Global solutions and blow-up for a class of strongly damped wave equations systems

Abstract

The initial-boundary value problem for semilinear wave equation systems with a strong dissipative term in bounded domain is studied. The existence of global solutions for this problem is proved by using potential well method, and the exponential decay of global solutions is given through introducing an appropriate Lyapunov function. Meanwhile, blow-up of solutions in the unstable set is also obtained.

Keywords

Nonlinear wave equations systems / global solutions / exponential decay / blow-up

1 Introduction

In this paper, we are concerned with the following damped wave equations systems

(1.1)u_{tt} -\mathrm{\Delta }u+{\mu }_1 u_t- {\omega }_1\mathrm{\Delta }{u}_t=f_1 (u,v ), (x,t)\in \mathrm{\Omega }\times R^{+},

(1.2)v_{tt}-\mathrm{\Delta }v+{\mu }_2{v}_t-{\omega }_2\mathrm{\Delta }{v}_t=f_2(u ,v), ( x,t) \in \mathrm{\Omega }\times R^{+}

with the initial-boundary value conditions

(1.3)u(x,0) =u_0(x) ,u_t(x, 0)=u_1( x),x\in \mathrm{\Omega },

(1.4)v(x,0) =v_0(x) ,v_t(x, 0)=v_1( x),x\in \mathrm{\Omega },

(1.5)u(x,t) =v(x ,t)= 0,(x,t)\in \mathrm{\partial }\mathrm{\Omega }\times R^{+},

where {\mu }_i, {\omega }_i> 0(i=1,2) are real numbers, \mathrm{\Omega }\subset R^n is a bounded domain with smooth boundary \mathrm{\partial }\mathrm{\Omega } so that divergence theorem can be applied. f_i( \cdot ,\cdot ) :R^2\to R(i=1 ,2) are given functions to be determined later.

Systems of nonlinear wave equations (1.1)−(1.5) go back to Reed [15] in 1976 who proposed a similar model in three space-dimensional space, but without the presence of the damping terms. When the strongly damped terms in (1.1) and (1.2) are replaced by nonlinear dissipative terms, problem (1.1)−(1.5) becomes the following initial-boundary value problem:

(1.6)u_{tt}-\mathrm{\Delta }u+a|u_t{|}^{r_1-2}{u}_t=f_1(u ,v), ( x,t) \in \mathrm{\Omega }\times R^{+},

(1.7)v_{tt}-\mathrm{\Delta }v+a|v_t{|}^{r_2-2}{v}_t=f_2(u ,v), ( x,t) \in \mathrm{\Omega }\times R^{+},

(1.8)u(x,0) =u_0(x) ,u_t(x, 0)=u_1( x),x\in \mathrm{\Omega },

(1.9)v(x,0) =v_0(x) ,v_t(x,0) =v_1(x) ,x\in \mathrm{\Omega },

(1.10)u(x,t) =v(x ,t)= 0,(x,t)\in \mathrm{\partial }\mathrm{\Omega }\times R^{+}.

Agre and Rammaha [1] showed the existence of global weak solutions and gave the blow-up of solutions under the condition of the negative initial energy. Alves et al. [2] studied the global existence, uniform decay rate and blow-up of solutions in finite time under some conditions on the parameters in the systems and with the nonnegative initial energy and careful analysis involving the Nehari Manifold. When the damping terms are degenerate, Rammaha and Sakuntasathien [14] obtained the existence and uniqueness of global and local solutions to (1.6)−(1.10). In addition, they proved that weak solutions for the systems blow up in finite time whenever the initial energy is negative and the exponent of the source terms is more dominant than the exponent of damping terms. Quite recently, Li, Sun and Liu [9] dealt with global existence, uniform decay and blow-up of solutions of problem (1.6)−(1.10). Said-Houari [16] obtained the global existence and decay of solutions for problem (1.6)−(1.10) under some restrictions on the nonlinearity of the damping and source terms.

Consider the initial-boundary value problem of a single damped wave equation

(1.11)u_{tt}-\mathrm{\Delta }u+\mu u_t- \omega \mathrm{\Delta }{u}_t=a | u{|}^{p-2}u,x\in \mathrm{\Omega }, t>0,

(1.12)u(x,0) =u_0(x) ,u_t(x, 0)=u_1( x),x\in \mathrm{\Omega },

(1.13)u(x,t) =0,x\in \mathrm{\partial }\mathrm{\Omega }, t\ge 0.

Gazzola and Squassina [4] studied the existence of local and global solutions of problem (1.11)−(1.13), and not only finite time blow-up for solutions starting in the unstable set was proved, but also high energy initial data for which the solution blows up were constructed. Later, Gerbi and Said-Houari [5-7] considered problem (1.11)−(1.13) or equation (1.11) with dynamic boundary conditions. They obtained the existence of local and global solutions and gave the asymptotic stability and blow-up result of solutions.

Introducing a strong damping term makes problem (1.1)−(1.5) different from the one considered in [1]. For this reason fewer results are, at the present time, known for the wave equation with the strong damping and many problems remain unsolved (see[4]).

Motivated by the above researches, in this paper, we prove the global existence for problem (1.1)−(1.5) by applying the potential well theory introduced by Sattinger [17] and Payne and Sattinger [13]. Moreover, we obtain the exponential decay and the blow-up result of solution for this problem.

We adopt the usual notations and convention. Let H^1(\mathrm{\Omega }) denote the Sobolev space with the usual scalar products and norm. H_0^1(\mathrm{\Omega }) denotes the closure in H^1(\mathrm{\Omega }) of C_0^{\mathrm{\infty }}(\mathrm{\Omega }). For simplicity of notations, hereafter we denote by \Vert \cdot {\Vert }_r the Lebesgue space L^r( \mathrm{\Omega }) norm and \Vert \cdot \Vert denotes L^2(\mathrm{\Omega }) norm, and we write equivalent norm \Vert \mathrm{\nabla }\cdot \Vert instead of H_0^1 (\mathrm{\Omega } ) norm \Vert \cdot {\Vert }_{H_0^1(\mathrm{\Omega })}. In addition, C_i(i =0,1, 2,\dots ) denote various positive constants which depend on the known constants and may be different at each appearance.

This paper is organized as follows. In the next section, we are going to give some preliminaries. In Section 3, we will study the existence and exponential decay of global solutions to problem (1.1)−(1.5). Then in Section 4, we are devoted to the proof of blow-up result of solutions in finite time.

2 Preliminaries

We make the following assumptions on the functions f_1(u, v),f_2(u,v ) and parameter p:

(2.1)

\mathbf{(}\mathbf{A}\mathbf{2}\mathbf{)}: f_1(u ,v) and f_2( u,v) are nonnegative real value functions from R^2 to R and satisfy

(2.2)\begin{array}{l}{f}_1(u,v)=b_1|u+v{|}^{2(p-1)}(u+v)+b_2|u{|}^{p-2}u|v{|}^p,\\ f_2(u,v)=b_1|u+v{|}^{2(p-1)}(u+v)+b_2|v{|}^{p-2}v|u{|}^p,\end{array}

where b_1,b_2>0 and p>1 are constants.

