Exponential decay rate for a quasilinear von Karman equation of memory type with acoustic boundary conditions

Abstract

In this paper, we show the exponential decay result of the quasilinear von Karman equation of memory type with acoustic boundary conditions. This work is devoted to investigating the influence of kernel function g and the effect of the nonlinear term $|u'|^{\rho}u''$ and to proving exponential decay rates of solutions when g does not necessarily decay exponentially. This result improves on earlier ones concerning the exponential decay.

1 Introduction

Let $\Omega\subset R^{2}$ be a bounded domain with sufficiently smooth boundary ∂Ω, $\Gamma_{0} \cup\Gamma_{1} =\partial\Omega$, $\Gamma_{0} \cap\Gamma_{1} =\emptyset$, $\Gamma_{0}$ and $\Gamma_{1}$ have positive measure and $\nu=(\nu_{1} , \nu _{2} )$ be the outward unit normal vector, and by $\tau=(-\nu_{2} , \nu_{1} )$ we denote the corresponding unit tangent vector on ∂Ω. We define $u' = \frac{\partial u}{\partial t}$, $\Delta u= \sum_{i=1}^{2} \frac{\partial^{2} u }{\partial x_{i}^{2} }$, $\Delta ^{2} u= \sum_{i=1}^{2} \frac{\partial^{4} u }{\partial x_{i}^{4} }$, where $x=(x_{1} , x_{2}) \in\Omega$. In this paper, we consider the quasilinear von Karman equation of memory type with acoustic boundary conditions:

$$\begin{aligned}& \bigl\vert u'\bigr\vert ^{\rho}u'' -\alpha\Delta u'' +\Delta^{2} u -\int _{0}^{t} g(t-s) \Delta ^{2} u(s)\, ds =[u,v]\quad \mbox{in } \Omega\times(0, \infty), \end{aligned}$$

(1.1)

$$\begin{aligned}& \Delta^{2} v=-[u,u]\quad \mbox{in } \Omega\times(0, \infty), \end{aligned}$$

(1.2)

$$\begin{aligned}& v=\frac{\partial v}{\partial\nu} =0 \quad \mbox{on } \Gamma\times(0, \infty), \end{aligned}$$

(1.3)

$$\begin{aligned}& u=\frac{\partial u}{\partial\nu} =0 \quad \mbox{on } \Gamma_{0} \times(0, \infty), \end{aligned}$$

(1.4)

$$\begin{aligned}& \mathcal{B}_{1} u -\mathcal{B}_{1} \biggl( \int _{0}^{t} g(t-s) u(s) \, ds \biggr) =0\quad \mbox{on } \Gamma_{1} \times(0, \infty), \end{aligned}$$

(1.5)

$$\begin{aligned}& \mathcal{B}_{2} u-\alpha\frac{\partial u''}{\partial\nu}- \mathcal {B}_{2} \biggl( \int_{0}^{t} g(t-s) u(s)\, ds \biggr) =-y' \quad \mbox{on } \Gamma_{1} \times(0, \infty), \end{aligned}$$

(1.6)

$$\begin{aligned}& u' +p(x) y' +q(x) y=0\quad \mbox{on } \Gamma_{1} \times(0, \infty), \end{aligned}$$

(1.7)

$$\begin{aligned}& u(0)=u_{0} ,\qquad u'(0)=u_{1}\quad \mbox{in } \Omega,\qquad y(0)=y_{0} \quad \mbox{on } \Gamma_{1}, \end{aligned}$$

(1.8)

where $\alpha>0$, $\rho>0$. The functions g, p and q satisfy some conditions to be specified later, the von Karman bracket $[\cdot, \cdot ]$ is given by

$$[u,\phi] \equiv u_{x_{1} x_{1}} \phi_{x_{2} x_{2}} +u_{x_{2} x_{2}} \phi_{x_{1} x_{1}} -2 u_{x_{1} x_{2}} \phi_{x_{1} x_{2}} $$

and

$$\begin{aligned}& \mathcal{B}_{1} u =\Delta u +(1-\mu)B_{1} u, \\& \mathcal{B}_{2} u =\frac {\partial}{\partial\nu} \Delta u +(1-\mu) B_{2} u, \end{aligned}$$

here $\mu\in(0, \frac{1}{2} )$ is Poisson’s ratio,

$$\begin{aligned}& B_{1} u= 2 \nu_{1} \nu_{2} u_{x_{1}}u_{x_{2}} -\nu_{1}^{2} u_{x_{2} x_{2}} -\nu_{2}^{2} u_{x_{1} x_{1}}, \\& B_{2} u=\frac{\partial}{\partial\tau}\bigl[\bigl(\nu_{1}^{2} - \nu_{2}^{2} \bigr)u_{x_{1} x_{2}} +\nu_{1} \nu_{2} (u_{x_{2} x_{2}}-u_{x_{1} x_{1}})\bigr]. \end{aligned}$$

The physical applications of the above system are related to the problem of noise control and suppression in practical applications. In this model, problem (1.1)-(1.8) describes small vibrations of a thin homogeneous isotropic plate of uniform thickness of α with acoustic boundary conditions on a portion of the boundary and the Dirichlet boundary condition on the rest, $u(x,t)$ denotes the transversal displacement of the plate, the Airy stress function, $v(x,t)$ a vibrating plate and $y(x,t)$ the normal displacement to the boundary; and Eq. (1.1) was interpreted by the stresses at any instant dependent on the complete history of strains. In this model, the portion of the boundary denoted by $\Gamma_{1}$ is a locally reacting plate, with each point on the plate acting like a damped harmonic oscillator in the response to excess stress from the fluid in the interior Eq. (1.7). The coupling between the acoustic stress and the displacement of the boundary is given by Eqs. (1.5)-(1.6). The noise sound propagates through some acoustic medium, for example, through air, in a room which is characterized by a bounded domain Ω and whose walls, ceiling and floor are described by the boundary conditions. This is the description of Wu in [1]. For more physical explanation of wave equations with acoustic boundary, we refer the reader to [2–10]. The acoustic boundary conditions were introduced by Beale and Rosencrans in [3, 11], where the authors proved the global existence and regularity of solutions in a Hilbert space of the linear problem

$$\begin{aligned}& u_{tt}=\Delta u \quad \mbox{in } \Omega\times(0,\infty), \\& \frac{\partial u}{\partial\nu}=z_{t}\quad \mbox{on } \partial\Omega \times(0, \infty), \\& u_{t} +m(x) z_{tt} +p(x) z_{t} +q(x) z=0 \quad \mbox{on } \partial\Omega \times(0, \infty), \end{aligned}$$

where m, p and q are nonnegative functions on the boundary with m and q being strictly positive.

