A blow-up result for a system of coupled viscoelastic equations with arbitrary positive initial energy

Abstract

This article is devoted to a study of the blow-up result for a system of coupled viscoelastic wave equations. By establishing a new auxiliary function and using the reduction to absurdity method, we obtain some sufficient conditions on initial data such that the solution blows up in finite time at arbitrarily high initial energy. This work generalizes and improves earlier results in the literature.

Introduction

In this article, we investigate the blow-up property of the coupled viscoelastic wave equations of the form

$$\begin{aligned} \textstyle\begin{cases} \vert u_{t} \vert ^{\rho }u_{tt}-\Delta u+\int _{0}^{t}g_{1}(t-\tau )\triangle u( \tau )\,d\tau + \vert u_{t} \vert ^{m-2}u_{t}=f_{1}(u, v), \quad x\in \Omega , t>0, \\ \vert v_{t} \vert ^{\rho }v_{tt}-\Delta v+\int _{0}^{t}g_{2}(t-\tau )\triangle v( \tau )\,d\tau + \vert v_{t} \vert ^{r-2}v_{t}=f_{2}(u, v), \quad x\in \Omega , t>0, \\ u(x,t)=v(x,t)=0, \quad x\in \partial \Omega , t>0, \\ u(x,0)=u_{0}(x),\qquad u_{t}(x,0)=u_{1}(x),\quad x\in \Omega , \\ v(x,0)=v_{0}(x), \qquad v_{t}(x,0)=v_{1}(x), \quad x\in \Omega . \end{cases}\displaystyle \end{aligned}$$

(1.1)

Here Ω is a bounded domain of $R^{n}$ with a smooth boundary ∂Ω. $\rho >0$, $g_{1}$ and $g_{2}$ are the kernel of memory terms, the nonlinear terms $f_{1}$ and $f_{2}$ will be specified later. The problem of (1.1) has been considered by many mathematics researchers and results in connection with blow-up and decay have been extensively established.

For single viscoelastic wave equation, Messaoudi [11] discussed the following equation:

$$ \textstyle\begin{cases} u_{tt}-\Delta u+\int _{0}^{t} g(t-\tau )\triangle u(\tau )\,d\tau +u_{t} \vert u_{t} \vert ^{m-2}=u \vert u \vert ^{p-2},\quad x \in \Omega , t>0, \\ u(x, t)=0,\quad x\in \partial \Omega , t\geq 0, \\ u(x, 0)=u_{0}(x),\qquad u_{t}(x,0)=u_{1}(x), \quad x\in \Omega , \end{cases} $$

(1.2)

where $m\geq 1$, $p\geq 2$, Ω is a bounded domain. The author studied the interaction between the weak damping term $u_{t}|u_{t}|^{m-2}$ and the nonlinear source term $u|u|^{p-2}$, which was first considered by Levine [9, 10] when $m=1$, and found, under suitable conditions on g and initial data, that the solutions exist globally for any initial data if $m\geq p$ and blow up in finite time with negative initial energy if $p>m$. This blow-up result has been pushed by the same author in [12], to certain solutions with positive initial energy. Recently, Song [19] proved, by using the reduction to absurdity method, that the solutions of Eq. (1.2) blow up in finite time with the initial data have arbitrarily high initial energy. In the case when the nonlinear damping term $u_{t}|u_{t}|^{m-2}$ is replaced by the strong damping term $-\triangle u_{t}$ in Eq. (1.2), Song and Zhong [21] showed, by using the potential well theory introduced by Payne and Sattinger [16], that a blow-up result for solutions with positive initial energy. Later, Song and Xue [20] improved the blow-up result in which the initial data have arbitrarily high initial energy. In [22], Xu and Lian studied a nonlinear wave equation with weak and strong damping terms and logarithmic source term, they established the local existence of weak solution, showed the global existence, energy decay in the framework of potential well and obtained the blow-up of the solution with sub-critical initial energy. Furthermore, they in parallel extend all the conclusions for the sub-critical case to the critical case by scaling technique. Besides, a high energy infinite time blow-up result is established. Within a similar potential well framework, the semilinear pseudo parabolic equation [24] and parabolic system [23] were discussed in depth.

In the same direction, Song [18] discussed the following initial-boundary value problem:

$$ \vert u_{t} \vert ^{\rho }u_{tt}- \triangle u+ \int _{0}^{t} g(t-\tau )\triangle u( \tau )\,d\tau +u_{t} \vert u_{t} \vert ^{m-2}=u \vert u \vert ^{p-2}, \quad x\in \Omega , t>0, $$

(1.3)

where Ω is a bounded domain and $\rho >0$, $m\geq 1$, $p>2$. By modifying the method used in Messaoudi [12], the author proved that the solution blows up in finite time with initial data have positive initial energy. Recently, He and Song [8] pushed the blow-up result to certain solutions with arbitrary positive initial energy. When the nonlinear damping term $u_{t}|u_{t}|^{m-2}$ is substituted by the strong damping term $-\triangle u_{t}$, Hao et al. [7], inspired by the method used in Song [19], proved that solutions with negative initial energy as well as positive initial energy blow-up in finite time provided $p>\rho +2$, and obtained, by using the perturbed energy functional technique, that solutions exist globally for any initial data provided $p\leq \rho +2$.

For a blow-up result in systems of hyperbolic equations, the coupled system

$$ \textstyle\begin{cases} u_{tt}-\Delta u+u_{t} \vert u_{t} \vert ^{m-1}=f_{1}(u,v), \quad x\in \Omega , t>0, \\ v_{tt}-\Delta v+v_{t} \vert v_{t} \vert ^{r-1}=f_{2}(u,v), \quad x\in \Omega , t>0, \\ u(x,t)=v(x,t)=0, \quad x\in \partial \Omega , t>0, \\ u(x,0)=u_{0}(x), \qquad u_{t}(x,0)=u_{1}(x),\quad x\in \Omega , \\ v(x,0)=v_{0}(x), \qquad v_{t}(x,0)=v_{1}(x),\quad x\in \Omega , \end{cases} $$

(1.4)

was considered by Agre and Rammaha [2], where $m,r\geq 1$ and Ω is a bounded domain with smooth boundary. The authors found, by using the same method as in [4], that any weak solution blows up in finite time with negative initial energy. Furthermore, Said-Houari [17] extended this blow-up result to positive initial energy.

In the presence of the memory term, Han and Wang [5] considered the following system of viscoelastic equations:

$$ \textstyle\begin{cases} u_{tt}-\Delta u+\int _{0}^{t} g(t-\tau )\triangle u(\tau )\,d\tau +u_{t} \vert u_{t} \vert ^{m-1}=f_{1}(u,v),\quad x \in \Omega , t>0, \\ v_{tt}-\Delta v+\int _{0}^{t} h(t-\tau )\triangle v(\tau )\,d\tau +v_{t} \vert v_{t} \vert ^{r-1}=f_{2}(u,v),\quad x \in \Omega , t>0, \\ u(x,t)=v(x,t)=0,\quad x\in \partial \Omega , t>0, \\ u(x,0)=u_{0}(x), \qquad u_{t}(x,0)=u_{1}(x),\quad x\in \Omega , \\ v(x,0)=v_{0}(x), \qquad v_{t}(x,0)=v_{1}(x), \quad x\in \Omega , \end{cases} $$

(1.5)

They obtained the local existence, global existence, uniqueness and a blow-up result for certain solutions with negative initial energy. In [13], Messaoudi and Said-Houari extended this blow-up result to certain solutions with positive initial energy. Later, Zhao and Wang [1] proved the finite time blow-up of solutions whose initial data have arbitrarily high initial energy. In the same nature, Mustafa [14] considered a coupled system of nonlinear viscoelastic equations, he proved the well-posedness and established a generalized stability result for this system. More relevant knowledge we refer the reader to the literature [15].

As far as we know, the problem of the blow-up phenomenon for system (1.1) with arbitrary positive initial energy has not been considered. Our aim in this paper is to extend the research method for the blow-up phenomena used in [8] to the couple viscoelastic wave system (1.1), while we should handle the additional difficulty caused by damping term, viscoelastic term and source term. In order to overcome the difficulty, we construct a suitable auxiliary functions $(u,\frac{u_{t}|u_{t}|^{\rho }}{\rho +1})+(v, \frac{v_{t}|v_{t}|^{\rho }}{\rho +1})-(\frac{1}{\varepsilon _{1}^{*}})^{ \frac{\gamma ^{*}}{\gamma ^{*}-1}}\frac{\gamma -1}{\gamma } E(t)$ and combine the reduction to absurdity method to derive contradiction, namely, we find suitable conditions on initial data such that the solution of (1.1) blows up in finite time at arbitrary high initial energy level.

This article is organized as follows. In Sect. 2, we present some material needed for our work. Section 3 is devoted to the blow-up result.

Preliminaries

In this section, we give some material needed for our work. Firstly, let us make the following assumptions.

