Blow up of positive initial-energy solutions for coupled nonlinear wave equations with degenerate damping and source terms

Abstract

In this work, we consider coupled nonlinear wave equations with degenerate damping and source terms. We will show the blow up of solutions in finite time with positive initial energy. This improves earlier results in the literature.

1 Introduction

In this work, we consider the following initial-boundary value problem:

$$ \left \{ \begin{array}{@{}l@{\quad}l} u_{tt}+ ( \vert u\vert ^{k}+\vert v\vert ^{l} ) \vert u_{t}\vert ^{p-1}u_{t}=\operatorname {div} ( \rho ( \vert \nabla u\vert ^{2} ) \nabla u ) +f_{1} ( u,v ) , & ( x,t ) \in\Omega\times ( 0,T ) , \\ v_{tt}+ ( \vert v\vert ^{\theta}+\vert u\vert ^{\varrho} ) \vert v_{t}\vert ^{q-1}v_{t}=\operatorname {div} ( \rho ( \vert \nabla v\vert ^{2} ) \nabla v ) +f_{2} ( u,v ) ,& ( x,t ) \in\Omega\times ( 0,T ) , \\ u ( x,t ) =v ( x,t ) =0, & ( x,t ) \in\partial\Omega\times ( 0,T ) , \\ u ( x,0 ) =u_{0} ( x ) ,\qquad u_{t} ( x,0 ) =u_{1} ( x ) ,& x\in\Omega, \\ v ( x,0 ) =v_{0} ( x ) ,\qquad v_{t} ( x,0 ) =v_{1} ( x ) ,& x\in\Omega, \end{array} \right . $$

(1.1)

where Ω is a bounded domain with smooth boundary ∂Ω in $R^{n}$ ($n=1,2,3$); $p,q\geq1$, $k,l,\theta,\varrho \geq0$; $f_{i} ( \cdot ,\cdot ) :R^{2}\longrightarrow R$ are given functions to be specified later.

In the case of $\rho=1$, equation (1.1) takes the form

$$ \left \{ \begin{array}{@{}l} u_{tt}-\bigtriangleup u+ ( \vert u\vert ^{k}+\vert v\vert ^{l} ) \vert u_{t}\vert ^{p-1}u_{t}=f_{1} ( u,v ) , \\ v_{tt}-\bigtriangleup v+ ( \vert v\vert ^{\theta}+\vert u\vert ^{\varrho} ) \vert v_{t}\vert ^{q-1}v_{t}=f_{2} ( u,v ) . \end{array} \right . $$

(1.2)

In [1] Rammaha and Sakuntasathien studied the global well posedness of the solution of problem (1.2). Agre and Rammaha [2] studied the global existence and the blow up of the solution of problem (1.2) for $k=l=\theta=\varrho=0$, and also Alves et al. [3] investigated the existence, uniform decay rates and blow up of the solution to systems. After that, the blow up result was improved by Houari [4]. Also, Houari [5] showed that the local solution obtained in [2] is global and decay of solutions.

When $k=l=\theta=\varrho=0$, equation (1.1) reduces to the following form:

$$ \left \{ \begin{array}{@{}l} u_{tt}+\vert u_{t}\vert ^{p-1}u_{t}=\operatorname{div} ( \rho ( \vert \nabla u\vert ^{2} ) \nabla u ) +f_{1} ( u,v ) , \\ v_{tt}+\vert v_{t}\vert ^{q-1}v_{t}=\operatorname{div} ( \rho ( \vert \nabla v\vert ^{2} ) \nabla v ) +f_{2} ( u,v ) . \end{array} \right . $$

(1.3)

Wu et al. [6] obtained the global existence and blow up of the solution of problem (1.3) under some suitable conditions. Also, Fei and Hongjun [7] considered problem (1.3) and improved the blow up result obtained in [6] for a large class of initial data in positive initial energy using some techniques as in Payne and Sattinger [8] and some estimates used firstly by Vitillaro [9]. Recently, Pişkin and Polat [10] studied the local and global existence, energy decay and blow up of the solution of problem (1.3).

In this work, we analyze the influence of degenerate damping terms and source terms on the solutions of problem (1.1). Blow up of the solution with positive initial energy was proved for $2 ( r+2 ) >\max \{ k+p+1,l+p+1,\theta+q+1,\varrho+q+1 \}$ by using the technique of [9] with a modification in the energy functional.

This work is organized as follows. In Section 2, we present some lemmas and the local existence theorem. In Section 3, the blow up of the solution is given.

2 Preliminaries

In this section, we shall give some assumptions and lemmas which will be used throughout this work. Let $\Vert \cdot \Vert$ and $\Vert \cdot \Vert _{p}$ denote the usual $L^{2} ( \Omega )$ norm and $L^{p} ( \Omega )$ norm, respectively.

Next, we give assumptions for problem (1.1).

• (A1) ρ is a positive $C^{1}$ function satisfying

  $$ \rho ( s ) =b_{1}+b_{2}s^{m},\quad m\geq0, $$

  where $b_{1}$, $b_{2}$ are nonnegative constants and $b_{1}+b_{2}>0$.

• (A2) For the nonlinearity, we suppose that

  $$ \left \{ \begin{array}{@{}l@{\quad}l} p,q\geq1 & \mbox{if }n=1,2, \\ 1\leq p,q\leq5 &\mbox{if }n=3. \end{array} \right . $$

Concerning the functions $f_{1} ( u,v )$ and $f_{2} ( u,v )$, we take

$$\begin{aligned}& f_{1} ( u,v ) =a\vert u+v\vert ^{2 ( r+1 ) } ( u+v ) +b \vert u\vert ^{r}u\vert v\vert ^{r+2},\\& f_{2} ( u,v ) =a\vert u+v\vert ^{2 ( r+1 ) } ( u+v ) +b\vert v\vert ^{r}v\vert u\vert ^{r+2}, \end{aligned}$$

where $a,b>0$ are constants and r satisfies

$$ \left \{ \begin{array}{@{}l@{\quad}l} -1< r &\mbox{if }n=1,2, \\ -1<r\leq1 &\mbox{if }n=3. \end{array} \right . $$

(2.1)

According to the above equalities they can easily verify that

$$ u f_{1} ( u,v ) +vf_{2} ( u,v ) =2 ( r+2 ) F ( u,v ) ,\quad \forall ( u,v ) \in R^{2}, $$

(2.2)

where

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

(2.3)

We have the following result.

Lemma 2.1

[11]

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

$$ c_{0} \bigl( \vert u\vert ^{2 ( r+2 ) }+\vert v\vert ^{2 ( r+2 ) } \bigr) \leq2 ( r+2 ) F ( u,v ) \leq c_{1} \bigl( \vert u\vert ^{2 ( r+2 ) }+\vert v\vert ^{2 ( r+2 ) } \bigr) $$

(2.4)

is satisfied.

