Energy decay of solutions of nonlinear viscoelastic problem with the dynamic and acoustic boundary conditions

Abstract

In this paper, we are concerned with the energy decay rate of the nonlinear viscoelastic problem with dynamic and acoustic boundary conditions.

1 Introduction

In this paper, we are concerned with the energy decay rate of the following nonlinear viscoelastic problem with a time-varying delay in the boundary feedback and acoustic boundary conditions:

$$\begin{aligned}& \begin{aligned}[b] &u_{tt} -\delta_{0} \Delta u + \int_{0}^{t} g(t-s) \operatorname{div} \bigl(a(x) \nabla u(s)\bigr) \,ds+ \bigl(\delta_{1} +b(x) \big|u_{t} (t)\big|^{m-2} \bigr) u_{t} \\ & \quad=|u|^{p-2} u\quad\hbox{in } \Omega\times(0,+\infty) ,\end{aligned} \end{aligned}$$

(1.1)

$$\begin{aligned}& u=0 \quad\hbox{on } \Gamma_{0} \times(0, \infty) , \end{aligned}$$

(1.2)

$$\begin{aligned}& \begin{aligned}[b]&u_{tt}+ \delta_{0} \frac{\partial u(t)}{\partial\nu} - \int_{0}^{t} g(t-s) \bigl(a(x) \nabla u(s)\bigr) \cdot\nu \,ds +\mu_{1} k_{1}\bigl( u_{t} (t)\bigr) + \mu_{2} k_{2} \bigl( u_{t} \bigl( t-\tau(t)\bigr) \bigr) \\ &\quad=h(x) y_{t}\quad\hbox{on } \Gamma_{1} \times(0, \infty), \end{aligned} \end{aligned}$$

(1.3)

$$\begin{aligned}& u_{t} +f(x) y_{t} +m(x) y=0 \quad \hbox{on } \Gamma_{1} \times (0,\infty), \end{aligned}$$

(1.4)

$$\begin{aligned}& u(x,0) =u_{0} (x), \qquad u_{t} (x,0) =u_{1} (x)\quad \hbox{in } \Omega , \end{aligned}$$

(1.5)

$$\begin{aligned}& y(x,0)=y_{0} (x) \quad\hbox{on } \Gamma_{1} , \end{aligned}$$

(1.6)

$$\begin{aligned}& u_{t} \bigl(x, t-\tau(t)\bigr)=j_{0} \bigl(x,t-\tau(0) \bigr) \quad \hbox{on } \Gamma _{1} \times\bigl(0, \tau(0)\bigr) , \end{aligned}$$

(1.7)

where Ω regular and is a bounded domain of $R^{n}$, $n\geq1$, $\partial\Omega=\Gamma_{0} \cup\Gamma_{1}$. Here $\Gamma_{0}$, $\Gamma_{1}$ are closed and disjoint with $\operatorname{meas} (\Gamma_{0} )>0$ and ν is the unit outward normal to ∂Ω, $\delta_{0} >0$, $\delta_{1} \geq0$, $m\geq2$, $p>2$, g denotes the memory kernel and a, b are real valued functions which satisfy appropriate conditions. The functions $f,m,h: \Gamma_{1} \to R^{+}$ are essentially bounded, $k_{1}, k_{2} : R\to R$ are given functions, $\tau(t)>0$ represents the time-varying delay, $\mu_{1}$, $\mu_{2}$ are real numbers with $\mu_{1} >0$, $\mu_{2} \neq0$ and the initial data $(u_{0}, u_{1}, y_{0} )$ belongs to a suitable space. This type of equation usually arises in the theory of viscoelasticity. It is well known that viscoelastic materials have memory effects, which is due to the mechanical response influenced by the history of the materials themselves. From the mathematical point of view, their memory effects are modeled by integrodifferential equations. Hence, equations related to the behavior of the solutions for the PDE system have attracted considerable attention in recent years. We can refer to recent work in [1–7].

The dynamic boundary conditions are not only important from the theoretical point of view but also arise in numerous practical problems. Among the early results dealing with this type of boundary conditions are those of [8, 9] in which the author has made contributions to this field. Recently, some authors have studied the existence and decay of solutions for a wave equation with dynamic boundary conditions [10–13].

