Global existence of solutions and energy decay for a Kirchhoff-type equation with nonlinear dissipation

Abstract

This paper deals with the initial boundary value problem for a class of nonlinear Kirchhoff-type equation with dissipative term

$$u_{tt}-\phi \left({\parallel \mathrm{\nabla }u\parallel }_2^2\right)\mathrm{\Delta }u+a{|u_t|}^{\alpha -2}{u}_t=b{|u|}^{\beta -2}u,x\in \mathrm{\Omega },t>0$$

in a bounded domain, where $a,b>0$ and $\alpha ,\beta >2$ are constants. We obtain the global existence of solutions by constructing a stable set in $H_0^1(\mathrm{\Omega })$ and show the energy decay estimate by applying a lemma of Komornik.

MSC:35B40, 35L70.

1 Introduction

In this paper, we investigate the existence and asymptotic stability of global solutions for the initial boundary value problem of the following Kirchhoff-type equation with nonlinear dissipative term in a bounded domain

$$u_{tt}-\phi \left({\parallel \mathrm{\nabla }u\parallel }_2^2\right)\mathrm{\Delta }u+a{|u_t|}^{\alpha -2}{u}_t=b{|u|}^{\beta -2}u,x\in \mathrm{\Omega },t>0,$$

(1.1)

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

(1.2)

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

(1.3)

where Ω is a bounded domain in $R^n$ with a smooth boundary ∂ Ω, $a,b>0$ and $\alpha ,\beta >2$ are constants, $\phi (s)$ is a $C^1$-class function on $[0,+\mathrm{\infty })$ satisfying

$$\phi (s)\ge m_0,s\phi (s)\ge {\int }_0^s\phi (\theta )d\theta ,\mathrm{\forall }s\in [0,+\mathrm{\infty })$$

(1.4)

with $m_0\ge 1$ is a constant.

If $\mathrm{\Omega }=[0,L]$ is an interval of the real line, equation (1.1) describes a small amplitude vibration of an elastic string with fixed endpoints. The original equation is

$$\rho h{u}_{tt}+\delta u_t+f=({\gamma }_0+\frac{Eh}{2L}{\int }_0^L{|u_x|}^{2}ds)u_{xx},$$

where L is the rest length, E is the Young modulus, ρ is the mass density, h is the cross-section area, ${\gamma }_0$ is the initial axial tension, δ is the resistance modulus and f is a nonlinear perturbation effect.

When $a=b=0$, $\phi (s)=s^r$, $r\ge 1$ and $u_0\ne 0$ (the mildly degenerate case), the local existence of solutions in Sobolev space was investigated by many author [1–6]. Concerning a global existence of solutions for mildly degenerate Kirchhoff equations, it is natural to add a dissipative term $u_t$ or $\mathrm{\Delta }{u}_t$.

For $a=1$, $b=0$, $\alpha =2$, $\phi (s)=s^r$, $r\ge 1$, the problem (1.1)-(1.3) was treated by Nishihara and Yamada [7]. They proved the existence and uniqueness of a global solution $u(t)$ for small data $(u_0,u_1)\in (H_0^1(\mathrm{\Omega })\cap H^2(\mathrm{\Omega }))\times H_0^1(\mathrm{\Omega })$ with $u_0\ne 0$ and the polynomial decay of the solution. Aassila and Benaissa [8] extended the global existence part of [7] to the case where $\phi (s)\ge 0$ with $\phi ({\parallel \mathrm{\nabla }{u}_0\parallel }^2)\ne 0$ and the case of nonlinear dissipative term case ($a\ne 0$).

In the case $a=0$, for large β and $\phi (s)\ge r>0$, D’Ancona and Spagnolo [9] proved that if $u_0,u_1\in C_0^{\mathrm{\infty }}(R^n)$ are small, then problem (1.1)-(1.3) has a global solution. The nondegenerate case with $\alpha =2$, $a>0$ and $b=0$ was considered by De Brito, Yamada and Nishihara [10–13], they proved that for small initial data $(u_0,u_1)\in (H_0^1(\mathrm{\Omega })\cap H^2(\mathrm{\Omega }))\times H_0^1(\mathrm{\Omega })$ there exists a unique global solution of (1.1)-(1.3) that decays exponentially as $t\to +\mathrm{\infty }$.

