Global behavior for a strongly coupled model of plankton allelopathy

Abstract

In this article, we consider a strongly coupled model of plankton allelopathy. Using the energy estimates and Gagliardo-Nirenberg-type inequalities, the existence and uniform boundedness of global solutions for the model are proved. Meanwhile, the sufficient conditions for global asymptotic stability of the positive equilibrium for this model are given by constructing a Lyapunov function.

AMS Subject Classification (2000): 35K57; 35B35; 92D25.

1 Introduction

The effects of toxic substances on ecological communities is an important problem from an environmental point of view. In 1996, Chattopadhyay [1] modified two species Lotka-Volterra competitive system by considering that each species produces a substance toxic to the other but only when the other is present and the modified model takes the following form

$$\left\{\begin{array}{c}\hfill \frac{du}{dt}=u\left(a_1-b_{11}u-b_{12}v-e_{1}uv\right),\hfill \\ \hfill \frac{dv}{dt}=v\left(a_2-b_{21}u-b_{22}v-e_{2}uv\right),\hfill \end{array}\right.$$

(1.1)

where a _i , b _ij , and e _i (i, j = 1, 2) are positive constants, u(t), v(t) denote the population density of two competing species; a₁, a₂ are the intrinsic growth rates of two competing species; b₁₁, b₂₂ are the rates of intra-specific competition of the first and the second species, respectively; b₁₂, b₂₁ are the rates of inter-specific competition of the first and the second species, respectively; e₁ and e₂ are, respectively, the rates of toxic inhibition of the first species by the second and vice versa. For more details on the backgrounds about this system, see [1].

The system (1.1) has a positive equilibrium E* = (u*, v*) if and only if

$$\frac{b_{12}}{b_{22}}<\frac{a_1}{a_2}<\frac{b_{11}}{b_{21}},\frac{b_{12}}{b_{22}}<\frac{e_1}{e_2}<\frac{b_{11}}{b_{21}},$$

(1.2)

where

$$u^{*}=\left(-q_{12}-\sqrt{q_{12}^2-4p_{12}{r}_{12}}\right)/(2p_{12}),v^{*}=\left(-q_{21}-\sqrt{q_{21}^2-4p_{21}{r}_{21}}\right)/(2p_{21}),$$

and

$$p_{ij}=b_{ij}{e}_i-b_{ii}{e}_j,q_{ij}=a_i{e}_j-a_j{e}_i-b_{ii}{b}_{jj}+b_{ij}{b}_{ji},r_{ij}=a_i{b}_{jj}-a_j{b}_{ij},i,j=1,2.$$

Chattopadhyay [1] proved that the equilibrium (u*, v*) is globally asymptotically stable if

$$4\left(b_{11}+e_{1}v\right)\left(b_{22}+e_{2}u\right)\ge {\left(b_{12}+b_{21}+e_1{u}^{*}+e_2{v}^{*}\right)}^2.$$

(1.3)

The corresponding weakly coupled reaction-diffusion system for (1.1) is as follows

$$\left\{\begin{array}{cc}\hfill u_t=d_1\text{Δ}u+u\left(a_1-b_{11}u-b_{12}v-e_{1}uv\right),\hfill & \hfill x\in \text{Ω},t>0,\hfill \\ v_t=d_2\text{Δ}v+v\left(a_2-b_{21}u-b_{22}v-e_{2}uv\right),\hfill & x\in \text{Ω},t>0,\hfill \\ {\partial }_{n}u={\partial }_{n}v=0,\hfill & \hfill x\in \partial \text{Ω},t>0,\hfill \\ u\left(x,0\right)=u_0\left(x\right),v\left(x,0\right)=v_0\left(x\right),\hfill & x\in \text{Ω},\hfill \end{array}\right.$$

(1.4)

where Ω ⊂ ℝ^N is bounded smooth domain, n is the outward unit normal vector of the boundary ∂ Ω, ∂ _n = ∂/∂n. The, constants d₁ and d₂, called diffusion coefficients, are positive, and u₀(x) and v₀(x) are non-negative functions which are not identically zero.

It is obvious that (u*, v*) is the unique positive equilibrium of the system (1.4) if (1.2) holds. Tian et al. [2] proved that the equilibrium (u*, v*) of the system (1.4) is locally asymptotically stable if (1.2) holds.

In recent years, the SKT-type cross-diffusion systems have attracted the attention of a great number of investigators and have been successfully developed on the theoretical backgrounds. The above work mainly concentrate on (1) The instability and stability induced by cross-diffusion, and the existence of non-constant positive steady-state solutions [3–5]; (2) the global existence of strong solutions [6–13]; (3) the global existence of weak solutions based on semi-discretization or finite element approximation [14–17]; and (4) the dynamical behaviors [9, 10], etc.

Tian et al. [2] considered the following SKT-type cross-diffusion system

$$\left\{\begin{array}{cc}{u}_t=d_1\text{Δ}u+u\left(a_1-b_{11}u-b_{12}v-e_{1}uv\right),\hfill & x\in \text{Ω},t>0,\hfill \\ v_t=d_2\text{Δ}\left(v+d_{3}uv\right)+v\left(a_2-b_{21}u-b_{22}v-e_{2}uv\right),\hfill & x\in \text{Ω},t>0,\hfill \\ {\partial }_{n}u={\partial }_{n}v=0,\hfill & x\in \partial \text{Ω},t>0,\hfill \\ u\left(x,0\right)=u_0\left(x\right),v\left(x,0\right)=v_0\left(x\right),\hfill & x\in \text{Ω},\hfill \end{array}\right.$$

(1.5)

and they proved that:

• (1) If ${\mu }_2<\tilde{\mu }$, then there exists a positive constant $d_2^{*}$ such that the equilibrium (u*, v*) of the system (1.5) is unstable provided that $d_2\ge d_2^{*}$, the (1.2) and the following condition

  $$\left(1+d_3{u}^{*}\right)\left(b_{11}+e_1{v}^{*}\right)<d_3{v}^{*}\left(b_{12}+e_1{u}^{*}\right)$$

  (1.6)

hold, where 0 = μ₀ < μ₁ < μ₂ < · · · are the eigenvalues of the operator -Δ on Ω with the homogeneous Neumann boundary condition, $\tilde{\mu }=v^{*}\left(b_{12}+e_1{u}^{*}\right)/d_1$;

• (2) the steady-state system of the system (1.4) has non-constant positive solution, if one of the following conditions is satisfied:

• (i) d₁ ≥ D₁ for some positive D₁(d₂);

• (ii) d₂ ≥ D₂ for some positive D₂(d₁); and

• (3) if $\tilde{\mu }\in \left({\mu }_n,{\mu }_{n+1}\right)$ for some n ≥ 1, and the sum ${\sigma }_n={\sum }_{i=2}^n\mathsf{\text{dim}}E\left({\mu }_i\right)$ is odd, then there exists a positive number $d_2^{*}$ such that the system (1.5) has at least one inhomogeneous positive steady-state solution if $d_2\ge d_2^{*}$, (1.2) and (1.6) hold, where d₁ and d₃ are fixed.

We are concerned with the following plankton allelopathy model with full cross-diffusion

$$\left\{\begin{array}{cc}{u}_t=\text{Δ}\left(d_{1}u+{\alpha }_{11}{u}^2+{\alpha }_{12}uv\right)+u\left(a_1-b_{11}u-b_{12}v-e_{1}uv\right),\hfill & x\in \text{Ω},t>0,\hfill \\ v_t=\text{Δ}\left(d_{2}v+{\alpha }_{21}uv+{\alpha }_{22}{v}^2\right)+v\left(a_2-b_{21}u-b_{22}v-e_{2}uv\right),\hfill & x\in \text{Ω},t>0,\hfill \\ {\partial }_{n}u={\partial }_{n}v=0,\hfill & x\in \partial \text{Ω},t>0,\hfill \\ u\left(x,0\right)=u_0\left(x\right),v\left(x,0\right)=v_0\left(x\right),\hfill & x\in \text{Ω},\hfill \end{array}\right.$$

(1.7)

where Ω is a bounded domain ℝ^N with smooth boundary ∂Ω, n is the outward unit normal vector of the boundary ∂Ω. d _i , α _ij , a _i , b _ij , e _i (i, j = 1, 2) are positive constants and the initial data u₀ and v₀ are continuous non-negative functions which are not identically zero. The homogeneous Neumann boundary condition indicates that the system is self-contained with zero population flux across the boundary. The parameters d₁, d₂ are the diffusion rates, α _ii (i = 1, 2) are referred as self-diffusion pressures, and α _ij (i, j = 1, 2, i ≠ j) are cross-diffusion pressures. For more details on the backgrounds about self-diffusion and cross-diffusion, one can see [8].

The local existence of solutions for the system (1.7) is an immediate consequence of a series of important articles by Amann [18–20]. Roughly speaking, if u₀(x) and v₀(x) in $W_p^1\left(\text{Ω}\right)$ with p > n, then (1.7) has a unique non-negative solution u, v ∈ C ([0, T ), $W_P^1\left(\text{Ω}\right))\cap C^{\infty }(\left(0,T\right)$, C^∞(Ω)), where T ∈ (0, ∞] is the maximal existence time for the solution. If the solution (u, v) satisfies the estimate

$$\mathsf{\text{sup}}\left\{{\left|\right|u\left(\cdot ,t\right)\left|\right|}_{W_p^1\left(\text{Ω}\right)},{\left|\right|v\left(\cdot ,t\right)\left|\right|}_{W_p^1\left(\text{Ω}\right)}:0<t<T\right\}<\infty ,$$

then T = +∞. Moreover, if u₀(x), $v_0\left(x\right)\in W_P^2\left(\text{Ω}\right)$, then u, v ∈ C ([0, ∞), $W_P^2\left(\text{Ω}\right))$.