Moreover, the acoustic boundary conditions was introduced by Morse and Ingard in [14] and developed by Beale and Rosencrans in [15], where the authors proved the global existence and regularity of the linear problem. Recently, some authors have studied the existence and decay of solutions for a viscoelastic wave equation with acoustic boundary conditions (see [16–19]). Time delay so often arises in many physical, chemical, biological and economical phenomena because these phenomena depend not only on the present state but also on the history of the system in a more complicated way.

In recent years, differential equations with time delay effects have become an active area of research; see for example [20–26] and the references therein. To stabilize a hyperbolic system involving input delay terms, additional control terms will be necessary. For instance in [22], the authors are proved the boundary stabilization of a nonlinear viscoelastic equation with interior time-varying delay and nonlinear dissipative boundary feedback. In particular, Wu and Chen [27] consider the nonlinear viscoelastic wave equation with boundary dissipation

$$\begin{gathered} u_{tt} (t) -K_{0} \Delta u(t) + \int_{0}^{t} g(t-s) \operatorname{div} \bigl(a(x) \nabla u(s)\bigr) \,ds +b(x) u_{t} =f(u) \quad\hbox{in } \Omega\times(0,\infty ), \\ u=0 \quad \hbox{on } \Gamma_{0} \times(0, \infty), \\ K_{0} \frac{\partial u}{\partial\nu}- \int_{0}^{t} g(t-s) \bigl(a(x) \nabla u(s)\bigr) \cdot\nu \,ds +h(u_{t} )=0\quad \hbox{on } \Gamma_{1} \times(0, \infty), \\ u(0)=0, \qquad u_{t} (0)=u_{1} , \quad x\in\Omega,\end{gathered} $$

where $K_{0} >0$ and Ω is a bounded domain in $R^{n}$ ($n\geq1$) with a smooth boundary $\Gamma=\Gamma_{0} \cup\Gamma_{1}$. The authors studied the uniform decay of solutions for a nonlinear viscoelastic wave equation with boundary dissipation. In [28], Boukhatem and Benabderrahmane have proved the existence and decay of solutions for a viscoelastic wave equation with acoustic boundary conditions as follows:

$$\begin{aligned}& u_{tt} +Lu - \int_{0}^{t} g(t-s) Lu(s) \,ds =|u|^{p-2}u \quad \hbox{in } \Omega\times(0, \infty), \\& u=0 \quad\hbox{on } \Gamma_{0} \times(0, \infty), \\& \frac{\partial u}{\partial\nu_{L}}- \int_{0}^{t} g(t-s) \frac {\partial u}{\partial\nu_{L}}(s) \,ds =h(x) z_{t} \quad\hbox{on } \Gamma _{1} \times(0,\infty), \\& u_{t} +f(x) z_{t} + m(x) z=0 \quad\hbox{on } \Gamma_{1} \times (0,\infty), \\& u(x,0)=u_{0} (x) , \qquad u_{t} (x,0)=u_{1} (x) \quad \hbox{in } \Omega , \\& z(x,0)=z_{0} (x)\quad \hbox{on } \Gamma_{1}, \end{aligned}$$

where $Lu =-\operatorname{div} (A\nabla u)=\sum_{i,j=1}^{N} \frac{\partial}{\partial x_{i} } ( a_{ij}(x) \frac{\partial u}{\partial x_{j} } )$ and $\frac{\partial u}{\partial\nu_{L}}=\sum_{i,j=1}^{N} a_{ij}(x) \frac{\partial u}{\partial x_{j} } \nu_{i}$.