• (A1) $g_{i}:R_{+}\rightarrow R_{+}$ (for $i=1,2$) are non-increasing differentiable functions satisfying

  $$ g_{i}(0)>0,\quad 1- \int ^{\infty }_{0}g_{i}(\tau )\,d\tau =l_{i}>0. $$

• (A2) For nonlinear terms, we assume that

  $$\begin{aligned} &0< \rho < +\infty , n=1,2,\quad 0< \rho \leq \frac{2}{n-2}, \quad n \geq 3, \\ &2\leq m,r< +\infty , n=1,2,\quad 2\leq m,r\leq \frac{2n}{n-2},\quad n \geq 3. \end{aligned}$$

• (A3) For the functions $f_{1}$ and $f_{2}$, we note that

  $$\begin{aligned} &f_{1}(u,v)= \bigl[a \vert u+v \vert ^{2(p+1)}(u+v)+b \vert u \vert ^{p}u \vert v \vert ^{p+2} \bigr], \\ &f_{2}(u,v)= \bigl[a \vert u+v \vert ^{2(p+1)}(u+v)+b \vert v \vert ^{p}v \vert u \vert ^{p+2} \bigr],\quad a, b>0, \end{aligned}$$

  where

  $$ -1< p< +\infty , n=1,2, \quad -1< p\leq \frac{3-n}{n-2},\quad n\geq 3. $$

  It is easy to verify that

  $$\begin{aligned} uf_{1}(u,v)+vf_{2}(u,v)=2(p+2)F(u,v),\quad \forall (u,v)\in \mathbb{R}^{2}, \end{aligned}$$

  where

  $$ F(u,v)=\frac{1}{2(p+2)} \bigl[a \vert u+v \vert ^{2(p+2)}+2b \vert uv \vert ^{p+2} \bigr]. $$

At present, we state the following local existence theorem which can be proved by combining the arguments in [3, 6]. Here we omit the proof.

Theorem 2.1

Suppose that (A1), (A2), (A3) hold and $u_{0}, v_{0} \in H^{1}_{0}(\Omega )$, $u_{1}, v_{1} \in L^{2}(\Omega )$ are given, then system (1.1) possesses a unique local solution $(u,v)$ such that

$$\begin{aligned} &(u,v)\in C \bigl([0,T];H^{1}_{0}(\Omega ) \bigr)\times C \bigl([0,T];H^{1}_{0}( \Omega ) \bigr), \\ & (u_{t},v_{t})\in C \bigl([0,T];L^{2}(\Omega ) \bigr)\cap L^{m} \bigl( \Omega \times [0,T] \bigr)\times C \bigl([0,T];L^{2}(\Omega ) \bigr)\cap L^{r} \bigl(\Omega \times [0,T] \bigr), \end{aligned}$$

for the maximum existence time $T>0$, where $T\in (0,\infty ]$.

The energy of the system (1.1) is

$$\begin{aligned} E(t)={}&\frac{1}{\rho +2} \bigl( \Vert u_{t} \Vert ^{\rho +2}_{\rho +2}+ \Vert v_{t} \Vert ^{ \rho +2}_{\rho +2} \bigr)+\frac{1}{2} \biggl(1- \int ^{t}_{0}g_{1}(\tau )\,d\tau \biggr) \Vert \nabla u \Vert ^{2}_{2}- \int _{\Omega }F(u,v)\,dx \\ &{} +\frac{1}{2} \bigl[(g_{1}\circ \nabla u)+(g_{2} \circ \nabla v) \bigr]+\frac{1}{2} \biggl(1- \int ^{t}_{0}g_{2}(\tau )\,d\tau \biggr) \Vert \nabla v \Vert ^{2}_{2}, \end{aligned}$$

where

$$ (g\circ \nabla v) (t)= \int _{0}^{t}g(t-\tau ) \bigl\Vert v(t)-v(\tau ) \bigr\Vert ^{2}_{2}\,d\tau . $$

Lemma 2.2

Assume (A1) holds, and let $(u,v)$ be a solution of system (1.1), then $E(t)$ is non-increasing, namely

$$ \frac{dE(t)}{dt}\leq - \Vert u_{t} \Vert ^{m}_{m}- \Vert v_{t} \Vert ^{r}_{r}\leq 0,\quad \forall t\geq 0. $$

(2.1)

Proof

Multiplying the first two equations in system (1.1) by $u_{t}$, $v_{t}$, respectively, and then integrating over Ω, we get

$$\begin{aligned} &\frac{d}{dt} \biggl[\frac{1}{\rho +2} \bigl( \Vert u_{t} \Vert ^{\rho +2}_{\rho +2}+ \Vert v_{t} \Vert ^{ \rho +2}_{\rho +2} \bigr)+\frac{1}{2} \bigl( \Vert \nabla u \Vert ^{2}_{2}+ \Vert \nabla v \Vert ^{2}_{2} \bigr)- \int _{\Omega }F(u,v)\,dx \biggr] \\ &\quad =- \Vert u_{t} \Vert ^{m}_{m}- \Vert v_{t} \Vert ^{r}_{r}+ \int ^{t}_{0}g_{1}(t-\tau ) \int _{\Omega }\nabla u_{t}(t)\cdot \nabla u(\tau )\,dx\,d\tau \\ &\qquad {} + \int ^{t}_{0}g_{2}(t-\tau ) \int _{\Omega }\nabla v_{t}(t)\cdot \nabla v(\tau )\,dx\,d\tau , \end{aligned}$$

(2.2)

where

$$\begin{aligned} & \int ^{t}_{0}g_{1}(t-\tau ) \int _{\Omega }\nabla u_{t}(t)\cdot \nabla u(\tau )\,dx\,d\tau \\ &\quad = \int ^{t}_{0}g_{1}(t-\tau ) \int _{\Omega }\nabla u_{t}(t)\cdot \bigl[ \nabla u(\tau )-\nabla u(t) \bigr]\,dx\,d\tau \\ &\qquad {} + \int ^{t}_{0}g_{1}(t-\tau ) \int _{\Omega }\nabla u_{t}(t)\cdot \nabla u(t)\,dx\,d\tau \\ &\quad =-\frac{1}{2} \int ^{t}_{0}g_{1}(t-\tau ) \biggl( \frac{d}{dt} \int _{ \Omega } \bigl\vert \nabla u(\tau )-\nabla u(t) \bigr\vert ^{2}\,dx \biggr)\,d\tau \\ &\qquad {} + \int ^{t}_{0}g_{1}( \tau ) \biggl( \frac{d}{dt}\frac{1}{2} \int _{\Omega } \bigl\vert \nabla u(t) \bigr\vert ^{2}\,dx \biggr)\,d\tau \\ &\quad =-\frac{1}{2}\frac{d}{dt} \biggl[ \int ^{t}_{0}g_{1}(t-\tau ) \int _{ \Omega } \bigl\vert \nabla u(\tau )-\nabla u(t) \bigr\vert ^{2}\,dx\,d\tau \biggr] \\ &\qquad {}+\frac{1}{2} \frac{d}{dt} \biggl[ \int ^{t}_{0}g_{1}(\tau ) \int _{\Omega } \bigl\vert \nabla u(t) \bigr\vert ^{2}\,dx\,d \tau \biggr] \\ &\qquad {} +\frac{1}{2} \int _{0}^{t}g_{1}'(t-\tau ) \int _{\Omega } \bigl\vert \nabla u( \tau )-\nabla u(t) \bigr\vert ^{2}\,dx\,d\tau -\frac{1}{2}g_{1}(t) \int _{\Omega } \bigl\vert \nabla u(t) \bigr\vert ^{2}\,dx. \end{aligned}$$

(2.3)

Similarly

$$\begin{aligned} & \int ^{t}_{0}g_{2}(t-\tau ) \int _{\Omega }\nabla v_{t}(t)\cdot \nabla v(\tau )\,dx\,d\tau \\ &\quad =-\frac{1}{2}\frac{d}{dt} \biggl[ \int ^{t}_{0}g_{2}(t-\tau ) \int _{ \Omega } \bigl\vert \nabla v(\tau )-\nabla v(t) \bigr\vert ^{2}\,dx\,d\tau \biggr] \\ &\qquad {}+\frac{1}{2} \frac{d}{dt} \biggl[ \int ^{t}_{0}g_{2}(\tau ) \int _{\Omega } \bigl\vert \nabla v(t) \bigr\vert ^{2}\,dx\,d \tau \biggr] \\ &\qquad {} +\frac{1}{2} \int _{0}^{t}g_{2}'(t-\tau ) \int _{\Omega } \bigl\vert \nabla v( \tau )-\nabla v(t) \bigr\vert ^{2}\,dx\,d\tau -\frac{1}{2}g_{2}(t) \int _{\Omega } \bigl\vert \nabla v(t) \bigr\vert ^{2}\,dx. \end{aligned}$$

(2.4)