When $\phi (s)\ge 0$, Ghisi and Gobbino [14] proved the existence and uniqueness of a global solution $u(t)$ of the problem (1.1)-(1.3) for small initial data $(u_0,u_1)\in (H_0^1(\mathrm{\Omega })\cap H^2(\mathrm{\Omega }))\times H_0^1(\mathrm{\Omega })$ with $m({\parallel \mathrm{\nabla }{u}_0\parallel }^2)\ne 0$ and the asymptotic behavior $(u(t),u_t(t),u_{tt}(t))\to (u_{\mathrm{\infty }},0,0)$ in $(H_0^1(\mathrm{\Omega })\cap H^2(\mathrm{\Omega }))\times H_0^1(\mathrm{\Omega })\times L^2(\mathrm{\Omega })$ as $t\to +\mathrm{\infty }$, where either $u_{\mathrm{\infty }}=0$ or $\phi ({\parallel \mathrm{\nabla }{u}_{\mathrm{\infty }}\parallel }^2)=0$.

The case $\phi (s)\ge r>0$ has been considered by Hosoya and Yamada [15] under the following condition:

$$0\le \beta <\frac{2}{n-4},n\ge 5;0\le \beta <+\mathrm{\infty },n\le 4.$$

They proved that, if the initial datas are small enough, the problem (1.1)-(1.3) has a global solution which decays exponentially as $t\to +\mathrm{\infty }$.

In this paper, we prove the global existence for the problem (1.1)-(1.3) by applying the potential well theory introduced by Sattinger [16] and Payne and Sattinger [17]. Meanwhile, we obtain the asymptotic stability of global solutions by use of the lemma of Komornik [18].

We adopt the usual notation and convention. Let $H^m(\mathrm{\Omega })$ denote the Sobolev space with the norm ${\parallel u\parallel }_{H^m(\mathrm{\Omega })}={({\sum }_{|\alpha |\le m}{\parallel D^{\alpha }u\parallel }_{L^2(\mathrm{\Omega })}^2)}^{\frac{1}{2}}$, $H_0^m(\mathrm{\Omega })$ denotes the closure in $H^m(\mathrm{\Omega })$ of $C_0^{\mathrm{\infty }}(\mathrm{\Omega })$. For simplicity of notation, hereafter we denote by ${\parallel \cdot \parallel }_p$ the Lebesgue space $L^p(\mathrm{\Omega })$ norm, $\parallel \cdot \parallel$ denotes $L^2(\mathrm{\Omega })$ norm and we write equivalent norm $\parallel \mathrm{\nabla }\cdot \parallel$ instead of $H_0^1(\mathrm{\Omega })$ norm ${\parallel \cdot \parallel }_{H_0^1(\mathrm{\Omega })}$. Moreover, M denotes various positive constants depending on the known constants and it may be difference in each appearance.

This paper is organized as follows: In the next section, we will give some preliminaries. Then in Section 3, we state the main results and give their proof.

2 Preliminaries

In order to state and prove our main results, we first define the following functionals:

$$K(u)=m_0{\parallel \mathrm{\nabla }u\parallel }^2-b{\parallel u\parallel }_{\beta }^{\beta },J(u)=\frac{m_0}{2}{\parallel \mathrm{\nabla }u\parallel }^2-\frac{b}{\beta }{\parallel u\parallel }_{\beta }^{\beta },$$

for $u\in H_0^1(\mathrm{\Omega })$. Then we define the stable set S by

$$S=\{u\in H_0^1(\mathrm{\Omega }),K(u)>0\}\cup \{0\}.$$

We denote the total energy functional associated with (1.1)-(1.3) by

$$E(t)=\frac{1}{2}{\parallel u_t\parallel }^2+\frac{1}{2}{\int }_0^{{\parallel \mathrm{\nabla }u\parallel }^2}\phi (s)ds-\frac{b}{\beta }{\parallel u\parallel }_{\beta }^{\beta }$$

(2.1)

for $u\in H_0^1(\mathrm{\Omega })$, $t\ge 0$, and $E(0)=\frac{1}{2}{\parallel u_1\parallel }^2+\frac{1}{2}{\int }_0^{{\parallel \mathrm{\nabla }{u}_0\parallel }^2}\phi (s)ds-\frac{b}{\beta }{\parallel u_0\parallel }_{\beta }^{\beta }$ is the total energy of the initial data.