It is easy to see from (A2) that

(2.3)u{f}_1(u, v)+v f_2(u,v)=2pF(u,v ),\mathrm{\forall }( u,v) \in R^2,

where

(2.4)F(u,v) =\frac{b_1}{2p}|u+v{|}^{2p} +\frac{b_2}{p}|uv{|}^p.

Moreover, a quick computation will show that there exist two positive constants C_0 and C_1 such that the following inequality holds (see [11]):

(2.5)\frac{C_0}{2p}(|u{|}^{2p} +|v{|}^{2p})\le F( u,v) \le \frac{C_1}{2p}(|u{|}^{2p} +|v{|}^{2p}).

Now, we define the following functionals:

(2.6)J(t)=J ([u(t),v(t) ])=\frac{1}{2}(\Vert \mathrm{\nabla }u{\Vert }^2 +\Vert \mathrm{\nabla }v{\Vert }^2 )-{\int }_{\mathrm{\Omega }}F(u,v )\mathrm{d}x,

(2.7)K(t)=K ([u(t),v(t) ])=(\Vert \mathrm{\nabla } u{\Vert }^2+\Vert \mathrm{\nabla }v {\Vert }^2)- 2p{\int }_{\mathrm{\Omega }}F(u,v )\mathrm{d}x,

for [u,v ]\in H_0^1(\mathrm{\Omega })\times H_0^1(\mathrm{\Omega }), and denote the total energy related to equations (1.1) and (1.2) by

(2.8)\begin{array}{rl}E(t)& ={\displaystyle \frac{1}{2}}(\Vert u_t|{\Vert }^2+\Vert v_t{\Vert }^2+\Vert \mathrm{\nabla }u{\Vert }^2+\Vert \mathrm{\nabla }v{\Vert }^2)-{\int }_{\mathrm{\Omega }}F(u,v)\mathrm{d}x\\ & ={\displaystyle \frac{1}{2}}(\Vert u_t{\Vert }^2+\Vert v_t{\Vert }^2)+J([u,v])\end{array}

for [u,v ]\in H_0^1(\mathrm{\Omega })\times H_0^1(\mathrm{\Omega }),t\ge 0 and

(2.9)E(0)=\frac{1}{2}(\Vert u_1{\Vert }^2+\Vert v_1{\Vert }^2+\Vert \mathrm{\nabla }{u}_0{\Vert }^2+ \Vert \mathrm{\nabla }{v}_0{\Vert }^2)- {\int }_{\mathrm{\Omega }}F(u_0,v_0)\mathrm{d}x

is the initial total energy.

As in [13], the mountain pass value of J(t) (also known as potential well depth) is defined as:

(2.10)d=inf\{\underset{\lambda \ge 0}{sup}J[\lambda u,\lambda v]:[u, v]\in H_0^1(\mathrm{\Omega })\times H_0^1( \mathrm{\Omega }) /\{0\}\} .

We introduce the so-called Nehari manifold (see [12, 18]) as follows:

\mathcal{N}=\{ [u,v ]\in H_0^1( \mathrm{\Omega }) \times H_0^1(\mathrm{\Omega })/\{0\};K (t)=0\}.

\mathcal{N} separates the two unbounded sets:

{\mathcal{N}}^{+}=\{[u ,v]\in H_0^1(\mathrm{\Omega })\times H_0^1( \mathrm{\Omega }) /\{0\};K (t)>0\}\cup \{0\}.

and

Then, the stable set \mathcal{W} and unstable set \mathcal{U} can be defined respectively by

\mathcal{W}=\{ [u,v ]\in H_0^1( \mathrm{\Omega }) \times H_0^1(\mathrm{\Omega }):J(t)\le d\}\cap {\mathcal{N}}^{+}

and

\mathcal{U}=\{ [u,v ]\in H_0^1( \mathrm{\Omega }) \times H_0^1(\mathrm{\Omega }):J(t)\le d\}\cap {\mathcal{N}}^{-}.

It is readily seen that the potential well depth d defined in (2.10) may also be characterized as

(2.11)d=\underset{[u,v]\in \mathcal{N}}{inf}J ([u,v] ).

As it was remarked by [7], this alternative characterization of d shows that

(2.12)\beta =dist( 0,\mathcal{N})=\underset{[u,v]\in \mathcal{N}}{inf}(\Vert \mathrm{\nabla }u{\Vert }^2+\Vert \mathrm{\nabla } v{\Vert }^2{)}^{\frac{1}{2}}=(\frac{2pd}{p-1}{)}^{\frac{1}{2}}>0.

For the applications through this paper, we introduce the definition of solutions to problem (1.1)−(1.5) given by Gazzola and Squassina in [4] and list up some known lemmas.

Definition 2.1　A pair of functions [u,v ] is said to be a weak solution of (1.1)−(1.5) on [0,T] if

[u,v ]\in C([0 ,T], H_0^1(\mathrm{\Omega })\times H_0^1(\mathrm{\Omega }))\cap C^2([0, T],H^{-1}\times H^{-1}(\mathrm{\Omega })),

and

[u_t,v_t ]\in C([0 ,T], L^2(\mathrm{\Omega })\times L^2(\mathrm{\Omega }))\cap L^2 ([0,T] ,H_0^1(\mathrm{\Omega })\times H_0^1(\mathrm{\Omega }))

satisfy

{\int }_{\mathrm{\Omega }}{u}_{tt}\phi \mathrm{d}x+{\int }_{\mathrm{\Omega }}\mathrm{\nabla }u\mathrm{\nabla }\phi \mathrm{d}x+{\mu }_1{\int }_{\mathrm{\Omega }}{u}_t\phi \mathrm{d}x+{\omega }_1{\int }_{\mathrm{\Omega }} \mathrm{\nabla }{u}_t\mathrm{\nabla }\phi \mathrm{d}x={\int }_{\mathrm{\Omega }}{f}_1(u,v )\phi \mathrm{d}x,

{\int }_{\mathrm{\Omega }}{v}_{tt}\psi \mathrm{d}x+{\int }_{\mathrm{\Omega }}\mathrm{\nabla }v\mathrm{\nabla }\psi \mathrm{d}x+{\mu }_2{\int }_{\mathrm{\Omega }}{v}_t\psi \mathrm{d}x+{\omega }_2{\int }_{\mathrm{\Omega }} \mathrm{\nabla }{v}_t\mathrm{\nabla }\psi \mathrm{d}x={\int }_{\mathrm{\Omega }}{f}_2(u,v )\psi \mathrm{d}x

for all test functions [\phi ,\psi ]\in H_0^1(\mathrm{\Omega })\times H_0^1(\mathrm{\Omega }) and almost all t\in [0, T].

Lemma 2.1　 Let r be a real number with if n\le 2 and 2\le r\le \frac{2n}{n-2} if n>2. Then there is a constant C depending on \mathrm{\Omega } and r such that

\Vert u{\Vert }_r\le C\Vert \mathrm{\nabla } u\Vert ,\mathrm{\forall } u\in H_0^1(\mathrm{\Omega }).