Liu [29] investigated the following viscoelastic wave equation with an interval time-varying delay term:

$$\begin{gathered} u_{tt}(x,t)-\Delta u(x,t) +\alpha(t) \int_{0}^{t} g(t-s) \Delta u(x,s) \,ds+a_{0} u_{t} (x,t) +a_{1} \bigl(x, t-\tau(t) \bigr)=0 \\ \quad \hbox{in } \Omega\times(0, \infty), \\ u(x,t)=0\quad \hbox{on } \partial\Omega\times(0,\infty), \\ u(x,0)=u_{0} (x), \qquad u_{t} (x,0)=u_{1} (x)\quad \hbox{in } \Omega, \\ u_{t} \bigl(x, t-\tau(0)\bigr)=f_{0} \bigl(x, t-\tau(0) \bigr)\quad \hbox{on } \Omega \times\bigl(0, \tau(0)\bigr),\end{gathered} $$

where Ω is a bounded domain $R^{n}$ ($n\geq2$) with a boundary ∂Ω of class $C^{2}$, α and g are positive non-increasing functions defined on $R^{+}$, $a_{0}$ and $a_{1}$ are real number with $a_{0} >0$, $\tau(t)>0$ represents the time-varying delay. He also proved the general decay rate for the energy of a weak viscoelastic wave equation with an interval time-varying delay term.

Recently, Li and Chai [30] have investigated the energy decay for a nonlinear wave equation of variable coefficients with acoustic boundary conditions and a time-varying delay in the boundary feedback form:

$$\begin{gathered} u_{tt} -\operatorname{div} \bigl(A(x) \nabla u\bigr) +\varphi(u_{t})=0\quad \hbox{in } \Omega\times(0,\infty), \\ u=0\quad \hbox{on } \Gamma_{1} \times(0, \infty), \\ u_{t} +f(x) z_{t} +k(x) z =0 \quad \hbox{on } \Gamma_{1} \times (0,\infty), \\ \frac{\partial u}{\partial\nu_{A}} -h(x) z_{t} +\mu_{1} \beta \bigl(u_{t} (x,t)\bigr) +\mu_{2} u_{t} \bigl(x, t- \tau(t)\bigr)=0 \quad \hbox{on } \Gamma_{1} \times(0,\infty), \\ u(x,0) =u_{0} (x) , \qquad u_{t} (x,0)=u_{1} (x)\quad \hbox{in } \Omega , \\ z(x,0)=z_{0} (x)\quad \hbox{on } \Gamma_{1}, \\ u_{t} \bigl( x, t-\tau(0)\bigr)=j_{0} \bigl(x, t- \tau(0)\bigr)\quad \hbox{on } \Gamma _{1} \times\bigl(0, \tau(0) \bigr),\end{gathered} $$

where divX denotes the divergence of the vector field X in the Euclidean metric, $A(x) =(a_{ij}(x))$ are symmetric and positive definite metrics for all $x\in R^{n}$ and $a_{ij}(x)$ are smooth functions on $R^{n}$, $\frac{\partial u}{\partial\nu_{A}}=\sum_{i,j=1}^{n} a_{ij} \frac{\partial u}{\partial x_{j}} \nu_{i}$, where $\nu=(\nu_{1},\nu_{2}, \ldots, \nu_{n})^{T}$ denotes the outward unit normal vector of the boundary and $\nu+A =A\mu$.

Motivated by previous work, in this paper, we study the energy decay rate of the nonlinear viscoelastic problem with a time-varying delay in the boundary conditions. Previously many authors have considered the uniform decay of solutions for a nonlinear viscoelastic wave equations with boundary dissipations. However, to our knowledge, there is no energy decay result of the nonlinear viscoelastic problem with a the dynamic, time-varying delay and acoustic boundary conditions. Thus this work is significant. The outline of the paper is the following. In Section 2, we give some notation and hypotheses for our result. In Section 3, we prove our main result.

2 Preliminary

In this section, we present some material that we shall use in order to present our results. We denote by $(u,v) =\int_{\Omega}u(x) v(x) \,dx$ the scalar product in $L^{2} (\Omega)$. We denote by $\|\cdot\|_{q}$ the $L^{q} (\Omega)$ norm for $1\leq q\leq\infty$ and $\|\cdot\| _{q,\Gamma_{1}}$ for $L^{q} (\Gamma_{1})$. We introduce by

$$V=\bigl\{ u\in H^{1} (\Omega) ; u=0 \hbox{ on } \Gamma_{0} \bigr\} $$

the closed subspace of $H^{1} (\Omega)$ equipped with the norm equivalent to the usual norm in $H^{1} (\Omega)$. The Poincaré inequality holds in V, i.e., there exists a constant $C_{*}$ such that

$$ \forall u\in V ,\quad \|u \|_{p} \leq C_{*} \|\nabla u\|,\quad 2 < p\leq\bar{p}, $$