Lemma 2.1 Let q be a number with $2\le q<+\mathrm{\infty }$, $n\le 2$ and $2\le q\le \frac{2n}{n-2}$, $n>2$. Then there is a constant C depending on Ω and q such that

$${\parallel u\parallel }_q\le C{\parallel u\parallel }_{H_0^1(\mathrm{\Omega })},\mathrm{\forall }u\in H_0^1(\mathrm{\Omega }).$$

Lemma 2.2 [18]

Let $y(t):R^{+}\to R^{+}$ be a nonincreasing function and assume that there are two constants $\mu \ge 1$ and $A>0$ such that

$${\int }_s^{+\mathrm{\infty }}y{(t)}^{\frac{\mu +1}{2}}dt\le Ay(s),0\le s<+\mathrm{\infty },$$

then $y(t)\le Cy(0){(1+t)}^{-\frac{2}{\mu -1}}$, $\mathrm{\forall }t\ge 0$, if $\mu >1$, where C is positive constants independent of $y(0)$.

Lemma 2.3 Let $u(t,x)$ be a solutions to the problem (1.1)-(1.3). Then $E(t)$ is a nonincreasing function for $t>0$ and

$$\frac{d}{dt}E(t)=-a{\parallel u_t(t)\parallel }_{\alpha }^{\alpha }.$$

(2.2)

Proof By multiplying equation (1.1) by $u_t$ and integrating over Ω, we get

$$\frac{d}{dt}E(u(t))=-a{\parallel u_t(t)\parallel }_{\alpha }^{\alpha }\le 0.$$

Therefore, $E(t)$ is a nonincreasing function on t. □

We state a local existence result, which is known as a standard one (see [6, 19]).

Theorem 2.1 Suppose that $\alpha ,\beta >2$ satisfy

$$2<\beta <+\mathrm{\infty },n\le 2;2<\beta \le \frac{2(n-1)}{n-2},n>2,$$

(2.3)

$$2<\alpha <+\mathrm{\infty },n\le 2;2<\alpha \le \frac{2n}{n-2},n>2,$$

(2.4)

and let $(u_0,u_1)\in H_0^1(\mathrm{\Omega })\times L^2(\mathrm{\Omega })$. Then there exists $T>0$ such that the problem (1.1)-(1.3) has a unique local solution $u(t)$ in the class

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

(2.5)

In order to prove the existence of global solutions of the problem (1.1)-(1.3), we need the following lemma.

Lemma 2.4 Supposed that (2.3) holds, If $u_0\in S$, $u_1\in L^2(\mathrm{\Omega })$ such that

$$\delta =b{C}^{\beta }{(\frac{2\beta }{(\beta -2)m_0}E(0))}^{\frac{\beta -2}{2}}<1,$$

(2.6)

then $u\in S$, for each $t\in [0,T)$.

Proof Assume that there exists a number $t^{\ast }\in [0,T)$ such that $u(t)\in S$ on $[0,t^{\ast })$ and $u(t^{\ast })\notin S$. Then we have

$$K\left(u\left(t^{\ast }\right)\right)=0,u\left(t^{\ast }\right)\ne 0.$$

(2.7)

Since $u(t)\in S$ on $[0,t^{\ast })$, it holds that

$$\begin{array}{rl}J(u(t))& =\frac{m_0}{2}{\parallel \mathrm{\nabla }u(t)\parallel }^2-\frac{b}{\beta }{\parallel u(t)\parallel }_{\beta }^{\beta }\\ \ge \frac{m_0}{2}{\parallel \mathrm{\nabla }u(t)\parallel }^2-\frac{m_0}{\beta }{\parallel \mathrm{\nabla }u(t)\parallel }^2=\frac{(\beta -2)m_0}{2\beta }{\parallel \mathrm{\nabla }u(t)\parallel }^2,\end{array}$$