Lemma 2.2 (Young inequality)　 Let X,Y and \epsilon be positive constants and . Then one has the inequality

XY\le \frac{{\epsilon }^{\xi }{X}^{\xi }}{\xi }+\frac{Y^{\zeta }}{\zeta {\epsilon }^{\zeta }}.

Lemma 2.3　 Let [u(t) ,v(t )] be a pair of solutions for problem (1.1)−(1.5). Then E(t) is a nonincreasing function for t>0 and

(2.13)E^{\prime } (t)=-({\mu }_1\Vert u_t{\Vert }_2^2 +{\mu }_2\Vert v_t{\Vert }_2^2 +{\omega }_1\Vert \mathrm{\nabla }{u}_t{\Vert }^2+{\omega }_2\Vert \mathrm{\nabla }{v}_t{\Vert }^2)\le 0.

Proof　Multiplying equation (1.1) by u_t and (1.2) by v_t, and integrating over \mathrm{\Omega } \times [0,t]. Then, adding them together, and integrating by parts, we get

(2.14)E(t)=E (0)-{\int }_0^t[{\mu }_1\Vert u_t(s) {\Vert }_2^2+ {\mu }_2\Vert v_t(s){\Vert }_2^2 +{\omega }_1\Vert \mathrm{\nabla }{u}_t(s) {\Vert }^2+{\omega }_2\Vert \mathrm{\nabla }{v}_t(s){\Vert }^2] \mathrm{d}s

for t\ge 0. Being the primitive of an integrable function, E(t) is absolutely continuous and equality (2.13) is fulfilled.

We conclude this section by stating a local existence result of problem (1.1)−(1.5), which can be established by using a similar way as done in combination of the arguments in [1,3,4,10,19].

Theorem 2.1 (Local existence)　 Suppose that (A1) holds. If [u_0,v_0]\in H_0^1(\mathrm{\Omega })\times H_0^1(\mathrm{\Omega }) and [u_1,v_1]\in L^2(\mathrm{\Omega })\times L^2(\mathrm{\Omega }), then there exists T>0 such that problem (1.1)−(1.5) has a unique local solution [u(t ),v(t) ], which satisfies

[u,v]\in C([0,T);H_0^1(\mathrm{\Omega })\times H_0^1(\mathrm{\Omega })),

u_t\in C([0,T );L^2( \mathrm{\Omega }) )\cap L^2([0 ,T); H_0^1(\mathrm{\Omega })),

v_t\in C([0,T );L^2( \mathrm{\Omega }) )\cap L^2([0 ,T); H_0^1(\mathrm{\Omega })).

Moreover, at least one of the following statements holds true: (1)\Vert u_t{\Vert }^2+\Vert v_t{\Vert }^2+\Vert \mathrm{\nabla }u {\Vert }^2+\Vert \mathrm{\nabla }v {\Vert }^2\to + \mathrm{\infty } as t\to T^{-}; (2) T=+\mathrm{\infty }.

3 Global existence and exponential decay

We can now proceed in study of the existence of global solution for problem (1.1)−(1.5). For this purpose, we need the following lemmas.

Lemma 3.1　 Suppose that (A1) and (A2) hold. If [u,v]\in H_0^1(\mathrm{\Omega })\times H_0^1(\mathrm{\Omega }), then

(3.1)d=\frac{p-1}{2p}(\frac{1}{C_2{B}^{\ast }} {)}^{\frac{2p}{p-1}}>0,

where B^{\ast } is the optimal Sobolev's constant from H_0^1(\mathrm{\Omega }) to L^{2p}(\mathrm{\Omega }) and the positive constant C_2 is determined later.

Proof　Since

J(\lambda [u,v])=\frac{{\lambda }^2}{2}(\Vert \mathrm{\nabla }u{\Vert }^2 +\Vert \mathrm{\nabla }v{\Vert }^2 )-{\lambda }^{2p}{\int }_{\mathrm{\Omega }}F(u ,v)\mathrm{d}x,

we get

\frac{\mathrm{d}}{\mathrm{d}\lambda }J(\lambda [u,v])=\lambda (\Vert \mathrm{\nabla }u{\Vert }^2 +\Vert \mathrm{\nabla }v{\Vert }^2 )-2p{\lambda }^{2p-1}{\int }_{\mathrm{\Omega }}F(u ,v)\mathrm{d}x,

\frac{{\mathrm{d}}^2}{\mathrm{d}{\lambda }^2}J(\lambda [u,v])=(\Vert \mathrm{\nabla }u{\Vert }^2 +\Vert \mathrm{\nabla }v{\Vert }^2 )-2p(2p -1) {\lambda }^{2(p-1)}{\int }_{\mathrm{\Omega }} F(u, v)\mathrm{d}x.

Let \frac{\mathrm{d}}{\mathrm{d}\lambda }J(\lambda [u,v])=0. Then we obtain

{\lambda }_1 =(\frac{\Vert \mathrm{\nabla }u{\Vert }^2+\Vert \mathrm{\nabla }v{\Vert }^2}{2p{\int }_{\mathrm{\Omega }}F(u,v)\mathrm{d}x}{)}^{\frac{1}{2(p-1)}}.

As \lambda ={\lambda }_1, an elementary calculation shows that

Therefore, we have

(3.2)\underset{\lambda \ge 0}{sup}J(\lambda [u,v] )=J({\lambda }_1[u, v])=\frac{p-1}{2p}[\frac{(\Vert \mathrm{\nabla }u{\Vert }^2+\Vert \mathrm{\nabla }v{\Vert }^2{)}^p}{2p{\int }_{\mathrm{\Omega }}F(u,v)\mathrm{d}x}{]}^{\frac{1}{p-1}}.

By Minkowski's inequality and Lemma 2.1, we attain

(3.3)\Vert u+v{\Vert }_{2p}^2\le 2(\Vert u{\Vert }_{2p}^2+\Vert v{\Vert }_{2p}^2)\le 2B^{\ast }{}^2(\Vert \mathrm{\nabla }u{\Vert }^2+ \Vert \mathrm{\nabla }v{\Vert }^2).

Also, we gain from Hölder inequality and Lemma 2.2 that

(3.4)\Vert uv{\Vert }_p\le \Vert u{\Vert }_{2p}\cdot \Vert v{\Vert }_{2p}\le \frac{1}{2}(\Vert u{\Vert }_{2p}^2+ \Vert v {\Vert }_{2p}^2)\le \frac{B^{\ast }{}^2}{2}(\Vert \mathrm{\nabla }u{\Vert }^2+\Vert \mathrm{\nabla }v{\Vert }^2).

From (2.4), (3.3) and (3.4), we deduce that

(3.5){\int }_{\mathrm{\Omega }}F( u,v) \mathrm{d}x\le \frac{(C_2{B}^{\ast }){}^{2p}}{2p}(\Vert \mathrm{\nabla }u{\Vert }^2+\Vert \mathrm{\nabla }v{\Vert }^2{)}^p,

where C_2=(2^p{b}_1 +\frac{b_2}{2^{p-1}}{)}^{\frac{1}{2p}}.