By inserting (2.3) and (2.4) into (2.2), and combining (A1), we can obtain

$$\begin{aligned} &\frac{d}{dt} \biggl[\frac{1}{\rho +2} \bigl( \Vert u_{t} \Vert ^{\rho +2}_{\rho +2}+ \Vert v_{t} \Vert ^{ \rho +2}_{\rho +2} \bigr)+\frac{1}{2}\biggl(1- \int ^{t}_{0}g_{1}(\tau )\,d\tau \biggr) \Vert \nabla u \Vert ^{2}_{2} \\ &\qquad {}+\biggl(1- \int ^{t}_{0}g_{2}(\tau )\,d\tau \biggr) \Vert \nabla v \Vert ^{2}_{2} \\ &\qquad {} +\frac{1}{2} \bigl[(g_{1}\circ \nabla u)+(g_{2} \circ \nabla v) \bigr]- \int _{\Omega }F(u,v)\,dx \biggr] \\ &\quad =- \Vert u_{t} \Vert ^{m}_{m}- \Vert v_{t} \Vert ^{r}_{r}+\frac{1}{2} \bigl[ \bigl(g_{1}'\circ \nabla u\bigr)+\bigl(g_{2}' \circ \nabla v\bigr) \bigr]-\frac{1}{2}g_{1}(t) \Vert \nabla u \Vert ^{2}_{2}- \frac{1}{2}g_{2}(t) \Vert \nabla v \Vert ^{2}_{2}, \end{aligned}$$

namely

$$\begin{aligned} E'(t)={}&{-} \Vert u_{t} \Vert ^{m}_{m}- \Vert v_{t} \Vert ^{r}_{r}+\frac{1}{2} \bigl[\bigl(g_{1}' \circ \nabla u\bigr)+\bigl(g_{2}'\circ \nabla v\bigr) \bigr]\-\frac{1}{2}g_{1}(t) \Vert \nabla u \Vert ^{2}_{2}-\frac{1}{2}g_{2}(t) \Vert \nabla v \Vert ^{2}_{2} \\ \leq{}& {-} \Vert u_{t} \Vert ^{m}_{m}- \Vert v_{t} \Vert ^{r}_{r}\leq 0. \end{aligned}$$

(2.5)

 □

Lemma 2.3

([1], Lemma 2)

There exist two positive constants $c_{0}$ and $c_{1}$ such that

$$ \frac{c_{0}}{2(p+2)} \bigl( \vert u \vert ^{2(p+2)}+ \vert v \vert ^{2(p+2)} \bigr)\leq F(u,v) \leq \frac{c_{1}}{2(p+2)} \bigl( \vert u \vert ^{2(p+2)}+ \vert v \vert ^{2(p+2)} \bigr). $$

Lemma 2.4

Assume $0<\pi <\alpha <\gamma$ hold, then we have the following inequality:

$$ \Vert u \Vert ^{\alpha }_{\alpha }\leq \Vert u \Vert ^{\pi }_{\pi }+ \Vert u \Vert ^{\gamma }_{\gamma }. $$

Proof

(1) When $|u|>1$, then $\int _{\Omega }|u|^{\alpha }\,dx\leq \int _{\Omega }|u|^{\gamma }\,dx\leq \int _{\Omega }|u|^{\pi }\,dx+\int _{\Omega }|u|^{\gamma }\,dx$ is ture.

(2) When $|u|\leq 1$, then $\int _{\Omega }|u|^{\alpha }\,dx\leq \int _{\Omega }|u|^{\pi }\,dx\leq \int _{\Omega }|u|^{\pi }\,dx+\int _{\Omega }|u|^{\gamma }\,dx$ is ture, where $|\cdot |$ represents the absolute value. □

Lemma 2.5

Assume $0<\varrho <\varsigma$ hold, then we have the following inequality:

$$ \Vert u \Vert ^{\varrho }_{\varsigma }< \Vert u \Vert ^{\varsigma }_{\varsigma }+1. $$

Proof

(1) Assume $\|u\|_{\varsigma }>1$ hold, then $\|u\|^{\varrho }_{\varsigma }<\|u\|^{\varsigma }_{\varsigma }<\|u\|^{\varsigma }_{\varsigma }+1$ is ture.

(2) Assume $\|u\|_{\varsigma }\leq 1$ hold, then $\|u\|^{\varrho }_{\varsigma }<\|u\|^{\varsigma }_{\varsigma }+1$ is ture. □

In order to obtain our main result, we need the following lemma which presents the same one of (He and Song [8] Lemma 2.2) with suitable modification.

Lemma 2.6

([8], Lemma 2.2)

Assume that $2<\rho +2<\min \{m,r\}$ and $\max \{m,r\}<2(p+2)$. Assume further that (A1)–(A3) hold and $g_{i}$ ($i=1,2$) satisfying (3.1). If there exists a number $t_{0}\geq 0$ such that $E(t_{0})<0$, then the solution of the system (1.1) blows up in finite time.

Blow-up result

In this section, we discuss the blow-up phenomenon.

Theorem 3.1

Assume the conditions (A1)–(A3) and $2<\rho +2<\min \{m,r\}$, $\max \{m,r\}<2(p+2)$ hold, and $g_{i}$ satisfies the condition

$$ \max \biggl\{ \int ^{\infty }_{0}g_{1}(\tau )\,d\tau , \int ^{\infty }_{0}g_{2}( \tau )\,d\tau \biggr\} < \frac{p+1}{p+1+\frac{1}{4(p+2)}}, $$

(3.1)

for $i=1,2$. Let $(u,v)$ be the solution of system (1.1), satisfying

$$ \int _{\Omega }u(0)\frac{u_{t}(0) \vert u_{t}(0) \vert ^{\rho }}{\rho +1}\,dx+ \int _{\Omega }v(0)\frac{v_{t}(0) \vert v_{t}(0) \vert ^{\rho }}{\rho +1}\,dx>ME(0)>0, $$

(3.2)

then $(u,v)$ blows up in finite time, where

$$ M=\biggl(\frac{1}{\varepsilon _{1}^{*}}\biggr)^{\frac{\gamma ^{*}}{\gamma ^{*}-1}} \frac{\gamma -1}{\gamma } $$

$\gamma =\max \{m,r\}$, $\gamma ^{*}=\min \{m,r\}$, $\varepsilon ^{*}_{1}>0$ is a constant such that $(\frac{1}{\varepsilon _{1}})^{\frac{\gamma ^{*}}{\gamma ^{*}-1}} \frac{\gamma -1}{\gamma }\geq \frac{2(p+2)(1-\varepsilon )}{\beta }$, $\varepsilon \in (0,1)$ is a small enough constant such that

$$\begin{aligned} & \kappa _{1}(\varepsilon ) = \bigl((p+2) (1-\varepsilon )-1 \bigr)l_{1} - \frac{1}{4(p+2)(1-\varepsilon )}(1-l_{1})>0, \\ &\kappa _{2}(\varepsilon )= \bigl((p+2) (1-\varepsilon )-1 \bigr)l_{2} - \frac{1}{4(p+2)(1-\varepsilon )}(1-l_{2})>0, \\ &\alpha =\min \biggl\{ \kappa _{1}(\varepsilon )\lambda - \frac{\varepsilon ^{m}_{1}}{m},\kappa _{2}(\varepsilon )\lambda - \frac{\varepsilon ^{r}_{1}}{r},c_{0}\varepsilon - \frac{\varepsilon ^{m}_{1}}{m},c_{0} \varepsilon - \frac{\varepsilon ^{r}_{1}}{r}\biggr\} , \\ &\beta =\min \biggl\{ \frac{1}{\rho +1}+ \frac{2(p+2)(1-\varepsilon )}{\rho +2},\alpha \biggr\} , \end{aligned}$$

and λ is the first eigenvalue of −Δ.