(2.8)

we have from $K(u(t^{\ast }))=0$ that

$$\begin{array}{rl}J\left(u\left(t^{\ast }\right)\right)& =\frac{m_0}{2}{\parallel \mathrm{\nabla }u\left(t^{\ast }\right)\parallel }^2-\frac{b}{\beta }{\parallel u\left(t^{\ast }\right)\parallel }_{\beta }^{\beta }\\ =\frac{m_0}{2}{\parallel \mathrm{\nabla }u\left(t^{\ast }\right)\parallel }^2-\frac{m_0}{\beta }{\parallel \mathrm{\nabla }u\left(t^{\ast }\right)\parallel }^2=\frac{(\beta -2)m_0}{2\beta }{\parallel \mathrm{\nabla }u\left(t^{\ast }\right)\parallel }^2,\end{array}$$

(2.9)

we conclude from (1.4) and (2.1) that

$$\begin{array}{rl}E(t)& \ge \frac{1}{2}{\parallel u_t(t)\parallel }^2+\frac{m_0}{2}{\parallel \mathrm{\nabla }u(t)\parallel }^2-\frac{b}{\beta }{\parallel u(t)\parallel }_{\beta }^{\beta }\\ =\frac{1}{2}{\parallel u_t(t)\parallel }^2+J(u(t)).\end{array}$$

(2.10)

Therefore, we obtain from (2.8), (2.9) and (2.10) that

$${\parallel \mathrm{\nabla }u(t)\parallel }^2\le \frac{2\beta }{(\beta -2)m_0}J(u(t))\le \frac{2\beta }{(\beta -2)m_0}E(t)\le \frac{2\beta }{(\beta -2)m_0}E(0),$$

(2.11)

for $\mathrm{\forall }t\in [0,t^{\ast }]$.

By exploiting Lemma 2.1, (2.6) and (2.11), we easily arrive at

$$\begin{array}{rl}b{\parallel u(t)\parallel }_{\beta }^{\beta }& \le b{C}^{\beta }{\parallel \mathrm{\nabla }u(t)\parallel }^{\beta }=b{C}^{\beta }{\parallel \mathrm{\nabla }u(t)\parallel }^{\beta -2}{\parallel \mathrm{\nabla }u(t)\parallel }^2\\ \le b{C}^{\beta }{(\frac{2\beta }{(\beta -2)m_0}E(0))}^{\frac{\beta -2}{2}}{\parallel \mathrm{\nabla }u(t)\parallel }^2<{\parallel \mathrm{\nabla }u(t)\parallel }^2,\end{array}$$

(2.12)

for all $t\in [0,t^{\ast }]$.

Therefore, we obtain

$$K\left(u\left(t^{\ast }\right)\right)=m_0{\parallel \mathrm{\nabla }u\left(t^{\ast }\right)\parallel }^2-b{\parallel u\left(t^{\ast }\right)\parallel }_{\beta }^{\beta }>0,$$

(2.13)

which contradicts (2.7). Thus, we conclude that $u(t)\in S$ on $[0,T)$. □

3 The global existence and nonexistence

Theorem 3.1 Suppose that (2.3) and (2.4) hold, and $u(t)$ is a local solution of problem (1.1)-(1.3) on $[0,T)$. If $u_0\in S$ and $u_1\in L^2(\mathrm{\Omega })$ satisfy (2.6), then $u(x,t)$ is a global solution of the problem (1.1)-(1.3).

Proof It suffices to show that ${\parallel \mathrm{\nabla }u(t)\parallel }^2+{\parallel u_t(t)\parallel }^2$ is bounded independently of t.

Under the hypotheses in Theorem 3.1, we get from Lemma 2.4 that $u(t)\in S$ on $[0,T)$. So the formula (2.8) holds on $[0,T)$.