It follows from (3.2) and (3.5) that

d=\frac{p-1}{2p}(\frac{1}{C_2{B}^{\ast }} {)}^{\frac{2p}{p-1}}>0.

Thus, we complete the proof of Lemma 3.1.

Lemma 3.2　 Assume that (A1) and (A2) hold. If [u_0,v_0]\in {\mathcal{N}}^{\mathcal{+}},[u_1, v_1]\in L^2 (\mathrm{\Omega } )\times L^2(\mathrm{\Omega }) and , then [u(t ),v(t) ]\in {\mathcal{N}}^{\mathcal{+}} for each t\in [0 ,T).

Proof　By [ u_0,v_0] \in {\mathcal{N}}^{\mathcal{+}}, we see that K([u_0,v_0])>0. Then, in virtue of the continuity, there exists t^{\ast } \in [0, T) such that K(t) \ge 0 for t\in [0, t^{\ast }). We get from (2.6) and (2.7) that

J(t)=\frac{p-1}{2p}( \Vert \mathrm{\nabla }u{\Vert }^2+ \Vert \mathrm{\nabla }v{\Vert }^2 )+\frac{1}{2p}K (t),

which implies that

(3.6)

Therefore, we have

(3.7)\Vert \mathrm{\nabla }u{\Vert }^2+ \Vert \mathrm{\nabla }v{\Vert }^2 \le \frac{2p}{p-1}J(t ).

From (2.8), (3.6), (3.7) and Lemma 2.3, we obtain

(3.8)\Vert \mathrm{\nabla }u{\Vert }^2+ \Vert \mathrm{\nabla }v{\Vert }^2 \le \frac{2p}{p-1}E(t )\le \frac{2p}{p-1}E(0), \mathrm{\forall }t\in [0 ,t^{\ast }).

It yields from (3.5) and (3.8) that

(3.9)\begin{array}{rl}2p{\int }_{\mathrm{\Omega }}F(u,v)\mathrm{d}x& \le (C_2{B}^{\ast }){}^{2p}(\Vert \mathrm{\nabla }u{\Vert }^2+\Vert \mathrm{\nabla }v{\Vert }^2{)}^p\\ & =(C_2{B}^{\ast }){}^{2p}(\Vert \mathrm{\nabla }u{\Vert }^2+\Vert \mathrm{\nabla }v{\Vert }^2{)}^{p-1}(\Vert \mathrm{\nabla }u{\Vert }^2+\Vert \mathrm{\nabla }v{\Vert }^2)\\ & \le \theta (\Vert \mathrm{\nabla }u{\Vert }^2+\Vert \mathrm{\nabla }v{\Vert }^2),\end{array}

where

\theta =(C_2 B^{\ast }){}^{2p}(\frac{2p}{p-1}E(0){)}^{p-1}.

By and (3.1), we find

(3.10)

We conclude from (3.9) and (3.10) that

(3.11)

Hence K(t)>0, \mathrm{\forall }t\in [0,t^{\ast }), which shows that [u(t) ,v(t )]\in {\mathcal{N}}^{+}, \mathrm{\forall }t\in [0 ,t^{\ast }).

Noting

we repeat steps (3.6)−(3.11) to extend t^{\ast } to 2t^{\ast }. We continue this procedure until [u(t ),v(t) ]\in {\mathcal{N}}^{+}, \mathrm{\forall } t\in [0 ,T).

The following theorem shows that the solution obtained in Theorem 2.1 is a global solution.

Theorem 3.1　 Assume that (A1) and (A2) hold. If [u_0,v_0]\in \mathcal{W},[ u_1,v_1] \in L^2(\mathrm{\Omega })\times L^2(\mathrm{\Omega }) and , then the local solution furnished in Theorem 2.1 is a global solution and T may be taken arbitrarily large.

Proof　It suffices to show that \Vert u_t{\Vert }^2+\Vert v_t{\Vert }^2+\Vert \mathrm{\nabla }u{\Vert }^2 +\Vert \mathrm{\nabla }v{\Vert }^2 is bounded independently of t. Under the hypotheses in Theorem 3.1, by Lemma 3.2, we have [u,v] \in \mathcal{W} on [0,T). So, the following formula holds on [0,T):

(3.12)J(t)=\frac{1}{2}(\Vert \mathrm{\nabla }u{\Vert }^2+\Vert \mathrm{\nabla } v{\Vert }^2)- {\int }_{\mathrm{\Omega }}F(u ,v)\mathrm{d}x\ge \frac{p-1}{2p}(\Vert \mathrm{\nabla } u{\Vert }^2+\Vert \mathrm{\nabla }v {\Vert }^2).

We get from (3.12) that

(3.13)

which implies that

The above inequality and the continuation principle lead to the global existence of solutions for problem (1.1)−(1.5).

The following theorem shows that the global solutions of problem (1.1)−(1.5) is exponential decay.

Theorem 3.2　 Assume that (A1) and (A2) are valid. If [ u_0,v_0] \in {\mathcal{N}}^{\mathcal{+}}, [u_1,v_1]\in L^2(\mathrm{\Omega })\times L^2(\mathrm{\Omega }) and , then there exist two positive constants M and \eta independent of t such that the global solution has the following exponential decay property:

Proof　By Lemma 3.2, we see that [u( t),v (t)]\in {\mathcal{N}}^{\mathcal{+}} for all t\ge 0. Thus, we have for all t\ge 0. In order to prove the exponential decay of global solutions, we define

(3.14)F(t)=E (t)+\epsilon {\int }_{\mathrm{\Omega }}(u{u}_t +v{v}_t)\mathrm{d}x+\frac{1}{2}\epsilon {\int }_{\mathrm{\Omega }}({\omega }_1|\mathrm{\nabla }u{|}^2+{\omega }_2|\mathrm{\nabla }v {|}^2)\mathrm{d}x,

where \epsilon >0 will be determined later. It is easy to prove that there exist two positive constants {\xi }_1 and {\xi }_2 depending on \epsilon such that

(3.15){\xi }_{1}E(t)\le F(t) \le {\xi }_{2}E(t)

for all t\ge 0.

In fact, from (2.1), Lemma 2.1 and (3.13), we get

(3.16)\begin{array}{rl}F(t)& \le E(t)+\frac{\epsilon }{2}(\Vert u_t{\Vert }^2+\Vert v_t{\Vert }^2+\Vert u{\Vert }^2+\Vert v{\Vert }^2)+\frac{\epsilon }{2}({\omega }_1\Vert \mathrm{\nabla }u{\Vert }^2+{\omega }_2\Vert \mathrm{\nabla }v{\Vert }^2)\\ & \le (1+\epsilon )E(t)+\frac{\epsilon }{2}[B^{\ast }{}^2+max\{{\omega }_1,{\omega }_2\}](\Vert \mathrm{\nabla }u{\Vert }^2+\Vert \mathrm{\nabla }v{\Vert }^2)\\ & \le [(1+\epsilon )+\frac{p}{p-1}\epsilon (B^{\ast }{}^2+max\{{\omega }_1,{\omega }_2\})]E(t)={\xi }_{2}E(t).\end{array}

On the other hand, by using Lemma 2.2, we have

(3.17){\int }_{\mathrm{\Omega }}u{u}_t\mathrm{d}x\le \frac{1}{4\gamma }\Vert u_t{\Vert }^2+\gamma \Vert u{\Vert }^2,{\int }_{\mathrm{\Omega }} v{v}_t\mathrm{d}x\le \frac{1}{4\gamma }\Vert v_t{\Vert }^2+ \gamma \Vert v{\Vert }^2.