Proof

Suppose that $(u,v)$ is a global solution of system (1.1). Multiplying the first two equations of system (1.1) by u and v, respectively, and integrating over Ω, we obtain

$$\begin{aligned}& \bigl( \vert u_{t} \vert ^{\rho }u_{tt},u \bigr)+ \Vert \nabla u \Vert ^{2}_{2}+ \int _{\Omega } \int ^{t}_{0}g_{1}(t- \tau )\Delta u(\tau )\,d\tau u(t)\,dx+ \int _{\Omega }uu_{t} \vert u_{t} \vert ^{m-2}\,dx \\& \quad = \int _{\Omega }uf_{1}(u,v)\,dx, \end{aligned}$$

(3.3)

$$\begin{aligned}& \bigl( \vert v_{t} \vert ^{\rho }v_{tt},v \bigr)+ \Vert \nabla v \Vert ^{2}_{2}+ \int _{\Omega } \int ^{t}_{0}g_{2}(t- \tau )\Delta v(\tau )\,d\tau v(t)\,dx+ \int _{\Omega }vv_{t} \vert v_{t} \vert ^{m-2}\,dx \\& \quad = \int _{\Omega }vf_{2}(u,v)\,dx. \end{aligned}$$

(3.4)

Taking the derivative of $(u,\frac{u_{t}|u_{t}|^{\rho }}{\rho +1})$ and $(v,\frac{v_{t}|v_{t}|^{\rho }}{\rho +1})$, respectively, and combining (3.3) and (3.4), we have

$$\begin{aligned} &\frac{d}{dt}\biggl(u,\frac{u_{t} \vert u_{t} \vert ^{\rho }}{\rho +1}\biggr)= \frac{1}{\rho +1} \bigl\Vert u_{t}(t) \bigr\Vert ^{ \rho +2}_{\rho +2} - \Vert \nabla u \Vert ^{2}_{2}+ \int ^{t}_{0} g_{1}(t-\tau ) \int _{\Omega }\nabla u(\tau )\nabla u(t)\,dx\,d\tau \\ & \hphantom{\frac{d}{dt}\biggl(u,\frac{u_{t} \vert u_{t} \vert ^{\rho }}{\rho +1}\biggr)=}{}- \int _{\Omega }uu_{t} \vert u_{t} \vert ^{m-2}\,dx+ \int _{\Omega }uf_{1}(u,v)\,dx, \end{aligned}$$

(3.5)

$$\begin{aligned} &\frac{d}{dt}\biggl(v,\frac{v_{t} \vert v_{t} \vert ^{\rho }}{\rho +1}\biggr)= \frac{1}{\rho +1} \bigl\Vert v_{t}(t) \bigr\Vert ^{ \rho +2}_{\rho +2} - \Vert \nabla v \Vert ^{2}_{2}+ \int ^{t}_{0} g_{2}(t-\tau ) \int _{\Omega }\nabla v(\tau )\nabla v(t)\,dx\,d\tau \\ &\hphantom{\frac{d}{dt}\biggl(v,\frac{v_{t} \vert v_{t} \vert ^{\rho }}{\rho +1}\biggr)=}{} - \int _{\Omega }vv_{t} \vert v_{t} \vert ^{r-2}\,dx+ \int _{\Omega }vf_{2}(u,v)\,dx. \end{aligned}$$

(3.6)

For the third term on the right side of (3.5), we get

$$\begin{aligned} \int ^{t}_{0} g_{1}(t-\tau ) \int _{\Omega }\nabla u(\tau )\nabla u(t)\,dx\,d \tau ={}& \int ^{t}_{0} g_{1}(t-\tau ) \int _{\Omega }\nabla u(t)\nabla \bigl( u(\tau )-u(t) \bigr)\,dx\,d\tau \\ &{} + \int _{0}^{t} g_{1}(t-\tau ) \bigl\Vert \nabla u(t) \bigr\Vert ^{2}_{2}\,d\tau \\ ={}& \int ^{t}_{0} g_{1}(t-\tau ) \int _{\Omega }\nabla u(t) \bigl(\nabla u( \tau )-\nabla u(t) \bigr)\,dx\,d\tau \\ &{} + \int _{0}^{t} g_{1}(\tau )\,d\tau \bigl\Vert \nabla u(t) \bigr\Vert ^{2}_{2}. \end{aligned}$$

(3.7)

In the same way, we have

$$\begin{aligned} \int ^{t}_{0} g_{2}(t-\tau ) \int _{\Omega }\nabla v(\tau )\nabla v(t)\,dx\,d \tau={}& \int ^{t}_{0} g_{2}(t-\tau ) \int _{\Omega }\nabla v(t) \bigl( \nabla v(\tau )-\nabla v(t) \bigr)\,dx\,d\tau \\ &{} + \int _{0}^{t} g_{2}(\tau )\,d\tau \bigl\Vert \nabla v(t) \bigr\Vert ^{2}_{2}. \end{aligned}$$

(3.8)

Then, inserting (3.7) and (3.8) into (3.5) and (3.6), respectively,

$$\begin{aligned} &\frac{d}{dt}\biggl(u,\frac{u_{t} \vert u_{t} \vert ^{\rho }}{\rho +1}\biggr) \\ &\quad = \frac{1}{\rho +1} \bigl\Vert u_{t}(t) \bigr\Vert ^{ \rho +2}_{\rho +2} - \Vert \nabla u \Vert ^{2}_{2}+ \int ^{t}_{0} g_{1}(t-\tau ) \int _{\Omega }\nabla u(t) \bigl(\nabla u(\tau )-\nabla u(t) \bigr)\,dx\,d\tau \\ &\qquad {} + \int _{0}^{t} g_{1}(\tau )\,d\tau \bigl\Vert \nabla u(t) \bigr\Vert ^{2}_{2}- \int _{\Omega }uu_{t} \vert u_{t} \vert ^{m-2}\,dx+ \int _{\Omega }uf_{1}(u,v)\,dx, \end{aligned}$$

(3.9)

$$\begin{aligned} &\frac{d}{dt}\biggl(v,\frac{v_{t} \vert v_{t} \vert ^{\rho }}{\rho +1}\biggr) \\ &\quad = \frac{1}{\rho +1} \bigl\Vert v_{t}(t) \bigr\Vert ^{ \rho +2}_{\rho +2} - \Vert \nabla v \Vert ^{2}_{2}+ \int ^{t}_{0} g_{2}(t-\tau ) \int _{\Omega }\nabla v(t) \bigl(\nabla v(\tau )-\nabla v(t) \bigr)\,dx\,d\tau \\ &\qquad {} + \int _{0}^{t} g_{2}(\tau )\,d\tau \bigl\Vert \nabla v(t) \bigr\Vert ^{2}_{2}- \int _{\Omega }vv_{t} \vert v_{t} \vert ^{r-2}\,dx+ \int _{\Omega }vf_{2}(u,v)\,dx. \end{aligned}$$

(3.10)

For the third term on the right side of (3.9), applying the Cauchy inequality, we obtain

$$\begin{aligned} & \int ^{t}_{0} g_{1}(t-\tau ) \int _{\Omega }\nabla u(t) \bigl(\nabla u( \tau )-\nabla u(t) \bigr)\,dx\,d\tau \\ & \quad \geq -\frac{2(p+2)(1-\varepsilon )}{2}(g_{1}\circ \nabla u) (t)- \frac{1}{4(p+2)(1-\varepsilon )} \int ^{t}_{0}g_{1}(\tau )\,d\tau \Vert \nabla u \Vert ^{2}_{2}, \end{aligned}$$

(3.11)

where $\varepsilon \in (0,1)$, similarly

$$\begin{aligned} & \int ^{t}_{0} g_{2}(t-\tau ) \int _{\Omega }\nabla v(t) \bigl(\nabla v( \tau )-\nabla v(t) \bigr)\,dx\,d\tau \\ & \quad \geq -\frac{2(p+2)(1-\varepsilon )}{2}(g_{2}\circ \nabla v) (t)- \frac{1}{4(p+2)(1-\varepsilon )} \int ^{t}_{0}g_{2}(\tau )\,d\tau \Vert \nabla v \Vert ^{2}_{2}. \end{aligned}$$

(3.12)

Inserting (3.11) and (3.12) into (3.9) and (3.10), respectively, we can get

$$\begin{aligned} &\frac{d}{dt}\biggl(u,\frac{u_{t} \vert u_{t} \vert ^{\rho }}{\rho +1}\biggr)\geq \frac{1}{\rho +1} \bigl\Vert u_{t}(t) \bigr\Vert ^{\rho +2}_{\rho +2} - \Vert \nabla u \Vert ^{2}_{2} -\frac{2(p+2)(1-\varepsilon )}{2}(g_{1}\circ \nabla u) (t) \\ &\hphantom{\frac{d}{dt}\biggl(u,\frac{u_{t} \vert u_{t} \vert ^{\rho }}{\rho +1}\biggr)\geq}{} + \int _{0}^{t} g_{1}(\tau )\,d\tau \bigl\Vert \nabla u(t) \bigr\Vert ^{2}_{2}- \int _{\Omega }uu_{t} \vert u_{t} \vert ^{m-2}\,dx+ \int _{\Omega }uf_{1}(u,v)\,dx \\ &\hphantom{\frac{d}{dt}\biggl(u,\frac{u_{t} \vert u_{t} \vert ^{\rho }}{\rho +1}\biggr)\geq}{} -\frac{1}{4(p+2)(1-\varepsilon )} \int ^{t}_{0}g_{1}(\tau )\,d\tau \Vert \nabla u \Vert ^{2}_{2}, \end{aligned}$$