Lemma 2.2

(Sobolev-Poincaré inequality) [12]

Let q be a number with $2\leq q<\infty$ ($n=1,2$) or $2\leq q\leq 2n/ ( n-2 )$ ($n\geq3$), then there is a constant $C_{\ast}=C_{\ast} ( \Omega, q )$ such that

$$ \Vert u\Vert _{q}\leq C_{\ast} \Vert \nabla u\Vert \quad\textit{for }u\in H_{0}^{1} ( \Omega ) . $$

Lemma 2.3

[13]

Suppose that

$$ p\leq2\frac{n-1}{n-2},\quad n\geq3 $$

holds. Then there exists a positive constant $C>1$ depending on Ω only such that

$$ \Vert u\Vert _{p}^{s}\leq C \bigl( \Vert \nabla u \Vert ^{2}+\Vert u\Vert _{p}^{p} \bigr) $$

for any $u\in H_{0}^{1} ( \Omega )$, $2\leq s\leq p$.

Lemma 2.4

$E ( t )$ is a nonincreasing function for $t\geq0$ and

$$ \frac{d}{dt}E ( t ) =-\int_{\Omega} \bigl( \vert u \vert ^{k}+\vert v\vert ^{l} \bigr) \vert u_{t}\vert ^{p+1}\,dx-\int_{\Omega} \bigl( \vert v\vert ^{\theta}+\vert u\vert ^{\varrho} \bigr) \vert v_{t}\vert ^{q+1}\,dx. $$

(2.5)

Proof

Multiplying the first equation of (1.1) by $u_{t}$, the second equation by $v_{t}$, and integrating them over Ω, then adding them together and integrating by parts, we obtain

$$\begin{aligned} &E ( t ) -E ( 0 ) =-\int_{0}^{t}\int _{\Omega} \bigl( \bigl( \vert u\vert ^{k}+\vert v \vert ^{l} \bigr) \vert u_{\tau} \vert ^{p+1}+ \bigl( \vert v\vert ^{\theta }+\vert u\vert ^{\varrho} \bigr) \vert v_{\tau} \vert ^{q+1} \bigr) \,dx\,d\tau \\ &\quad\mbox{for }t\geq0. \end{aligned}$$

(2.6)

 □

Next, we state the local existence theorem that can be established by combining arguments of [1, 10]. Firstly, we give the definition of a weak solution to problem (1.1).

Definition 2.1

A pair of functions $( u,v )$ is said to be a weak solution of (1.1) on $[ 0,T ]$ if $u,v\in C ( [ 0,T ] ; W_{0}^{1,2 ( m+1 ) } ( \Omega ) \cap L^{r+1} ( \Omega ) )$, $u_{t}\in C ( [ 0,T ] ;L^{2} ( \Omega ) ) \cap L^{p+1} ( \Omega\times ( 0,T ) )$ and $v_{t}\in C ( [ 0,T ] ;L^{2} ( \Omega ) ) \cap L^{q+1} ( \Omega\times ( 0,T ) )$. In addition, $( u,v )$ satisfies

$$\begin{aligned}& \begin{aligned}[b] &\int_{\Omega}u^{\prime} ( t ) \phi \,dx-\int _{\Omega }u_{1} ( t ) \phi \,dx+\int_{\Omega} \bigl( \rho \bigl( \vert \nabla u\vert ^{2} \bigr) \nabla u \bigr) \nabla\phi \,dx \\ &\qquad{}+\int_{0}^{t}\int_{\Omega} \bigl( \vert u\vert ^{k}+\vert v\vert ^{l} \bigr) \bigl\vert u^{\prime}\bigr\vert ^{p-1}u^{\prime }\phi \,dx\,d\tau \\ &\quad=\int_{0}^{t}\int_{\Omega}f_{1} \bigl( u ( \tau ) ,v ( \tau ) \bigr) \phi \,dx\,d\tau, \end{aligned} \end{aligned}$$

(2.7)

$$\begin{aligned}& \begin{aligned}[b] &\int_{\Omega}v^{\prime} ( t ) \varphi \,dx-\int _{\Omega }v_{1} ( t ) \varphi \,dx+\int _{\Omega} \bigl( \rho \bigl( \vert \nabla v\vert ^{2} \bigr) \nabla v \bigr) \nabla\varphi \,dx \\ &\qquad{}+\int_{0}^{t}\int_{\Omega} \bigl( \vert v\vert ^{\theta }+\vert u\vert ^{\varrho} \bigr) \bigl\vert v^{\prime }\bigr\vert ^{q-1}v^{\prime}\varphi \,dx\,d\tau \\ &\quad=\int_{0}^{t}\int_{\Omega}f_{2} \bigl( u ( \tau ) ,v ( \tau ) \bigr) \varphi \,dx\,d\tau \end{aligned} \end{aligned}$$

(2.8)

for all test functions $\phi\in W_{0}^{1,2 ( m+1 ) } ( \Omega ) \cap L^{p+1} ( \Omega )$, $\varphi\in W_{0}^{1,2 ( m+1 ) } ( \Omega ) \cap L^{q+1} ( \Omega )$ and for almost all $t\in [ 0,T ]$.

Theorem 2.1

(Local existence)

Assume that (A1), (A2) and (2.1) hold. Then, for any initial data $u_{0},v_{0}\in W_{0}^{1,2 ( m+1 ) } ( \Omega ) \cap L^{r+1} ( \Omega )$ and $u_{1},v_{1}\in L^{2} ( \Omega )$, there exists a unique local weak solution $( u,v )$ of problem (1.1) (in the sense of Definition  2.1) defined in $[ 0,T ]$ for some $T>0$, and satisfies the energy identity

$$ E ( t ) +\int_{0}^{t}\int_{\Omega} \bigl( \bigl( \vert u\vert ^{k}+\vert v\vert ^{l} \bigr) \vert u_{\tau }\vert ^{p+1}+ \bigl( \vert v\vert ^{\theta}+\vert u\vert ^{\varrho} \bigr) \vert v_{\tau} \vert ^{q+1} \bigr) \,dx\,d\tau=E ( 0 ) , $$

(2.9)

where

$$ E ( t ) =\frac{1}{2} \bigl( \Vert u_{t}\Vert ^{2}+\Vert v_{t}\Vert ^{2} \bigr) + \frac{1}{2}\int_{\Omega } \bigl( P \bigl( \vert \nabla u \vert ^{2} \bigr) +P \bigl( \vert \nabla v\vert ^{2} \bigr) \bigr)\,dx-\int_{\Omega }F ( u,v )\,dx, $$

(2.10)

where $P ( s ) =\int_{0}^{s}\rho ( \xi )\,d\xi$, $s\geq 0$.