Therefore, we obtain the following inequality:

(3.18)\begin{array}{rl}F(t)& \ge E(t)-\frac{\epsilon }{4\gamma }(\Vert u_t{\Vert }^2+\Vert v_t{\Vert }^2)-\epsilon \gamma (\Vert u{\Vert }^2+\Vert v{\Vert }^2)+\frac{\epsilon }{2}({\omega }_1\Vert \mathrm{\nabla }u{\Vert }^2+{\omega }_2\Vert \mathrm{\nabla }v{\Vert }^2)\\ & \ge E(t)-\frac{\epsilon }{4\gamma }(\Vert u_t{\Vert }^2+\Vert v_t{\Vert }^2)+\epsilon [\frac{1}{2}min\{{\omega }_1,{\omega }_2\}-B^{\ast }{}^2\gamma ](\Vert \mathrm{\nabla }u{\Vert }^2+\Vert \mathrm{\nabla }v{\Vert }^2).\end{array}

By choosing \gamma small enough such that \gamma \le \frac{min\{{\omega }_1,{\omega }_2\}}{2B_{\ast }^2}, it follows from (2.8) and (3.18) that

F(t)\ge E (t)-\frac{\epsilon }{4\gamma }(\Vert u_t{\Vert }^2+ \Vert v_t{\Vert }^2)\ge J(t) +(\frac{1}{2}-\frac{\epsilon }{4\gamma })(\Vert u_t{\Vert }^2+\Vert v_t{\Vert }^2).

We then pick \epsilon >0 so small (i.e., \epsilon \le 2 \gamma) that

(3.19)F(t)\ge min\{1,1-\frac{\epsilon }{2\gamma })\}(J (t)+\frac{1}{2} \Vert u_t{\Vert }^2)={\xi }_{1}E(t ).

From (3.16) and (3.19), inequality (3.15) holds.

By differentiating (3.14) and using equation (1.1) and (1.2), and combining Lemma 2.3, we obtain

(3.20)\begin{array}{rl}{F}^{\prime }(t)=& -({\omega }_1\Vert \mathrm{\nabla }{u}_t{\Vert }^2+{\omega }_2\Vert \mathrm{\nabla }{v}_t{\Vert }^2)+(\epsilon -{\mu }_1)\Vert u_t{\Vert }^2+(\epsilon -{\mu }_2)\Vert v_t{\Vert }^2\\ & -\epsilon (\Vert \mathrm{\nabla }u{\Vert }^2+\Vert \mathrm{\nabla }v{\Vert }^2)+2p\epsilon {\int }_{\mathrm{\Omega }}F(u,v)\mathrm{d}x-\epsilon {\int }_{\mathrm{\Omega }}({\mu }_{1}u{u}_t+{\mu }_{2}v{v}_t)\mathrm{d}x.\end{array}

It yields form (3.9), (3.17) and (3.20) that

(3.21)\begin{array}{rl}{F}^{\prime }(t)=& -({\omega }_1\Vert \mathrm{\nabla }{u}_t{\Vert }^2+{\omega }_2\Vert \mathrm{\nabla }{v}_t{\Vert }^2)\\ & +(\epsilon \left(1+\frac{{\mu }_1}{4\delta }\right)-{\mu }_1)\Vert u_t{\Vert }^2+(\epsilon \left(1+\frac{{\mu }_2}{4\delta }\right)-{\mu }_2)\Vert v_t{\Vert }^2\\ & +\epsilon [\delta B^{\ast }{}^{2}max\{{\mu }_1,{\mu }_2\}+\theta -1](\Vert \mathrm{\nabla }u{\Vert }^2+\Vert \mathrm{\nabla }v{\Vert }^2).\end{array}

By (3.10), we find that . Choosing such that .

Let \eta =1-\theta -\delta B^{\ast }{}^{2}max\{{\mu }_1, {\mu }_2 \}. Then \eta >0. Thus, for any positive constant \mathrm{\Lambda }>0, from (2.8) and (3.21), we have

(3.22)\begin{array}{rl}{F}^{\prime }(t)\le & -\mathrm{\Lambda }\epsilon E(t)+[\epsilon \left(1+\frac{\mathrm{\Lambda }}{2}+\frac{{\mu }_1}{4\delta }\right)-{\mu }_1]\Vert u_t{\Vert }^2\\ & +[\epsilon \left(1+\frac{\mathrm{\Lambda }}{2}+\frac{{\mu }_2}{4\delta }\right)-{\mu }_2]\Vert v_t{\Vert }^2-({\omega }_1\Vert \mathrm{\nabla }{u}_t{\Vert }^2+{\omega }_2\Vert \mathrm{\nabla }{v}_t{\Vert }^2)\\ & +\epsilon (\frac{\mathrm{\Lambda }}{2}-\eta )(\Vert \mathrm{\nabla }u{\Vert }^2+\Vert \mathrm{\nabla }v{\Vert }^2).\end{array}

Choosing \mathrm{\Lambda }\le 2\eta and \epsilon so small enough that

inequality (3.22) implies that

(3.23)F^{\prime } (t)\le -\mathrm{\Lambda }\epsilon E(t), \mathrm{\forall }t\ge 0.

We conclude from (3.15) and (3.23) that

(3.24)F^{\prime } (t)\le -kF(t),\mathrm{\forall }t\ge 0,

where k=\mathrm{\Lambda }\epsilon / {\xi }_2>0.

Integrating the differential inequality (3.24) from 0 to t gives the following exponential decay estimate for function F(t):

(3.25)F(t)\le F (0){\mathrm{e}}^{-kt}, \mathrm{\forall }t\ge 0.

Consequently, we obtain from (3.15) once again that

E(t)\le M {\mathrm{e}}^{-kt},\mathrm{\forall }t\ge 0,

where M=F( 0)/{\xi }_1. This completes the proof of Theorem 3.2.

4 Blow-up result

In this section, we are concerned with the blow-up property of solutions for problem (1.1)−(1.5) and give the estimate of lifespan of solutions. For this purpose, we give the following lemmas.

Lemma 4.1 [8]　 Suppose that P(t) \in C^2, P(t)\ge 0 satisfies the inequality

P(t)P^{″}(t )-(1+\rho )P^{\prime 2}(t) \ge 0

for certain real number \rho >0 and P(0)>0 ,P^{\prime }(0)>0. Then there exists a real number T_{\ast } such that and P(t) \to +\mathrm{\infty } as t\to T_{\ast }^{-}.