Therefore, we have from (2.8) that

$$\frac{1}{2}{\parallel u_t\parallel }^2+\frac{(\beta -2)m_0}{2\beta }{\parallel \mathrm{\nabla }u(t)\parallel }^2\le \frac{1}{2}{\parallel u_t(t)\parallel }^2+J(u(t))=E(t)\le E(0).$$

(3.1)

Hence, we get

$${\parallel u_t(t)\parallel }^2+{\parallel \mathrm{\nabla }u(t)\parallel }^2\le max(2,\frac{2\beta }{(\beta -2)m_0})E(0)<+\mathrm{\infty }.$$

The above inequality and the continuation principle lead to the global existence of the solution, that is, $T=+\mathrm{\infty }$. Thus, the solution $u(t)$ is a global solution of the problem (1.1)-(1.3). □

Now we employ the analysis method to discuss the solution of the problem (1.1)-(1.3) occurs blow-up in finite time. Our result reads as follows.

Theorem 3.2 Assume that (i) $2<\beta <\frac{2n}{n-2}$, if $n>2$; (ii) $0<\beta <+\mathrm{\infty }$, if $n\le 2$. If $u_0\in S$ and $u_1\in L^2(\mathrm{\Omega })$ such that

$$E(0)<Q_0,{\parallel u_0\parallel }_{\beta }>S_0,$$

where

$$Q_0=\frac{(\beta -2)b}{2\beta }{\left(\frac{m_0}{b{C}^2}\right)}^{\frac{\beta }{\beta -2}},S_0={\left(\frac{m_0}{b{C}^2}\right)}^{\frac{1}{\beta -2}}$$

with $C>0$ is a positive Sobolev constant. Then the solution of the problem (1.1)-(1.3) does not exist globally in time.

Proof On the contrary, under the conditions in Theorem 3.2, suppose that $u(x,t)$ is a global solution of the problem (1.1)-(1.3); then by Lemma 2.1, it is well known that there exists a constant $C>0$ depending only n, β such that ${\parallel u\parallel }_{\beta }\le C\parallel \mathrm{\nabla }u\parallel$ for all $u\in H_0^1(\mathrm{\Omega })$.

From the above inequality, we conclude that

$${\parallel \mathrm{\nabla }u\parallel }^2\ge C^{-2}{\parallel u\parallel }_{\beta }^2.$$

(3.2)

It follows from (1.4), (2.1) and (3.2) that

$$\begin{array}{rl}E(t)& =\frac{1}{2}{\parallel u_t\parallel }^2+\frac{1}{2}{\int }_0^{{\parallel \mathrm{\nabla }u\parallel }^2}\phi (s)ds-\frac{b}{\beta }{\parallel u\parallel }_{\beta }^{\beta }\\ \ge \frac{m_0}{2}{\parallel \mathrm{\nabla }u\parallel }^2-\frac{b}{\beta }{\parallel u\parallel }_{\beta }^{\beta }\ge \frac{m_0}{2C^2}{\parallel u\parallel }_{\beta }^2-\frac{b}{\beta }{\parallel u\parallel }_{\beta }^{\beta }.\end{array}$$

(3.3)

Setting

$$s=s(t)={\parallel u(t)\parallel }_{\beta }={\left\{{\int }_{\mathrm{\Omega }}{|u(x,t)|}^{\beta }dx\right\}}^{\frac{1}{\beta }}.$$

We denote the right side of (3.3) by $Q(s)=Q({\parallel u(t)\parallel }_{\beta })$, then

$$Q(s)=\frac{m_0}{2C^2}{s}^2-\frac{b}{\beta }{s}^{\beta },s\ge 0.$$

(3.4)

By (3.4), we obtain

$$Q^{\prime }(s)=\frac{m_0}{C^2}s-b{s}^{\beta -1}.$$

Let $Q^{\prime }(s)=0$, then we obtain $S_0={(\frac{m_0}{b{C}^2})}^{\frac{1}{\beta -2}}$.