Frota and Larkin [12] eliminated the term $z_{tt}$ and established global solvability and decay estimates for a linear wave equation with boundary conditions

$$\begin{aligned}& u' +p(x) z' +q(x) z=0 \quad \mbox{in } \Gamma_{1} \times(0, \infty), \\ & \frac{\partial u}{\partial\nu} =h(x) z'\quad \mbox{on } \Gamma_{1} \times (0, \infty), \\ & u=0\quad \mbox{on } \Gamma_{0} \times(0, \infty). \end{aligned}$$

The decay rate estimated for wave equations of memory type with acoustic boundary conditions was studied by Park and Park [9], and Park et al. [13] investigated the general decay for a von Karman equation of memory type with acoustic boundary conditions. Park and Ha [14] considered the Klein-Gordon equation with damping $|u_{t} |^{\rho}u_{t}$ and acoustic boundary conditions. Wu [1] also proved the well-posedness for variable-coefficient wave equation with nonlinear damped acoustic boundary conditions. Recently Boukhatem and Benabderrahmane [15] proved the existence and decay of solutions for a viscoelastic wave equation with acoustic boundary conditions. The semilinear wave equation with porous acoustic boundary conditions was studied by Graber and Said-Houari [16], and Graber [17] investigated the strong stability and uniform decay of solutions to a wave equation with semilinear porous acoustic boundary conditions. The uniform decay for a von Karman plate equation with a boundary memory condition was studied by Park and Park [18]. Park and Kang [19] considered the uniform decay of solutions for von Karman equations of dynamic viscoelasticity with memory. The asymptotic behavior and energy decay of the solutions for a quasilinear viscoelastic problems were studied by many authors [20–22], and Kang [23] proved the exponential decay for quasilinear von Karman equation with memory.

Motivated by [13] and [23], in this paper we prove the exponential decay of a quasilinear von Karman equation of memory type with acoustic boundary conditions for problem (1.1)-(1.8) satisfying

$$\frac{d}{dt} \mathcal{L} (t) \leq-C \xi(t) \mathcal{L}(t)\quad \mbox{for some } C>0 \mbox{ and for all } t\geq t_{0}. $$

This is done by applying the idea presented in [4, 5] with some necessary modification due to the nature of the problem treated here. To the best of our knowledge, there are no results for a quasilinear von Karman equation of memory type with acoustic boundary conditions. Thus this work is meaningful. In particular, the nonlinear term $|u'|^{\rho} u''$ is difficult to analyze, and the result of the energy decay is dependent on the kernel g. So we overcome the issue using the change of Lyapunov functional. The structure of this paper is as follows. In Section 2, we give some notation and material needed for our work. In Section 3, we prove the main results.

2 Preliminaries

In this section, we present some material needed in the proof of our result. Throughout this paper, we define

$$\begin{aligned}& V=\bigl\{ u\in H^{1} (\Omega): u=0 \mbox{ on } \Gamma_{0} \bigr\} , \\& W=\biggl\{ u\in H^{2} (\Omega): u=\frac{\partial u}{\partial\nu}=0 \mbox{ on } \Gamma_{0} \biggr\} , \\& (u,v) =\int_{\Omega}u(x)v(x)\, dx , \\& (u,v)_{\Gamma_{1}} =\int_{\Gamma_{1}} u(x) v(x)\, d\Gamma. \end{aligned}$$

For a Banach space X, $\|\cdot\|_{X}$ denotes the norm of X. For simplicity, we denote $\|\cdot\|_{L^{2} (\Omega)}$ by norm $\|\cdot\|$ and $\|\cdot\|_{L^{1} (\Gamma_{1})}$ by $\|\cdot\|_{\Gamma_{1}}$, respectively. We define, for all $1\leq p<\infty$,

$$\|u\|_{p}^{p} =\int_{\Omega}\bigl\vert u(x)\bigr\vert ^{p}\, dx. $$

For $0< \mu<\frac{1}{2}$, the bilinear form $a(\cdot, \cdot)$ is defined by

$$\begin{aligned} a(u,v) =&\int_{\Omega}\bigl\{ u_{x_{1} x_{1}} v_{x_{1} x_{1}} +u_{x_{2} x_{2}} v_{x_{2} x_{2}}+\mu(u_{x_{1} x_{1}} v_{x_{2} x_{2}} \\ & +u_{x_{2} x_{2}} v_{x_{1} x_{1}}) +2 (1-\mu) u_{x_{1} x_{2}} v_{x_{1} x_{2}} \bigr\} \, dx. \end{aligned}$$

(2.1)

A simple calculation, based on the integration by part formula, yields

$$\int_{\Omega}\Delta^{2} u v\, dx = a(u,v) - \biggl( \mathcal{B}_{1} u , \frac {\partial v}{\partial\nu} \biggr)_{\Gamma} +( \mathcal{B}_{2} u, v)_{\Gamma}. $$

Thus, for $(u, v) \in(H^{4} (\Omega)\cap W)\times W$, it holds

$$\int_{\Omega}\Delta^{2} u v \, dx = a(u,v) - \biggl( \mathcal{B}_{1} u , \frac {\partial v}{\partial\nu} \biggr)_{\Gamma_{1}} +( \mathcal{B}_{2} u, v)_{\Gamma_{1}}. $$

Since $\Gamma_{0} \neq\emptyset$, we have (see [6]) that $\sqrt {a(u,u)}$ is equivalent to the $H^{2} (\Omega)$ norm in W, that is,

$$ C_{1} \|u\|_{H^{2} (\Omega)}^{2} \leq a(u,u) \leq C_{2} \|u\|_{H^{2} (\Omega)}^{2}\quad \mbox{for some } C_{1}, C_{2} >0. $$

(2.2)

This and the Sobolev imbedding theorem imply that for some positive constants $C_{p}$, $\tilde{C}_{p}$ and $C_{s}$,

$$ \begin{aligned} &\|u\|^{2} \leq C_{p} a(u,u) , \qquad \|u \|_{\Gamma_{1}}^{2} \leq\tilde{C}_{p} a(u,u)\quad \mbox{and} \\ &\|\nabla u\|^{2} \leq C_{s} a(u,u) \quad \mbox{for all } u\in W. \end{aligned} $$

(2.3)

And since $V \hookrightarrow L^{\rho+2}(\Omega)$, $\rho>0$, there exists a positive constant K such that

$$ \|u\|_{\rho+2} \leq K\|\nabla u\|_{2},\quad u\in V. $$

(2.4)

By (2.1) and Young’s inequality, we deduce that

$$a(u,v)\leq\delta\|u\|_{H^{2} (\Omega)}^{2} +\frac{5}{8\delta}\|v \|_{H^{2} (\Omega)}^{2} \quad \mbox{for all } \delta>0. $$

From this and (2.2), it holds that

$$ a(u,v)\leq\delta\|u\|_{H^{2} (\Omega)}^{2} + \frac{5}{8C_{1}\delta}a(v,v)\quad \mbox{for all } \delta>0. $$

(2.5)

Now we introduce the relative results of the Airy stress function and von Karman bracket $[\cdot, \cdot]$.