(2.1)

where

$$\bar{p} = \left \{ \textstyle\begin{array}{l@{\quad}l} \frac{2(n-2)}{n-2} , &\hbox{if } n\geq3,\\ \infty, & \hbox{if } n=1,2, \end{array}\displaystyle \right . $$

and there exists a constant $\tilde{C_{*}}>0$ such that

$$ \|u\|_{\Gamma_{1}} \leq\tilde{C_{*}} \|\nabla u\|, \quad \forall u \in V. $$

(2.2)

For studying the problem (1.1)-(1.7) we will need the following assumptions:

$(H1)$ :

The kernel function $g: R^{+} \to R^{+}$ is a bounded $C^{1}$ function satisfying

$$ g(0)>0,\quad \delta_{0}-\|a\|_{\infty} \int_{0}^{\infty}g(s) \,ds := l>0, $$

(2.3)

and there exists a non-increasing $C^{1}$ positive differentiable function $\zeta:R^{+} \to R^{+}$ satisfying

$$ g'(t) \leq-\zeta(t) g(t) ,\qquad \int_{0}^{\infty}\zeta(s) \,ds =\infty, \quad \forall t \geq0. $$

(2.4)

$(H2)$ :

$a : \Omega\to R$ is a nonnegative function and $a\in C^{1} (\bar{\Omega})$ such that

$$\begin{aligned}& a(x)\geq a_{0} >0, \end{aligned}$$

(2.5)

$$\begin{aligned}& \big\| \nabla a(x) \big\| ^{2} \leq\alpha_{1}^{2} \|a \|_{\infty}^{2} \end{aligned}$$

(2.6)

for some positive constant $\alpha_{1}$, $b: \Omega\to R$ is a nonnegative functions and $b\in C^{1} (\bar{\Omega})$ such that $b(x) \geq b_{0} >0$.

$(H3)$ :

Similarly to [31] $k_{1} :R\to R$ is a nondecreasing $C^{1}$ function such that there exist $\varepsilon_{1}, C_{1} ,C_{2} >0$ and a convex, increasing function $K_{1} : R^{+} \to R^{+}$, of the class $C^{1} (R^{+} )\cap C^{2} (R^{+})$ satisfying $K_{1} (0)=0$, $K_{1}$ is linear (or $(K_{1}'(0)=0)$ and $K_{1} '' >0$ on $[0,\varepsilon_{1} ]$) such that

$$\begin{aligned}& C_{1} |s| \leq\big|k_{1} (s)\big|\leq C_{2} |s| \quad\hbox{for all } |s|\geq \varepsilon_{1}, \end{aligned}$$

(2.7)

$$\begin{aligned}& s^{2}+ k_{1}^{2} (s) \leq K_{1}^{-1} \bigl(s k_{1} (s)\bigr) \quad \hbox{for all } |s|\leq \varepsilon_{1} , \end{aligned}$$

(2.8)

$k_{2} : R \to R$ is an odd nondecreasing $C^{1}$ function such that there exist $C_{3}, C_{4}, C_{5} >0$,

$$\begin{aligned}& \big|k'_{2} (s)\big|\leq C_{3} , \end{aligned}$$

(2.9)

$$\begin{aligned}& C_{4} sk_{2} (s) \leq K_{2} (s) \leq C_{5} sk_{1} (s) \quad\hbox{for } s\in R, \end{aligned}$$

(2.10)

where

$$K_{2} (s) = \int_{0}^{s} k_{2} (r) \,dr. $$

$(H4)$ :

For the time-varying delay, we assume as in [22] that $\tau \in W^{2, \infty} ([0,T])$ for $T>0$ and there exist positive constants $\tau_{0}$, $\tau_{1}$ such that

$$ 0< \tau_{0} \leq\tau(t) \leq\tau_{1},\quad \forall t>0. $$

(2.11)