For the following SKT system

$$\left\{\begin{array}{cc}{u}_t=d_1\text{Δ}\left[\left(1+\alpha v+\gamma u\right)u\right]+au\left(1-u-cv\right),\hfill & x\in \text{Ω},t>0,\hfill \\ v_t=d_2\text{Δ}\left[\left(1+\delta v\right)v\right]+bv\left(1-du-v\right),\hfill & x\in \text{Ω},t>0,\hfill \\ {\partial }_{n}u={\partial }_{n}v=0,\hfill & x\in \partial \text{Ω},t>0,\hfill \\ u\left(x,0\right)=u_0\left(x\right),v\left(x,0\right)=v_0\left(x\right),\hfill & x\in \text{Ω}\hfill \end{array}\right.$$

(P)

Yamada [13] proposed four open problems:

• (1) The global existence of solutions of (P) in the case δ > 0 and the space dimension N ≥ 6;

• (2) the global existence in the case γ = 0;

• (3) in order to study the asymptotic behavior of u, v as t → ∞, need to establish the uniform boundedness of global solutions; and

• (4) the global existence of solutions for the following full SKT system

  $$\left\{\begin{array}{cc}{u}_t=d_1\text{Δ}\left[\left(1+\alpha v+\gamma u\right)u\right]+au\left(1-u-cv\right),\hfill & x\in \text{Ω},t>0,\hfill \\ v_t=d_2\text{Δ}\left[\left(1+\beta u+\delta v\right)v\right]+bv\left(1-du-v\right),\hfill & x\in \text{Ω},t>0,\hfill \\ {\partial }_{n}u={\partial }_{n}v=0,\hfill & x\in \partial \text{Ω},t>0,\hfill \\ u\left(x,0\right)=u_0\left(x\right),v\left(x,0\right)=v_0\left(x\right),\hfill & x\in \text{Ω}\hfill \end{array}\right.$$

with α, γ, β, δ > 0.

Very few global existence results for (1.7) are known. The main purpose of this article is to establish the uniform boundedness of global solutions for the system (1.7) in one space dimension. For convenience, we consider the following system

$$\left\{\begin{array}{cc}{u}_t={\left(d_{1}u+{\alpha }_{11}{u}^2+{\alpha }_{12}uv\right)}_{xx}+u\left(a_1-b_{11}u-b_{12}v-e_{1}uv\right),\hfill & 0<x<1,t>0,\hfill \\ v_t={\left(d_{2}v+{\alpha }_{21}uv+{\alpha }_{22}{v}^2\right)}_{xx}+v\left(a_2-b_{21}u-b_{22}v-e_{2}uv\right),\hfill & 0<x<1,t>0,\hfill \\ u_x\left(x,t\right)=v_x\left(x,t\right)=0,\hfill & x=0,1,t>0,\hfill \\ u\left(x,0\right)=u_0\left(x\right),v\left(x,0\right)=v_0\left(x\right),\hfill & 0<x<1.\hfill \end{array}\right.$$

(1.8)

We firstly investigate the global existence and the uniform boundedness of the solutions for (1.8), then prove the global asymptotic stability of the positive equilibrium (u*, v*) of (1.8) by an important lemma from [21]. The proof is complete and complement to the uniform convergence theorems in [22–24].

It is obvious that (u*, v*) is the unique positive equilibrium of the system (1.8) if (1.2) holds.

For simplicity, we denote $\left|\right|\cdot |{|}_{W_p^k\left(0,1\right)}$ by | · | _k,p and $\left|\right|\cdot |{|}_{L^p\left(0,1\right)}$ by | · | _p . Our main results are as follows.

Theorem 1.1. Let $u_0,v_0\in W_2^2\left(0,1\right)$, (u, v) is the unique non-negative solution of system (1.2) in the maximal existence interval [0, T ). Assume that

$$8{\alpha }_{11}{\alpha }_{21}>{\alpha }_{12}^2,8{\alpha }_{22}{\alpha }_{12}>{\alpha }_{21}^2.$$

(1.9)

Then there exist t₀ > 0 and positive constants M, M' which depend on d _i , α _ij , a _i , b _ij , e _i (i, j = 1, 2), such that

$$\mathsf{\text{sup}}\left\{{\left|u\left(\cdot ,t\right)\right|}_{1,2},{\left|v\left(\cdot ,t\right)\right|}_{1,2}:t\in \left(t_0,T\right)\right\}\le M^{\prime },$$

(1.10)

$$\mathsf{\text{max}}\left\{u\left(x,t\right),v\left(x,t\right):\left(x,t\right)\in \left[0,1\right]\times \left(t_0,T\right)\right\}\le M,$$

(1.11)

and T = +∞. Moreover, in the case that d₁, d₂ ≥ 1, $d_2/d_1\in \left[\underset{-}{d},\bar{d}\right]$, where $\underset{¯}{d}$ and $\bar{d}$ are positive constants, M', M depend on $\underset{-}{d},\bar{d}$, but do not depend on d₁, d₂.

Remark 1.1. Since the continuous embedding H¹(Ω) ↪ L^∞(Ω) holds only in one space dimension, we can only establish the uniform maximum-norm estimates about time for the solution in one space dimension.

Theorem 1.2 Assume that all conditions in Theorem 1.1 are satisfied. Assume further that,

$$4d_1{d}_2{u}^{*}{v}^{*}>M^2{\left({\alpha }_{12}{u}^{*}+{\alpha }_{21}{v}^{*}\right)}^2,$$

(1.12)

$$4b_{11}{b}_{22}>{\left(b_{12}+b_{21}+e_1{u}^{*}+e_2{v}^{*}\right)}^2$$

(1.13)

and (1.2) hold, M is given by (1.11). Then the unique positive equilibrium (u*, v*) of (1.8) is globally asymptotically stable.

Remark 1.2. The system (1.8) has no non-constant positive steady-state solution if all conditions of Theorem 1.2 hold.

2 Global solutions

In order to establish the uniform $W_2^1$-estimates of the solutions for system (1.2), the following Gagliardo-Nirenberg- type inequalities and the corresponding corollary play important roles (see [25]).

Theorem 2.1. Let Ω ⊂ ℝⁿ be a bounded domain with ∂Ω ∈ C^m. For every function $u\in W_r^m\left(\text{Ω}\right)$, 1 ≤ q, r ≤ ∞, the derivative D^ju (0 ≤ j < m) satisfies the inequality

$${\left|D^{j}u\right|}_p\le C\left({\left|D^{m}u\right|}_r^a{\left|u\right|}_q^{1-a}+{\left|u\right|}_q\right),$$

provided one of the following three conditions is satisfied: (1) r ≤ q, (2) 0 < n(r-q)/(mrq) < 1, or (3) n(r-q)/(mrq) = 1 and m - n/q is not a non-negative integer, where 1/p = j/n + a(1/r - m/n) + (1 - a)/q for all a ∈ [j/m, 1), and the positive constant C depends on n, m, j, q, r, a.

Corollary 1. There exists a positive constant C such that

$${\left|u\right|}_2\le C\left({\left|u_x\right|}_2^{1/3}{\left|u\right|}_1^{2/3}+{\left|u\right|}_1\right),\forall u\in W_2^1\left(0,1\right),$$

(2.1)

$${\left|u\right|}_4\le C\left({\left|u_x\right|}_2^{1/2}{\left|u\right|}_1^{1/2}+{\left|u\right|}_1\right),\forall u\in W_2^1\left(0,1\right),$$

(2.2)

$${\left|u\right|}_{\frac{7}{2}}\le C\left({\left|u_x\right|}_2^{10/21}{\left|u\right|}_1^{11/21}+{\left|u\right|}_1\right),\forall u\in W_2^1\left(0,1\right),$$

(2.3)

$${\left|u_x\right|}_2\le C\left({\left|u_{xx}\right|}_2^{3/5}{\left|u\right|}_1^{2/5}+{\left|u\right|}_1\right),\forall u\in W_2^2\left(0,1\right).$$

(2.4)

Throughout this article, we always denote that C is Sobolev embedding constant or other kind of universal constant, A _j , B _j , C _j are some positive constants which depend only on α _ij , a _i , b _ij , e _i (i, j = 1, 2), K _j are positive constants depending on d _i , α _ij , a _i , b _ij , e _i (i, j = 1, 2). When d₁, d₂ ≥ 1, $d_2/d_1\in \left[\underset{-}{d},\bar{d}\right]$, K _j depend on $\underset{-}{d},\bar{d}$, but do not depend on d₁ and d₂.

Proof of Theorem 1.1. Step 1, estimate |u|₁, |v|₁.

Taking integration of the first, the first and the second equations in (1.8) over the domain (0,1), respectively, we have

$$\begin{array}{c}\frac{d}{dt}\underset{0}{\overset{1}{\int }}udx=\underset{0}{\overset{1}{\int }}u\left(a_1-b_{11}u-b_{12}v-e_{1}uv\right)dx\le a_1\underset{0}{\overset{1}{\int }}udx-b_{11}{\left(\underset{0}{\overset{1}{\int }}udx\right)}^2,\\ \frac{d}{dt}\underset{0}{\overset{1}{\int }}vdx=\underset{0}{\overset{1}{\int }}v\left(a_2-b_{21}u-b_{22}v-e_{2}uv\right)dx\le a_2\underset{0}{\overset{1}{\int }}vdx-b_{22}{\left(\underset{0}{\overset{1}{\int }}vdx\right)}^2.\end{array}$$

So, there exists a positive constant M₀ which depends on a _i , b _ij (i, j = 1, 2), such that

$$\underset{0}{\overset{1}{\int }}udx,\underset{0}{\overset{1}{\int }}vdx\le M_0,t\ge {\tau }_0.$$

(2.5)

Moreover, there exists a positive constant $M_0^{\prime }$ which depends on a _i , b _ij (i, j = 1, 2) and L¹-norm of u₀, v₀, such that

$$\underset{0}{\overset{1}{\int }}udx,\underset{0}{\overset{1}{\int }}vdx\le M_0^{\prime },t\ge 0.$$

(2.5a)

Step 2, estimate |u|₂, |v|₂.