(3.13)

$$\begin{aligned} & \frac{d}{dt}\biggl(v,\frac{v_{t} \vert v_{t} \vert ^{\rho }}{\rho +1}\biggr)\geq \frac{1}{\rho +1} \bigl\Vert v_{t}(t) \bigr\Vert ^{\rho +2}_{\rho +2} - \Vert \nabla v \Vert ^{2}_{2} -\frac{2(p+2)(1-\varepsilon )}{2}(g_{2}\circ \nabla v) (t) \\ &\hphantom{\frac{d}{dt}\biggl(v,\frac{v_{t} \vert v_{t} \vert ^{\rho }}{\rho +1}\biggr)\geq}{} + \int _{0}^{t} g_{2}(\tau )\,d\tau \bigl\Vert \nabla v(t) \bigr\Vert ^{2}_{2}- \int _{\Omega }vv_{t} \vert v_{t} \vert ^{r-2}\,dx+ \int _{\Omega }vf_{2}(u,v)\,dx \\ &\hphantom{\frac{d}{dt}\biggl(v,\frac{v_{t} \vert v_{t} \vert ^{\rho }}{\rho +1}\biggr)\geq}{} -\frac{1}{4(p+2)(1-\varepsilon )} \int ^{t}_{0}g_{2}(\tau )\,d\tau \Vert \nabla v \Vert ^{2}_{2}. \end{aligned}$$

(3.14)

Adding (3.13) and (3.14), we derive that

$$\begin{aligned} &\frac{d}{dt}\biggl(u,\frac{u_{t} \vert u_{t} \vert ^{\rho }}{\rho +1}\biggr)+ \frac{d}{dt}\biggl(v, \frac{v_{t} \vert v_{t} \vert ^{\rho }}{\rho +1}\biggr) \\ &\quad \geq \frac{1}{\rho +1} \bigl( \bigl\Vert u_{t}(t) \bigr\Vert ^{ \rho +2}_{\rho +2}+ \bigl\Vert v_{t}(t) \bigr\Vert ^{\rho +2}_{\rho +2} \bigr)+2(p+2) \int _{\Omega }F(u,v)\,dx \\ &\qquad {} -\biggl(1- \int ^{t}_{0}g_{1}(\tau )\,d\tau \biggr) \Vert \nabla u \Vert ^{2}_{2}-\biggl(1- \int _{0}^{t}g_{2}(\tau )\,d\tau \biggr) \Vert \nabla v \Vert ^{2}_{2} \\ &\qquad {} -\frac{2(p+2)(1-\varepsilon )}{2} \bigl((g_{1}\circ \nabla u) (t)+(g_{2} \circ \nabla v) (t) \bigr) \\ &\qquad {} -\frac{1}{4(p+2)(1-\varepsilon )} \biggl( \int ^{t}_{0}g_{1}(\tau )\,d\tau \Vert \nabla u \Vert ^{2}_{2}+ \int ^{t}_{0}g_{2}(\tau ) \,d\tau \Vert \nabla v \Vert ^{2}_{2} \biggr) \\ &\qquad {} - \int _{\Omega }uu_{t} \vert u_{t} \vert ^{m-2}\,dx- \int _{\Omega }vv_{t} \vert v_{t} \vert ^{r-2}\,dx. \end{aligned}$$

(3.15)

Adding $2(p+2)(1-\varepsilon )E(t)$ on the right side of (3.15), we can get

$$\begin{aligned} &\frac{d}{dt}\biggl(u,\frac{u_{t} \vert u_{t} \vert ^{\rho }}{\rho +1}\biggr)+ \frac{d}{dt}\biggl(v, \frac{v_{t} \vert v_{t} \vert ^{\rho }}{\rho +1}\biggr) \\ &\quad \geq \biggl( \frac{1}{\rho +1}+ \frac{2(p+2)(1-\varepsilon )}{\rho +2} \biggr) \bigl( \bigl\Vert u_{t}(t) \bigr\Vert ^{\rho +2}_{ \rho +2}+ \bigl\Vert v_{t}(t) \bigr\Vert ^{\rho +2}_{\rho +2} \bigr) \\ &\qquad {} +2(p+2)\varepsilon \int _{\Omega }F(u,v) -2(p+2) (1-\varepsilon )E(t) \\ &\qquad {} + \bigl((p+2) (1-\varepsilon )-1 \bigr) \biggl(1- \int ^{t}_{0}g_{1}(\tau )\,d\tau \biggr) \Vert \nabla u \Vert ^{2}_{2} \\ &\qquad {} + \bigl((p+2) (1-\varepsilon )-1 \bigr) \biggl(1- \int _{0}^{t}g_{2}(\tau )\,d\tau \biggr) \Vert \nabla v \Vert ^{2}_{2} \\ &\qquad {} -\frac{1}{4(p+2)(1-\varepsilon )} \biggl( \int ^{t}_{0}g_{1}(\tau )\,d\tau \Vert \nabla u \Vert ^{2}_{2}+ \int ^{t}_{0}g_{2}(\tau )\,d\tau \Vert \nabla v \Vert ^{2}_{2} \biggr) \\ &\qquad {} - \int _{\Omega }uu_{t} \vert u_{t} \vert ^{m-2}\,dx- \int _{\Omega }vv_{t} \vert v_{t} \vert ^{r-2}\,dx. \end{aligned}$$

(3.16)

For the last two terms on the right side of (3.16), applying the Hölder inequality and the Young inequality, we arrive at

$$\begin{aligned} \biggl\vert \int _{\Omega } \vert u_{t} \vert ^{m-2}u_{t}u\,dx \biggr\vert &\leq \frac{\varepsilon ^{m}_{1} \Vert u \Vert ^{m}_{m}}{m}+\biggl( \frac{1}{\varepsilon _{1}} \biggr)^{\frac{m}{m-1}} \frac{(m-1) \Vert u_{t} \Vert ^{m}_{m}}{m}, \end{aligned}$$

(3.17)

where $\varepsilon _{1}>0$. By Lemma 2.4 and the conditions of the theorem, one can deduce that $\|u\|^{m}_{m}\leq \|u\|^{2}_{2}+\|u\|^{2(p+2)}_{2(p+2)}$, then the inequality (3.17) can be rewritten as

$$\begin{aligned} \biggl\vert \int _{\Omega } \vert u_{t} \vert ^{m-2}u_{t}u\,dx \biggr\vert &\leq \frac{\varepsilon ^{m}_{1}}{m} \bigl( \Vert u \Vert ^{2}_{2}+ \Vert u \Vert ^{2(p+2)}_{2(p+2)} \bigr)+\biggl(\frac{1}{\varepsilon _{1}}\biggr)^{\frac{m}{m-1}} \frac{(m-1) \Vert u_{t} \Vert ^{m}_{m}}{m}, \end{aligned}$$

(3.18)

similarly, we have

$$\begin{aligned} \biggl\vert \int _{\Omega } \vert v_{t} \vert ^{r-2}v_{t}v\,dx \biggr\vert &\leq \frac{\varepsilon ^{r}_{1}}{r} \bigl( \Vert v \Vert ^{2}_{2}+ \Vert v \Vert ^{2(p+2)}_{2(p+2)} \bigr)+\biggl(\frac{1}{\varepsilon _{1}}\biggr)^{\frac{r}{r-1}} \frac{(r-1) \Vert v_{t} \Vert ^{r}_{r}}{r}. \end{aligned}$$

(3.19)

Substituting (3.18) and (3.19) into (3.16), then we have

$$\begin{aligned} &\frac{d}{dt}\biggl(u,\frac{u_{t} \vert u_{t} \vert ^{\rho }}{\rho +1}\biggr)+ \frac{d}{dt}\biggl(v, \frac{v_{t} \vert v_{t} \vert ^{\rho }}{\rho +1}\biggr) +\biggl(\frac{1}{\varepsilon _{1}} \biggr)^{ \frac{m}{m-1}}\frac{(m-1) \Vert u_{t} \Vert ^{m}_{m}}{m}+ \biggl( \frac{1}{\varepsilon _{1}} \biggr)^{\frac{r}{r-1}} \frac{(r-1) \Vert v_{t} \Vert ^{r}_{r}}{r} \\ &\quad \geq \biggl(\frac{1}{\rho +1}+ \frac{2(p+2)(1-\varepsilon )}{\rho +2} \biggr) \bigl( \bigl\Vert u_{t}(t) \bigr\Vert ^{\rho +2}_{ \rho +2}+ \bigl\Vert v_{t}(t) \bigr\Vert ^{\rho +2}_{\rho +2} \bigr) \\ & \qquad {}+2(p+2)\varepsilon \int _{\Omega }F(u,v) -2(p+2) (1-\varepsilon )E(t) \\ &\qquad {} + \bigl((p+2) (1-\varepsilon )-1 \bigr) \biggl(1- \int ^{t}_{0}g_{1}(\tau )\,d\tau \biggr) \Vert \nabla u \Vert ^{2}_{2} \\ &\qquad {} + \bigl((p+2) (1-\varepsilon )-1 \bigr) \biggl(1- \int _{0}^{t}g_{2}(\tau )\,d\tau \biggr) \Vert \nabla v \Vert ^{2}_{2} \\ &\qquad {} -\frac{1}{4(p+2)(1-\varepsilon )} \biggl( \int ^{t}_{0}g_{1}(\tau )\,d\tau \Vert \nabla u \Vert ^{2}_{2}+ \int ^{t}_{0}g_{2}(\tau )\,d\tau \Vert \nabla v \Vert ^{2}_{2} \biggr) \\ &\qquad {} - \frac{\varepsilon ^{r}_{1}}{r} \bigl( \Vert v \Vert ^{2}_{2}+ \Vert v \Vert ^{2(p+2)}_{2(p+2)} \bigr) - \frac{\varepsilon ^{m}_{1}}{m} \bigl( \Vert u \Vert ^{2}_{2}+ \Vert u \Vert ^{2(p+2)}_{2(p+2)} \bigr). \end{aligned}$$