Lemma 2.1

[24]

Let u, w be functions in $H^{2} (\Omega)$ and v in $H_{0}^{2} (\Omega )$, where Ω is an open bounded and connected set of $R^{2}$ with regular boundary. Then

$$\int_{\Omega}w[u,v] \,dx =\int_{\Omega}v[w,u] \,dx. $$

Lemma 2.2

[25]

Let $u\in H^{2} (\Omega)$ and v be the Airy stress function satisfying (1.2) and (1.3). Then the following relations hold:

$$ [u,v]\in L^{2} (\Omega) \quad \textit{and}\quad \bigl\Vert [u,v]\bigr\Vert \leq C \|u\|_{H^{2} (\Omega )}\|v\|_{W^{2,\infty} (\Omega)} \leq C' \|u\|_{H^{2} (\Omega)}\|u\|_{H^{2} (\Omega)}^{2}, $$

(2.6)

where C and $C'$ are some positive constants.

Now we state the assumptions for problem (1.1)-(1.8). For the relaxation function g, we assume that $g: R_{+} \to R_{+}$ is continuously differentiable verifying that

$$ g(0)>0,\qquad l:=\int_{0}^{\infty}g(s) \,ds < 1 $$

(2.7)

and

$$ g'(t)\leq-\xi(t) g(t) \quad \mbox{for } t\geq0, $$

(2.8)

where $\xi: R_{+} \to R_{+}$ is a nonincreasing differentiable function. Condition (2.8) was considered by Messaoudi and Mustafa [26] when studying the stability of a memory-type Timoshenko system. For the functions p and q, we assume that $p, q \in C (\Gamma_{1} )$ and $p(x), q(x) >0$ on $\Gamma_{1}$. This assumption implies that there exist positive constants $p_{i}$, $q_{i}$ ($i=0,1$) such that

$$ p_{0} \leq p(x) \leq p_{1} ,\qquad q_{0} \leq q(x)\leq q_{1}\quad \mbox{for all } x\in \Gamma_{1} . $$

(2.9)

The existence of solution can be proved by the Faedo-Galerkin method (see [14, 27, 28]).

Theorem 2.1

Let the initial data $(u_{0}, u_{1}, y_{0} )\in(H^{4} (\Omega) \cap W) \times (H^{3} (\Omega)\cap W ) \times L^{2} (\Gamma_{1})$ and the conditions above on g, p and q hold. Then problem (1.1)-(1.8) admits a unique solution $(u,y)$ in the class

$$u\in C\bigl(0,T;W\cap H^{4} (\Omega)\bigr)\cap C^{1} \bigl(0,T; V\cap H^{3} (\Omega)\bigr),\qquad y, y' \in L^{2} \bigl(0,T; L^{2} (\Gamma_{1})\bigr). $$

3 Main result

In this section, we shall prove the general decay rate of the solution for problem (1.1)-(1.8). For simplicity of notation, we define

$$\begin{aligned} \begin{aligned} &(g\ast u) (t) =\int_{0}^{t} g(t-s) u(s) \,ds, \\ &(g \mathbin{\square} u) (t) = \int_{0}^{t} g(t-s) \bigl\Vert u(t)-u(s)\bigr\Vert ^{2} \,ds \end{aligned} \end{aligned}$$

and

$$\bigl(g \mathbin{\square}\partial^{2} u\bigr) (t)=\int _{0}^{t} g(t-s) a\bigl(u(t)-u(s), u(t)-u(s)\bigr) \,ds. $$

From (2.3), we deduce

$$ (g\mathbin{\square} u) (t) \leq C_{p} \bigl(g \mathbin{ \square}\partial^{2} u\bigr) (t) \quad \mbox{for some constant } C_{p} . $$

(3.1)

From (1.1)-(1.8), we have

$$ E'(t)=a\bigl((g\ast u) (t) , u'(t) \bigr)-\int_{\Gamma_{1}} p(x) \bigl(y'(x,t) \bigr)^{2} \, d \Gamma, $$

(3.2)

where

$$\begin{aligned} E(t) =&\frac{1}{\rho+2} \bigl\Vert u'(t)\bigr\Vert _{\rho+2}^{\rho+2} +\frac{1}{2} a\bigl(u(t),u(t)\bigr) + \frac{\alpha}{2} \bigl\Vert \nabla u'(t)\bigr\Vert ^{2} \\ &{}+\frac{1}{4} \bigl\Vert \Delta v(t)\bigr\Vert ^{2} + \frac{1}{2} \int_{\Gamma_{1}} q(x) \bigl(y(x,t) \bigr)^{2} \, d \Gamma. \end{aligned}$$

(3.3)

A direct calculation gives

$$\begin{aligned} a\bigl((g\ast u) (t) , u' (t) \bigr) =&- \frac{1}{2} g(t) a\bigl(u(t),u(t)\bigr) +\frac {1}{2} \bigl(g'\mathbin{\square}\partial^{2} u\bigr) (t) \\ &{} -\frac{1}{2} \frac{d}{dt} \biggl[ \bigl(g\mathbin{\square} \partial^{2} u\bigr) (t) - \biggl(\int_{0}^{t} g(s) \,ds \biggr) a\bigl(u(t),u(t)\bigr) \biggr]. \end{aligned}$$

(3.4)

Moreover, (2.5) gives

$$\begin{aligned} a\bigl((g \ast u) (t) ,u(t) \bigr) =&\int_{0}^{t} g(t-s) a\bigl( u(s)-u(t), u(t)\bigr) \,ds + \biggl( \int_{0}^{t} g(s) \,ds \biggr) a\bigl(u(t),u(t)\bigr) \\ \leq&2 l a\bigl(u(t),u(t)\bigr)+ \frac{5}{8C_{1}} \bigl(g \mathbin{\square} \partial^{2} u\bigr) (t). \end{aligned}$$

(3.5)

Define a modified energy by

$$\begin{aligned} \mathcal{E}(t) =& \frac{1}{\rho+2} \bigl\Vert u' (t)\bigr\Vert _{\rho+2}^{\rho+2} +\frac {\alpha}{2} \bigl\Vert \nabla u'(t)\bigr\Vert ^{2}+\frac{1}{2} \biggl( 1- \int_{0}^{t} g(s) \,ds \biggr) a\bigl(u(t),u(t) \bigr) \\ &{} + \bigl(g\mathbin{\square}\partial^{2} u\bigr) (t) + \frac{1}{4}\bigl\Vert \Delta v(t)\bigr\Vert ^{2} + \frac {1}{2}\int_{\Gamma_{1}} q(x) \bigl\vert y(x,t)\bigr\vert ^{2}\, d\Gamma. \end{aligned}$$

(3.6)

Then applying (3.4) to (3.2), we derive

$$ \mathcal{E}'(t) =-\frac{1}{2} g(t) a \bigl(u(t),u(t)\bigr)+\frac{1}{2} \bigl(g'\mathbin{\square} \partial^{2} u\bigr) (t) -\int_{\Gamma_{1}} p(x) \bigl(y'(x,t)\bigr)^{2}\, d \Gamma< 0. $$

(3.7)