Lemma 4.2　 Let [u(t) ,v(t )] be a pair of solutions of (1.1)−(1.5) which is given by Theorem 2.1. If [ u_0,v_0] \in \mathcal{U} and , then [u(t),v(t) ]\in \mathcal{U} and for all t\ge 0.

Proof　It follows from the conditions in Lemma 4.2 and Lemma 2.3 that

Therefore, by (2.8), we have

(4.1)

Next, let us assume by contradiction that there exists t^{\ast }\in [0,T ) such that u(t^{\ast })\notin \mathcal{U}. Then by continuity, we have K(t^{\ast })=0. This implies that [u( t^{\ast }),v(t^{\ast })] \in \mathcal{N}. We get from (2.11) that J(t^{\ast })\ge d, which is contradiction with (4.1). Consequently, Lemma 4.2 holds.

Theorem 4.1　 Assume that (A1) and (A2) are valid. If [ u_0,v_0] \in \mathcal{U},[u_1,v_1]\in L^2(\mathrm{\Omega })\times L^2(\mathrm{\Omega }) and . Then the solutions [u(t),v(t) ] in Theorem 2.1 of problem (1.1)−(1.5) blow up in finite time , which means that

\underset{t\to T_{\ast }^{-}}{lim}(\Vert \mathrm{\nabla }u(t){\Vert }^2+ \Vert \mathrm{\nabla }v(t){\Vert }^2 )=+\mathrm{\infty }.

Proof　From (2.10) and (3.2), we have

(4.2)d\le \underset{\lambda \ge 0}{sup}J(\lambda [u,v])=\frac{p-1}{2p}[\frac{(\Vert \mathrm{\nabla }u{\Vert }^2+\Vert \mathrm{\nabla }v{\Vert }^2{)}^p}{2p{\int }_{\mathrm{\Omega }}F(u,v)\mathrm{d}x}{]}^{\frac{1}{p-1}}.

By [u_0 ,v_0]\in \mathcal{U}, and Lemma 4.2, we obtain [u,v]\in \mathcal{U} for all t\in [0,T]. Thus, we get

for all t\in [0,T]. Consequently, by using this last inequality, (4.2) becomes

d\le \frac{p-1}{2p}( \Vert \mathrm{\nabla }u{\Vert }^2+ \Vert \mathrm{\nabla }v{\Vert }^2 ),\mathrm{\forall }t\in [0,T ],

which implies that

(4.3)\Vert \mathrm{\nabla }u{\Vert }^2+ \Vert \mathrm{\nabla }v{\Vert }^2 \ge \frac{2\mathrm{d}p}{p-1},\mathrm{\forall }t\in [0,T].

Assume by contradiction that solution [u(t) ,v(t )] is global. Then for any T>0, we define \mathrm{\Theta }(t) :[0, T]\to [0, +\mathrm{\infty }) by

(4.4)\begin{array}{rl}\mathrm{\Theta }(t)=& (\Vert u(t){\Vert }^2+\Vert v(t){\Vert }^2)+{\int }_0^t({\mu }_1\Vert u(s){\Vert }^2+{\mu }_2\Vert v(s){\Vert }^2)\mathrm{d}s\\ & +{\int }_0^t({\omega }_1\Vert \mathrm{\nabla }u(s){\Vert }^2+{\omega }_2\Vert \mathrm{\nabla }v(s){\Vert }^2)\mathrm{d}s\\ & +(T-t)({\mu }_1\Vert u_0{\Vert }^2+{\mu }_2\Vert v_0{\Vert }^2+{\omega }_1\Vert \mathrm{\nabla }{u}_0{\Vert }^2+{\omega }_2\Vert \mathrm{\nabla }{v}_0{\Vert }^2).\end{array}

Note that \mathrm{\Theta }(t) >0 for all t\in [0, T]. By the continuity of the function \mathrm{\Theta }(t), there exists \lambda >0 (independent of the choice of T) such that

(4.5)\mathrm{\Theta }(t) \ge \lambda , \mathrm{\forall }t\in [0,T ].

By differentiating both sides of (4.4) on t, we get

(4.6)\begin{array}{rl}{\mathrm{\Theta }}^{\prime }(t)& =2{\int }_{\mathrm{\Omega }}(u{u}_t+v{v}_t)\mathrm{d}x+({\mu }_1\Vert u(t){\Vert }^2+{\mu }_2\Vert v(t){\Vert }^2)\\ & +({\omega }_1\Vert \mathrm{\nabla }u(t){\Vert }^2+{\omega }_2\Vert \mathrm{\nabla }v(t){\Vert }^2)\\ & -({\mu }_1\Vert u_0{\Vert }^2+{\mu }_2\Vert v_0{\Vert }^2+{\omega }_1\Vert \mathrm{\nabla }{u}_0{\Vert }^2+{\omega }_2\Vert \mathrm{\nabla }{v}_0{\Vert }^2)\\ & =2{\int }_{\mathrm{\Omega }}(u{u}_t+v{v}_t)\mathrm{d}x+2{\mu }_1{\int }_0^t{\int }_{\mathrm{\Omega }}u(s)u_t(s)\mathrm{d}x\mathrm{d}s\\ & +2{\mu }_2{\int }_0^t{\int }_{\mathrm{\Omega }}v(s)v_t(s)\mathrm{d}x\mathrm{d}s+2{\omega }_1{\int }_0^t{\int }_{\mathrm{\Omega }}\mathrm{\nabla }u(s)\mathrm{\nabla }{u}_t(s)\mathrm{d}x\mathrm{d}s\\ & +2{\omega }_2{\int }_0^t{\int }_{\mathrm{\Omega }}\mathrm{\nabla }v(s)\mathrm{\nabla }{v}_t(s)\mathrm{d}x\mathrm{d}s.\end{array}

Taking the derivative of the function {\mathrm{\Theta }}^{\prime }(t) in (4.6), we obtain

(4.7)\begin{array}{rl}{\mathrm{\Theta }}^{″}(t)=& 2(\Vert u_t{\Vert }^2+\Vert v_t{\Vert }^2)+2{\int }_{\mathrm{\Omega }}(u{u}_{tt}+v{v}_{tt})\mathrm{d}x\\ & +2{\mu }_1{\int }_{\mathrm{\Omega }}u{u}_t\mathrm{d}x+2{\mu }_2{\int }_{\mathrm{\Omega }}v{v}_t\mathrm{d}x+2{\omega }_1{\int }_{\mathrm{\Omega }}\mathrm{\nabla }u\mathrm{\nabla }{u}_t\mathrm{d}x+2{\omega }_2{\int }_{\mathrm{\Omega }}\mathrm{\nabla }v\mathrm{\nabla }{v}_t\mathrm{d}x.\end{array}

It deduces from (1.1), (1.2) and (4.7) that

(4.8){\mathrm{\Theta }}^{″}(t)=2 \left(\Vert u_t(t){\Vert }^2+\Vert v_t(t){\Vert }^2+2p{\int }_{\mathrm{\Omega }}F(u,v)\mathrm{d}x-\Vert \mathrm{\nabla }u(t){\Vert }^2-\Vert \mathrm{\nabla }v(t){\Vert }^2\right) .