Moreover, we assume that there exists $d>0$ such that

$$ \tau' (t) \leq d < 1 \quad\hbox{for } t>0, $$

(2.12)

and that $\mu_{1}$, $\mu_{2}$ satisfy

$$ |\mu_{2} |< \frac{C_{4} (1-d)}{C_{5} (1-C_{4} d)}\mu_{1}. $$

(2.13)

$(H5)$ :

The functions $f, m, h >0$ are essentially bounded such that $f(x), m(x), h(x) >0$. Furthermore, there exist positive constants $f_{0}$, $m_{0}$ and $h_{0}$ such that

$$f(x) \geq f_{0} , \qquad m(x) \geq m_{0} ,\qquad h(x) \geq h_{0} \quad \hbox{for all a.e. } x\in\Gamma_{1} . $$

Remark 2.1

By the mean value theorem for integrals and the monotonicity of $k_{2}$, we find that

$$K_{2} (s) = \int_{0}^{s} k_{2} (r) \,dr \leq sk_{2} (s). $$

Then from (2.10), we obtain $C_{4} \leq1$.

For studying problem (1.1)-(1.7), we introduce a new variable z as in [22],

$$ z(x,\rho, t) =u_{t} \bigl(x, t-\rho\tau(t) \bigr), \quad x\in \Gamma_{1} , \rho\in(0,1), t>0. $$

(2.14)

Then problem (1.1)-(1.7) is equivalent to

$$\begin{aligned}& \begin{aligned}[b] &u_{tt} -\delta_{0} \Delta u + \int_{0}^{t} g(t-s) \operatorname{div}\bigl(a(x) \nabla u(s)\bigr) \,ds+\bigl(\delta_{1} +b(x) \big|u_{t} (t)\big|^{m-2} \bigr) u_{t} (t) \\ &\quad =|u|^{p-2} u \quad\hbox{in } \Omega\times(0,\infty), \end{aligned} \end{aligned}$$

(2.15)

$$\begin{aligned}& u=0 \quad \hbox{on } \Gamma_{0} \times(0, \infty), \end{aligned}$$

(2.16)

$$\begin{aligned}& \begin{aligned}[b]& u_{tt}+ \delta_{0} \frac{\partial u}{\partial\nu} - \int_{0}^{t} g(t-s) \bigl(a(x) \nabla u(s)\bigr) \cdot\nu \,ds +\mu_{1} k_{1} \bigl( u_{t} (x,t)\bigr) +\mu_{2} k_{2} \bigl( z (x,1,t)\bigr) \\ &\quad =h(x) y_{t} \quad \hbox{on } \Gamma_{1} \times(0, \infty), \end{aligned} \end{aligned}$$

(2.17)

$$\begin{aligned}& u_{t} +f(x) y_{t} +m(x) y=0\quad \hbox{on } \Gamma_{1} \times (0,\infty), \end{aligned}$$

(2.18)

$$\begin{aligned}& \tau(t) z_{t} (x,\rho, t) +\bigl(1-\rho\tau' (t) \bigr) z_{\rho}(x,\rho, t) =0 \quad\hbox{on } \Gamma_{1} \times(0,1)\times(0, \infty), \end{aligned}$$

(2.19)

$$\begin{aligned}& z(x,0,t)=u_{t} (x,t) \quad \hbox{on } \Gamma_{1} \times(0,\infty), \end{aligned}$$

(2.20)

$$\begin{aligned}& u(x,0) =u_{0} (x), \qquad u_{t} (x,0) =u_{1} (x)\quad \hbox{in } \Omega, \end{aligned}$$

(2.21)

$$\begin{aligned}& y(x,0)=y_{0} (x) \quad\hbox{on } \Gamma_{1} , \end{aligned}$$

(2.22)

$$\begin{aligned}& z_{t} (x, \rho, 0) =j_{0} \bigl(x, -\rho\tau(0)\bigr)\quad \hbox{on } \Gamma_{1} \times\bigl(0,\tau(0)\bigr). \end{aligned}$$

(2.23)

Now we are in a position to state the local existence result of problem (2.15)-(2.23) which can be established by combining with the argument of [15, 32].