Multiplying the first two equations in system (1.2) by u, v, respectively, and integrating over (0, 1) we have

$$\begin{array}{c}\frac{1}{2}\frac{d}{dt}\underset{0}{\overset{1}{\int }}{u}^{2}dx\le -d_1\underset{0}{\overset{1}{\int }}{u}_x^{2}dx-\underset{0}{\overset{1}{\int }}\left[\left(2{\alpha }_{11}u+{\alpha }_{12}v\right)u_x^2+{\alpha }_{12}{v}_{x}u{u}_x\right]dx+a_1\underset{0}{\overset{1}{\int }}{u}^{2}dx,\\ \frac{1}{2}\frac{d}{dt}\underset{0}{\overset{1}{\int }}{v}^{2}dx\le -d_2\underset{0}{\overset{1}{\int }}{v}_x^{2}dx-\underset{0}{\overset{1}{\int }}\left[\left({\alpha }_{21}u+2{\alpha }_{22}v\right)v_x^2+{\alpha }_{21}{u}_{x}v{v}_x\right]dx+a_2\underset{0}{\overset{1}{\int }}{v}^{2}dx,\end{array}$$

from which it follows that

$$\begin{array}{cc}\hfill \frac{1}{2}\frac{d}{dt}\underset{0}{\overset{1}{\int }}\left(u^2+v^2\right)dx& \le -d\underset{0}{\overset{1}{\int }}\left(u_x^2+v_x^2\right)dx+a_1\underset{0}{\overset{1}{\int }}{u}^{2}dx+a_2\underset{0}{\overset{1}{\int }}{v}^{2}dx-\underset{0}{\overset{1}{\int }}q\left(u_x,v_x\right)dx\hfill \\ \le -d\underset{0}{\overset{1}{\int }}\left(u_x^2+v_x^2\right)dx+\left(a_1+a_2\right)\underset{0}{\overset{1}{\int }}\left(u^2+v^2\right)dx-\underset{0}{\overset{1}{\int }}q\left(u_x,v_x\right)dx,\hfill \end{array}$$

where d = min{d₁, d₂}. Some tedious calculations yield that

$$q\left(u_x,v_x\right)=\left(2{\alpha }_{11}u+{\alpha }_{12}v\right)u_x^2+\left({\alpha }_{12}u+{\alpha }_{21}v\right)u_x{v}_x+\left({\alpha }_{21}u+2{\alpha }_{22}v\right)v_x^2$$

is positive definite quadratic form of u _x , v _x if (1.9) holds. So (1.9) implies that

$$\frac{1}{2}\frac{d}{dt}\underset{0}{\overset{1}{\int }}\left(u^2+v^2\right)dx\le -d\underset{0}{\overset{1}{\int }}\left(u_x^2+v_x^2\right)dx+\left(a_1+a_2\right)\underset{0}{\overset{1}{\int }}\left(u^2+v^2\right)dx.$$

(2.6)

Now, we proceed in the following two cases.

• (i) t ≥ τ₀.The inequality (2.1) implies that $|u{|}_2^6\le C\left({\left|u_x\right|}_2^2{\left|u\right|}_1^4+{\left|u\right|}_1^6\right)\le C{M}_0^4\left({\left|u_x\right|}_2^2+M_0^2\right)$. So we have ${\int }_0^1{u}_x^{2}dx\ge \frac{1}{C{M}_0^4}{\left({\int }_0^1{u}^{2}dx\right)}^3-M_0^2,$ and

  $$-\underset{0}{\overset{1}{\int }}\left(u_x^2+v_x^2\right)dx\le -\frac{1}{9C{M}_0^4}{\left[\underset{0}{\overset{1}{\int }}\left(u^2+v^2\right)dx\right]}^3+2M_0^2.$$

  (2.7)

It follows from (2.5) and (2.6) that

$$\frac{1}{2}\frac{d}{dt}\underset{0}{\overset{1}{\int }}\left(u^2+v^2\right)dx\le d\left\{-C_2{\left[\underset{0}{\overset{1}{\int }}\left(u^2+v^2\right)dx\right]}^3+2M_0^2+\frac{1}{d}\left(a_1+a_2\right)\underset{0}{\overset{1}{\int }}\left(u^2+v^2\right)dx\right\}.$$

(2.8)

This means that there exist positive constants τ₁ and M₁ depending on d _i , a _i , b _ij (i, j = 1, 2), such that

$$\underset{0}{\overset{1}{\int }}{u}^{2}dx,\underset{0}{\overset{1}{\int }}{v}^{2}dx\le M_1,t\ge {\tau }_1.$$

(2.9)

When d ≥ 1, M₁ is independent of d because the zero point of the right-hand side in (2.11) can be estimated by positive constants independent of d.

• (ii) t ≥ 0. Repeating estimates in (i) by (2.8)', we can obtain that there exists a positive constant $M_1^{\prime }$ depending on d _i , a _i , b _ij (i, j = 1, 2) and the L¹, L²-norm of u₀, v₀, such that

  $$\underset{0}{\overset{1}{\int }}{u}^{2}dx,\underset{0}{\overset{1}{\int }}{v}^{2}dx\le M_1^{\prime },t\ge 0.$$

  (2.9a)

When d ≥ 1, $M_1^{\prime }$ is independent of d.

Step 3, estimate |u _x |₂, |v _x |₂.

Introducing the following scaling

$$ũ=\frac{u}{d_1},ṽ=\frac{v}{d_1},\tilde{t}=d_{1}t.$$

(2.10)

Denoting ξ = d₂/d₁, and using u, v, t instead of $ũ,ṽ,\tilde{t}$, respectively, then system (1.2) reduces to

$$\left\{\begin{array}{cc}{u}_t=P_{xx}+f\left(u,v\right),\hfill & 0<x<1,t>0,\hfill \\ v_t=Q_{xx}+g\left(u,v\right),\hfill & 0<x<1,t>0,\hfill \\ u_x\left(x,t\right)=v_x\left(x,t\right)=0,\hfill & x=0,1,t>0,\hfill \\ u\left(x,0\right)={ũ}_0\left(x\right),v\left(x,0\right)={ṽ}_0\left(x\right),\hfill & 0<x<1,\hfill \end{array}\right.$$

(2.11)

where P = u + α₁₁u² + α₁₂uv, Q = ξv + α₂₁uv + α₂₂v², $f\left(u,v\right)=a_1{d}_1^{-1}u-b_{11}{u}^2-b_{12}uv-e_1{d}_1{u}^{2}v$, $g\left(u,v\right)=a_2{d}_1^{-1}v-b_{21}uv-b_{22}{v}^2-e_2{d}_{1}u{v}^2$.

We still proceed in the following two cases.

• (i) $t\ge {\tau }_1^{*}=d_1{\tau }_1$. It is clear that

  $$\begin{array}{c}\underset{0}{\overset{1}{\int }}udx,\underset{0}{\overset{1}{\int }}vdx\le M_0{d}_1^{-1},\hfill \\ \underset{0}{\overset{1}{\int }}{u}^{2}dx,\underset{0}{\overset{1}{\int }}{v}^{2}dx\le M_1{d}_1^{-2},\hfill \\ |P{|}_1,|Q{|}_1\le A_1{K}_1{d}_1^{-1},\hfill \end{array}$$

  (2.12)

where $K_1=\left(1+\xi \right)+M_1{d}_1^{-1}$ and A₁ = max{M₀, α₁₁ + α₁₂, α₂₁ + α₂₂}.

Multiplying the first two equations in (2.11) by P _t , Q _t , integrating them over the domain (0, 1), respectively, and then adding up the two integration equalities, we have

$$\begin{array}{cc}\hfill \frac{1}{2}{ȳ}^{\prime }\left(t\right)=& -\underset{0}{\overset{1}{\int }}{u}_t^{2}dx-\xi \underset{0}{\overset{1}{\int }}{v}_t^{2}dx-\underset{0}{\overset{1}{\int }}q\left(u_t,v_t\right)dx\hfill \\ +\underset{0}{\overset{1}{\int }}\left[\left(1+2{\alpha }_{11}u+{\alpha }_{12}v\right)u_{t}f+{\alpha }_{12}u{v}_{t}f\right]dx+\underset{0}{\overset{1}{\int }}\left[\left(\xi +{\alpha }_{21}u+2{\alpha }_{22}v\right)v_{t}g+{\alpha }_{21}v{u}_{t}g\right]dx\hfill \end{array}$$

where $ȳ\left(t\right)={\int }_0^1\left(P_x^2+Q_x^2\right)dx$. It is not hard verify by (1.9) that there exists a positive constant C₃ depending only on α _ij (i, j = 1, 2), such that

$$q\left(u_t,v_t\right)\ge C_3\left(u+v\right)\left(u_t^2+v_t^2\right).$$

Thus,

$$\begin{array}{cc}\hfill \frac{1}{2}{ȳ}^{\prime }\left(t\right)\le & -\underset{0}{\overset{1}{\int }}{u}_t^{2}dx-\xi \underset{0}{\overset{1}{\int }}{v}_t^{2}dx-C_3\underset{0}{\overset{1}{\int }}\left(u+v\right)\left(u_t^2+v_t^2\right)dx\hfill \\ +\underset{0}{\overset{1}{\int }}\left[\left(1+2{\alpha }_{11}u+{\alpha }_{12}v\right)u_{t}f+{\alpha }_{12}u{v}_{t}f\right]dx+\underset{0}{\overset{1}{\int }}\left[\left(\xi +{\alpha }_{21}u+2{\alpha }_{22}v\right)v_{t}g+{\alpha }_{21}v{u}_{t}g\right]dx.\hfill \end{array}$$

(2.13)