3 Blow up of solutions

In this section, we are going to consider the blow up of the solution for problem (1.1).

Lemma 3.1

Suppose that (2.1) holds. Then there exists $\eta>0$ such that for any $( u,v ) \in ( H^{2m} ( \Omega ) \cap H_{0}^{m} ( \Omega ) ) \times ( H^{2m} ( \Omega ) \cap H_{0}^{m} ( \Omega ) )$ the inequality

$$ \Vert u+v\Vert _{2 ( r+2 ) }^{2 ( r+2 ) }+2\Vert uv\Vert _{r+2}^{r+2}\leq\eta \biggl( \int_{\Omega } \bigl( P \bigl( \vert \nabla u\vert ^{2} \bigr) +P \bigl( \vert \nabla v\vert ^{2} \bigr) \bigr) \biggr) ^{r+2} $$

(3.1)

holds.

Proof

The proof is almost the same as that of [11], so we omit it here. □

In order to state and prove our result and for the sake of simplicity, we take $a=b=1$. We introduce the following:

$$ \begin{aligned} &B=\eta^{\frac{1}{2 ( r+2 ) }},\qquad \alpha_{1}=B^{-\frac {r+2}{r+1}},\\ &E_{1}= \biggl( \frac{1}{2}-\frac{1}{2 ( r+2 ) } \biggr) \alpha _{1}^{2},\qquad E_{2}= \biggl( \frac{1}{2 ( m+1 ) }- \frac{1}{2 ( r+2 ) } \biggr) \alpha_{1}^{2}, \end{aligned} $$

(3.2)

where η is the optimal constant in (3.1).

The following lemma will play an essential role in the proof of our main result, and it is similar to the lemma used firstly by Vitillaro [9].

Lemma 3.2

[7]

Assume that (A1) and (2.1) hold. Let $( u,v )$ be a solution of (1.1). Assume further that $E ( 0 ) < E_{1}$ and

$$ \biggl( \int_{\Omega} \bigl( P \bigl( \vert \nabla u_{0}\vert ^{2} \bigr) +P \bigl( \vert \nabla v_{0}\vert ^{2} \bigr) \bigr)\,dx \biggr) ^{\frac{1}{2}}> \alpha_{1}. $$

(3.3)

Then there exists a constant $\alpha_{2}>\alpha_{1}$ such that

$$\begin{aligned}& \biggl( \int_{\Omega} \bigl( P \bigl( \vert \nabla u\vert ^{2} \bigr) +P \bigl( \vert \nabla v\vert ^{2} \bigr) \bigr)\,dx \biggr) ^{\frac{1}{2}}>\alpha_{2}\quad\textit{for }t>0, \end{aligned}$$

(3.4)

$$\begin{aligned}& \bigl( \Vert u+v\Vert _{2 ( r+2 ) }^{2 ( r+2 ) }+2\Vert uv\Vert _{r+2}^{r+2} \bigr) ^{\frac{1}{2 ( r+2 ) }}\geq B\alpha_{2} \quad\textit{for }t>0. \end{aligned}$$

(3.5)

Theorem 3.1

Assume that (A1), (A2) and (2.1) hold. Assume further that

$$ 2 ( r+2 ) >\max \{ k+p+1,l+p+1,\theta+q+1,\varrho +q+1 \} . $$

Then any solution of problem (1.1) with initial data satisfying

$$\biggl( \int_{\Omega} \bigl( P \bigl( \vert \nabla u_{0}\vert ^{2} \bigr) +P \bigl( \vert \nabla v_{0}\vert ^{2} \bigr) \bigr)\,dx \biggr) ^{\frac{1}{2}}>\alpha_{1}, \qquad E ( 0 ) < E_{2}, $$

cannot exist for all time.

Proof

We suppose that the solution exists for all time and we reach a contradiction.

Set

$$ H ( t ) =E_{2}-E ( t ) . $$

(3.6)

By using (2.10) and (3.6) we get

$$\begin{aligned} 0 < &H ( 0 ) \leq H ( t ) =E_{2}-\frac{1}{2} \bigl( \Vert u_{t}\Vert ^{2}+\Vert v_{t}\Vert ^{2} \bigr) \\ &{}-\frac{1}{2}\int_{\Omega} \bigl( P \bigl( \vert \nabla u\vert ^{2} \bigr) +P \bigl( \vert \nabla v\vert ^{2} \bigr) \bigr)\,dx+\int_{\Omega}F ( u,v )\,dx. \end{aligned}$$

(3.7)

From (3.4) and (2.4) we have

$$\begin{aligned} &E_{2}-\frac{1}{2} \bigl( \Vert u_{t}\Vert ^{2}+\Vert v_{t}\Vert ^{2} \bigr) - \frac{1}{2}\int_{\Omega} \bigl( P \bigl( \vert \nabla u \vert ^{2} \bigr) +P \bigl( \vert \nabla v\vert ^{2} \bigr) \bigr)\,dx+\int_{\Omega}F ( u,v )\,dx \\ &\quad\leq E_{2}-\frac{1}{2}\alpha_{1}^{2}+ \frac{c_{1}}{2 ( r+2 ) } \bigl( \Vert u\Vert _{2 ( r+2 ) }^{2 ( r+2 ) }+ \Vert v\Vert _{2 ( r+2 ) }^{2 ( r+2 ) } \bigr) \\ &\quad\leq E_{1}-\frac{1}{2}\alpha_{1}^{2}+ \frac{c_{1}}{2 ( r+2 ) } \bigl( \Vert u\Vert _{2 ( r+2 ) }^{2 ( r+2 ) }+ \Vert v\Vert _{2 ( r+2 ) }^{2 ( r+2 ) } \bigr) \\ &\quad\leq-\frac{1}{2 ( r+2 ) }\alpha_{1}^{2}+\frac{c_{1}}{2 ( r+2 ) } \bigl( \Vert u\Vert _{2 ( r+2 ) }^{2 ( r+2 ) }+\Vert v\Vert _{2 ( r+2 ) }^{2 ( r+2 ) } \bigr) \\ &\quad\leq\frac{c_{1}}{2 ( r+2 ) } \bigl( \Vert u\Vert _{2 ( r+2 ) }^{2 ( r+2 ) }+ \Vert v\Vert _{2 ( r+2 ) }^{2 ( r+2 ) } \bigr) . \end{aligned}$$

(3.8)

Combining (3.7) and (3.8) we have

$$ 0< H ( 0 ) \leq H ( t ) \leq\frac{c_{1}}{2 ( r+2 ) } \bigl( \Vert u\Vert _{2 ( r+2 ) }^{2 ( r+2 ) }+\Vert v\Vert _{2 ( r+2 ) }^{2 ( r+2 ) } \bigr) . $$

(3.9)