Combining (4.4), (4.6) with (4.8), we receive that

(4.9)\begin{array}{rl}& \mathrm{\Theta }{\mathrm{\Theta }}^{″}(t)-\frac{p+3}{4}{\mathrm{\Theta }}^{\prime }(t{)}^2\\ & =2\mathrm{\Theta }(t)[\Vert u_t(t){\Vert }^2+\Vert v_t(t){\Vert }^2\\ & +2p{\int }_{\mathrm{\Omega }}F(u,v)\mathrm{d}x-\Vert \mathrm{\nabla }u(t){\Vert }^2-\Vert \mathrm{\nabla }v(t){\Vert }^2]\\ & -(p+3)[\mathrm{\Theta }(t)-(T-t)({\mu }_1\Vert u_0{\Vert }^2+{\mu }_2\Vert v_0{\Vert }^2+{\omega }_1\Vert \mathrm{\nabla }{u}_0{\Vert }^2+{\omega }_2\Vert \mathrm{\nabla }{v}_0{\Vert }^2)]\\ & \times [\Vert u_t(t){\Vert }^2+\Vert v_t(t){\Vert }^2+{\mu }_1{\int }_0^t\Vert u_t(s){\Vert }^2\mathrm{d}s+{\mu }_2{\int }_0^t\Vert v_t(s){\Vert }^2\mathrm{d}s\\ & +{\omega }_1{\int }_0^t\Vert \mathrm{\nabla }{u}_t(s){\Vert }^2\mathrm{d}s+{\omega }_2{\int }_0^t\Vert \mathrm{\nabla }{v}_t(s){\Vert }^2\mathrm{d}s]+(r+2)\chi (t),\end{array}

where

(4.10)\begin{array}{rl}\chi (t)=& [\Vert u(t){\Vert }^2+\Vert v(t){\Vert }^2+{\int }_0^t({\mu }_1\Vert u(s){\Vert }^2+{\mu }_2\Vert v(s){\Vert }^2\\ & +{\omega }_1\Vert \mathrm{\nabla }u(s){\Vert }^2+{\omega }_2\Vert \mathrm{\nabla }v(s){\Vert }^2)\mathrm{d}s]\times [\Vert u_t(t){\Vert }^2+\Vert v_t(t){\Vert }^2\\ & +{\int }_0^t({\mu }_1\Vert u_t(s){\Vert }^2+{\mu }_2\Vert v_t(s){\Vert }^2+{\omega }_1\Vert \mathrm{\nabla }{u}_t(s){\Vert }^2+{\omega }_2\Vert \mathrm{\nabla }{v}_t(s){\Vert }^2)\mathrm{d}s]\\ & -[{\int }_{\mathrm{\Omega }}(u{u}_t+v{v}_t)\mathrm{d}x+{\mu }_1{\int }_0^t{\int }_{\mathrm{\Omega }}u(s)u_t(s)\mathrm{d}x\mathrm{d}s+{\mu }_2{\int }_0^t{\int }_{\mathrm{\Omega }}v(s)v_t(s)\mathrm{d}x\mathrm{d}s\\ & +{\omega }_1{\int }_0^t{\int }_{\mathrm{\Omega }}\mathrm{\nabla }u(s)\mathrm{\nabla }{u}_t(s)\mathrm{d}x\mathrm{d}s+{\omega }_2{\int }_0^t{\int }_{\mathrm{\Omega }}\mathrm{\nabla }v(s)\mathrm{\nabla }{v}_t(s)\mathrm{d}x\mathrm{d}s{]}^2.\end{array}

By applying Schwarz inequality, we gain that

(4.11)({\int }_{\mathrm{\Omega }}u{u}_t\mathrm{d}x{)}^2\le \Vert u(t){\Vert }^2\Vert u_t(t){\Vert }^2, ({\int }_{\mathrm{\Omega }}v{v}_t\mathrm{d}x{)}^2\le \Vert v(t) {\Vert }^2\Vert v_t(t){\Vert }^2 ,

(4.12)({\mu }_1{\int }_0^t{\int }_{\mathrm{\Omega }}u{u}_t\mathrm{d}x\mathrm{d}s{)}^2\le {\mu }_1^2{\int }_0^t\Vert u(s){\Vert }^2\mathrm{d}s{\int }_0^t\Vert u_t(s) {\Vert }^2\mathrm{d}s,

(4.13)({\mu }_2{\int }_0^t{\int }_{\mathrm{\Omega }}v{v}_t\mathrm{d}x\mathrm{d}s{)}^2\le {\mu }_2^2{\int }_0^t\Vert v(s){\Vert }^2\mathrm{d}s{\int }_0^t\Vert v_t(s) {\Vert }^2\mathrm{d}s,

(4.14)({\omega }_1{\int }_0^t{\int }_{\mathrm{\Omega }}\mathrm{\nabla }u(s)\mathrm{\nabla } u_t(s)\mathrm{d}x\mathrm{d}s{)}^2\le {\omega }_1^2{\int }_0^t\Vert \mathrm{\nabla }u(s){\Vert }^2\mathrm{d}s{\int }_0^t\Vert \mathrm{\nabla }{u}_t(s) {\Vert }^2\mathrm{d}s,

(4.15)({\omega }_2{\int }_0^t{\int }_{\mathrm{\Omega }}\mathrm{\nabla }v(s)\mathrm{\nabla } v_t(s)\mathrm{d}x\mathrm{d}s{)}^2\le {\omega }_2^2{\int }_0^t\Vert \mathrm{\nabla }v(s){\Vert }^2\mathrm{d}s{\int }_0^t\Vert \mathrm{\nabla }{v}_t(s) {\Vert }^2\mathrm{d}s,

(4.16)\begin{array}{rl}& 2{\omega }_1{\int }_0^t{\int }_{\mathrm{\Omega }}\mathrm{\nabla }u(s)\mathrm{\nabla }{u}_t(s)\mathrm{d}x\mathrm{d}s{\int }_{\mathrm{\Omega }}u{u}_t\mathrm{d}x\\ & \le {\omega }_1\Vert u_t(t){\Vert }^2{\int }_0^t\Vert \mathrm{\nabla }u(s){\Vert }^2\mathrm{d}s+{\omega }_1\Vert u(t){\Vert }^2{\int }_0^t\Vert \mathrm{\nabla }{u}_t(s){\Vert }^2\mathrm{d}s,\end{array}

