then by application of standard results based on the regularizing effects of the heat equation (see M. Pierre [31], Lemma 3.4) which gives

$$ (46) \qquad \|w\|_{L^p(Q_t)} \le C_2 (1 + \|u\|_{L^p(Q_t)}), \quad \text{for all } p \ge 1, $$
then $w$ is uniformly bounded on $[0, T_{\max}]$ and the solution is global.

2. Take $i_1 = i_2 = 2$, $i_3 = 1$ and homogenous Neumann boundary conditions. We can write the reactions as follows

$$ (47) \qquad \begin{aligned} \partial_t u - d_1 \Delta u &= u^{p_1} v^{q_1} w^{r_1} - u^{p_2} v^{q_2} w^{r_2}, \\ \partial_t v - d_2 \Delta v &= k (u^{p_1} v^{q_1} w^{r_1} - u^{p_2} v^{q_2} w^{r_2}), \\ \partial_t w - d_3 \Delta w &= l (u^{p_2} v^{q_2} w^{r_2} - u^{p_1} v^{q_1} w^{r_1}). \end{aligned} $$

Multiplying equation 3 by $w^{r-1}$ and integrating over $\Omega$, we obtain

$$ (48) \qquad \partial_t \int_{\Omega} w^r dx + C \int_{\Omega} w^{p-2} |\nabla w|^2 dx + C_3 \int_{\Omega} u^{p_1} v^{q_1} w^{r+r_1-1} dx \le C_3 \int_{\Omega} u^{p_2} v^{q_2} w^{r+r_2-1} dx. $$

By integrating 48 from 0 to $t$, we get

$$ (49) \qquad \int_{\Omega} w^r dx + C \int_{0}^{t} \int_{\Omega} w^{p-2} |\nabla w|^2 dx dt + C_3 \int_{0}^{t} \int_{\Omega} u^{p_1} v^{q_1} w^{r+r_1-1} dx dt \le C_4 + C_3 \int_{0}^{t} \int_{\Omega} u^{p_2} v^{q_2} w^{r+r_2-1} dx dt. $$

We have, by Holder inequality

$$ (50) \qquad \int_{(Q_t)} u^{p_2} v^{q_2} w^{r+r_2-1} \le C_5 \|u\|_{L^{\alpha p_2}(Q_t)}^{p_2} \|v\|_{L^{\beta q_2}(Q_t)}^{q_2} \|w\|_{L^{(r+r_1-1)}(Q_t)}^{(r+r_2-1)}, $$

where

$$ \frac{1}{\alpha} + \frac{1}{\beta} + \frac{r+r_2-1}{r+r_1-1} = 1. $$

For $r$ sufficiently large, we can choose $\alpha > 1$ and $\beta > 1$ such that $\alpha p_2$ and $\beta q_2 < r + r_1 - 1$ which gives

$$ (51) \qquad L^{(r+r_1-1)}(Q_t) \subset L^{\alpha p_2}(Q_t), L^{\beta q_2}(Q_t). $$

Since

$$ (\partial_t u - d_1 \Delta u) + (\partial_t v - d_3 \Delta v) = (\partial_t v - d_1 \Delta v) + (\partial_t w - d_3 \Delta w) = 0, $$

then by application of standard results based on the regularizing effects of the heat equation (see M. Pierre [31], Lemma 3.4) which gives

$$ \|u\|_{L^p(Q_t)}, \|v\|_{L^p(Q_t)} \le C_2 (1 + \|w\|_{L^p(Q_t)}), \text{ for all } p \ge 1. $$

Consequently inequality 50 becomes

$$ (52) \qquad \int_{(Q_t)} u^{p_2} v^{q_2} w^{r+r_2-1} \le C_6 (1 + \|w\|_{L^{(r+r_1-1)}(Q_t)}^{(p_2+q_2+r+r_2-1)}). $$

So inequality 49 gives

$$ (53) \qquad C_3 \int_{0}^{t} \int_{\Omega} u^{p_1} v^{q_1} w^{r+r_1-1} dx dt \le C_7 (1 + \|w\|_{L^{(r+r_1-1)}(Q_t)}^{(p_2+q_2+r+r_2-1)}). $$

Also we can show by comparison arguments for parabolic equations (D. Henry [5] and F. Rothe [32]) that if the initial data are nonnegative; then the solutions are nonnegative on $[0, T_{\max}] \times \Omega$. More then, from similar arguments using maximum principle, we have

$$ (54) \qquad u(t, x) > \underline{u}^0 > 0 \text{ and } v(t, x) > \overline{u}^0 > 0, \text{ on } ]0, T_{\max}[ \times \Omega, i = 1, \dots, m, $$
where $u^0$ and $v^0$ are positive constants depending on the initial data and $T_{\max}$. The inequality (10) becomes

$$ \left(1 + \|w\|_{L^{(r+r_1-1)}(Q_t)}\right)^{(r+r_1-1)} \le C_8(1 + \|w\|_{L^{(r+r_1-1)}(Q_t)}^{(p_2+q_2+r+r_2-1)}). $$

Since $p_2 + q_2 + r_2 < r_1$ for all $r \ge 1$, then by Young's inequality we deduce

$$ (55) \qquad \|w\|_{L^{(r+r_1-1)}(Q_t)} \le C_{10}, \text{ for all } r \ge 1. $$

From the system we have

$$ \int_{\Omega} u^p dx \le C_{11} + C_{12} \int_{(Q_t)} u^{p+p_1-1} v^{q_1} w^{r_1} dxdt, \quad \int_{\Omega} v^q dx \le C_{13} + C_{14} \int_{(Q_t)} u^{p_1} v^{q+q_1-1} w^{r_1} dx. $$

Using Holder's inequality, with 48 and 55 together we deduce

$$ \|u\|_{L^p(\Omega)}, \|v\|_{L^q(\Omega)} \le C_{15}, \text{ for all } p, q \ge 1. $$

This ends the proof of the Lemma.

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

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DEPARTMENT OF MATHEMATICS AND INFORMATICS, COLLEGE OF SCIENCE AND TECHNOLOGY, UNIVERSITY OF ABBES LAGHROUR KHENCHELA. ALGERIA.

E-mail address: kouachi@univ-khenchela.dz