Theorem 2.1

Assume that $(H1)$-$(H5)$ hold. Then given $(u_{0} , u_{1}) \in V \times L^{2} (\Omega)$, $y_{0} \in L^{2} (\Gamma_{1})$ and $j_{0} \in L^{2} (\Gamma_{1} \times(0,1))$, there exist $T>0$ and a unique weak solution $(u,y,z)$ of problem (2.15)-(2.23) such that

$$\begin{gathered} u\in C (0,T; V), \qquad u_{t} \in C \bigl(0,T; L^{2} ( \Omega)\bigr)\cap L^{2} \bigl(\Gamma_{1} \times(0,1)\bigr), \\ h^{1/2} y \in L^{2} \bigl(0,T; L^{2} ( \Gamma_{1} )\bigr),\qquad h^{1/2} y_{t} \in L^{2} \bigl(0,T; L^{2} (\Gamma_{1})\bigr).\end{gathered} $$

3 Global existence and asymptotic behavior

In order to study the global existence of solution for problem (2.15)-(2.23) given by Theorem 2.1, we define the functions

$$ \begin{aligned}[b]J(t) ={}&\frac{1}{2} \biggl(\delta_{0} -a(x) \int_{0}^{t} g(s) \,ds \biggr)\big\| \nabla u(t ) \big\| ^{2} +\frac{1}{2} (g\circ\nabla u) (t) \\ & -\frac{1}{p} \big\| u(t)\big\| _{p}^{p} + \frac{1}{2} \int_{\Gamma _{1}} m(x) h(x) y^{2} (t)\, d \Gamma + \frac{\xi\tau(t)}{2} \int_{\Gamma_{1}} \int_{0}^{1} K_{2} \bigl(z(x,\rho, t) \bigr)\,d \rho \,d\Gamma\end{aligned} $$

(3.1)

and

$$ \begin{aligned}[b] I(t) ={}& \biggl(\delta_{0} -a(x) \int_{0}^{t} g(s) \,ds \biggr)\big\| \nabla u(t ) \big\| ^{2} +(g\circ\nabla u) (t) \\ & - \big\| u(t)\big\| _{p}^{p} + \int_{\Gamma_{1}} m(x) h(x) y^{2} (t) \,d \Gamma +\xi\tau(t) \int_{\Gamma_{1}} \int_{0 }^{1} K_{2} \bigl(z(x,\rho, t) \bigr) \,d \rho \,d\Gamma ,\end{aligned} $$

(3.2)

where $(g\circ\nabla u)(t)= \int_{\Omega}\int_{0}^{t} g(t-s) |\nabla u(t)-\nabla u(s)|^{2} \,ds \,dx$. Adopting the proof of [33], we still have the following results.

Lemma 3.1

For any $u\in C^{1} (0,t;H^{1} (\Omega))$, we have

$$\begin{aligned}[b] & \int_{\Omega}a(x) \int_{0}^{t} g(t-s) \nabla u(s) \nabla u_{t} (t) \,ds \,dx \\ &\quad =-\frac{1}{2} \int_{\Omega}a(x) g(t) \big|\nabla u(t)\big|^{2} \,dx + \frac {1}{2} \bigl(g'\circ\nabla u\bigr) (t) \\ & \qquad{} -\frac{1}{2}\frac{d}{dt} \biggl[ (g\circ\nabla u) (t) - \int _{\Omega}a(x) \int_{0}^{t} g(s) \,ds \big|\nabla u(t)\big|^{2} \,dx \biggr].\end{aligned} $$

(3.3)

We define the modified energy functional $E(t)$ associated with problem (2.15)-(2.23) by

$$\begin{aligned} E(t) ={}& \frac{1}{2} \big\| u_{t} (t)\big\| ^{2} + \frac{1}{2} \biggl(\delta_{0} -a(x) \int_{0}^{t} g(s) \,ds \biggr)\big\| \nabla u(t ) \big\| ^{2} +\frac{1}{2} (g\circ \nabla u) (t) \\ & -\frac{1}{p} \big\| u(t)\big\| _{p}^{p} + \frac{1}{2} \int_{\Gamma _{1}} m(x) h(x) y^{2} (t)\, d \Gamma+ \frac{\xi\tau(t)}{2} \int_{\Gamma _{1}} \int_{0 }^{1} K_{2} \bigl(z(x,\rho, t) \bigr) \,d\rho \,d\Gamma \\ & +\frac{1}{2} \big\| u_{t} (t)\big\| _{\Gamma_{1}}^{2} \\ ={}&\frac{1}{2} \big\| u_{t} (t)\big\| ^{2} + J(t)+ \frac{1}{2} \big\| u_{t} (t)\big\| _{\Gamma_{1}}^{2} ,\end{aligned} $$