(4.17)\begin{array}{rl}& 2{\omega }_2{\int }_0^t{\int }_{\mathrm{\Omega }}\mathrm{\nabla }v(s)\mathrm{\nabla }{v}_t(s)\mathrm{d}x\mathrm{d}s{\int }_{\mathrm{\Omega }}v{v}_t\mathrm{d}x\\ & \le {\omega }_2\Vert v_t(t){\Vert }^2{\int }_0^t\Vert \mathrm{\nabla }v(s){\Vert }^2\mathrm{d}s+{\omega }_2\Vert v(t){\Vert }^2{\int }_0^t\Vert \mathrm{\nabla }{v}_t(s){\Vert }^2\mathrm{d}s,\end{array}

and

(4.18)\begin{array}{rl}& 2{\mu }_1{\int }_0^t{\int }_{\mathrm{\Omega }}u(s)u_t(s)\mathrm{d}x\mathrm{d}s{\int }_{\mathrm{\Omega }}u{u}_t\mathrm{d}x\\ & \le {\mu }_1\Vert u_t(t){\Vert }^2{\int }_0^t\Vert u(s){\Vert }^2\mathrm{d}s+{\mu }_1\Vert u(t){\Vert }^2{\int }_0^t\Vert u_t(s){\Vert }^2\mathrm{d}s.\end{array}

(4.19)\begin{array}{rl}& 2{\mu }_2{\int }_0^t{\int }_{\mathrm{\Omega }}v(s)v_t(s)\mathrm{d}x\mathrm{d}s{\int }_{\mathrm{\Omega }}v{v}_t\mathrm{d}x\\ & \le {\mu }_2\Vert v_t(t){\Vert }^2{\int }_0^t\Vert v(s){\Vert }^2\mathrm{d}s+{\mu }_2\Vert v(t){\Vert }^2{\int }_0^t\Vert v_t(s){\Vert }^2\mathrm{d}s.\end{array}

These inequalities (4.11)−(4.19) entail \chi (t) \ge 0 for all t\in [0, T]. Therefore, we reach the following differential inequality from (4.9) that

(4.20)\mathrm{\Theta }{\mathrm{\Theta }}^{″} (t)-\frac{p+3}{4}{\mathrm{\Theta }}^{\prime }(t{)}^2\ge \mathrm{\Theta }(t)\mathrm{\Gamma }(t),\mathrm{\forall }t\in [0,T],

where

(4.21)\begin{array}{rl}\mathrm{\Gamma }(t)=& 2[\Vert u_t(t){\Vert }^2+\Vert v_t(t){\Vert }^2+2p{\int }_{\mathrm{\Omega }}F(u,v)\mathrm{d}x-\Vert \mathrm{\nabla }u(t){\Vert }^2-\Vert \mathrm{\nabla }v(t){\Vert }^2]\\ & -(p+3)[\Vert u_t(t){\Vert }^2+\Vert v_t(t){\Vert }^2+{\mu }_1{\int }_0^t\Vert u_t(s){\Vert }^2\mathrm{d}s+{\mu }_2{\int }_0^t\Vert v_t(s){\Vert }^2\mathrm{d}s\\ & +{\omega }_1{\int }_0^t\Vert \mathrm{\nabla }{u}_t(s){\Vert }^2\mathrm{d}s+{\omega }_2{\int }_0^t\Vert \mathrm{\nabla }{v}_t(s){\Vert }^2\mathrm{d}s].\end{array}

From (2.8), we attain that

(4.22)\begin{array}{rl}\mathrm{\Gamma }(t)=& -2pE(t)+(p-1)(\Vert \mathrm{\nabla }u(t){\Vert }^2-\Vert \mathrm{\nabla }v(t){\Vert }^2)]\\ & -(p+3)[{\mu }_1{\int }_0^t\Vert u_t(s){\Vert }^2\mathrm{d}s+{\mu }_2{\int }_0^t\Vert v_t(s){\Vert }^2\mathrm{d}s\\ & +{\omega }_1{\int }_0^t\Vert \mathrm{\nabla }{u}_t(s){\Vert }^2\mathrm{d}s+{\omega }_2{\int }_0^t\Vert \mathrm{\nabla }{v}_t(s){\Vert }^2\mathrm{d}s].\end{array}

By (2.14), we get

(4.23)\begin{array}{rl}\mathrm{\Gamma }(t)=& -2pE(0)+(p-1)(\Vert \mathrm{\nabla }u(t){\Vert }^2-\Vert \mathrm{\nabla }v(t){\Vert }^2)]\\ & +(p-1)[{\mu }_1{\int }_0^t\Vert u_t(s){\Vert }^2\mathrm{d}s+{\mu }_2{\int }_0^t\Vert v_t(s){\Vert }^2\mathrm{d}s\\ & +{\omega }_1{\int }_0^t\Vert \mathrm{\nabla }{u}_t(s){\Vert }^2\mathrm{d}s+{\omega }_2{\int }_0^t\Vert \mathrm{\nabla }{v}_t(s){\Vert }^2\mathrm{d}s].\end{array}

We conclude from (4.3) and that

(4.24)\begin{array}{rl}\mathrm{\Gamma }(t)& >2p(d-E(0))+(p-1)[{\mu }_1{\int }_0^t\Vert u_t(s){\Vert }^2\mathrm{d}s+{\mu }_2{\int }_0^t\Vert v_t(s){\Vert }^2\mathrm{d}s\\ & +{\omega }_1{\int }_0^t\Vert \mathrm{\nabla }{u}_t(s){\Vert }^2\mathrm{d}s+{\omega }_2{\int }_0^t\Vert \mathrm{\nabla }{v}_t(s){\Vert }^2\mathrm{d}s]\\ & >(p-1)[{\mu }_1{\int }_0^t\Vert u_t(s){\Vert }^2\mathrm{d}s+{\mu }_2{\int }_0^t\Vert v_t(s){\Vert }^2\mathrm{d}s\\ & +{\omega }_1{\int }_0^t\Vert \mathrm{\nabla }{u}_t(s){\Vert }^2\mathrm{d}s+{\omega }_2{\int }_0^t\Vert \mathrm{\nabla }{v}_t(s){\Vert }^2\mathrm{d}s]\ge 0.\end{array}

Hence, there exists \vartheta >0 which is independent of T such that

(4.25)\mathrm{\Gamma }(t) \ge \vartheta , \mathrm{\forall }t\ge 0.

It yields from (4.3), (4.20) and (4.25) that

(4.26)\mathrm{\Theta }{\mathrm{\Theta }}^{″} (t)-\frac{p+3}{4}{\mathrm{\Theta }}^{\prime }(t{)}^2\ge \lambda \vartheta ,\mathrm{\forall }t\in [0,T ].

Let \rho =\frac{p-1}{4}>0. Then, by Lemma 4.1, there is T_{\ast } such that

(4.27)

and

(4.28)\underset{t\to T_{\ast }^{-}}{lim}\mathrm{\Theta } (t)=+\mathrm{\infty } .

From Lemma 2.1 and (4.4) (the definition of \mathrm{\Theta } (t)), (4.28) implies that

\underset{t\to T_{\ast }^{-}}{lim}(\Vert \mathrm{\nabla }u(t){\Vert }^2+ \Vert \mathrm{\nabla }v(t){\Vert }^2 )=+\mathrm{\infty }.

Thus, we cannot suppose that solutions of (1.1)−(1.5) is global.

This finishes the proof of Theorem 4.1.

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