Using Young inequality, Hölder inequality, and (2.12), we can obtain the following estimates

$$\begin{array}{c}\underset{0}{\overset{1}{\int }}{u}^3\mathsf{\text{d}}x\le {\left(\underset{0}{\overset{1}{\int }}{u}^7\mathsf{\text{d}}x\right)}^{1/5}{\left(\underset{0}{\overset{1}{\int }}{u}^2\mathsf{\text{d}}x\right)}^{4/5}\le M_1^{4/5}{d}_1^{-8/5}{\left(\underset{0}{\overset{1}{\int }}{u}^7\mathsf{\text{d}}x\right)}^{1/5},\hfill \\ \underset{0}{\overset{1}{\int }}{u}^4\mathsf{\text{d}}x\le {\left(\underset{0}{\overset{1}{\int }}{u}^7\mathsf{\text{d}}x\right)}^{2/5}{\left(\underset{0}{\overset{1}{\int }}{u}^2\mathsf{\text{d}}x\right)}^{3/5}\le M_1^{3/5}{d}_1^{-6/5}{\left(\underset{0}{\overset{1}{\int }}{u}^7\mathsf{\text{d}}x\right)}^{2/5},\hfill \\ \underset{0}{\overset{1}{\int }}{u}^5\mathsf{\text{d}}x\le {\left(\underset{0}{\overset{1}{\int }}{u}^7\mathsf{\text{d}}x\right)}^{3/5}{\left(\underset{0}{\overset{1}{\int }}{u}^2\mathsf{\text{d}}x\right)}^{2/5}\le M_1^{2/5}{d}_1^{-4/5}{\left(\underset{0}{\overset{1}{\int }}{u}^7\mathsf{\text{d}}x\right)}^{3/5},\hfill \\ \underset{0}{\overset{1}{\int }}{u}^6\mathsf{\text{d}}x\le {\left(\underset{0}{\overset{1}{\int }}{u}^7\mathsf{\text{d}}x\right)}^{4/5}{\left(\underset{0}{\overset{1}{\int }}{u}^2\mathsf{\text{d}}x\right)}^{1/5}\le M_1^{1/5}{d}_1^{-2/5}{\left(\underset{0}{\overset{1}{\int }}{u}^7\mathsf{\text{d}}x\right)}^{4/5},\hfill \\ \underset{0}{\overset{1}{\int }}{u}^{2}v\mathsf{\text{d}}x\le {\left(\underset{0}{\overset{1}{\int }}{u}^7\mathsf{\text{d}}x\right)}^{1/5}{\left(\underset{0}{\overset{1}{\int }}{u}^2\mathsf{\text{d}}x\right)}^{3/10}{\left(\underset{0}{\overset{1}{\int }}{v}^2\mathsf{\text{d}}x\right)}^{1/2}\le M_1^{4/5}{d}_1^{-8/5}{\left(\underset{0}{\overset{1}{\int }}{u}^7\mathsf{\text{d}}x\right)}^{1/5},\hfill \\ \underset{0}{\overset{1}{\int }}{u}^{3}v\mathsf{\text{d}}x\le {\left(\underset{0}{\overset{1}{\int }}{u}^7\mathsf{\text{d}}x\right)}^{2/5}{\left(\underset{0}{\overset{1}{\int }}{u}^2\mathsf{\text{d}}x\right)}^{1/10}{\left(\underset{0}{\overset{1}{\int }}{v}^2\mathsf{\text{d}}x\right)}^{1/2}\le M_1^{3/5}{d}_1^{-6/5}{\left(\underset{0}{\overset{1}{\int }}{u}^7\mathsf{\text{d}}x\right)}^{2/5},\hfill \\ \underset{0}{\overset{1}{\int }}{u}^{4}v\mathsf{\text{d}}x\le \frac{4}{5}\underset{0}{\overset{1}{\int }}{u}^5\mathsf{\text{d}}x+\frac{1}{5}\underset{0}{\overset{1}{\int }}{v}^5\mathsf{\text{d}}x\le \frac{4}{5}{M}_1^{2/5}{d}_1^{-4/5}\left[{\left(\underset{0}{\overset{1}{\int }}{u}^7\mathsf{\text{d}}x\right)}^{3/5}+{\left(\underset{0}{\overset{1}{\int }}{v}^7\mathsf{\text{d}}x\right)}^{3/5}\right],\hfill \\ \underset{0}{\overset{1}{\int }}{u}^2{v}^2\mathsf{\text{d}}x\le \frac{1}{2}\left(\underset{0}{\overset{1}{\int }}{u}^4\mathsf{\text{d}}x+\underset{0}{\overset{1}{\int }}{v}^4\mathsf{\text{d}}x\right)\le \frac{1}{2}{M}_1^{3/5}{d}_1^{-4/5}\left[{\left(\underset{0}{\overset{1}{\int }}{u}^7\mathsf{\text{d}}x\right)}^{2/5}+{\left(\underset{0}{\overset{1}{\int }}{v}^7\mathsf{\text{d}}x\right)}^{2/5}\right],\hfill \\ \underset{0}{\overset{1}{\int }}{u}^3{v}^2\mathsf{\text{d}}x\le \frac{3}{5}\underset{0}{\overset{1}{\int }}{u}^5\mathsf{\text{d}}x+\frac{2}{5}\underset{0}{\overset{1}{\int }}{v}^5\mathsf{\text{d}}x\le \frac{3}{5}{M}_1^{2/5}{d}_1^{-4/5}\left[{\left(\underset{0}{\overset{1}{\int }}{u}^7\mathsf{\text{d}}x\right)}^{3/5}+{\left(\underset{0}{\overset{1}{\int }}{v}^7\mathsf{\text{d}}x\right)}^{3/5}\right],\hfill \\ \underset{0}{\overset{1}{\int }}{u}^4{v}^2\mathsf{\text{d}}x\le \frac{2}{3}\underset{0}{\overset{1}{\int }}{u}^6\mathsf{\text{d}}x+\frac{1}{3}\underset{0}{\overset{1}{\int }}{v}^6\mathsf{\text{d}}x\le \frac{2}{3}{M}_1^{1/5}{d}_1^{-4/5}\left[{\left(\underset{0}{\overset{1}{\int }}{u}^7\mathsf{\text{d}}x\right)}^{4/5}+{\left(\underset{0}{\overset{1}{\int }}{v}^7\mathsf{\text{d}}x\right)}^{4/5}\right],\hfill \\ \underset{0}{\overset{1}{\int }}{u}^4{v}^3\mathsf{\text{d}}x\le \frac{4}{7}\underset{0}{\overset{1}{\int }}{u}^7\mathsf{\text{d}}x+\frac{3}{7}\underset{0}{\overset{1}{\int }}{v}^7\mathsf{\text{d}}x\le \frac{4}{7}\left(\underset{0}{\overset{1}{\int }}{u}^7\mathsf{\text{d}}x+\underset{0}{\overset{1}{\int }}{v}^7\mathsf{\text{d}}x\right),\hfill \\ \underset{0}{\overset{1}{\int }}{u}^5{v}^2\mathsf{\text{d}}x\le \frac{5}{7}\underset{0}{\overset{1}{\int }}{u}^7\mathsf{\text{d}}x+\frac{2}{7}\underset{0}{\overset{1}{\int }}{v}^7\mathsf{\text{d}}x\le \frac{5}{7}\left(\underset{0}{\overset{1}{\int }}{u}^7\mathsf{\text{d}}x+\underset{0}{\overset{1}{\int }}{v}^7\mathsf{\text{d}}x\right).\hfill \end{array}$$

(2.14)

Applying the above estimates and Gagliardo-Nirenberg-type inequalities to the terms on the right-hand side of (2.13), we have

$$\begin{array}{c}-\underset{0}{\overset{1}{\int }}{u}_t^{2}dx\le -\frac{1}{2}\underset{0}{\overset{1}{\int }}{P}_{xx}^{2}dx+\underset{0}{\overset{1}{\int }}{f}^{2}dx,\hfill \\ -\xi \underset{0}{\overset{1}{\int }}{v}_t^{2}dx\le -\frac{\xi }{2}\underset{0}{\overset{1}{\int }}{Q}_{xx}^{2}dx+\xi \underset{0}{\overset{1}{\int }}{g}^{2}dx,\end{array}$$

$$\begin{array}{cc}\hfill \underset{0}{\overset{1}{\int }}{f}^{2}dx& \le \underset{0}{\overset{1}{\int }}\left(a_1^2{d}_1^{-2}{u}^2+b_{11}^2{u}^4+b_{12}^2{u}^2{v}^2+e_1^2{d}_1^2{u}^4{v}^2+2b_{11}{b}_{12}{u}^{3}v+2b_{11}{e}_1{d}_1{u}^{4}v+2b_{12}{e}_1{d}_1{u}^3{v}^2\right)dx\hfill \\ \le a_1^2{M}_1{d}_1^{-4}+\left(b_{11}^2+\frac{1}{2}{b}_{12}^2+2b_{11}{b}_{12}\right)M_1^{3/5}{d}_1^{-6/5}\left[{\left(\underset{0}{\overset{1}{\int }}{u}^7\mathsf{\text{d}}x\right)}^{2/5}+{\left(\underset{0}{\overset{1}{\int }}{v}^7\mathsf{\text{d}}x\right)}^{2/5}\right]\hfill \\ +\left(\frac{8}{5}{b}_{11}+\frac{6}{5}{b}_{12}\right)e_1{M}_1^{2/5}{d}_1^{1/5}\left[{\left(\underset{0}{\overset{1}{\int }}{u}^7\mathsf{\text{d}}x\right)}^{3/5}+{\left(\underset{0}{\overset{1}{\int }}{v}^7\mathsf{\text{d}}x\right)}^{3/5}\right]\hfill \\ +\frac{2}{3}{e}_1^2{M}_1^{1/5}{d}_1^{8/5}\left[{\left(\underset{0}{\overset{1}{\int }}{u}^7\mathsf{\text{d}}x\right)}^{4/5}+{\left(\underset{0}{\overset{1}{\int }}{v}^7\mathsf{\text{d}}x\right)}^{4/5}\right],\hfill \end{array}$$