This and the assumptions g and p imply that $\mathcal{E} (t)$ is nonincreasing and one easily sees that

$$ E(t)\leq C \mathcal{E}(t) \quad \mbox{for any } t>0 \mbox{ and for some positive constant } C. $$

(3.8)

Therefore, it is enough to obtain the desired energy decay for the modified energy $\mathcal{E}(t)$, which will be done in what follows. For this object, let us define the functional

$$ L(t)= N \mathcal{E}(t) +\varepsilon\Phi(t)+\Psi(t), $$

(3.9)

where

$$\begin{aligned} \Phi(t) =& \biggl( {1 \over {\rho+1}}\bigl\vert u'(t)\bigr\vert ^{\rho}u'(t), u(t)\biggr) + \alpha \bigl(\nabla u'(t),\nabla u(t)\bigr) \\ &{} +\bigl(u(t),y(t)\bigr)_{\Gamma_{1}} +\frac{1}{2} \int _{\Gamma_{1}} p(x) y(x,t)^{2} \, d\Gamma \end{aligned}$$

(3.10)

and

$$\begin{aligned} \Psi(t) =&-\int_{0}^{t} g(t-s) \biggl(u(t)-u(s),{1 \over {\rho+1}}\bigl\vert u'(t)\bigr\vert ^{\rho }u'(t)\biggr) \,ds \\ &{} -\alpha\int_{0}^{t} g(t-s) \bigl(\nabla u(t) -\nabla u(s), \nabla u'(t)\bigr)\,ds. \end{aligned}$$

(3.11)

Lemma 3.1

For $N>0$ large enough, there exist $\beta_{1} >0$ and $\beta_{2} >0$ such that

$$ \beta_{1} \mathcal{E} (t)\leq L(t) \leq \beta_{2} \mathcal{E}(t) \quad \textit{for any } t\geq0. $$

(3.12)

Proof

From (3.9), we have

$$\begin{aligned} L(t)-N \mathcal{E}(t) =&\frac{\varepsilon}{\rho+1} \int_{\Omega} \bigl\vert u'(t)\bigr\vert ^{\rho}u'(t) u(t) \,dx+ \varepsilon\alpha \int_{\Omega}\nabla u'(t) \nabla u(t) \,dx \\ &{} +\varepsilon\int_{\Gamma_{1}}u(t)y(t)\, d\Gamma+\varepsilon\int _{\Gamma _{1}} p(x) \bigl(y(x,t)\bigr)^{2}\, d\Gamma \\ &{} -\frac{1}{\rho+1}\int_{\Omega}\bigl\vert u'(t)\bigr\vert ^{\rho}u'(t) \biggl( \int _{0}^{t} g(t-s) \bigl(u(t)-u(s)\bigr)\,ds \biggr) \,dx \\ &{} -\alpha\int_{\Omega}\nabla u'(t) \biggl( \int _{0}^{t} g(t-s) \bigl(\nabla u(t)-\nabla u(s)\bigr) \,ds \biggr) \,dx \\ =& J_{1} +J_{2} +J_{3} +J_{4} +J_{5} + J_{6} . \end{aligned}$$

Using Hölder’s inequality and Young’s inequality, (2.3), (2.4), (2.9), (3.6), (3.7) and after some calculation, we obtain

$$\begin{aligned}& J_{1} = \frac{\varepsilon}{\rho+1} \int_{\Omega} \bigl\vert u'(t)\bigr\vert ^{\rho}u'(t) u(t) \,dx \\& \hphantom{J_{1}}\leq\frac{\varepsilon}{\rho+1} \biggl(\int_{\Omega} \bigl\vert u'(t)\bigr\vert ^{\rho+2} \,dx \biggr)^{\frac{\rho+1}{\rho+2}} \biggl( \int_{\Omega}\bigl\vert u(t)\bigr\vert ^{\rho+2} \,dx\biggr)^{\frac{1}{\rho+2}} \\& \hphantom{J_{1}}\leq{\frac{\varepsilon}{\rho+2}}\bigl\Vert u'(t)\bigr\Vert _{\rho+2}^{\rho+2} +{\frac {\varepsilon}{(\rho+1)(\rho+2)}}\bigl\Vert u(t)\bigr\Vert _{\rho+2}^{\rho+2} \\& \hphantom{J_{1}}\leq{\frac{\varepsilon}{\rho+2}}\bigl\Vert u'(t)\bigr\Vert _{\rho+2}^{\rho+2} +{\frac {\varepsilon}{(\rho+1)(\rho+2)}}K^{\rho+2} C_{s}^{\frac{\rho+2}{2}}\bigl( 2E(0)\bigr)^{\frac{\rho}{2}} a\bigl(u(t),u(t) \bigr) , \\& J_{2} =\varepsilon\alpha \int_{\Omega} \nabla u'(t) \cdot\nabla u(t) \,dx \leq\frac{\varepsilon\alpha }{2} \bigl\Vert \nabla u'(t)\bigr\Vert ^{2} +\frac {\varepsilon\alpha }{2} \bigl\Vert \nabla u(t)\bigr\Vert ^{2} \\& \hphantom{J_{2}}\leq\frac{\varepsilon\alpha }{2} \bigl\Vert \nabla u'(t) \bigr\Vert ^{2} +\frac {\varepsilon\alpha C_{s} }{2} a\bigl(u(t),u(t)\bigr), \\& J_{3} = \varepsilon\int_{\Gamma_{1}} u(t) y(t) \, d \Gamma \leq\frac {\varepsilon}{2} \int_{\Gamma_{1}} \bigl\vert u(t)\bigr\vert ^{2} \, d \Gamma+ \frac{\varepsilon }{2} \int_{\Gamma_{1}} y(t)^{2} \, d \Gamma \\& \hphantom{J_{3}}\leq\frac{\varepsilon\tilde{C}_{p}}{2} a\bigl(u(t),u(t)\bigr) + \frac {\varepsilon}{2q_{0}} \int_{\Gamma_{1}} q(x) \bigl(y(x,t) \bigr)^{2} \, d \Gamma \end{aligned}$$

and

$$ J_{4} = \varepsilon\int_{\Gamma_{1}} p(x) \bigl(y(x,t) \bigr)^{2} \, d \Gamma\leq\frac {\varepsilon p_{1}}{q_{0}} \int_{\Gamma_{1}} q(x) \bigl(y(x,t)\bigr)^{2} \, d \Gamma. $$