(3.20)

Take $\gamma =\max \{m,r\}$, $\gamma ^{*}=\min \{m,r\}$. Combining with (2.5), we know $E'(t)\leq -\|u_{t}\|^{m}_{m}-\|v_{t}\|^{r}_{r}$, that is, $-E'(t)\geq \|u_{t}\|^{m}_{m}+\|v_{t}\|^{r}_{r}$, then (3.20) can be rewritten as

$$\begin{aligned} &\frac{d}{dt} \biggl(\biggl(u,\frac{u_{t} \vert u_{t} \vert ^{\rho }}{\rho +1}\biggr)+ \biggl(v, \frac{v_{t} \vert v_{t} \vert ^{\rho }}{\rho +1}\biggr) -\biggl(\frac{1}{\varepsilon _{1}} \biggr)^{ \frac{\gamma ^{*}}{\gamma ^{*}-1}}\frac{\gamma -1}{\gamma }E(t) \biggr) \\ &\quad \geq \frac{d}{dt} \biggl(\biggl(u,\frac{u_{t} \vert u_{t} \vert ^{\rho }}{\rho +1}\biggr)+\biggl(v, \frac{v_{t} \vert v_{t} \vert ^{\rho }}{\rho +1}\biggr) \biggr)+\biggl(\frac{1}{\varepsilon _{1}}\biggr)^{ \frac{\gamma ^{*}}{\gamma ^{*}-1}} \frac{\gamma -1}{\gamma }\bigl( \Vert u_{t} \Vert ^{m}_{m}+ \Vert v_{t} \Vert ^{r}_{r}\bigr) \\ &\quad \geq \frac{d}{dt}\biggl(u,\frac{u_{t} \vert u_{t} \vert ^{\rho }}{\rho +1}\biggr)+\frac{d}{dt} \biggl(v, \frac{v_{t} \vert v_{t} \vert ^{\rho }}{\rho +1}\biggr) +\biggl(\frac{1}{\varepsilon _{1}} \biggr)^{ \frac{m}{m-1}}\frac{(m-1) \Vert u_{t} \Vert ^{m}_{m}}{m} \\ &\qquad {}+ \biggl( \frac{1}{\varepsilon _{1}} \biggr)^{\frac{r}{r-1}} \frac{(r-1) \Vert v_{t} \Vert ^{r}_{r}}{r} \\ &\quad \geq \biggl(\frac{1}{\rho +1}+\frac{2(p+2)(1-\varepsilon )}{\rho +2} \biggr) \bigl( \bigl\Vert u_{t}(t) \bigr\Vert ^{\rho +2}_{\rho +2}+ \bigl\Vert v_{t}(t) \bigr\Vert ^{\rho +2}_{ \rho +2} \bigr) \\ &\qquad {} +2(p+2)\varepsilon \int _{\Omega }F(u,v)\,dx-2(p+2) (1-\varepsilon )E(t) \\ &\qquad {} + \bigl((p+2) (1-\varepsilon )-1 \bigr) \biggl(1- \int ^{t}_{0}g_{1}(\tau )\,d\tau \biggr) \Vert \nabla u \Vert ^{2}_{2} \\ &\qquad {} + \bigl((p+2) (1-\varepsilon )-1 \bigr) \biggl(1- \int _{0}^{t}g_{2}(\tau )\,d\tau \biggr) \Vert \nabla v \Vert ^{2}_{2} \\ & \qquad {}-\frac{1}{4(p+2)(1-\varepsilon )} \biggl( \int ^{t}_{0}g_{1}(\tau )\,d\tau \Vert \nabla u \Vert ^{2}_{2}+ \int ^{t}_{0}g_{2}(\tau )\,d\tau \Vert \nabla v \Vert ^{2}_{2} \biggr) \\ &\qquad {} - \frac{\varepsilon ^{r}_{1}}{r} \bigl( \Vert v \Vert ^{2}_{2}+ \Vert v \Vert ^{2(p+2)}_{2(p+2)} \bigr) - \frac{\varepsilon ^{m}_{1}}{m} \bigl( \Vert u \Vert ^{2}_{2}+ \Vert u \Vert ^{2(p+2)}_{2(p+2)} \bigr). \end{aligned}$$

(3.21)

Combining Lemma 2.3 and the Poincaré inequality, we can deduce

$$\begin{aligned} &\frac{d}{dt} \biggl(\biggl(u,\frac{u_{t} \vert u_{t} \vert ^{\rho }}{\rho +1}\biggr)+ \biggl(v, \frac{v_{t} \vert v_{t} \vert ^{\rho }}{\rho +1}\biggr) -\biggl(\frac{1}{\varepsilon _{1}} \biggr)^{ \frac{\gamma ^{*}}{\gamma ^{*}-1}}\frac{\gamma -1}{\gamma }E(t) \biggr) \\ &\quad \geq \biggl(\frac{1}{\rho +1}+\frac{2(p+2)(1-\varepsilon )}{\rho +2} \biggr) \bigl( \bigl\Vert u_{t}(t) \bigr\Vert ^{\rho +2}_{\rho +2}+ \bigl\Vert v_{t}(t) \bigr\Vert ^{\rho +2}_{ \rho +2} \bigr) \\ &\qquad {} +c_{0}\varepsilon \bigl( \Vert u \Vert ^{2(p+2)}_{2(p+2)}+ \Vert v \Vert ^{2(p+2)}_{2(p+2)} \bigr)-2(p+2) (1-\varepsilon )E(t) \\ &\qquad {} + \biggl[ \bigl((p+2) (1-\varepsilon )-1 \bigr) \biggl(1- \int ^{t}_{0}g_{1}( \tau )\,d\tau \biggr) - \frac{1}{4(p+2)(1-\varepsilon )} \int _{0}^{t} g_{1}( \tau )\,d\tau \biggr] \lambda \Vert u \Vert ^{2}_{2} \\ &\qquad {} + \biggl[ \bigl((p+2) (1-\varepsilon )-1 \bigr) \biggl(1- \int _{0}^{t}g_{2}( \tau )\,d\tau \biggr)- \frac{1}{4(p+2)(1-\varepsilon )} \int ^{t}_{0}g_{2}( \tau )\,d\tau \biggr] \lambda \Vert v \Vert ^{2}_{2} \\ &\qquad {} - \frac{\varepsilon ^{r}_{1}}{r} \bigl( \Vert v \Vert ^{2}_{2}+ \Vert v \Vert ^{2(p+2)}_{2(p+2)} \bigr) - \frac{\varepsilon ^{m}_{1}}{m} \bigl( \Vert u \Vert ^{2}_{2}+ \Vert u \Vert ^{2(p+2)}_{2(p+2)} \bigr) \\ &\quad \geq \biggl(\frac{1}{\rho +1}+\frac{2(p+2)(1-\varepsilon )}{\rho +2} \biggr) \bigl( \bigl\Vert u_{t}(t) \bigr\Vert ^{\rho +2}_{\rho +2}+ \bigl\Vert v_{t}(t) \bigr\Vert ^{\rho +2}_{ \rho +2} \bigr) \\ &\qquad {} +c_{0}\varepsilon \bigl( \Vert u \Vert ^{2(p+2)}_{2(p+2)}+ \Vert v \Vert ^{2(p+2)}_{2(p+2)} \bigr)-2(p+2) (1-\varepsilon )E(t) \\ &\qquad {} + \biggl[ \bigl((p+2) (1-\varepsilon )-1 \bigr)l_{1} - \frac{1}{4(p+2)(1-\varepsilon )}(1-l_{1}) \biggr]\lambda \Vert u \Vert ^{2}_{2} \\ &\qquad {} + \biggl[ \bigl((p+2) (1-\varepsilon )-1 \bigr)l_{2}- \frac{1}{4(p+2)(1-\varepsilon )}(1-l_{2}) \biggr]\lambda \Vert v \Vert ^{2}_{2} \\ &\qquad {} - \frac{\varepsilon ^{r}_{1}}{r} \bigl( \Vert v \Vert ^{2}_{2}+ \Vert v \Vert ^{2(p+2)}_{2(p+2)} \bigr) - \frac{\varepsilon ^{m}_{1}}{m} \bigl( \Vert u \Vert ^{2}_{2}+ \Vert u \Vert ^{2(p+2)}_{2(p+2)} \bigr), \end{aligned}$$