We then define

$$ \Psi ( t ) =H^{1-\sigma} ( t ) +\varepsilon \biggl( \int _{\Omega}uu_{t}\,dx+\int_{\Omega}vv_{t}\,dx \biggr) , $$

(3.10)

where ε is small to be chosen later and

$$\begin{aligned} 0 < &\sigma\leq\min \biggl\{ \frac{r+1}{2 ( r+2 ) },\frac{2r+3- ( k+p ) }{2p ( r+2 ) }, \frac{2r+3- ( l+p ) }{2p ( r+2 ) }, \\ &{} \frac{2r+3- ( \varrho+q ) }{2q ( r+2 ) },\frac{2r+3- ( \theta+q ) }{2q ( r+2 ) } \biggr\} . \end{aligned}$$

(3.11)

Our goal is to show that $\Psi ( t )$ satisfies a differential inequality of the form

$$ \Psi^{\prime} ( t ) \geq\xi\Psi^{\zeta} ( t ) , \quad\zeta>1. $$

(3.12)

This, of course, will lead to a blow up in finite time.

By taking a derivative of (3.10) and using equation (1.1) we obtain

$$\begin{aligned} \Psi^{\prime} ( t ) =& ( 1-\sigma ) H^{-\sigma } ( t ) H^{\prime} ( t ) +\varepsilon \bigl( \Vert u_{t}\Vert ^{2}+\Vert v_{t}\Vert ^{2} \bigr) \\ &{}-\varepsilon \int_{\Omega} \bigl( \rho \bigl( \vert \nabla u\vert ^{2} \bigr) \vert \nabla u\vert ^{2}+\rho \bigl( \vert \nabla v\vert ^{2} \bigr) \vert \nabla v\vert ^{2} \bigr)\,dx \\ &{}-\varepsilon \biggl( \int_{\Omega}u \bigl( \vert u\vert ^{k}+\vert v\vert ^{l} \bigr) u_{t}\vert u_{t}\vert ^{p-1}\,dx+\int_{\Omega}v \bigl( \vert v\vert ^{\theta}+\vert u\vert ^{\varrho} \bigr) v_{t}\vert v_{t}\vert ^{q-1}\,dx \biggr) \\ &{}+\varepsilon\int_{\Omega} \bigl( u f_{1} ( u,v ) +vf_{2} ( u,v ) \bigr)\,dx \\ =& ( 1-\sigma ) H^{-\sigma} ( t ) H^{\prime} ( t ) +\varepsilon \bigl( \Vert u_{t}\Vert ^{2}+\Vert v_{t} \Vert ^{2} \bigr) -\varepsilon b_{1} \bigl( \Vert \nabla u\Vert ^{2}+\Vert \nabla v\Vert ^{2} \bigr) \\ &{}-\varepsilon b_{2} \bigl( \Vert \nabla u\Vert _{2 ( r+2 ) }^{2 ( r+2 ) }+\Vert \nabla v\Vert _{2 ( r+2 ) }^{2 ( r+2 ) } \bigr) +\varepsilon \bigl( \Vert u+v\Vert _{2 ( r+2 ) }^{2 ( r+2 ) }+2 \Vert uv\Vert _{r+2}^{r+2} \bigr) \\ &{}-\varepsilon \biggl( \int_{\Omega}u \bigl( \vert u\vert ^{k}+\vert v\vert ^{l} \bigr) u_{t}\vert u_{t}\vert ^{p-1}\,dx+\int_{\Omega}v \bigl( \vert v\vert ^{\theta}+\vert u\vert ^{\varrho} \bigr) v_{t}\vert v_{t}\vert ^{q-1}\,dx \biggr) . \end{aligned}$$

(3.13)

From the definition of $H ( t )$, it follows that

$$\begin{aligned} &{-}b_{2} \bigl( \Vert \nabla u\Vert _{2 ( r+2 ) }^{2 ( r+2 ) }+ \Vert \nabla v\Vert _{2 ( r+2 ) }^{2 ( r+2 ) } \bigr) \\ &\quad=2 ( m+1 ) H ( t ) -2 ( m+1 ) E_{2}+ ( m+1 ) \bigl( \Vert u_{t}\Vert ^{2}+\Vert v_{t}\Vert ^{2} \bigr) \\ &\qquad{}+ ( m+1 ) b_{1} \bigl( \Vert \nabla u\Vert ^{2}+ \Vert \nabla v\Vert ^{2} \bigr) -2 ( m+1 ) \int_{\Omega}F ( u,v )\,dx. \end{aligned}$$

(3.14)

Substituting (3.14) into (3.13), we obtain

$$\begin{aligned} \Psi^{\prime} ( t ) =& ( 1-\sigma ) H^{-\sigma } ( t ) H^{\prime} ( t ) +\varepsilon ( m+2 ) \bigl( \Vert u_{t} \Vert ^{2}+\Vert v_{t}\Vert ^{2} \bigr) \\ &{}+ \varepsilon b_{1}m \bigl( \Vert \nabla u\Vert ^{2}+ \Vert \nabla v\Vert ^{2} \bigr) \\ &{}+2\varepsilon ( m+1 ) H ( t ) -2\varepsilon ( m+1 ) E_{2}+ \varepsilon \biggl( 1-\frac{m+1}{r+2} \biggr) \bigl( \Vert u+v\Vert _{2 ( r+2 ) }^{2 ( r+2 ) }+2\Vert uv\Vert _{r+2}^{r+2} \bigr) \\ &{}-\varepsilon \biggl( \int_{\Omega}u \bigl( \vert u\vert ^{k}+\vert v\vert ^{l} \bigr) u_{t}\vert u_{t}\vert ^{p-1}\,dx+\int_{\Omega}v \bigl( \vert v\vert ^{\theta}+\vert u\vert ^{\varrho} \bigr) v_{t}\vert v_{t}\vert ^{q-1}\,dx \biggr) . \end{aligned}$$

(3.15)

In order to estimate the last two terms in (3.15), we make use of the following Young inequality:

$$ XY\leq\frac{\delta^{k}X^{k}}{k}+\frac{\delta^{-l}Y^{l}}{l}, $$

where $X,Y\geq0$, $\delta>0$, $k,l\in R^{+}$ such that $\frac {1}{k}+\frac{1}{l}=1$. Consequently, applying the above inequality we have

$$ \int_{\Omega}uu_{t}\vert u_{t}\vert ^{p-1}\,dx\leq\frac {\delta _{1}^{p+1}}{p+1}\Vert u\Vert _{p+1}^{p+1}+ \frac{p\delta _{1}^{-\frac{p+1}{p}}}{p+1}\Vert u_{t}\Vert _{p+1}^{p+1}, $$

and therefore

$$\begin{aligned} \int_{\Omega} \bigl( \vert u\vert ^{k}+\vert v \vert ^{l} \bigr) uu_{t}\vert u_{t}\vert ^{p-1}\,dx \leq&\frac {\delta _{1}^{p+1}}{p+1}\int_{\Omega} \bigl( \vert u\vert ^{k}+\vert v\vert ^{l} \bigr) \vert u \vert ^{p+1}\,dx \\ &{}+\frac{p\delta_{1}^{-\frac{p+1}{p}}}{p+1}\int_{\Omega} \bigl( \vert u\vert ^{k}+\vert v\vert ^{l} \bigr) \vert u_{t} \vert ^{p+1}\,dx. \end{aligned}$$