Using Fubini’s theorem, Hölder’s inequality and Young’s inequality, (2.3), (2.4), (3.6), (3.7), and after some calculation, we obtain

$$\begin{aligned} J_{5} =& -\frac{1}{\rho+1} \int_{\Omega} \bigl\vert u'(t)\bigr\vert ^{\rho}u'(t) \int _{0}^{t} g(t-s) \bigl(u(t)-u(s)\bigr)\,ds \,dx \\ =& -\frac{1}{\rho+1} \int_{0}^{t} g(t-s)\int _{\Omega} \bigl\vert u'(t)\bigr\vert ^{\rho}u'(t) \bigl(u(t)-u(s)\bigr) \,dx \,ds \\ \leq&\frac{1}{\rho+1} \bigl\Vert u'(t)\bigr\Vert _{\rho+2}^{\rho+1} \int_{0}^{t} g(t-s) \bigl\Vert u(t)-u(s)\bigr\Vert _{\rho+2} \,ds \\ \leq&\frac{1}{\rho+2} \bigl\Vert u'(t)\bigr\Vert _{\rho+2}^{\rho+2}+ \frac{1}{(\rho +1)(\rho+2)} \biggl(\int _{0}^{t} g(t-s) \bigl\Vert u(t)-u(s)\bigr\Vert _{\rho+2} \,ds \biggr)^{\rho+2} \\ \leq&\frac{1}{\rho+2} \bigl\Vert u'(t)\bigr\Vert _{\rho+2}^{\rho+2} \\ &{} + \frac{1}{(\rho+1)(\rho+2)}\biggl[\biggl(\int_{0}^{t} g(t-s)\,ds\biggr)^{1 \over 2} \biggl(\int_{0}^{t} g(t-s) \bigl\Vert u(t)-u(s)\bigr\Vert _{\rho+2}^{2} \,ds \biggr)^{1 \over 2} \biggr]^{\rho+2} \\ \leq&\frac{1}{\rho+2} \bigl\Vert u'(t)\bigr\Vert _{\rho+2}^{\rho+2} \\ &{} + \frac{1}{(\rho+1)(\rho+2)}l^{\rho+2 \over 2}K^{\rho+2}\biggl(\int _{0}^{t} g(t-s) \bigl\Vert \nabla u(t)-\nabla u(s)\bigr\Vert ^{2} \,ds\biggr)^{\rho+2 \over 2} \\ \leq&\frac{1}{\rho+2} \bigl\Vert u'(t)\bigr\Vert _{\rho+2}^{\rho+2} \\ &{} + \frac{1}{(\rho+1)(\rho+2)}l^{\rho+2 \over 2}K^{\rho+2} C_{s} ^{\rho+2 \over 2}\biggl(\int_{0}^{t} g(t-s) a\bigl( u(t)-u(s)\bigr) \,ds\biggr)^{\rho+2 \over 2} \\ \leq&\frac{1}{\rho+2} \bigl\Vert u'(t)\bigr\Vert _{\rho+2}^{\rho+2} \\ &{} + \frac{1}{(\rho+1)(\rho+2)}l^{\rho+2 \over 2}K^{\rho+2} C_{s} ^{\rho+2 \over 2}\bigl(\mathcal{E}(0)\bigr)^{\rho\over 2}\bigl( g\mathbin{\square} \partial^{2} u\bigr) (t) \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} J_{6} &= - \alpha\int_{\Omega} \nabla u'(t) \cdot\int_{0}^{t} g(t-s) \bigl( \nabla u(t)-\nabla u(s)\bigr)\,ds \,dx \\ &\leq\frac{\alpha}{2} \bigl\Vert \nabla u'(t)\bigr\Vert ^{2} +\frac{\alpha}{2} l\int_{0}^{t} g(t-s)\bigl\Vert \nabla u(t)-\nabla u(s)\bigr\Vert ^{2} \,ds \\ &\leq\frac{\alpha}{2} \bigl\Vert \nabla u'(t)\bigr\Vert ^{2} +\frac{\alpha l C_{s} }{2} \bigl( g\mathbin{\square}\partial^{2} u\bigr) (t). \end{aligned} \end{aligned}$$

From the results of $J_{1},J_{2}, \ldots,J_{6}$, we get

$$\begin{aligned} \bigl\vert L(t)- N\mathcal{E}(t)\bigr\vert \leq&\frac{\varepsilon+1}{\rho+2}\bigl\Vert u'(t)\bigr\Vert _{\rho+2}^{\rho+2} + \frac{\alpha}{2} (\varepsilon+1)\bigl\Vert \nabla u'(t)\bigr\Vert ^{2} \\ &{} + \biggl[ \frac{\varepsilon}{(\rho+1)(\rho+2)}K^{\rho+2} C_{s}^{\rho +2 \over 2} \bigl( 2E(0)\bigr)^{\rho\over 2} +\frac{\varepsilon\alpha C_{s} }{2} +\frac{\varepsilon\tilde{C}_{p} }{2} \biggr]a\bigl(u(t),u(t)\bigr) \\ &{} + \biggl[ \frac{1}{(\rho+1)(\rho+2)} l^{\rho+2 \over 2}K^{\rho+2} C_{s}^{\rho+2 \over 2} \bigl(\mathcal{E}(0) \bigr)^{\rho\over 2} + \frac{\alpha l C_{s} }{2} \biggr] \bigl(g\mathbin{\square}\partial^{2} u \bigr) (t) \\ &{}+ \biggl(\frac{\varepsilon}{2q_{0}} +\frac{p_{1} \varepsilon}{q_{0}} \biggr) \int _{\Gamma_{1}} q(x) \bigl(y(x,t)\bigr)^{2} \, d\Gamma. \end{aligned}$$

Therefore, for N is sufficiently large,

$$\bigl\vert L(t)-N \mathcal{E}(t)\bigr\vert \leq C \mathcal{E}(t). $$

Here we can take

$$\begin{aligned} C :=& \max \biggl( \varepsilon+1, \frac{2}{1-l} \biggl[ \frac{\varepsilon }{(\rho+1)(\rho+2)}K^{\rho+2} C_{s}^{\rho+2 \over 2} \bigl( 2E(0) \bigr)^{\rho \over 2} +\frac{\varepsilon\alpha C_{s} }{2} +\frac{\varepsilon\tilde {C}_{p} }{2} \biggr], \\ & \biggl[ \frac{1}{(\rho+1)(\rho+2)} l^{\rho+2 \over 2}K^{\rho+2} C_{s}^{\rho+2 \over 2} \bigl(\mathcal{E}(0) \bigr)^{\rho\over 2} + \frac{\alpha l C_{s} }{2} \biggr], \biggl[\frac{\varepsilon}{q_{0}}(1 +2p_{1} ) \biggr] \biggr) >0. \end{aligned}$$

So that we have

$$ \beta_{1} \mathcal{E}(t) \leq L(t) \leq\beta_{2} \mathcal{E}(t), $$

where $\beta_{1} =N-C$, $\beta_{2} = N+C$. We complete the proof of Lemma 3.1. □

Lemma 3.2

For any $t_{0} >0$ and sufficiently large $N>0$, there exist positive constants $\alpha_{0}$ and $\alpha_{1}$ such that

$$ L'(t) \leq-\alpha_{0} \mathcal{E} (t) + \alpha_{1} \bigl(g\mathbin{\square}\partial^{2} u\bigr) (t) \quad \textit{for all } t\geq t_{0} . $$

(3.13)