$$\begin{array}{cc}\hfill \xi \underset{0}{\overset{1}{\int }}{g}^{2}dx& \le \xi \underset{0}{\overset{1}{\int }}\left(a_2^2{d}_1^{-2}{v}^2+b_{21}^2{u}^2{v}^2+b_{22}^2{v}^4+e_2^2{d}_1^2{u}^2{v}^4+2b_{21}{b}_{22}u{v}^3+2b_{21}{e}_2{d}_1{u}^2{v}^3+2b_{22}{e}_2{d}_{1}u{v}^4\right)dx\hfill \\ \le \xi a_2^2{M}_1{d}_1^{-4}+\xi \left(b_{22}^2+\frac{1}{2}{b}_{21}^2+2b_{21}{b}_{22}\right)M_1^{3/5}{d}_1^{-6/5}\left[{\left(\underset{0}{\overset{1}{\int }}{u}^7\mathsf{\text{d}}x\right)}^{2/5}+{\left(\underset{0}{\overset{1}{\int }}{v}^7\mathsf{\text{d}}x\right)}^{2/5}\right]\hfill \\ +2\xi \left(\frac{3}{5}{b}_{21}+\frac{4}{5}{b}_{22}\right)e_2{M}_1^{2/5}{d}_1^{1/5}\left[{\left(\underset{0}{\overset{1}{\int }}{u}^7\mathsf{\text{d}}x\right)}^{3/5}+{\left(\underset{0}{\overset{1}{\int }}{v}^7\mathsf{\text{d}}x\right)}^{3/5}\right]\hfill \\ +\frac{2}{3}\xi e_2^2{M}_1^{1/5}{d}_1^{8/5}\left[{\left(\underset{0}{\overset{1}{\int }}{u}^7\mathsf{\text{d}}x\right)}^{4/5}+{\left(\underset{0}{\overset{1}{\int }}{v}^7\mathsf{\text{d}}x\right)}^{4/5}\right],\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill -\underset{0}{\overset{1}{\int }}{u}_t^{2}dx-\xi \underset{0}{\overset{1}{\int }}{v}_t^{2}dx\le & -\frac{1}{2}\underset{0}{\overset{1}{\int }}{P}_{xx}^{2}dx-\frac{\xi }{2}\underset{0}{\overset{1}{\int }}{Q}_{xx}^{2}dx+\left(a_1^2+\xi a_2^2\right)M_1{d}_1^{-4}\hfill \\ +\left[\left(b_{11}^2+\frac{1}{2}{b}_{12}^2+2b_{11}{b}_{12}\right)+\xi \left(b_{22}^2+\frac{1}{2}{b}_{21}^2+2b_{21}{b}_{22}\right)\right]M_1^{3/5}{d}_1^{-6/5}\left[{\left(\underset{0}{\overset{1}{\int }}{u}^7\mathsf{\text{d}}x\right)}^{2/5}+{\left(\underset{0}{\overset{1}{\int }}{v}^7\mathsf{\text{d}}x\right)}^{2/5}\right]\hfill \\ +\left[\left(\frac{8}{5}{b}_{11}+\frac{6}{5}{b}_{12}\right)e_1+2\xi \left(\frac{3}{5}{b}_{21}+\frac{4}{5}{b}_{22}\right)e_2\right]M_1^{2/5}{d}_1^{1/5}\left[{\left(\underset{0}{\overset{1}{\int }}{u}^7\mathsf{\text{d}}x\right)}^{3/5}+{\left(\underset{0}{\overset{1}{\int }}{v}^7\mathsf{\text{d}}x\right)}^{3/5}\right]\hfill \\ +\frac{2}{3}\left(e_1^2+\xi e_2^2\right)M_1^{1/5}{d}_1^{8/5}\left[{\left(\underset{0}{\overset{1}{\int }}{u}^7\mathsf{\text{d}}x\right)}^{4/5}+{\left(\underset{0}{\overset{1}{\int }}{v}^7\mathsf{\text{d}}x\right)}^{4/5}\right].\hfill \end{array}$$

(2.15)

Similarly, we can obtain

$$\begin{array}{cc}\hfill \underset{0}{\overset{1}{\int }}{u}_{t}fdx& \le \underset{0}{\overset{1}{\int }}{u}_t\left(a_1{d}_1^{-1}u+b_{11}{u}^2+b_{12}uv+e_1{d}_1{u}^{2}v\right)dx\hfill \\ \le \frac{a_1^2{d}_1^{-2}}{2\epsilon }\underset{0}{\overset{1}{\int }}udx+\frac{\epsilon }{2}\underset{0}{\overset{1}{\int }}u{u}_t^{2}dx+\frac{b_{11}^2}{2\epsilon }\underset{0}{\overset{1}{\int }}{u}^{3}dx+\frac{\epsilon }{2}\underset{0}{\overset{1}{\int }}u{u}_t^{2}dx\hfill \\ +\frac{b_{12}^2}{2\epsilon }\underset{0}{\overset{1}{\int }}u{v}^{2}dx+\frac{\epsilon }{2}\underset{0}{\overset{1}{\int }}u{u}_t^{2}dx+\frac{{\left(e_1{d}_1\right)}^2}{2\epsilon }\underset{0}{\overset{1}{\int }}{u}^3{v}^{2}dx+\frac{\epsilon }{2}\underset{0}{\overset{1}{\int }}u{u}_t^{2}dx\hfill \\ \le \frac{a_1^2}{2\epsilon }{M}_0{d}_1^{-3}+\frac{b_{11}^2+b_{12}^2}{2\epsilon }{M}_1^{4/5}{d}_1^{-8/5}\left[{\left(\underset{0}{\overset{1}{\int }}{u}^7\mathsf{\text{d}}x\right)}^{1/5}+{\left(\underset{0}{\overset{1}{\int }}{v}^7\mathsf{\text{d}}x\right)}^{1/5}\right]\hfill \\ +\frac{3e_1^2}{10\epsilon }{M}_1^{2/5}{d}_1^{6/5}\left[{\left(\underset{0}{\overset{1}{\int }}{u}^7\mathsf{\text{d}}x\right)}^{3/5}+{\left(\underset{0}{\overset{1}{\int }}{v}^7\mathsf{\text{d}}x\right)}^{3/5}\right]+2\epsilon \underset{0}{\overset{1}{\int }}u{u}_t^{2}dx,\hfill \end{array}$$