Similarly

$$ \int_{\Omega}vv_{t}\vert v_{t}\vert ^{q-1}\,dx\leq\frac {\delta _{2}^{q+1}}{q+1}\Vert v\Vert _{q+1}^{q+1}+ \frac{q\delta _{2}^{-\frac{q+1}{q}}}{q+1}\Vert v_{t}\Vert _{q+1}^{q+1}, $$

and therefore

$$\begin{aligned} \int_{\Omega}v \bigl( \vert v\vert ^{\theta}+\vert u \vert ^{\varrho} \bigr) v_{t}\vert v_{t}\vert ^{q-1}\,dx \leq&\frac{\delta_{2}^{q+1}}{q+1}\int_{\Omega} \bigl( \vert v\vert ^{\theta}+\vert u\vert ^{\varrho} \bigr) \vert v \vert ^{q+1}\,dx \\ &{}+\frac{q\delta_{2}^{-\frac{q+1}{q}}}{q+1}\int_{\Omega} \bigl( \vert v\vert ^{\theta}+\vert u\vert ^{\varrho} \bigr) \vert v_{t} \vert ^{q+1}\,dx, \end{aligned}$$

where $\delta_{1}$, $\delta_{2}$ are constants depending on the time t that will be specified later. Therefore, (3.15) becomes

$$\begin{aligned} \Psi^{\prime} ( t ) \geq& ( 1-\sigma ) H^{-\sigma } ( t ) H^{\prime} ( t ) +\varepsilon ( m+2 ) \bigl( \Vert u_{t} \Vert ^{2}+\Vert v_{t}\Vert ^{2} \bigr) + \varepsilon b_{1}m \bigl( \Vert \nabla u\Vert ^{2}+ \Vert \nabla v\Vert ^{2} \bigr) \\ &{}+2\varepsilon ( m+1 ) H ( t ) -2\varepsilon ( m+1 ) E_{2}+ \varepsilon \biggl( 1-\frac{m+1}{r+2} \biggr) \bigl( \Vert u+v\Vert _{2 ( r+2 ) }^{2 ( r+2 ) }+2\Vert uv\Vert _{r+2}^{r+2} \bigr) \\ &{}-\varepsilon\frac{\delta_{1}^{p+1}}{p+1}\int_{\Omega} \bigl( \vert u \vert ^{k}+\vert v\vert ^{l} \bigr) \vert u\vert ^{p+1}\,dx-\varepsilon\frac{p\delta_{1}^{-\frac {p+1}{p}}}{p+1}\int _{\Omega} \bigl( \vert u\vert ^{k}+\vert v\vert ^{l} \bigr) \vert u_{t}\vert ^{p+1}\,dx \\ &{}-\varepsilon\frac{\delta_{2}^{q+1}}{q+1}\int_{\Omega} \bigl( \vert v \vert ^{\theta}+\vert u\vert ^{\varrho} \bigr) \vert v\vert ^{q+1}\,dx-\varepsilon\frac{q\delta_{2}^{-\frac {q+1}{q}}}{q+1}\int_{\Omega} \bigl( \vert v\vert ^{\theta}+\vert u\vert ^{\varrho} \bigr) \vert v_{t}\vert ^{q+1}\,dx. \end{aligned}$$

(3.16)

Therefore, by taking $\delta_{1}$ and $\delta_{2}$ so that $\delta _{1}^{-\frac{p+1}{p}}=k_{1}H^{-\sigma} ( t )$, $\delta_{2}^{-\frac {q+1}{q}}=k_{2}H^{-\sigma} ( t )$ where $k_{1}, k_{2}>0$ are specified later, we get

$$\begin{aligned} \Psi^{\prime} ( t ) \geq& \bigl( ( 1-\sigma ) -K\varepsilon \bigr) H^{-\sigma} ( t ) H^{\prime} ( t ) +\varepsilon ( m+2 ) \bigl( \Vert u_{t}\Vert ^{2}+\Vert v_{t}\Vert ^{2} \bigr) +\varepsilon b_{1}m \bigl( \Vert \nabla u \Vert ^{2}+\Vert \nabla v\Vert ^{2} \bigr) \\ &{}+2\varepsilon ( m+1 ) H ( t ) +\varepsilon c^{\prime } \bigl( \Vert u \Vert _{2 ( r+2 ) }^{2 ( r+2 ) }+\Vert v\Vert _{2 ( r+2 ) }^{2 ( r+2 ) } \bigr) \\ &{}-\varepsilon\frac{k_{1}^{-p}H^{\sigma p} ( t ) }{p+1}\int_{\Omega} \bigl( \vert u\vert ^{k}+\vert v\vert ^{l} \bigr) \vert u \vert ^{p+1}\,dx \\ &{}-\varepsilon\frac{k_{2}^{-q}H^{\sigma q} ( t ) }{q+1}\int_{\Omega} \bigl( \vert v\vert ^{\theta}+\vert u\vert ^{\varrho} \bigr) \vert v\vert ^{q+1}\,dx, \end{aligned}$$

(3.17)

where $K=\frac{k_{1}p}{p+1}+\frac{k_{2}q}{q+1}$ and $c^{\prime }=c_{0} ( 1-\frac{m+1}{r+2}-2 ( m+1 ) E_{2} ( B\alpha _{2} ) ^{-2 ( r+2 ) } )$. It is clear $c^{\prime}>0$ since $\alpha_{2}>\alpha_{1}=B^{-\frac{r+2}{r+1}}$.