As $s=S_0$, we have

$$Q^{″}(s){|}_{s=S_0}=(\frac{m_0}{C^2}-b(\beta -1)s^{\beta -2}){|}_{s=S_0}=-\frac{m_0(\beta -2)}{C^2}<0.$$

Consequently, the function $Q(s)$ has a single maximum value $Q_0$ at $S_0$, where

$$Q_0=Q(S_0)=\frac{(\beta -2)b}{2\beta }{\left(\frac{m_0}{b{C}^2}\right)}^{\frac{\beta }{\beta -2}}.$$

Since the initial data is such that $E(0)$, $s(0)$ satisfies $E(0)<Q_0$, ${\parallel u_0\parallel }_{\beta }>S_0$.

Therefore, we have from Lemma 2.3 that

$$E(t)\le E(0)<Q_0,\mathrm{\forall }t>0.$$

At the same time, by (3.3) and (3.4) it is evident that there can be no time $t>0$ for which

$$E(t)<Q_0,s(t)=S_0.$$

Hence, we have also $s(t)>S_0$ for all $t>0$ from the continuity of $E(t)$ and $s(t)$.

According to the above contradiction we know that the global solution of the problem (1.1)-(1.3) does not exist, i.e., the solution blows up in some finite time.

This completes the proof of Theorem 3.2. □

4 Energy decay estimate

The following theorem shows the asymptotic behavior of global solutions of the problem (1.1)-(1.3).

Theorem 4.1 If the hypotheses in Theorem  3.2 are valid, then the global solutions of the problem (1.1)-(1.3) has the following asymptotic property:

$$E(t)\le M{(1+t)}^{-\frac{2}{\alpha -2}},$$

where $M>0$ is a constant depending on initial energy $E(0)$.

Proof Multiplying by $E{(t)}^{\frac{\alpha -2}{2}}u$ on both sides of the equation (1.1) and integrating over $\mathrm{\Omega }\times [S,T]$, we obtain that

$$0={\int }_S^T{\int }_{\mathrm{\Omega }}E{(t)}^{\frac{\alpha -2}{2}}u[u_{tt}-\phi \left({\parallel \mathrm{\nabla }u\parallel }^2\right)\mathrm{\Delta }u+a{|u_t|}^{\alpha -2}{u}_t-bu{|u|}^{\beta -2}]dxdt,$$

(4.1)

where $0\le S<T<+\mathrm{\infty }$.

Since

$$\begin{array}{rl}{\int }_S^T{\int }_{\mathrm{\Omega }}E{(t)}^{\frac{\alpha -2}{2}}u{u}_{tt}dxdt=& {\int }_{\mathrm{\Omega }}E{(t)}^{\frac{\alpha -2}{2}}u{u}_{t}dx{|}_S^T-{\int }_S^T{\int }_{\mathrm{\Omega }}E{(t)}^{\frac{\alpha -2}{2}}{|u_t|}^{2}dxdt\\ -\frac{\alpha -2}{2}{\int }_S^T{\int }_{\mathrm{\Omega }}E{(t)}^{\frac{\alpha -4}{2}}{E}^{\prime }(t)u{u}_{t}dxdt.\end{array}$$

(4.2)

So, substituting the formula (4.2) into the right-hand side of (4.1), we get that

$$\begin{array}{rl}0=& {\int }_S^{T}E{(t)}^{\frac{\alpha -2}{2}}({\parallel u_t\parallel }^2+\phi \left({\parallel \mathrm{\nabla }u\parallel }^2\right){\parallel \mathrm{\nabla }u\parallel }^2-\frac{2b}{\beta }{\parallel u\parallel }_{\beta }^{\beta })dt\\ -{\int }_S^T{\int }_{\mathrm{\Omega }}E{(t)}^{\frac{\alpha -2}{2}}[2{|u_t|}^2-a{|u_t|}^{\alpha -2}{u}_{t}u]dxdt\\ -\frac{\alpha -2}{2}{\int }_S^T{\int }_{\mathrm{\Omega }}E{(t)}^{\frac{\alpha -4}{2}}{E}^{\prime }(t)u{u}_{t}dxdt+{\int }_{\mathrm{\Omega }}E{(t)}^{\frac{\alpha -2}{2}}u{u}_{t}dx{|}_S^T\\ +(\frac{2}{\beta }-1)b{\int }_S^{T}E{(t)}^{\frac{\alpha -2}{2}}{\parallel u\parallel }_{\beta }^{\beta }dt.\end{array}$$