$$\begin{array}{cc}\hfill 2{\alpha }_{11}\underset{0}{\overset{1}{\int }}u{u}_{t}fdx& \le 2{\alpha }_{11}\underset{0}{\overset{1}{\int }}u{u}_t\left(a_1{d}_1^{-1}u+b_{11}{u}^2+b_{12}uv+e_1{d}_1{u}^{2}v\right)dx\hfill \\ \le \frac{{\alpha }_{11}^2{a}_1^2{d}_1^{-2}}{\epsilon }\underset{0}{\overset{1}{\int }}{u}^{3}dx+\epsilon \underset{0}{\overset{1}{\int }}u{u}_t^{2}dx+\frac{{\alpha }_{11}^2{b}_{11}^2}{\epsilon }\underset{0}{\overset{1}{\int }}{u}^{5}dx+\epsilon \underset{0}{\overset{1}{\int }}u{u}_t^{2}dx\hfill \\ +\frac{{\alpha }_{11}^2{b}_{12}^2}{\epsilon }\underset{0}{\overset{1}{\int }}{u}^3{v}^{2}dx+\epsilon \underset{0}{\overset{1}{\int }}u{u}_t^{2}dx+\frac{{\alpha }_{11}^2{e}_1^2{d}_1^2}{\epsilon }\underset{0}{\overset{1}{\int }}{u}^5{v}^{2}dx+\epsilon \underset{0}{\overset{1}{\int }}u{u}_t^{2}dx\hfill \\ \le \frac{{\alpha }_{11}^2{a}_1^2}{\epsilon }{M}_1^{4/5}{d}_1^{-18/5}{\left(\underset{0}{\overset{1}{\int }}{u}^{7}dx\right)}^{1/5}+\frac{{\alpha }_{11}^2\left(b_{11}^2+b_{12}^2\right)}{\epsilon }{M}_1^{2/5}{d}_1^{-4/5}\left[{\left(\underset{0}{\overset{1}{\int }}{u}^7\mathsf{\text{d}}x\right)}^{3/5}+{\left(\underset{0}{\overset{1}{\int }}{v}^7\mathsf{\text{d}}x\right)}^{3/5}\right]\hfill \\ +\frac{5{\alpha }_{11}^2{e}_1^2}{7\epsilon }{d}_1^2\left(\underset{0}{\overset{1}{\int }}{u}^7\mathsf{\text{d}}x+\underset{0}{\overset{1}{\int }}{v}^7\mathsf{\text{d}}x\right)+2\epsilon \underset{0}{\overset{1}{\int }}u{u}_t^{2}dx,\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\alpha }_{12}\underset{0}{\overset{1}{\int }}v{u}_{t}fdx& \le {\alpha }_{12}\underset{0}{\overset{1}{\int }}v{u}_t\left(a_1{d}_1^{-1}u+b_{11}{u}^2+b_{12}uv+e_1{d}_1{u}^{2}v\right)dx\hfill \\ \le \frac{{\alpha }_{12}^2{a}_1^2{d}_1^{-2}}{2\epsilon }\underset{0}{\overset{1}{\int }}u{v}^{2}dx+\frac{\epsilon }{2}\underset{0}{\overset{1}{\int }}u{u}_t^{2}dx+\frac{{\alpha }_{12}^2{b}_{11}^2}{2\epsilon }\underset{0}{\overset{1}{\int }}{u}^3{v}^{2}dx+\frac{\epsilon }{2}\underset{0}{\overset{1}{\int }}u{u}_t^{2}dx\hfill \\ +\frac{{\alpha }_{12}^2{b}_{12}^2}{2\epsilon }\underset{0}{\overset{1}{\int }}u{v}^{4}dx+\frac{\epsilon }{2}\underset{0}{\overset{1}{\int }}u{u}_t^{2}dx+\frac{{\alpha }_{12}^2{e}_1^2{d}_1^2}{2\epsilon }\underset{0}{\overset{1}{\int }}{u}^3{v}^{4}dx+\frac{\epsilon }{2}\underset{0}{\overset{1}{\int }}u{u}_t^{2}dx\hfill \\ \le \frac{{\alpha }_{12}^2{a}_1^2}{2\epsilon }{M}_1^{4/5}{d}_1^{-18/5}{\left(\underset{0}{\overset{1}{\int }}{v}^{7}dx\right)}^{1/5}+\frac{{\alpha }_{12}^2\left(b_{11}^2+b_{12}^2\right)}{2\epsilon }{M}_1^{2/5}{d}_1^{-4/5}\left[{\left(\underset{0}{\overset{1}{\int }}{u}^7\mathsf{\text{d}}x\right)}^{3/5}+{\left(\underset{0}{\overset{1}{\int }}{v}^7\mathsf{\text{d}}x\right)}^{3/5}\right]\hfill \\ +\frac{2{\alpha }_{12}^2{e}_1^2}{7\epsilon }{d}_1^2\left(\underset{0}{\overset{1}{\int }}{u}^7\mathsf{\text{d}}x+\underset{0}{\overset{1}{\int }}{v}^7\mathsf{\text{d}}x\right)+2\epsilon \underset{0}{\overset{1}{\int }}u{u}_t^{2}dx,\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\alpha }_{12}\underset{0}{\overset{1}{\int }}u{v}_{t}fdx& \le {\alpha }_{12}\underset{0}{\overset{1}{\int }}u{v}_t\left(a_1{d}_1^{-1}u+b_{11}{u}^2+b_{12}uv+e_1{d}_1{u}^{2}v\right)dx\hfill \\ \le \frac{{\alpha }_{12}^2{a}_1^2{d}_1^{-2}}{2\epsilon }\underset{0}{\overset{1}{\int }}{u}^{3}dx+\frac{\epsilon }{2}\underset{0}{\overset{1}{\int }}u{v}_t^{2}dx+\frac{{\alpha }_{12}^2{b}_{11}^2}{2\epsilon }\underset{0}{\overset{1}{\int }}{u}^{5}dx+\frac{\epsilon }{2}\underset{0}{\overset{1}{\int }}u{v}_t^{2}dx\hfill \\ +\frac{{\alpha }_{12}^2{b}_{12}^2}{2\epsilon }\underset{0}{\overset{1}{\int }}{u}^3{v}^{2}dx+\frac{\epsilon }{2}\underset{0}{\overset{1}{\int }}u{v}_t^{2}dx+\frac{{\alpha }_{12}^2{e}_1^2{d}_1^2}{2\epsilon }\underset{0}{\overset{1}{\int }}{u}^5{v}^{2}dx+\frac{\epsilon }{2}\underset{0}{\overset{1}{\int }}u{v}_t^{2}dx\hfill \\ \le \frac{{\alpha }_{12}^2{a}_1^2}{2\epsilon }{M}_1^{4/5}{d}_1^{-18/5}{\left(\underset{0}{\overset{1}{\int }}{u}^{7}dx\right)}^{1/5}+\frac{{\alpha }_{12}^2\left(b_{11}^2+b_{12}^2\right)}{2\epsilon }{M}_1^{2/5}{d}_1^{-4/5}\left[{\left(\underset{0}{\overset{1}{\int }}{u}^7\mathsf{\text{d}}x\right)}^{3/5}+{\left(\underset{0}{\overset{1}{\int }}{v}^7\mathsf{\text{d}}x\right)}^{3/5}\right]\hfill \\ +\frac{5{\alpha }_{12}^2{e}_1^2}{14\epsilon }{d}_1^2\left(\underset{0}{\overset{1}{\int }}{u}^7\mathsf{\text{d}}x+\underset{0}{\overset{1}{\int }}{v}^7\mathsf{\text{d}}x\right)+2\epsilon \underset{0}{\overset{1}{\int }}u{v}_t^{2}dx,\hfill \end{array}$$

$$\begin{array}{cc}\hfill \xi \underset{0}{\overset{1}{\int }}{v}_{t}gdx& \le \xi \underset{0}{\overset{1}{\int }}{v}_t\left(a_2{d}_1^{-1}v+b_{21}uv+b_{22}{v}^2+e_2{d}_{1}u{v}^2\right)dx\hfill \\ \le \frac{{\xi }^2{a}_2^2{d}_1^{-2}}{2\epsilon }\underset{0}{\overset{1}{\int }}vdx+\frac{\epsilon }{2}\underset{0}{\overset{1}{\int }}v{v}_t^{2}dx+\frac{{\xi }^2{b}_{21}^2}{2\epsilon }\underset{0}{\overset{1}{\int }}{u}^{2}vdx+\frac{\epsilon }{2}\underset{0}{\overset{1}{\int }}v{v}_t^{2}dx\hfill \\ +\frac{{\xi }^2{b}_{22}^2}{2\epsilon }\underset{0}{\overset{1}{\int }}{v}^{3}dx+\frac{\epsilon }{2}\underset{0}{\overset{1}{\int }}v{v}_t^{2}dx+\frac{{\xi }^2{e}_2^2{d}_1^2}{2\epsilon }\underset{0}{\overset{1}{\int }}{u}^2{v}^{3}dx+\frac{\epsilon }{2}\underset{0}{\overset{1}{\int }}v{v}_t^{2}dx\hfill \\ \le \frac{{\xi }^2{a}_2^2}{2\epsilon }{M}_0{d}_1^{-3}+\frac{{\xi }^2\left(b_{21}^2+b_{22}^2\right)}{2\epsilon }{M}_1^{4/5}{d}_1^{-8/5}\left[{\left(\underset{0}{\overset{1}{\int }}{u}^7\mathsf{\text{d}}x\right)}^{1/5}+{\left(\underset{0}{\overset{1}{\int }}{v}^7\mathsf{\text{d}}x\right)}^{1/5}\right]\hfill \\ +\frac{3{\xi }^2{e}_2^2}{10\epsilon }{M}_1^{2/5}{d}_1^{6/5}\left[{\left(\underset{0}{\overset{1}{\int }}{u}^7\mathsf{\text{d}}x\right)}^{3/5}+{\left(\underset{0}{\overset{1}{\int }}{v}^7\mathsf{\text{d}}x\right)}^{3/5}\right]+2\epsilon \underset{0}{\overset{1}{\int }}v{v}_t^{2}dx,\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\alpha }_{21}\underset{0}{\overset{1}{\int }}u{v}_{t}gdx& \le {\alpha }_{21}\underset{0}{\overset{1}{\int }}u{v}_t\left(a_2{d}_1^{-1}v+b_{21}uv+b_{22}{v}^2+e_2{d}_{1}u{v}^2\right)dx\hfill \\ \le \frac{{\alpha }_{21}^2{a}_2^2{d}_1^{-2}}{2\epsilon }\underset{0}{\overset{1}{\int }}{u}^{2}vdx+\frac{\epsilon }{2}\underset{0}{\overset{1}{\int }}v{v}_t^{2}dx+\frac{{\alpha }_{21}^2{b}_{21}^2}{2\epsilon }\underset{0}{\overset{1}{\int }}{u}^{4}vdx+\frac{\epsilon }{2}\underset{0}{\overset{1}{\int }}v{v}_t^{2}dx\hfill \\ +\frac{{\alpha }_{21}^2{b}_{22}^2}{2\epsilon }\underset{0}{\overset{1}{\int }}{u}^2{v}^{3}dx+\frac{\epsilon }{2}\underset{0}{\overset{1}{\int }}v{v}_t^{2}dx+\frac{{\alpha }_{21}^2{e}_2^2{d}_1^2}{2\epsilon }\underset{0}{\overset{1}{\int }}{u}^4{v}^{3}dx+\frac{\epsilon }{2}\underset{0}{\overset{1}{\int }}v{v}_t^{2}dx\hfill \\ \le \frac{{\alpha }_{21}^2{a}_2^2}{2\epsilon }{M}_1^{4/5}{d}_1^{-18/5}{\left(\underset{0}{\overset{1}{\int }}{u}^{7}dx\right)}^{1/5}+\frac{{\alpha }_{21}^2\left(b_{21}^2+b_{22}^2\right)}{2\epsilon }{M}_1^{2/5}{d}_1^{-4/5}\left[{\left(\underset{0}{\overset{1}{\int }}{u}^7\mathsf{\text{d}}x\right)}^{3/5}+{\left(\underset{0}{\overset{1}{\int }}{v}^7\mathsf{\text{d}}x\right)}^{3/5}\right]\hfill \\ +\frac{5{\alpha }_{21}^2{e}_2^2}{14\epsilon }{d}_1^2\left(\underset{0}{\overset{1}{\int }}{u}^7\mathsf{\text{d}}x+\underset{0}{\overset{1}{\int }}{v}^7\mathsf{\text{d}}x\right)+2\epsilon \underset{0}{\overset{1}{\int }}v{v}_t^{2}dx,\hfill \end{array}$$