Applying the Young inequality, we have

$$\begin{aligned} \int_{\Omega} \bigl( \vert u\vert ^{k}+\vert v \vert ^{l} \bigr) \vert u\vert ^{p+1}\,dx \leq&\int _{\Omega} \vert u\vert ^{k+p+1}\,dx+\int _{\Omega} \vert v\vert ^{l}\vert u\vert ^{p+1}\,dx \\ \leq&\int_{\Omega} \vert u\vert ^{k+p+1}\,dx+ \frac{l}{l+p+1}\delta_{1}^{\frac{l+p+1}{l}}\int _{\Omega} \vert v\vert ^{l+p+1}\,dx \\ &{}+\frac{p+1}{l+p+1}\delta_{1}^{-\frac{l+p+1}{p+1}}\int _{\Omega} \vert u\vert ^{l+p+1}\,dx \\ =&\Vert u\Vert _{k+p+1}^{k+p+1}+\frac{l}{l+p+1} \delta_{1}^{ \frac{l+p+1}{l}}\Vert v\Vert _{l+p+1}^{l+p+1} \\ &{}+\frac{p+1}{l+p+1}\delta_{1}^{-\frac{l+p+1}{p+1}}\Vert u\Vert _{l+p+1}^{l+p+1} \end{aligned}$$

(3.18)

and

$$\begin{aligned} \int_{\Omega} \bigl( \vert v\vert ^{\theta}+\vert u \vert ^{\varrho} \bigr) \vert v\vert ^{q+1}\,dx \leq &\int _{\Omega} \vert v\vert ^{\theta+q+1}\,dx+\int _{\Omega }\vert u\vert ^{\varrho} \vert v\vert ^{q+1}\,dx \\ \leq&\int_{\Omega} \vert v\vert ^{\theta+q+1}\,dx+ \frac {\varrho }{\varrho+q+1}\delta_{2}^{\frac{\varrho+q+1}{\varrho}}\int_{\Omega } \vert u\vert ^{\varrho+q+1}\,dx \\ &{}+\frac{q+1}{\varrho+q+1}\delta_{2}^{-\frac{\varrho+q+1}{q+1}}\int _{\Omega} \vert v\vert ^{\varrho+q+1}\,dx \\ =&\Vert v\Vert _{\theta+q+1}^{\theta+q+1}+\frac{\varrho}{ \varrho+q+1} \delta_{2}^{\frac{\varrho+q+1}{\varrho}} \Vert u\Vert _{\varrho+q+1}^{\varrho+q+1} \\ &{}+\frac{q+1}{\varrho+q+1}\delta_{2}^{-\frac{\varrho+q+1}{q+1}}\Vert v\Vert _{\varrho+q+1}^{\varrho+q+1}. \end{aligned}$$

(3.19)

Substituting (3.18) and (3.19) into (3.17), we have

$$\begin{aligned} \Psi^{\prime} ( t ) \geq& \bigl( ( 1-\sigma ) -K\varepsilon \bigr) H^{-\sigma} ( t ) H^{\prime} ( t ) +\varepsilon ( m+2 ) \bigl( \Vert u_{t}\Vert ^{2}+\Vert v_{t}\Vert ^{2} \bigr) +\varepsilon b_{1}m \bigl( \Vert \nabla u \Vert ^{2}+\Vert \nabla v\Vert ^{2} \bigr) \\ &{}+2\varepsilon ( m+1 ) H ( t ) +\varepsilon c^{\prime } \bigl( \Vert u \Vert _{2 ( r+2 ) }^{2 ( r+2 ) }+\Vert v\Vert _{2 ( r+2 ) }^{2 ( r+2 ) } \bigr) \\ &{}-\varepsilon\frac{k_{1}^{-p}H^{\sigma p} ( t ) }{p+1} \biggl( \Vert u\Vert _{k+p+1}^{k+p+1}+ \frac{l}{l+p+1}\delta _{1}^{\frac{l+p+1}{l}}\Vert v\Vert _{l+p+1}^{l+p+1}+\frac {p+1}{l+p+1}\delta _{1}^{-\frac{l+p+1}{p+1}} \Vert u\Vert _{l+p+1}^{l+p+1} \biggr) \\ &{}-\varepsilon\frac{k_{2}^{-q}H^{\sigma q} ( t ) }{q+1} \biggl( \Vert v\Vert _{\theta+q+1}^{\theta+q+1}+ \frac{\varrho }{\varrho +q+1}\delta_{2}^{\frac{\varrho+q+1}{\varrho}} \Vert u\Vert _{\varrho+q+1}^{\varrho+q+1} \\ &{}+\frac{q+1}{\varrho+q+1}\delta _{2}^{-\frac{\varrho+q+1}{q+1}} \Vert v\Vert _{\varrho+q+1}^{\varrho +q+1} \biggr). \end{aligned}$$

(3.20)

Since $2 ( r+2 ) >\max \{ k+p+1,l+p+1,\theta+q+1,\varrho +q+1 \}$, we obtain

$$\begin{aligned}& H^{\sigma p} ( t ) \Vert u\Vert _{k+p+1}^{k+p+1}\leq C \bigl( \Vert u\Vert _{2 ( r+2 ) }^{2\sigma p ( r+2 ) +k+p+1}+\Vert v\Vert _{2 ( r+2 ) }^{2\sigma p ( r+2 ) }\Vert u\Vert _{k+p+1}^{k+p+1} \bigr) , \end{aligned}$$

(3.21)

$$\begin{aligned}& H^{\sigma q} ( t ) \Vert v\Vert _{\theta +q+1}^{\theta +q+1}\leq C \bigl( \Vert v\Vert _{2 ( r+2 ) }^{2\sigma q ( r+2 ) +\theta+q+1}+\Vert u\Vert _{2 ( r+2 ) }^{2\sigma q ( r+2 ) }\Vert v\Vert _{\theta +q+1}^{\theta+q+1} \bigr) , \end{aligned}$$

(3.22)

$$\begin{aligned}& \begin{aligned}[b] &\frac{l}{l+p+1}\delta_{1}^{\frac{l+p+1}{l}}H^{\sigma p} ( t ) \Vert v\Vert _{l+p+1}^{l+p+1}\\ &\quad\leq C\frac{l}{l+p+1}\delta _{1}^{\frac{l+p+1}{l}} \bigl( \Vert v\Vert _{2 ( r+2 ) }^{2\sigma p ( r+2 ) +l+p+1}+ \Vert u\Vert _{2 ( r+2 ) }^{2\sigma p ( r+2 ) }\Vert v\Vert _{l+p+1}^{l+p+1} \bigr) , \end{aligned} \end{aligned}$$

(3.23)

and

$$ \begin{aligned}[b] &\frac{\varrho}{\varrho+q+1}\delta_{2}^{\frac{\varrho+q+1}{\varrho}}H^{\sigma q} ( t ) \Vert u\Vert _{\varrho +q+1}^{\varrho+q+1}\\ &\quad\leq C\frac{\varrho}{\varrho+q+1}\delta _{2}^{\frac{\varrho+q+1}{\varrho}} \bigl( \Vert u\Vert _{2 ( r+2 ) }^{2\sigma q ( r+2 ) +\varrho+q+1}+ \Vert v\Vert _{2 ( r+2 ) }^{2\sigma q ( r+2 ) }\Vert u\Vert _{\varrho+q+1}^{\varrho+q+1} \bigr) . \end{aligned} $$

(3.24)