(4.3)

We obtain from (2.12) and (2.11) that

$$b(1-\frac{2}{\beta }){\parallel u\parallel }_{\beta }^{\beta }\le \delta \frac{\beta -2}{\beta }{\parallel \mathrm{\nabla }u\parallel }^2\le \delta \frac{\beta -2}{\beta }\cdot \frac{2\beta }{(\beta -2)m_0}E(t)=\frac{2\delta }{m_0}E(t).$$

(4.4)

We derive from (1.4) that

$${\int }_0^{{\parallel \mathrm{\nabla }u\parallel }^2}\phi (s)ds\le \phi \left({\parallel \mathrm{\nabla }u\parallel }^2\right){\parallel \mathrm{\nabla }u\parallel }^2.$$

(4.5)

It follows from (4.3), (4.4) and (4.5) that

$$\begin{array}{r}2(1-\frac{\delta }{m_0}){\int }_S^{T}E{(t)}^{\frac{\alpha }{2}}dt\\ \le {\int }_S^T{\int }_{\mathrm{\Omega }}E{(t)}^{\frac{\alpha -2}{2}}[2{|u_t|}^2-a{|u_t|}^{\alpha -2}{u}_{t}u]dxdt\\ +\frac{\alpha -2}{2}{\int }_S^T{\int }_{\mathrm{\Omega }}E{(t)}^{\frac{\alpha -4}{2}}{E}^{\prime }(t)u{u}_{t}dxdt-{\int }_{\mathrm{\Omega }}E{(t)}^{\frac{\alpha -2}{2}}u{u}_{t}dx{|}_S^T.\end{array}$$

(4.6)

We have from Lemma 2.1 and (3.1) that

$$\begin{array}{r}|\frac{\alpha -2}{2}{\int }_S^T{\int }_{\mathrm{\Omega }}E{(t)}^{\frac{\alpha -4}{2}}{E}^{\prime }(t)u{u}_{t}dxdt|\\ \le \frac{\alpha -2}{2}{\int }_S^{T}E{(t)}^{\frac{\alpha -4}{2}}(-E^{\prime }(t))(\frac{1}{2}{\parallel u\parallel }^2+\frac{1}{2}{\parallel u_t\parallel }^2)dt\\ \le -\frac{\alpha -2}{2}{\int }_S^{T}E{(t)}^{\frac{\alpha -4}{2}}{E}^{\prime }(t)(\frac{\beta C^2}{(\beta -2)m_0}\cdot \frac{(\beta -2)m_0}{2\beta }{\parallel \mathrm{\nabla }u\parallel }^2+\frac{1}{2}{\parallel u_t\parallel }^2)dt\\ \le -\frac{\alpha -2}{2}max(\frac{\beta C^2}{(\beta -2)m_0},1){\int }_S^{T}E{(t)}^{\frac{\alpha -2}{2}}{E}^{\prime }(t)dt\\ =-\frac{\alpha -2}{\alpha }max(\frac{\beta C^2}{(\beta -2)m_0},1)E{(t)}^{\frac{\alpha }{2}}{|}_S^T\le ME{(S)}^{\frac{\alpha }{2}},\end{array}$$

(4.7)

similarly, we have

$$\begin{array}{rl}{|-{\int }_{\mathrm{\Omega }}E{(t)}^{\frac{\alpha -2}{2}}u{u}_{t}dx|}_S^T|& \le max(\frac{\beta C^2}{(\beta -2)m_0},1)E{(t)}^{\frac{\alpha }{2}}{|}_S^T\\ \le ME{(S)}^{\frac{\alpha }{2}}.\end{array}$$

(4.8)

Substituting the estimates (4.7) and (4.8) into (4.6), we conclude that

$$\begin{array}{r}2(1-\frac{\delta }{m_0}){\int }_S^{T}E{(t)}^{\frac{\alpha }{2}}dt\\ \le {\int }_S^T{\int }_{\mathrm{\Omega }}E{(t)}^{\frac{\alpha -2}{2}}[2{|u_t|}^2-a{|u_t|}^{\alpha -2}{u}_{t}u]dxdt+ME{(S)}^{\frac{\alpha }{2}}.\end{array}$$