Proof

Using (3.10), (1.1) and (2.3), we have

$$\begin{aligned} \Phi'(t) =&\int_{\Omega}\bigl( \bigl\vert u'(t)\bigr\vert ^{\rho}u''(t) -\alpha\Delta u''(t) \bigr) u(t) \,dx+ \frac{1}{\rho+1} \bigl\Vert u'(t)\bigr\Vert _{\rho+2}^{\rho+2} +\alpha\bigl\Vert \nabla u'(t)\bigr\Vert ^{2} \\ &{}+ \alpha \biggl( \frac{\partial u''(t)}{\partial\nu} , u(t) \biggr)_{\Gamma_{1}} +\bigl( u'(t) , y(t)\bigr)_{\Gamma_{1}}+\bigl(u(t), y'(t) \bigr)_{\Gamma _{1}} \\ &{}+\int_{\Gamma_{1}} p(x) y(x,t) y'(x,t) \, d\Gamma \\ =& \frac{1}{\rho+1} \bigl\Vert u'(t)\bigr\Vert _{\rho+2}^{\rho+2} +\alpha\bigl\Vert \nabla u'(t)\bigr\Vert ^{2} -a\bigl(u(t),u(t)\bigr) \\ &{}+ \biggl( \mathcal{B}_{1} u(t), \frac{\partial u(t)}{\partial\nu} \biggr)_{\Gamma_{1}}-\bigl(\mathcal{B}_{2} u(t),u(t) \bigr)_{\Gamma_{1}} + a\bigl((g\ast u) (t),u(t)\bigr) \\ &{}-\int_{0}^{t} g(t-s) \biggl[ \biggl( \mathcal{B}_{1} u(s),\frac{\partial u(t)}{\partial\nu} \biggr)_{\Gamma_{1}}-\bigl( \mathcal{B}_{2} u(s), u(t)\bigr)_{\Gamma _{1}} \biggr] \,ds \\ &{}+ \bigl(\bigl[u(t),v(t)\bigr], v(t)\bigr) +\alpha \biggl(\frac{\partial u''(t)}{\partial \nu}, u(t) \biggr)_{\Gamma_{1}} +\bigl(u'(t), y(t)\bigr)_{\Gamma_{1}} +\bigl(u(t), y'(t)\bigr)_{\Gamma_{1}} \\ &{}+\int_{\Gamma_{1}} p(x) y(x,t) y'(x,t) \, d\Gamma \\ =&\frac{1}{\rho+1} \bigl\Vert u'(t)\bigr\Vert _{\rho+2}^{\rho+2} +\alpha\bigl\Vert \nabla u'(t)\bigr\Vert ^{2} -a\bigl(u(t),u(t)\bigr) +a\bigl((g\ast u) (t) , u(t)\bigr) \\ &{}+2\bigl(u(t), y'(t) \bigr)_{\Gamma_{1}}-\bigl( \Delta^{2} v(t), v(t)\bigr) + \bigl(u'(t), y(t) \bigr)_{\Gamma_{1}} \\ &{}+\int_{\Gamma_{1}} p(x) y(x,t) y'(x,t) \, d \Gamma \\ \leq&\frac{1}{\rho+1} \bigl\Vert u'(t)\bigr\Vert _{\rho+2}^{\rho+2} +\alpha\bigl\Vert \nabla u'(t)\bigr\Vert ^{2} -(1-\delta\tilde{C}_{p} ) a\bigl(u(t),u(t)\bigr) \\ &{}+a\bigl((g\ast u) (t) ,u(t)\bigr) -\bigl\Vert \Delta v(t)\bigr\Vert ^{2} +\frac{1}{\delta}\bigl\Vert y'(t)\bigr\Vert _{\Gamma_{1}}^{2} \\ &{}-\int_{\Gamma_{1}} q(x) \bigl\vert y(x,t)\bigr\vert ^{2} \, d\Gamma. \end{aligned}$$

(3.14)

Adapting (3.5) to (3.14), we get

$$\begin{aligned} \Phi'(t) \leq&\frac{1}{\rho+1} \bigl\Vert u'(t)\bigr\Vert _{\rho+2}^{\rho+2}+\alpha\bigl\Vert \nabla u'(t)\bigr\Vert ^{2} -(1-2l-\delta \tilde{C}_{p} )a\bigl(u(t),u(t)\bigr) \\ &{} +\frac{1}{\delta} \bigl\Vert y'(t)\bigr\Vert _{\Gamma_{1}}^{2} -\int_{\Gamma_{1}} q(x) \bigl(y(x,t) \bigr)^{2} \, d\Gamma+\frac{5}{8C_{1} }\bigl(g\mathbin{\square} \partial^{2} u\bigr) (t) -\bigl\Vert \Delta v(t)\bigr\Vert ^{2} . \end{aligned}$$

(3.15)

Moreover, from (3.11) and (1.1), we derive

$$\begin{aligned} \Psi'(t) =& - \biggl( \bigl\vert u'(t)\bigr\vert ^{\rho}u''(t), \int_{0}^{t} g(t-s) \bigl(u(t)-u(s)\bigr) \,ds \biggr) \\ &{} -\alpha \biggl( \nabla u''(t), \int _{0}^{t} g(t-s) \bigl(\nabla u(t)-\nabla u(s)\bigr) \,ds \biggr) \\ &{} - \biggl( \frac{1}{\rho+1}\bigl\vert u'(t)\bigr\vert ^{\rho}u'(t), \int_{0}^{t} g'(t-s) \bigl(u(t)-u(s)\bigr) \,ds \biggr) \\ &{} -\alpha \biggl( \nabla u'(t), \int_{0}^{t} g'(t-s) \bigl(\nabla u(t)-\nabla u(s)\bigr) \,ds \biggr) \\ &{} -\alpha \biggl( \nabla u'(t), \int_{0}^{t} g(t-s) \nabla u'(t) \,ds \biggr) \\ &{} - \biggl( \frac{1}{\rho+1} \bigl\vert u'(t)\bigr\vert ^{\rho}u'(t), \int_{0}^{t} g(t-s) u'(t) \,ds \biggr) \\ =& \biggl(\Delta^{2} u(t)-\int_{0}^{t} g(t-s) \Delta^{2} u(s) \,ds-[u,v], \int_{0}^{t} g(t-s) \bigl(u(t)-u(s)\bigr) \,ds \biggr) \\ &{} - \alpha \biggl(\frac{\partial u''(t)}{\partial\nu}, \int_{0}^{t} g(t-s) \bigl(u(t)-u(s)\bigr) \,ds \biggr) \\ &{} - \biggl( \frac{1}{\rho+1}\bigl\vert u'(t)\bigr\vert ^{\rho}u'(t), \int_{0}^{t} g'(t-s) \bigl(u(t)- u(s)\bigr) \,ds \biggr) \\ &{} -\frac{1}{\rho+1}\biggl(\int_{0}^{t} g(s) \,ds\biggr) \bigl\Vert u'(t)\bigr\Vert _{\rho+2}^{\rho +2}- \alpha\biggl(\int_{0}^{t} g(s) \,ds\biggr) \bigl\Vert \nabla u'(t)\bigr\Vert ^{2} \\ &{} -\alpha \biggl( \nabla u'(t), \int_{0}^{t} g'(t-s) \bigl(\nabla u(t)-\nabla u(s)\bigr) \,ds \biggr) \\ =&\int_{0}^{t} g(t-s) a\bigl(u(t), u(t)-u(s) \bigr) \,ds \\ &{} -\int_{0}^{t} g(t-s) \int _{0}^{t} g(t-\tau) a\bigl(u(\tau), u(t)-u(s)\bigr) \, d\tau \,ds \\ &{} - \biggl( y', \int_{0}^{t} g(t-s) \bigl(u(t)-u(s)\bigr) \,ds \biggr)_{\Gamma _{1}} \\ &{} -\alpha \biggl( \nabla u'(t), \int_{0}^{t} g'(t-s) \bigl(\nabla u(t)-\nabla u(s)\bigr) \,ds \biggr) \\ &{} - \biggl( \frac{1}{\rho+1}\bigl\vert u'(t)\bigr\vert ^{\rho}u'(t), \int_{0}^{t} g'(t-s) \bigl(u(t)- u(s)\bigr) \,ds \biggr) \\ &{} - \biggl( [u,v], \int_{0}^{t} g(t-s) \bigl(u(t)-u(s)\bigr) \,ds \biggr) \\ &{} -\alpha\biggl(\int_{0}^{t} g(s) \,ds\biggr) \bigl\Vert \nabla u'(t)\bigr\Vert ^{2} -\frac{1}{\rho+1} \biggl(\int_{0}^{t} g(s) \,ds \biggr)\bigl\Vert u'(t)\bigr\Vert _{\rho +2}^{\rho+2} \\ =&I_{1} +I_{2} +\cdots+I_{6} \\ &{} -\alpha \biggl( \int_{0}^{t} g(s) \,ds \biggr) \bigl\Vert \nabla u'(t)\bigr\Vert ^{2} - \frac {1}{\rho+1} \biggl( \int_{0}^{t} g(s) \,ds \biggr) \bigl\Vert u'(t)\bigr\Vert _{\rho+2}^{\rho+2}. \end{aligned}$$