By using (3.11) and the algebraic inequality

$$ z^{\upsilon}\leq z+1\leq \biggl( 1+\frac{1}{a} \biggr) ( z+a ) , \quad \forall z\geq0, 0< \upsilon\leq1, a\geq0, $$

(3.25)

we have, for all $t\geq0$,

$$\begin{aligned}& \begin{aligned}[b] \Vert u\Vert _{2 ( r+2 ) }^{2\sigma p ( r+2 ) +k+p+1} &\leq d \bigl( \Vert u\Vert _{2 ( r+2 ) }^{2 ( r+2 ) }+H ( 0 ) \bigr) \\ &\leq d \bigl( \Vert u\Vert _{2 ( r+2 ) }^{2 ( r+2 ) }+H ( t ) \bigr) , \end{aligned} \end{aligned}$$

(3.26)

$$\begin{aligned}& \Vert v\Vert _{2 ( r+2 ) }^{2\sigma q ( r+2 ) +\theta+q+1}\leq d \bigl( \Vert v\Vert _{2 ( r+2 ) }^{2 ( r+2 ) }+H ( t ) \bigr), \end{aligned}$$

(3.27)

where $d=1+\frac{1}{H ( 0 ) }$. Similarly

$$\begin{aligned}& \Vert u\Vert _{2 ( r+2 ) }^{2\sigma q ( r+2 ) +\varrho+q+1}\leq d \bigl( \Vert u\Vert _{2 ( r+2 ) }^{2 ( r+2 ) }+H ( t ) \bigr) , \end{aligned}$$

(3.28)

$$\begin{aligned}& \Vert v\Vert _{2 ( r+2 ) }^{2\sigma p ( r+2 ) +l+p+1}\leq d \bigl( \Vert v\Vert _{2 ( r+2 ) }^{2 ( r+2 ) }+H ( t ) \bigr) . \end{aligned}$$

(3.29)

Also, since $( a+b ) ^{\lambda}\leq C ( a^{\lambda }+b^{\lambda} )$, $a,b>0$, by the Young inequality and using (3.11) and (3.25), we conclude that

$$\begin{aligned}& \begin{aligned}[b] \Vert v\Vert _{2 ( r+2 ) }^{2\sigma p ( r+2 ) }\Vert u\Vert _{k+p+1}^{k+p+1} &\leq C \bigl( \Vert v\Vert _{2 ( r+2 ) }^{2 ( r+2 ) }+\Vert u\Vert _{k+p+1}^{2 ( r+2 ) } \bigr) \\ &\leq C \bigl( \Vert v\Vert _{2 ( r+2 ) }^{2 ( r+2 ) }+\Vert u\Vert _{2 ( r+2 ) }^{2 ( r+2 ) } \bigr) , \end{aligned} \end{aligned}$$

(3.30)

$$\begin{aligned}& \Vert u\Vert _{2 ( r+2 ) }^{2\sigma q ( r+2 ) }\Vert v\Vert _{\theta+q+1}^{\theta+q+1}\leq C \bigl( \Vert u\Vert _{2 ( r+2 ) }^{2 ( r+2 ) }+ \Vert v\Vert _{2 ( r+2 ) }^{2 ( r+2 ) } \bigr) , \end{aligned}$$

(3.31)

$$\begin{aligned}& \Vert u\Vert _{2 ( r+2 ) }^{2\sigma p ( r+2 ) }\Vert v\Vert _{l+p+1}^{l+p+1}\leq C \bigl( \Vert u\Vert _{2 ( r+2 ) }^{2 ( r+2 ) }+ \Vert v\Vert _{2 ( r+2 ) }^{2 ( r+2 ) } \bigr) \end{aligned}$$

(3.32)

and

$$ \Vert v\Vert _{2 ( r+2 ) }^{2\sigma q ( r+2 ) }\Vert u\Vert _{\varrho+q+1}^{\varrho+q+1}\leq C \bigl( \Vert v\Vert _{2 ( r+2 ) }^{2 ( r+2 ) }+ \Vert u\Vert _{2 ( r+2 ) }^{2 ( r+2 ) } \bigr) . $$

(3.33)

Combining (3.21)-(3.24) and (3.26)-(3.33) into (3.20), we have

$$\begin{aligned} \Psi^{\prime} ( t ) \geq& \bigl( ( 1-\sigma ) -K\varepsilon \bigr) H^{-\sigma} ( t ) H^{\prime} ( t ) +\varepsilon ( m+2 ) \bigl( \Vert u_{t}\Vert ^{2}+\Vert v_{t}\Vert ^{2} \bigr) +\varepsilon b_{1}m \bigl( \Vert \nabla u \Vert ^{2}+\Vert \nabla v\Vert ^{2} \bigr) \\ &{}+\varepsilon \biggl[ 2 ( m+1 ) -Ck_{1}^{-p} \biggl( 1+ \frac {l}{l+p+1}\delta_{1}^{\frac{l+p+1}{l}}+ \frac{p+1}{l+p+1}\delta_{1}^{-\frac {l+p+1}{p+1}} \biggr) \\ &{} -Ck_{2}^{-q} \biggl( 1+\frac{\varrho}{\varrho+q+1} \delta_{2}^{ \frac{\varrho+q+1}{\varrho}}+\frac{q+1}{\varrho+q+1}\delta _{2}^{-\frac{\varrho+q+1}{q+1}} \biggr) \biggr] H ( t ) \\ &{}+\varepsilon \biggl[ c^{\prime}-Ck_{1}^{-p} \biggl( 1+\frac {l}{l+p+1}\delta _{1}^{\frac{l+p+1}{l}}+\frac{p+1}{l+p+1} \delta_{1}^{-\frac{l+p+1}{p+1}} \biggr) \\ &{} -Ck_{2}^{-q} \biggl( 1+\frac{\varrho}{\varrho+q+1} \delta_{2}^{ \frac{\varrho+q+1}{\varrho}}+\frac{q+1}{\varrho+q+1}\delta _{2}^{-\frac{\varrho+q+1}{q+1}} \biggr) \biggr] \\ &{}\times\bigl( \Vert u\Vert _{2 ( r+2 ) }^{2 ( r+2 ) }+\Vert v \Vert _{2 ( r+2 ) }^{2 ( r+2 ) } \bigr) . \end{aligned}$$

(3.34)