(3.22)

where λ is the first eigenvalue of −Δ, now we take

$$\begin{aligned} &\kappa _{1}(\varepsilon )= \bigl((p+2) (1-\varepsilon )-1 \bigr)l_{1} - \frac{1}{4(p+2)(1-\varepsilon )}(1-l_{1}), \\ &\kappa _{2}(\varepsilon )= \bigl((p+2) (1-\varepsilon )-1 \bigr)l_{2} - \frac{1}{4(p+2)(1-\varepsilon )}(1-l_{2}). \end{aligned}$$

Then (3.22) can be rewritten as

$$\begin{aligned} &\frac{d}{dt} \biggl(\biggl(u,\frac{u_{t} \vert u_{t} \vert ^{\rho }}{\rho +1}\biggr)+ \biggl(v, \frac{v_{t} \vert v_{t} \vert ^{\rho }}{\rho +1}\biggr) -\biggl(\frac{1}{\varepsilon _{1}} \biggr)^{ \frac{\gamma ^{*}}{\gamma ^{*}-1}}\frac{\gamma -1}{\gamma }E(t) \biggr) \\ &\quad \geq \biggl(\frac{1}{\rho +1}+\frac{2(p+2)(1-\varepsilon )}{\rho +2} \biggr) \bigl( \bigl\Vert u_{t}(t) \bigr\Vert ^{\rho +2}_{\rho +2}+ \bigl\Vert v_{t}(t) \bigr\Vert ^{\rho +2}_{ \rho +2} \bigr) \\ &\qquad {} + \biggl(\kappa _{1}(\varepsilon )\lambda - \frac{\varepsilon ^{m}_{1}}{m} \biggr) \Vert u \Vert ^{2}_{2}+ \biggl(\kappa _{2}( \varepsilon )\lambda - \frac{\varepsilon ^{r}_{1}}{r} \biggr) \Vert v \Vert ^{2}_{2} \\ &\qquad {} + \biggl(c_{0}\varepsilon - \frac{\varepsilon ^{r}_{1}}{r} \biggr) \Vert v \Vert ^{2(p+2)}_{2(p+2)} + \biggl(c_{0}\varepsilon - \frac{\varepsilon ^{m}_{1}}{m} \biggr) \Vert u \Vert ^{2(p+2)}_{2(p+2)} \\ & \qquad {}-2(p+2) (1-\varepsilon )E(t). \end{aligned}$$

(3.23)

By the condition

$$ \max \biggl\{ \int ^{\infty }_{0}g_{1}(\tau )\,d\tau , \int ^{\infty }_{0}g_{2}( \tau )\,d\tau \biggr\} < \frac{p+1}{p+1+\frac{1}{4(p+2)}}, $$

we can obtain

$$\begin{aligned} & \bigl((p+2)-1 \bigr)l_{1} -\frac{1}{4(p+2)}(1-l_{1})>0, \\ & \bigl((p+2)-1 \bigr)l_{2} -\frac{1}{4(p+2)}(1-l_{2})>0. \end{aligned}$$

Then we choose ε small enough such that

$$\begin{aligned} &\kappa _{1}(\varepsilon )= \bigl((p+2) (1-\varepsilon )-1 \bigr)l_{1} - \frac{1}{4(p+2)(1-\varepsilon )}(1-l_{1})>0, \\ &\kappa _{2}(\varepsilon )= \bigl((p+2) (1-\varepsilon )-1 \bigr)l_{2} - \frac{1}{4(p+2)(1-\varepsilon )}(1-l_{2})>0. \end{aligned}$$

And we pick $\varepsilon _{1}$ small enough such that

$$\begin{aligned} &\min \biggl\{ \kappa _{1}(\varepsilon )\lambda - \frac{\varepsilon ^{m}_{1}}{m},\kappa _{2}(\varepsilon )\lambda - \frac{\varepsilon ^{r}_{1}}{r}\biggr\} >0, \\ & \min \biggl\{ c_{0}\varepsilon - \frac{\varepsilon ^{m}_{1}}{m},c_{0} \varepsilon - \frac{\varepsilon ^{r}_{1}}{r}\biggr\} >0. \end{aligned}$$

(3.24)

Then we choose

$$\begin{aligned} &\alpha=\min \biggl\{ \kappa _{1}(\varepsilon )\lambda - \frac{\varepsilon ^{m}_{1}}{m},\kappa _{2}(\varepsilon )\lambda - \frac{\varepsilon ^{r}_{1}}{r},c_{0}\varepsilon - \frac{\varepsilon ^{m}_{1}}{m},c_{0} \varepsilon - \frac{\varepsilon ^{r}_{1}}{r}\biggr\} , \\ &\beta=\min \biggl\{ \frac{1}{\rho +1}+ \frac{2(p+2)(1-\varepsilon )}{\rho +2},\alpha \biggr\} . \end{aligned}$$

Using Lemma 2.4, (3.23) can be deduced as

$$\begin{aligned} &\frac{d}{dt} \biggl(\biggl(u,\frac{u_{t} \vert u_{t} \vert ^{\rho }}{\rho +1}\biggr)+ \biggl(v, \frac{v_{t} \vert v_{t} \vert ^{\rho }}{\rho +1}\biggr) -\biggl(\frac{1}{\varepsilon _{1}} \biggr)^{ \frac{\gamma ^{*}}{\gamma ^{*}-1}}\frac{\gamma -1}{\gamma }E(t) \biggr) \\ &\quad \geq \biggl(\frac{1}{\rho +1}+\frac{2(p+2)(1-\varepsilon )}{\rho +2} \biggr) \bigl( \bigl\Vert u_{t}(t) \bigr\Vert ^{\rho +2}_{\rho +2}+ \bigl\Vert v_{t}(t) \bigr\Vert ^{\rho +2}_{ \rho +2} \bigr) \\ &\qquad {} +\alpha \bigl( \Vert u \Vert ^{2}_{2}+ \Vert u \Vert ^{2(p+2)}_{2(p+2)} \bigr)+ \alpha \bigl( \Vert v \Vert ^{2}_{2}+ \Vert v \Vert ^{2(p+2)}_{2(p+2)} \bigr) \\ &\qquad {} -2(p+2) (1-\varepsilon )E(t) \\ &\quad \geq \beta \bigl( \bigl\Vert u_{t}(t) \bigr\Vert ^{\rho +2}_{\rho +2}+ \bigl\Vert v_{t}(t) \bigr\Vert ^{ \rho +2}_{\rho +2} \bigr) \\ &\qquad {} +\beta \bigl( \Vert u \Vert ^{2}_{2}+ \Vert u \Vert ^{2(p+2)}_{2(p+2)} \bigr)+\beta \bigl( \Vert v \Vert ^{2}_{2}+ \Vert v \Vert ^{2(p+2)}_{2(p+2)} \bigr) \\ &\qquad {} -2(p+2) (1-\varepsilon )E(t) \\ &\quad \geq \beta \bigl( \bigl\Vert u_{t}(t) \bigr\Vert ^{\rho +2}_{\rho +2}+ \bigl\Vert v_{t}(t) \bigr\Vert ^{ \rho +2}_{\rho +2} \bigr) +\beta \Vert u \Vert ^{\rho +2}_{\rho +2}+\beta \Vert v \Vert ^{ \rho +2}_{\rho +2} -2(p+2) (1-\varepsilon )E(t). \end{aligned}$$

(3.25)

By applying the Hölder and Young inequalities, we can deduce that

$$ \biggl(u,\frac{u_{t} \vert u_{t} \vert ^{\rho }}{\rho +1}\biggr)\leq \Vert u \Vert ^{\rho +2}_{\rho +2}+ \Vert u_{t} \Vert ^{ \rho +2}_{\rho +2}, $$

then we have

$$\begin{aligned} &\frac{d}{dt} \biggl(\biggl(u,\frac{u_{t} \vert u_{t} \vert ^{\rho }}{\rho +1}\biggr)+ \biggl(v, \frac{v_{t} \vert v_{t} \vert ^{\rho }}{\rho +1}\biggr) -\biggl(\frac{1}{\varepsilon _{1}} \biggr)^{ \frac{\gamma ^{*}}{\gamma ^{*}-1}}\frac{\gamma -1}{\gamma }E(t) \biggr) \\ &\quad \geq \beta \biggl(\biggl(u,\frac{u_{t} \vert u_{t} \vert ^{\rho }}{\rho +1}\biggr)+\biggl(v, \frac{v_{t} \vert v_{t} \vert ^{\rho }}{\rho +1}\biggr)- \frac{2(p+2)(1-\varepsilon )}{\beta }E(t) \biggr). \end{aligned}$$

(3.26)