(3.16)

In what follows we will estimate the terms on the right-hand side of (3.16). From (2.5) and (3.5) we deduce that

$$|I_{1} | \leq\eta l a\bigl(u(t),u(t)\bigr) +\frac{5}{8C_{1} \eta}\bigl(g \mathbin{\square}\partial ^{2} u\bigr) (t) $$

and

$$\begin{aligned} |I_{2}| \leq&\biggl\vert \int_{0}^{t} g(t-s) \int_{0}^{t} g(t-\tau) a\bigl( u(t)-u(\tau ), u(t)-u(s)\bigr) \, d\tau \,ds \biggr\vert \\ &{} +\biggl\vert \int_{0}^{t} g(t-s) \int _{0}^{t} g(t-\tau) a\bigl( u(t), u(t)-u(s)\bigr) \, d\tau \,ds \biggr\vert \\ \leq&\int_{0}^{t} g(t-s) \int _{0}^{t} g(t-\tau) \biggl[ a\bigl(u(t)-u(\tau), u(t)-u(\tau)\bigr) \\ &{} +\frac{5}{8C_{1} }a\bigl(u(t)-u(s), u(t)-u(s)\bigr) \biggr]\, d\tau \,ds \\ &{} + \biggl( \int_{0}^{t} g(\tau)\, d\tau \biggr) \biggl\vert \int_{0}^{t} g(t-s) a\bigl( u(t), u(t)-u(s)\bigr) \,ds \biggr\vert \\ \leq&\eta l^{2} a\bigl(u(t),u(t)\bigr) + \biggl( l+ \frac{5l}{8C_{1} }+\frac {5l}{8C_{1} \eta} \biggr) \bigl(g\mathbin{\square} \partial^{2} u\bigr) (t). \end{aligned}$$

Also, using Young’s inequality, (2.3) and (3.8), we obtain

$$\begin{aligned}& |I_{3} | \leq\eta\bigl\Vert y'(t)\bigr\Vert _{\Gamma_{1}}^{2} +\frac{C_{p} l}{4\eta} \bigl(g\mathbin{\square} \partial^{2} u\bigr) (t), \\& |I_{4}| \leq\eta\alpha\bigl\Vert \nabla u'(t)\bigr\Vert ^{2} +\frac{\alpha}{4\eta} \int_{0}^{t} -g'(s) \,ds \int_{0}^{t} -g'(t-s) \bigl\Vert \nabla u(t)-\nabla u(s)\bigr\Vert ^{2} \,ds \\& \hphantom{|I_{4}|}\leq\eta\alpha\bigl\Vert \nabla u'(t)\bigr\Vert ^{2} -\frac{\alpha g(0) C_{s} }{4\eta} \bigl(g'\mathbin{\square} \partial^{2} u\bigr) (t), \\& |I_{5}| \leq\frac{\eta}{\rho+1}\bigl\Vert u'(t)\bigr\Vert _{2(\rho+1)}^{2(\rho+1)}-\frac {g(0) C_{s}}{4\eta(\rho+1)} \bigl(g' \mathbin{\square}\partial^{2} u\bigr) (t) \\& \hphantom{|I_{5}|}\leq\frac{\eta K^{2(\rho+1)}}{\rho+1} \bigl(2\alpha^{-1} \mathcal {E}(0)\bigr)^{\rho}\bigl\Vert \nabla u'(t)\bigr\Vert ^{2} -\frac{g(0) C_{s} }{4\eta(\rho +1)}\bigl(g'\mathbin{\square} \partial^{2} u\bigr) (t), \\& |I_{6}| \leq\eta\bigl\Vert \bigl[u(t),v(t)\bigr]\bigr\Vert ^{2} +\frac{1}{4\eta}\biggl\Vert \int_{0}^{t} g(t-s) \bigl(u(t)-u(s)\bigr) \,ds \biggr\Vert ^{2} \\& \hphantom{|I_{6}|}\leq\eta C^{\prime} \bigl\Vert u(t)\bigr\Vert _{H^{2} (\Omega)}^{2} \bigl\Vert v(t)\bigr\Vert _{H^{2} (\Omega )}^{2} +\frac{C_{p} l}{4\eta} \bigl(g\mathbin{\square}\partial^{2} u\bigr) (t) \\& \hphantom{|I_{6}|}\leq C_{1}^{-1}\eta C^{\prime}4E(0)a \bigl(u(t),u(t)\bigr) +\frac{C_{p} l}{4\eta} \bigl(g\mathbin{\square} \partial^{2} u\bigr) (t). \end{aligned}$$