At this point, and for large values of $k_{1}$ and $k_{2}$, we can find positive constants $K_{1}$ and $K_{2}$ such that (3.34) becomes

$$\begin{aligned} \Psi^{\prime} ( t ) \geq& \bigl( ( 1-\sigma ) -K\varepsilon \bigr) H^{-\sigma} ( t ) H^{\prime} ( t ) +\varepsilon ( m+2 ) \bigl( \Vert u_{t}\Vert ^{2}+\Vert v_{t}\Vert ^{2} \bigr) +\varepsilon b_{1}m \bigl( \Vert \nabla u \Vert ^{2}+\Vert \nabla v\Vert ^{2} \bigr) \\ &{}+\varepsilon K_{1}H ( t ) +\varepsilon K_{2} \bigl( \Vert u\Vert _{2 ( r+2 ) }^{2 ( r+2 ) }+\Vert v\Vert _{2 ( r+2 ) }^{2 ( r+2 ) } \bigr) \\ \geq&\beta \bigl( \Vert u_{t}\Vert ^{2}+\Vert v_{t}\Vert ^{2}+H ( t ) +\Vert \nabla u\Vert ^{2}+\Vert \nabla v\Vert ^{2}+\Vert u\Vert _{2 ( r+2 ) }^{2 ( r+2 ) }+\Vert v\Vert _{2 ( r+2 ) }^{2 ( r+2 ) } \bigr), \end{aligned}$$

(3.35)

where $\beta=\min \{ \varepsilon ( m+2 ) ,\varepsilon b_{1}m,\varepsilon K_{1},\varepsilon K_{2} \}$, and we pick ε small enough so that $( 1-\sigma ) - K\varepsilon \geq0$. Consequently, we have

$$ \Psi ( t ) \geq\Psi ( 0 ) =H^{1-\sigma} ( 0 ) +\varepsilon \biggl( \int _{\Omega}u_{0}u_{1}\,dx+\int _{\Omega }v_{0}v_{1}\,dx \biggr) >0, \quad\forall t\geq0. $$

(3.36)

On the other hand, applying the Hölder inequality, we obtain

$$\begin{aligned} \biggl\vert \int_{\Omega}uu_{t}\,dx+\int _{\Omega}vv_{t}\,dx\biggr\vert ^{\frac{1}{1-\sigma}} \leq& \Vert u\Vert ^{\frac{1}{1-\sigma}} \Vert u_{t}\Vert ^{\frac{1}{1-\sigma}}+ \Vert v\Vert ^{\frac{1}{1-\sigma}} \Vert v_{t}\Vert ^{\frac{1}{1-\sigma}} \\ \leq&C \bigl( \Vert u\Vert _{2 ( r+2 ) }^{\frac{1}{ 1-\sigma}} \Vert u_{t}\Vert ^{\frac{1}{1-\sigma}}+\Vert v\Vert _{2 ( r+2 ) }^{\frac{1}{1-\sigma}} \Vert v_{t}\Vert ^{\frac{1}{1-\sigma}} \bigr) . \end{aligned}$$

(3.37)

The Young inequality gives

$$ \biggl\vert \int_{\Omega}uu_{t}\,dx+\int _{\Omega}vv_{t}\,dx\biggr\vert ^{\frac{1}{1-\sigma}}\leq C \bigl( \Vert u\Vert _{2 ( r+2 ) }^{\frac{\mu_{1}}{1-\sigma}}+\Vert u_{t} \Vert ^{\frac{\mu _{2}}{1-\sigma}}+\Vert v\Vert _{2 ( r+2 ) }^{\frac{\mu _{1}}{1-\sigma}}+\Vert v_{t}\Vert ^{\frac{\mu_{2}}{1-\sigma }} \bigr) $$

(3.38)

for $\frac{1}{\mu_{1}}+\frac{1}{\mu_{2}}=1$. We take $\mu_{2}=2 ( 1-\sigma )$ to get $\mu_{1}=\frac{2 ( 1-\sigma ) }{1-2\sigma}$ by (3.11). Therefore (3.38) becomes

$$ \biggl\vert \int_{\Omega}uu_{t}\,dx+\int _{\Omega}vv_{t}\,dx\biggr\vert ^{\frac{1}{1-\sigma}}\leq C \bigl( \Vert u_{t}\Vert ^{2}+\Vert v_{t} \Vert ^{2}+\Vert u\Vert _{2 ( r+2 ) }^{\frac{2}{1-2\sigma}}+\Vert v \Vert _{2 ( r+2 ) }^{\frac{2}{ 1-2\sigma}} \bigr) . $$

(3.39)

By using Lemma 2.3, we obtain

$$ \begin{aligned}[b] &\biggl\vert \int_{\Omega}uu_{t}\,dx+\int _{\Omega}vv_{t}\,dx\biggr\vert ^{\frac{1}{1-\sigma}}\\ &\quad\leq C \bigl( \Vert u_{t}\Vert ^{2}+\Vert v_{t} \Vert ^{2}+\Vert u\Vert _{2 ( r+2 ) }^{2 ( r+2 ) }+\Vert v \Vert _{2 ( r+2 ) }^{2 ( r+2 ) }+\Vert \nabla u\Vert ^{2}+ \Vert \nabla v\Vert ^{2} \bigr) . \end{aligned} $$

(3.40)

Thus

$$\begin{aligned} \Psi^{\frac{1}{1-\sigma}} ( t ) =& \biggl[ H^{1-\sigma} ( t ) +\varepsilon \biggl( \int_{\Omega}uu_{t}\,dx+\int _{\Omega }vv_{t}\,dx \biggr) \biggr] ^{\frac{1}{1-\sigma}} \\ \leq&2^{\frac{\sigma}{1-\sigma}} \biggl( H ( t ) +\varepsilon^{\frac{1}{1-\sigma}}\biggl\vert \int_{\Omega}uu_{t}\,dx+\int _{\Omega }vv_{t}\,dx\biggr\vert ^{\frac{1}{1-\sigma}} \biggr) \\ \leq&C \bigl( \Vert u_{t}\Vert ^{2}+\Vert v_{t}\Vert ^{2}+H ( t ) +\Vert u\Vert _{2 ( r+2 ) }^{2 ( r+2 ) }+\Vert v\Vert _{2 ( r+2 ) }^{2 ( r+2 ) }+ \Vert \nabla u\Vert ^{2}+\Vert \nabla v\Vert ^{2} \bigr) . \end{aligned}$$

(3.41)

By combining (3.35) and (3.41), we arrive at

$$ \Psi^{\prime} ( t ) \geq\xi\Psi^{\frac{1}{1-\sigma}} ( t ) , $$

(3.42)

where ξ is a positive constant.

A simple integration of (3.42) over $( 0,t )$ yields $\Psi^{ \frac{\sigma}{1-\sigma}} ( t ) \geq\frac{1}{\Psi^{-\frac {\sigma }{1-\sigma}} ( 0 ) -\frac{\xi\sigma t}{1-\sigma}}$, which implies that the solution blows up in a finite time $T^{\ast}$ with

$$ T^{\ast}\leq\frac{1-\sigma}{\xi\sigma\Psi^{\frac{\sigma }{1-\sigma}} ( 0 ) }. $$

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