It is easy to see that

$$ \biggl(\frac{1}{\varepsilon _{1}}\biggr)^{\frac{\gamma ^{*}}{\gamma ^{*}-1}} \frac{\gamma -1}{\gamma }\rightarrow + \infty ,\quad \varepsilon _{1} \rightarrow 0^{+}, $$

and $\frac{2(p+2)(1-\varepsilon )}{\beta }$ is a positive constant, hence there exists a constant $\varepsilon _{1}^{*}$ such that

$$ \biggl(\frac{1}{\varepsilon _{1}}\biggr)^{\frac{\gamma ^{*}}{\gamma ^{*}-1}} \frac{\gamma -1}{\gamma }\geq \frac{2(p+2)(1-\varepsilon )}{\beta }. $$

Therefore, we have

$$\begin{aligned} & \frac{d}{dt} \biggl(\biggl(u,\frac{u_{t} \vert u_{t} \vert ^{\rho }}{\rho +1}\biggr)+ \biggl(v, \frac{v_{t} \vert v_{t} \vert ^{\rho }}{\rho +1}\biggr) -\biggl(\frac{1}{\varepsilon _{1}^{*}}\biggr)^{ \frac{\gamma ^{*}}{\gamma ^{*}-1}} \frac{\gamma -1}{\gamma }E(t) \biggr) \\ &\quad \geq \beta \biggl(\biggl(u,\frac{u_{t} \vert u_{t} \vert ^{\rho }}{\rho +1}\biggr)+\biggl(v, \frac{v_{t} \vert v_{t} \vert ^{\rho }}{\rho +1}\biggr)-\biggl(\frac{1}{\varepsilon _{1}^{*}}\biggr)^{ \frac{\gamma ^{*}}{\gamma ^{*}-1}} \frac{\gamma -1}{\gamma } E(t) \biggr). \end{aligned}$$

(3.27)

Take

$$ H(t)=\biggl(u,\frac{u_{t} \vert u_{t} \vert ^{\rho }}{\rho +1}\biggr)+\biggl(v, \frac{v_{t} \vert v_{t} \vert ^{\rho }}{\rho +1}\biggr)- \biggl(\frac{1}{\varepsilon _{1}^{*}}\biggr)^{ \frac{\gamma ^{*}}{\gamma ^{*}-1}}\frac{\gamma -1}{\gamma } E(t), $$

from (3.2), we know

$$ H(0)=\biggl(u(0),\frac{u_{t}(0) \vert u_{t}(0) \vert ^{\rho }}{\rho +1}\biggr)+\biggl(v(0), \frac{v_{t}(0) \vert v_{t}(0) \vert ^{\rho }}{\rho +1} \biggr)-\biggl( \frac{1}{\varepsilon _{1}^{*}}\biggr)^{\frac{\gamma ^{*}}{\gamma ^{*}-1}} \frac{\gamma -1}{\gamma } E(0)>0. $$

By calculating $H'(t)\geq \beta H(t)$, we can get

$$ H(t)\geq e^{\beta t}H(0), \quad \forall t\geq 0. $$

(3.28)

Since $(u,v)$ shows global existence, by Lemma 2.2 and Lemma 2.6, we have $0< E(t)\leq E(0)$, $t\in [0,+\infty )$, then

$$\begin{aligned} \Vert u \Vert ^{\rho +2}_{\rho +2}+ \Vert u_{t} \Vert ^{\rho +2}_{\rho +2}+ \Vert v \Vert ^{\rho +2}_{ \rho +2}+ \Vert v_{t} \Vert ^{\rho +2}_{\rho +2} \geq& \biggl(u, \frac{u_{t} \vert u_{t} \vert ^{\rho }}{\rho +1}\biggr)+\biggl(u, \frac{u_{t} \vert u_{t} \vert ^{\rho }}{\rho +1}\biggr) \\ \geq& e^{\beta t}H(0). \end{aligned}$$

(3.29)

Using the Hölder inequality, Lemma 2.2 and Lemma 2.6, we have

$$\begin{aligned} & \bigl\Vert u(t) \bigr\Vert _{\rho +2}+ \Vert v \Vert _{\rho +2} \\ &\quad \leq \bigl\Vert u(0) \bigr\Vert _{\rho +2}+ \bigl\Vert v(0) \bigr\Vert _{ \rho +2}+ \int ^{t}_{0} \bigl\Vert u_{t}(\tau ) \bigr\Vert _{\rho +2}\,d\tau + \int ^{t}_{0} \bigl\Vert v_{t}( \tau ) \bigr\Vert _{\rho +2}\,d\tau \\ &\quad \leq \bigl\Vert u(0) \bigr\Vert _{\rho +2}+ \bigl\Vert v(0) \bigr\Vert _{\rho +2}+C_{1} \int ^{t}_{0} \bigl\Vert u_{t}( \tau ) \bigr\Vert _{m}\,d\tau +C_{2} \int ^{t}_{0} \bigl\Vert v_{t}(\tau ) \bigr\Vert _{r}\,d\tau \\ & \quad \leq \bigl\Vert u(0) \bigr\Vert _{\rho +2}+ \bigl\Vert v(0) \bigr\Vert _{\rho +2}+C_{1}t^{ \frac{m-1}{m}}\biggl( \int ^{t}_{0} \bigl\Vert u_{t}(\tau ) \bigr\Vert _{m}\,d\tau \biggr)^{\frac{1}{m}} \\ &\qquad {}+C_{2}t^{ \frac{r-1}{r}} \biggl( \int ^{t}_{0} \bigl\Vert v_{t}(\tau ) \bigr\Vert _{r}\,d\tau \biggr)^{\frac{1}{r}} \\ &\quad \leq \bigl\Vert u(0) \bigr\Vert _{\rho +2}+ \bigl\Vert v(0) \bigr\Vert _{\rho +2}+C_{1}t^{ \frac{m-1}{m}} \bigl(E(0)-E(t) \bigr)^{\frac{1}{m}}+C_{2}t^{\frac{r-1}{r}} \bigl(E(0)-E(t) \bigr)^{\frac{1}{r}} \\ &\quad \leq \bigl\Vert u(0) \bigr\Vert _{\rho +2}+ \bigl\Vert v(0) \bigr\Vert _{\rho +2}+C_{1}t^{ \frac{m-1}{m}} \bigl(E(0) \bigr)^{\frac{1}{m}}+C_{2}t^{\frac{r-1}{r}} \bigl(E(0) \bigr)^{\frac{1}{r}}, \end{aligned}$$

(3.30)

where $C_{1}$ and $C_{2}$ are positive constants. By combining (3.29) and (3.30), we know $\|u\|^{\rho +2}_{\rho +2}+\|v\|^{\rho +2}_{\rho +2}$ shows polynomial growth and $\|u_{t}\|^{\rho +2}_{\rho +2}+\|v_{t}\|^{\rho +2}_{\rho +2}$ shows exponential growth. By (2.5) and $E(t)$ being nonnegative, we can deduce

$$\begin{aligned} \int ^{t}_{0} \bigl\Vert u_{t}(\tau ) \bigr\Vert ^{m}_{m}\,d\tau + \int ^{t}_{0} \bigl\Vert v_{t}( \tau ) \bigr\Vert ^{r}_{r}\,d\tau \leq E(0), \end{aligned}$$

(3.31)

and thanks to Lemma 2.5 and the assumption $2<\rho +2<\min \{m,r\}$, we have

$$\begin{aligned} & \bigl\Vert u_{t}(\tau ) \bigr\Vert ^{\rho +2}_{m}< \bigl\Vert u_{t}(\tau ) \bigr\Vert ^{m}_{m}+1, \\ &\bigl\Vert v_{t}(\tau ) \bigr\Vert ^{\rho +2}_{r}< \bigl\Vert v_{t}(\tau ) \bigr\Vert ^{r}_{r}+1. \end{aligned}$$

(3.32)

By using the Sobolev embedding theorem and combining (3.31) and (3.32), we can get

$$\begin{aligned} &\int ^{t}_{0} \bigl\Vert u_{t}(\tau ) \bigr\Vert ^{\rho +2}_{\rho +2}\,d\tau + \int ^{t}_{0} \bigl\Vert v_{t}( \tau ) \bigr\Vert ^{\rho +2}_{\rho +2}\,d\tau \\ &\quad \leq C \biggl( \int ^{t}_{0} \bigl\Vert u_{t}( \tau ) \bigr\Vert ^{\rho +2}_{m}\,d\tau + \int ^{t}_{0} \bigl\Vert v_{t}(\tau ) \bigr\Vert ^{\rho +2}_{r}\,d\tau \biggr) \\ &\quad \leq C \biggl( \int ^{t}_{0}\bigl( \bigl\Vert u_{t}( \tau ) \bigr\Vert ^{m}_{m}+1\bigr)\,d\tau + \int ^{t}_{0}\bigl( \bigl\Vert v_{t}( \tau ) \bigr\Vert ^{r}_{r}+1\bigr)\,d\tau \biggr) \\ &\quad \leq CE(0)+2Ct, \end{aligned}$$

which contradicts with $\|u_{t}\|^{\rho +2}_{\rho +2}+\|v_{t}\|^{\rho +2}_{\rho +2}$ showing exponential growth. Hence the theorem is proved. □