$$\begin{array}{cc}\hfill 2{\alpha }_{22}\underset{0}{\overset{1}{\int }}v{v}_{t}gdx\le & 2{\alpha }_{22}\underset{0}{\overset{1}{\int }}v{v}_t\left(a_2{d}_1^{-1}v+b_{21}uv+b_{22}{v}^2+e_2{d}_{1}u{v}^2\right)dx\hfill \\ \le \frac{{\alpha }_{22}^2{a}_2^2{d}_1^{-2}}{\epsilon }\underset{0}{\overset{1}{\int }}{v}^{3}dx+\epsilon \underset{0}{\overset{1}{\int }}v{v}_t^{2}dx+\frac{{\alpha }_{22}^2{b}_{21}^2}{\epsilon }\underset{0}{\overset{1}{\int }}{u}^2{v}^{3}dx+\epsilon \underset{0}{\overset{1}{\int }}v{v}_t^{2}dx\hfill \\ +\frac{{\alpha }_{22}^2{b}_{22}^2}{\epsilon }\underset{0}{\overset{1}{\int }}{v}^{5}dx+\epsilon \underset{0}{\overset{1}{\int }}v{v}_t^{2}dx+\frac{{\alpha }_{22}^2{e}_2^2{d}_1^2}{\epsilon }\underset{0}{\overset{1}{\int }}{u}^2{v}^{5}dx+\epsilon \underset{0}{\overset{1}{\int }}v{v}_t^{2}dx\hfill \\ \le \frac{{\alpha }_{22}^2{a}_2^2}{\epsilon }{M}_1^{4/5}{d}_1^{-18/5}{\left(\underset{0}{\overset{1}{\int }}{v}^{7}dx\right)}^{1/5}+\frac{{\alpha }_{22}^2\left(b_{21}^2+b_{22}^2\right)}{\epsilon }{M}_1^{2/5}{d}_1^{-4/5}\left[{\left(\underset{0}{\overset{1}{\int }}{u}^7\mathsf{\text{d}}x\right)}^{3/5}+{\left(\underset{0}{\overset{1}{\int }}{v}^7\mathsf{\text{d}}x\right)}^{3/5}\right]\hfill \\ +\frac{5{\alpha }_{22}^2{e}_2^2}{7\epsilon }{d}_1^2\left(\underset{0}{\overset{1}{\int }}{u}^7\mathsf{\text{d}}x+\underset{0}{\overset{1}{\int }}{v}^7\mathsf{\text{d}}x\right)+2\epsilon \underset{0}{\overset{1}{\int }}v{v}_t^{2}dx,\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\alpha }_{21}\underset{0}{\overset{1}{\int }}v{u}_{t}gdx& \le {\alpha }_{21}\underset{0}{\overset{1}{\int }}v{u}_t\left(a_2{d}_1^{-1}v+b_{21}uv+b_{22}{v}^2+e_2{d}_{1}u{v}^2\right)dx\hfill \\ \le \frac{{\alpha }_{21}^2{a}_2^2{d}_1^{-2}}{2\epsilon }\underset{0}{\overset{1}{\int }}{v}^{3}dx+\frac{\epsilon }{2}\underset{0}{\overset{1}{\int }}v{u}_t^{2}dx+\frac{{\alpha }_{21}^2{b}_{21}^2}{2\epsilon }\underset{0}{\overset{1}{\int }}{u}^2{v}^{3}dx+\frac{\epsilon }{2}\underset{0}{\overset{1}{\int }}v{u}_t^{2}dx\hfill \\ +\frac{{\alpha }_{21}^2{b}_{22}^2}{2\epsilon }\underset{0}{\overset{1}{\int }}{v}^{5}dx+\frac{\epsilon }{2}\underset{0}{\overset{1}{\int }}v{u}_t^{2}dx+\frac{{\alpha }_{21}^2{e}_2^2{d}_1^2}{2\epsilon }\underset{0}{\overset{1}{\int }}{u}^2{v}^{5}dx+\frac{\epsilon }{2}\underset{0}{\overset{1}{\int }}v{u}_t^{2}dx\hfill \\ \le \frac{{\alpha }_{21}^2{a}_2^2}{2\epsilon }{M}_1^{4/5}{d}_1^{-18/5}{\left(\underset{0}{\overset{1}{\int }}{v}^{7}dx\right)}^{1/5}+\frac{{\alpha }_{21}^2\left(b_{21}^2+b_{22}^2\right)}{2\epsilon }{M}_1^{2/5}{d}_1^{-4/5}\left[{\left(\underset{0}{\overset{1}{\int }}{u}^7\mathsf{\text{d}}x\right)}^{3/5}+{\left(\underset{0}{\overset{1}{\int }}{v}^7\mathsf{\text{d}}x\right)}^{3/5}\right]\hfill \\ +\frac{5{\alpha }_{21}^2{e}_2^2}{14\epsilon }{d}_1^2\left(\underset{0}{\overset{1}{\int }}{u}^7\mathsf{\text{d}}x+\underset{0}{\overset{1}{\int }}{v}^7\mathsf{\text{d}}x\right)+2\epsilon \underset{0}{\overset{1}{\int }}v{v}_t^{2}dx.\hfill \end{array}$$

By the above inequalities and the condition (1.9), we have

$$\begin{array}{c}\underset{0}{\overset{1}{\int }}\left[\left(1+2{\alpha }_{11}u+{\alpha }_{12}v\right)u_{t}f+{\alpha }_{12}u{v}_{t}f\right]dx+\underset{0}{\overset{1}{\int }}\left[\left(\xi +{\alpha }_{21}u+2{\alpha }_{22}v\right)v_{t}g+{\alpha }_{21}v{u}_{t}g\right]dx\hfill \\ \le \bar{\lambda }\epsilon \underset{0}{\overset{1}{\int }}\left(u+v\right)\left(u_t^2+v_t^2\right)dx+\frac{C_4}{\epsilon }\left(1+{\xi }^2\right)M_0{d}_1^{-3}+\frac{C_5}{\epsilon }\left(1+{\xi }^2+d_1^{-2}\right)M_1^{4/5}{d}_1^{-8/5}{\left[\underset{0}{\overset{1}{\int }}\left(u^7+v^7\right)dx\right]}^{1/5}\hfill \\ +\frac{C_5}{\epsilon }\left(1+{\xi }^2+d_1^{-2}\right)M_1^{2/5}{d}_1^{6/5}{\left[\underset{0}{\overset{1}{\int }}\left(u^7+v^7\right)dx\right]}^{3/5}+\frac{C_6}{\epsilon }{d}_1^2\underset{0}{\overset{1}{\int }}\left(u^7+v^7\right)dx,\hfill \end{array}$$

(2.16)

where $\bar{\lambda }$ is a constant. Choose a small enough positive number ε(α _ij , a _i , b _ij , e _i )(i, j = 1, 2), such that $\bar{\lambda }\epsilon <C_3$.

Substituting inequalities (2.15) and (2.16) into (2.13), one can obtain

$$\frac{1}{2}{ȳ}^{\prime }\left(t\right)\le -\frac{1}{2}\underset{0}{\overset{1}{\int }}{P}_{xx}^{2}dx-\frac{\xi }{2}\underset{0}{\overset{1}{\int }}{Q}_{xx}^{2}dx+B_1{K}_2{d}_1^{-3}+B_2{d}_1^{2}Y+B_3{K}_3{d}_1^{6/5}{Y}^{3/5}+B_4{K}_4{d}_1^{-8/5}{Y}^{1/5},$$

(2.17)

where $Y={\int }_0^1\left(u^7+v^7\right)dx$, $K_2=\left(1+{\xi }^2\right)M_0$, $K_3=\left(1+{\xi }^2+d_1^{-2}\right)M_1^{2/5}$, $K_4=\left(1+{\xi }^2+d_1^{-2}\right)M_1^{4/5}$.

Clearly,

$$P\ge {\alpha }_{11}{u}^2,Q\ge {\alpha }_{22}{v}^2.$$

It follows from (2.12) and (2.3) to functions P, Q that

$$\begin{array}{c}Y\le B_5\underset{0}{\overset{1}{\int }}\left(P^{7/2}+Q^{7/2}\right)dx\le B_6{K}_1^{11/6}{d}_1^{-11/6}{ȳ}^{5/6}+B_6{K}_1^{7/2}{d}_1^{-7/2},\hfill \\ Y^{1/5}\le B_7{K}_1^{11/30}{d}_1^{-11/30}{ȳ}^{1/6}+B_7{K}_1^{7/10}{d}_1^{-7/10},\hfill \\ Y^{3/5}\le B_8{K}_1^{11/10}{d}_1^{-11/10}{ȳ}^{1/2}+B_8{K}_1^{21/10}{d}_1^{-21/10}.\hfill \end{array}$$

(2.18)

Moreover, one can obtain by (2.4) and (2.12)

$$-\frac{1}{2}\underset{0}{\overset{1}{\int }}{P}_{xx}^{2}dx-\frac{\xi }{2}\underset{0}{\overset{1}{\int }}{Q}_{xx}^{2}dx\le -B_9\mathsf{\text{ min}}\left\{1,\xi \right\}{K}_1^{-4/3}{d}_1^{4/3}{ȳ}^{5/3}+\left(1+\xi \right)K_1^2{d}_1^{-2}.$$

(2.19)

Combining (2.16), (2.18), and (2.19), we have

$$\begin{array}{cc}\hfill \frac{1}{2}{ȳ}^{\prime }\left(t\right)\le & -A_1\mathsf{\text{min}}\left\{1,\xi \right\}{K}_1^{-4/3}{d}_1^{4/3}{ȳ}^{5/3}\hfill \\ +A_2\left[\left(1+\xi \right)K_1^2{d}_1^{-2}+K_2{d}_1^{-3}+K_1^{7/2}{d}_1^{-3/2}+K_1^{21/10}{K}_3{d}_1^{-9/10}+K_1^{7/10}{K}_4{d}_1^{-23/10}\right]\hfill \\ +A_3{K}_1^{11/6}{d}_1^{1/6}{ȳ}^{5/6}+A_4{K}_1^{11/10}{K}_3{d}_1^{1/10}{ȳ}^{1/2}+A_5{K}_1^{11/30}{K}_4{d}_1^{-59/30}{ȳ}^{1/6}.\hfill \end{array}$$

(2.20)

Multiplying inequality (2.20) by $d_1^2$, we have

$$\begin{array}{cc}\hfill \frac{1}{2}{y}^{\prime }\left(t\right)\le & -A_1\mathsf{\text{min}}\left\{1,\xi \right\}{K}_1^{-4/3}{y}^{5/3}\hfill \\ +A_2\left[\left(1+\xi \right)K_1^2+K_2{d}_1^{-1}+K_1^{7/2}{d}_1^{1/2}+K_1^{21/10}{K}_3{d}_1^{11/10}+K_1^{7/10}{K}_4{d}_1^{-3/10}\right]\hfill \\ +A_3{K}_1^{11/6}{d}_1^{1/2}{y}^{5/6}+A_4{K}_1^{11/10}{K}_3{d}_1^{11/10}{y}^{1/2}+A_5{K}_1^{11/30}{K}_4{d}_1^{-3/10}{y}^{1/6},\hfill \end{array}$$