(4.9)

We get from Young inequality and Lemma 2.3 that

$$\begin{array}{rcl}2{\int }_S^T{\int }_{\mathrm{\Omega }}E{(t)}^{\frac{\alpha -2}{2}}{|u_t|}^{2}dxdt& \le & {\int }_S^T{\int }_{\mathrm{\Omega }}({\epsilon }_{1}E{(t)}^{\frac{\alpha }{2}}+M({\epsilon }_1){|u_t|}^{\alpha })dxdt\\ \le & M{\epsilon }_1{\int }_S^{T}E{(t)}^{\frac{\alpha }{2}}dt+M({\epsilon }_1){\int }_S^T{\parallel u_t\parallel }_{\alpha }^{\alpha }dt\\ =& M{\epsilon }_1{\int }_S^{T}E{(t)}^{\frac{\alpha }{2}}dt-\frac{M({\epsilon }_1)}{a}(E(T)-E(S))\\ \le & M{\epsilon }_1{\int }_S^{T}E{(t)}^{\frac{\alpha }{2}}dt+ME(S).\end{array}$$

(4.10)

From Young inequality, Lemma 2.1, Lemma 2.3 and (2.11), We receive that

$$\begin{array}{r}-a{\int }_S^T{\int }_{\mathrm{\Omega }}E{(t)}^{\frac{\alpha -2}{2}}u{u}_t{|u_t|}^{\alpha -2}dxdt\\ \le a{\int }_S^{T}E{(t)}^{\frac{\alpha -2}{2}}({\epsilon }_2{\parallel u\parallel }_{\alpha }^{\alpha }+M({\epsilon }_2){\parallel u_t\parallel }_{\alpha }^{\alpha })dt\\ \le a{C}^{\alpha }{\epsilon }_{2}E{(0)}^{\frac{\alpha -2}{2}}{\int }_S^T{\parallel \mathrm{\nabla }u\parallel }^{\alpha }dt+aM({\epsilon }_2)E{(S)}^{\frac{\alpha -2}{2}}{\int }_S^T{\parallel u_t\parallel }_{\alpha }^{\alpha }dt\\ =a{C}^{\alpha }{\epsilon }_{2}E{(0)}^{\frac{\alpha -2}{2}}{\int }_S^T{(\frac{2\beta }{(\beta -2)m_0}E(t))}^{\frac{\alpha }{2}}dt+M({\epsilon }_2)E{(S)}^{\frac{\alpha -2}{2}}(E(S)-E(T))\\ \le C^{\alpha }{\epsilon }_{2}E{(0)}^{\frac{\alpha -2}{2}}{\left(\frac{2\beta }{(\beta -2)m_0}\right)}^{\frac{\alpha }{2}}{\int }_S^{T}E{(t)}^{\frac{\alpha }{2}}dt+ME{(S)}^{\frac{\alpha }{2}}.\end{array}$$

(4.11)

Choosing small enough ${\epsilon }_1$ and ${\epsilon }_2$, such that

$$\frac{1}{2}[M{\epsilon }_1+E{(0)}^{\frac{\alpha -2}{2}}{\left(\frac{2\beta C^2}{(\beta -2)m_0}\right)}^{\frac{\alpha }{2}}{\epsilon }_2]+\frac{\delta }{m_0}<1,$$

then, substituting (4.10) and (4.11) into (4.9), we get

$${\int }_S^{T}E{(t)}^{\frac{\alpha }{2}}dt\le ME(S)+ME{(S)}^{\frac{\alpha }{2}}\le M{(1+E(0))}^{\frac{\alpha -2}{2}}E(S).$$

Therefore, we have from Lemma 2.2 that

$$E(t)\le M{(1+t)}^{-\frac{\alpha -2}{2}},t\in [0,+\mathrm{\infty }).$$

The proof of Theorem 4.1 is thus finished. □