Combining these estimates $I_{i}$, $i=1,\ldots,6$ and (3.16), we get

$$\begin{aligned} \Psi'(t) \leq&- \biggl( \int_{0}^{t} g(s) \,ds \biggr) \frac{\|u'(t)\|_{\rho +2}^{\rho+2}}{\rho+1} +\eta\bigl(l+l^{2} +C^{\prime}C_{1}^{-1}4E(0)\bigr) a\bigl(u(t),u(t)\bigr) \\ &{} - \biggl[ \alpha \biggl( \int_{0}^{t} g(s) \,ds \biggr) -\eta\biggl(\alpha+\frac {K^{2(\rho+1)}}{\rho+1} \bigl(2\alpha^{-1} \mathcal{E}(0)\bigr)^{\rho}\biggr) \biggr] \bigl\Vert \nabla u'(t)\bigr\Vert ^{2} \\ &{} + \biggl( \frac{5}{8C_{1} \eta}+ l+\frac{5l}{8C_{1} }+\frac{5l}{8C_{1} \eta } + \frac{C_{p} l}{2\eta} \biggr) \bigl(g\mathbin{\square}\partial^{2} u\bigr) (t) \\ &{} -\frac{g(0)C_{s}}{4\eta} \biggl( \alpha+\frac{1}{\rho+1} \biggr) \bigl(g'\mathbin{\square}\partial^{2} u\bigr) (t) +\eta \bigl\Vert y'(t)\bigr\Vert _{\Gamma_{1}}^{2}. \end{aligned}$$

(3.17)

Since g is continuous and positive for any $t\geq t_{0} >0$, we have

$$ \int_{0}^{t} g(s) \,ds \geq\int _{0}^{t_{0}} g(s) \,ds := g_{0} >0. $$

(3.18)

Thus, making use of (3.18) and combining (3.7), (3.9), (3.15) and (3.17), we arrive at

$$\begin{aligned} L'(t) \leq&-\biggl(g_{0} - \frac{\varepsilon}{\rho+1}\biggr) \bigl\Vert u'(t)\bigr\Vert _{\rho+2}^{\rho +2} \\ &{}- \biggl[ \frac{N}{2} g(t)+\varepsilon(1-2l-\delta \tilde{C}_{p} ) -\eta\bigl(l+l^{2} + C^{\prime}C_{1}^{-1}4 E(0) \bigr) \biggr] a\bigl(u(t),u(t)\bigr) \\ &{} - \biggl[ \alpha g_{0} -\eta\biggl(\alpha+\frac{K^{2(\rho+1)}}{\rho+1} \bigl(2\alpha^{-1} \mathcal{E}(0)\bigr)^{\rho}\biggr) - \varepsilon\alpha \biggr] \bigl\Vert \nabla u'(t)\bigr\Vert ^{2} -\varepsilon\bigl\Vert \Delta v(t)\bigr\Vert ^{2} \\ &{} + \biggl( \frac{5\varepsilon}{8C_{1}}+\frac{5}{8C_{1} \eta}+l+\frac {5l}{8C_{1}}+ \frac{5l}{8C_{1} \eta}+\frac{C_{p} \cdot l}{2\eta} \biggr) \bigl(g\mathbin{\square} \partial^{2} u\bigr) (t) \\ &{} + \biggl[ \frac{N}{2} -\frac{g(0) C_{s} }{4\eta} \biggl( \alpha+ \frac {1}{\rho+1} \biggr) \biggr] \bigl(g'\mathbin{\square} \partial^{2} u\bigr) (t) -\varepsilon\int_{\Gamma_{1} } q(x) \bigl(y(x,t)\bigr)^{2} \, d\Gamma \\ &{} - \biggl( N -\frac{\varepsilon}{\delta p_{0}}-\frac{\eta}{p_{0}} \biggr)\int _{\Gamma_{1}}p(x) \bigl(y^{\prime}(x,t)\bigr)^{2}\, d \Gamma\quad \mbox{for any } t\geq t_{0}. \end{aligned}$$

(3.19)

We first choose $\varepsilon>0$ and $\delta>0$ so small such that $g_{0} -\varepsilon>0$ and $1-2l-\delta\tilde{C}_{p} >0$, respectively. We also take $\eta>0$ sufficiently small and $N>0$ large enough so that $\frac{N}{2} g(t) +\varepsilon(1-2l-\delta\tilde{C}_{p} ) -\eta(l+l^{2} +C^{\prime}C_{1}^{-1}4E(0))>0$, $\alpha g_{0} -\eta(\alpha+\frac {K^{2(\rho+1)}}{\rho+1} (2\alpha^{-1} \mathcal{E}(0))^{\rho})- \varepsilon\alpha>0$, $\frac{N}{2}-\frac{g(0)C_{s} }{4\eta} ( \alpha+\frac{1}{\rho+1} ) >0$ and $N p_{0} -\varepsilon/\delta -\eta>0$ for any $t\geq t_{0}$. Then we deduce that desired result. We complete the proof of Lemma 3.2. □

Our main result is the following.

Theorem 3.1

Assume that $l<1/2$. Then, for each $t_{0} > 0$, there exist positive constants $C_{0}$ and C such that

$$E(t)\leq C_{0} e^{-C \int_{t_{0}}^{t} \xi(s) \,ds} \quad \textit{for all }t\geq t_{0}. $$

Proof

Multiplying (3.13) by $\xi(t)$, noting that ξ is nonincreasing, and using (2.8) and (3.7), we see that

$$\begin{aligned} \xi(t) L'(t) \leq&-\alpha_{0} \xi(t) \mathcal{E}(t)+ \alpha_{1} \xi(t) \bigl( g\mathbin{\square}\partial^{2} u \bigr) (t) \leq-\alpha_{0} \xi(t) \mathcal{E}(t)-\alpha_{1} \bigl(g'\mathbin{\square}\partial^{2} u\bigr) (t) \\ \leq&-\alpha_{0} \xi(t) \mathcal{E}(t)-2\alpha_{1} \mathcal{E}'(t)\quad \mbox{for all } t\geq t_{0}. \end{aligned}$$

This and the fact $\xi'(t)\leq0$ yield

$$ \frac{d}{dt} \bigl( \xi(t) L(t) +2\alpha_{1} \mathcal{E} (t) \bigr) \leq-\alpha_{0} \xi(t) \mathcal{E}(t) \quad \mbox{for all } t\geq t_{0}. $$

(3.20)

Now, we define

$$\mathcal{L} (t)= \xi(t) L(t) +2\alpha_{1} \mathcal{E}(t). $$

Since $\xi(t)$ is a nonincreasing positive function, we can easily observe that $\mathcal{L}(t)$ is equivalent to $\mathcal{E}(t)$. Thus (3.20) implies that

$$\frac{d}{dt} \mathcal{L} (t) \leq-C \xi(t) \mathcal{L}(t) \quad \mbox{for some } C>0 \mbox{ and for all } t\geq t_{0}. $$

Integrating the above inequality, we get

$$\mathcal{L}(t) \leq\mathcal{L}(t_{0})e^{-C \int_{t_{0}}^{t} \xi(s) \,ds}\quad \mbox{for all } t\geq t_{0}. $$

Consequently, from the equivalent relations of $\mathcal{L}$, L, $\mathcal {E}$ and (3.8), we obtain the result in Theorem 3.1. □