(2.21)

where $y={\int }_0^1\left[{\left(d_1{P}_x\right)}^2+{\left(d_1{Q}_x\right)}^2\right]dx$. The inequality (2.21) implies that there exist ${\tilde{\tau }}_2>0$ and positive constant $\widetilde{M_2}$ depending on d _i , α _ij , a _i , b _ij , e _i (i, j = 1, 2), such that

$$\underset{0}{\overset{1}{\int }}{\left(d_1{P}_x\right)}^{2}dx,\underset{0}{\overset{1}{\int }}{\left(d_1{Q}_x\right)}^{2}dx\le \widetilde{M_2},t\ge {\tilde{\tau }}_2.$$

(2.22)

In the case that d₁, d₂ ≥ 1, $\xi \in \left[\underset{-}{d},\bar{d}\right]$, the coefficients of inequality (2.20) can be estimated by some constants which depend on $\underset{-}{d},\bar{d}$, but do not depend on d₁, d₂. So $\widetilde{M_2}$ depends on α _ij , a _i , b _ij , e _i (i, j = 1, 2), $\underset{¯}{d}$ and $\bar{d}$, but it is irrelevant to d₁, d₂, when d₁, d₂ ≥ 1 and $\xi \in \left[\underset{-}{d},\bar{d}\right]$. Since

$$\left(\begin{array}{c}\hfill P_x\hfill \\ \hfill Q_x\hfill \end{array}\right)=\left(\begin{array}{cc}\hfill P_u\hfill & \hfill P_v\hfill \\ \hfill Q_u\hfill & \hfill Q_v\hfill \end{array}\right)\left(\begin{array}{c}\hfill u_x\hfill \\ \hfill v_x\hfill \end{array}\right),$$

we can transform the formulations of u _x , v _x into fraction representations, then distribute the denominators of the absolute value of the fractions to the numerators item and enlarge the term concerning with u _x , v _x to obtain

$$\left|d_1{u}_x\right|+\left|d_1{v}_x\right|\le L\left(\left|d_1{P}_x\right|+\left|d_1{Q}_x\right|\right),0<x<1,t>0,$$

(2.23)

where L is a constant depending only on ξ, α _ij (i, j = 1, 2). After scaling back and contacting estimates (2.22) and (2.23), there exist positive constant M₂ depending on d _i , α _ij , a _i , b _ij , e _i (i, j = 1, 2) and τ₂ > 0, such that

$$\underset{0}{\overset{1}{\int }}{u}_x^{2}dx,\underset{0}{\overset{1}{\int }}{v}_x^{2}dx\le M_2,t\ge {\tau }_2.$$

(2.24)

When d₁, d₂ ≥ 1 and $\xi \in \left[\underset{-}{d},\bar{d}\right]$, M₂ is independent of d₁, d₂.

• (ii) t ≥ 0. Modifying the dependency of the coefficients in inequalities (2.12)-(2.14), namely replacing M₀, M₁ with $M_0^{\prime },M_1^{\prime }$, there exists a positive constant $M_2^{\prime }$ depending on d _i , α _ij , a _i , b _ij , e _i (i, j = 1, 2) and the $W_2^1$-norm of u₀, v₀, such that

  $$\underset{0}{\overset{1}{\int }}{u}_x^{2}dx,\underset{0}{\overset{1}{\int }}{v}_x^{2}dx\le M_2^{\prime },t\ge 0.$$

  (2.24a)

Furthermore, in the case that d₁, d₂ ≥ 1, $\xi \in \left[\underset{-}{d},\bar{d}\right]$, $M_2^{\prime }$, depends on $\underset{-}{d},\bar{d}$, but does not depend on d₁, d₂.

Summarizing estimates (2.5), (2.9), (2.24), and Sobolev embedding theorem, there exist positive constants M, M' depending only on d _i , α _ij , a _i , b _ij , e _i (i, j = 1, 2), such that (1.10) and (1.11) hold. In particular, M, M' depend only on α _ij , a _i , b _ij , e _i (i, j = 1, 2), $\underset{¯}{d}$ and $\bar{d}$, but do not depend on d₁, d₂, when d₁, d₂ ≥ 1 and $\xi \in \left[\underset{-}{d},\bar{d}\right]$. Similarly, according to (2.5)', (2.9)', (2.24)', there exists a positive constant M'' depending on d _i , α _ij , a _i , b _ij , e _i (i, j = 1, 2) and the initial functions u₀, v₀, such that

$${\left|u\left(\cdot ,t\right)\right|}_{1,2},{\left|v\left(\cdot ,t\right)\right|}_{1,2}\le M^{″},t\ge 0.$$

Further, in the case that d₁, d₂ ≥ 1, $\xi \in \left[\underset{-}{d},\bar{d}\right]$, M" depends only on $\underset{-}{d},\bar{d}$, but do not depend on d₁, d₂. Thus, T = +∞.

This completes the proof of Theorem 1.1.

3 Global stability

In order to obtain the uniform convergence of the solution to system (1.2), we recall the following result which can be found in [21].

Lemma 3.1. Let a and b be positive constants. Assume that φ, ψ ∈ C¹ ([a, +∞)), ψ(t) ≥ 0 and φ is bounded from below. If φ'(t) ≤ -bψ(t) and ψ'(t) is bounded from above in [a, +∞), then $\underset{t\to \infty }{\mathsf{\text{lim}}}\psi \left(t\right)=0$.

Proof of Theorem 1.2. Let (u, v) be a solution for the system (1.8) with initial functions u₀(x), v₀(x) ≥ (≢)0. From the strong maximum principle for parabolic equations, it is not hard to verify that u, v > 0 for t > 0. Define the function

$$H\left(u,v\right)=\underset{0}{\overset{1}{\int }}\left(u-u^{*}-u^{*}\mathsf{\text{ ln}}\frac{u}{u^{*}}\right)dx+\underset{0}{\overset{1}{\int }}\left(v-v^{*}-v^{*}\mathsf{\text{ ln}}\frac{v}{v^{*}}\right)dx.$$

Then the time derivative of H(u, v) for the system (1.8) satisfies

$$\begin{array}{cc}\hfill \frac{dH}{dt}& =\underset{0}{\overset{1}{\int }}\frac{u-u^{*}}{u}{u}_{t}dx+\underset{0}{\overset{1}{\int }}\frac{v-v^{*}}{v}{v}_{t}dx\hfill \\ =-\underset{0}{\overset{1}{\int }}\left[\frac{u^{*}}{u^2}\left(d_1+2{\alpha }_{11}u+{\alpha }_{12}v\right)u_x^2+\left(\frac{{\alpha }_{12}{u}^{*}}{u}+\frac{{\alpha }_{21}{v}^{*}}{v}\right)u_x{v}_x+\frac{\alpha v^{*}}{v^2}\left(d_2+{\alpha }_{21}u+2{\alpha }_{22}v\right)v_x^2\right.]dx\hfill \\ -\underset{0}{\overset{1}{\int }}\left[{\left(u-u^{*}\right)}^2\left(b_{11}+e_{1}v\right)+\left(b_{12}+b_{21}+e_1{u}^{*}+e_2{v}^{*}\right)\left(u-u^{*}\right)\left(v-v^{*}\right)+{\left(v-v^{*}\right)}^2\left(b_{22}+e_{2}u\right)\right]dx.\hfill \end{array}$$

(3.1)

The first integrand in the right-hand side of (3.1) is positive definite if

$$4u^{*}{v}^{*}\left(d_1+2{\alpha }_{11}u+{\alpha }_{12}v\right)\left(d_2+{\alpha }_{21}u+2{\alpha }_{22}v\right)>{\left({\alpha }_{12}{u}^{*}v+{\alpha }_{21}{v}^{*}u\right)}^2,$$

(3.2)

and the condition (1.12) implies (3.2). The second integrand in the right-hand side of (3.1) is positive definite if

$$4\left(b_{11}+e_{1}v\right)\left(b_{22}+e_{2}u\right)>{\left(b_{12}+b_{21}+e_1{u}^{*}+e_2{v}^{*}\right)}^2,$$

(3.3)

and the condition (1.13) implies (3.3). Consequently, there exists δ > 0, such that

$$\frac{dH}{dt}\le -\delta \underset{0}{\overset{1}{\int }}\left[{\left(u-u^{*}\right)}^2+{\left(v-v^{*}\right)}^2\right]dx,\frac{dH}{dt}\le 0,\left(u,v\right)\ne \left(u^{*},v^{*}\right).$$

(3.4)

By the maximum-norm estimate in Theorem 1.1 and some tedious calculations, we can verify $\left(d/dt\right){\int }_0^1\left[{\left(u-u^{*}\right)}^2+{\left(v-v^{*}\right)}^2\right]dx$ is bounded from above. Then from Lemmas 3.1 and (3.4), we obtain

$$\underset{t\to \infty }{\mathsf{\text{lim}}}\underset{0}{\overset{1}{\int }}{\left(u-u^{*}\right)}^{2}dx=\underset{t\to \infty }{\mathsf{\text{lim}}}\underset{0}{\overset{1}{\int }}{\left(v-v^{*}\right)}^{2}dx=0.$$

(3.5)

It follows from (3.5) and Gagliardo-Nirenberg-type inequality $|u{|}_{\infty }\le C|u{|}_{1,2}^{1/2}|u{|}_2^{1/2}$ that (u, v) converges uniformly to (u*, v*). By the fact that H(u, v) is decreasing for t ≥ 0, it is obvious that (u*, v*) is globally asymptotically stable. So the proof of Theorem 1.2 is completed.