(3.4)

where ξ is a positive constant such that

$$ \frac{ |\mu_{2} |(1-C_{4} )}{C_{4} (1-d)} \leq\xi\leq\frac{\mu_{1} -|\mu _{2} |C_{5} }{C_{5}}. $$

(3.5)

Lemma 3.2

Let $(u, y,z)$ be the solution of (2.15)-(2.23). Then the energy functional defined by (3.4) is a non-increasing function and for all $t>0$, we have

$$ \begin{aligned}[b]\frac{d}{dt} E(t) \leq{}&{ -}\biggl(\mu_{1} -| \mu_{2}|C_{5} -\frac{\xi C_{5}}{2}\biggr) \int_{\Gamma_{1}} k_{1} \bigl(u_{t} (t) \bigr)u_{t} (t) \,d\Gamma \\ & - \biggl[ \frac{ \xi C_{4}}{2} \bigl(1-\tau' (t)\bigr)-| \mu_{2} |(1-C_{4} )\biggr] \int_{\Gamma_{1}} z(x,1,t) k_{2} \bigl(z(x,1,t)\bigr)\,d\Gamma \\ & - \int_{\Gamma_{1}} h(x) f(x) y_{t}^{2} (t) \,d\Gamma+ \frac{1}{2} \bigl(g' \circ\nabla u\bigr) (t) \\ & - \frac{a_{0} }{2} g(t) \big\| \nabla u(t)\big\| ^{2}-\delta_{1} \big\| u_{t} (t)\big\| ^{2} -b_{0} \big\| u_{t} (t)\big\| _{m}^{m} \\ \leq{}&0.\end{aligned} $$

(3.6)

Proof

Multiplying in (2.15) by $u_{t}$, integrating over Ω, using Green’s formula and exploiting the conditions (2.16) and (2.17), we have

$$ \begin{aligned}[b]&\frac{d}{dt} \biggl[ \frac{1}{2} \big\| u_{t} (t) \big\| ^{2} +\frac {1}{2}\delta_{0} \big\| \nabla u (t) \big\| ^{2}-\frac{1}{p}\big\| u(t)\big\| _{p}^{p} + \frac {1}{2} \big\| u_{t} (t)\big\| _{\Gamma_{1}}^{2} \biggr] - \int_{\Gamma_{1}} h(x) y_{t} (t) u_{t} (t) \,d\Gamma \\ & \quad =- \int_{\Omega}\int_{0}^{t} g(t-s) \bigl(a(x) \nabla u(s)\bigr) \nabla u_{t} (t) \,ds\,dx- \int_{\Omega}\bigl(\delta_{1} +b(x) \big|u_{t} (t)\big|^{m-2}\bigr) \big|u_{t} (t)\big|^{2} \,dx \\ & \qquad{} -\mu_{1} \int_{\Gamma_{1} } k_{1} \bigl(u_{t} (t)\bigr) u_{t} (t) \,d \Gamma-\mu_{2} \int_{\Gamma_{1}} k_{2} \bigl(z(x,1,t)\bigr) u_{t} (t) \,d\Gamma.\end{aligned} $$

(3.7)

On the other hand, from (2.18), we see that

$$ - \int_{\Gamma_{1}} h(x) y_{t} (t) u_{t} (t) \,d\Gamma= \int_{\Gamma_{1}} h(x) f(x) y_{t}^{2} (t) \,d\Gamma+ \int_{\Gamma_{1}} h(x) m(x) y(t) y_{t} (t) \,d\Gamma. $$

(3.8)

We also multiply the equation in (2.19) by $\xi k_{2} (z(x,\rho ,t))$ and integrate over $\Gamma_{1} \times(0,1)$ to obtain

$$ \begin{aligned}[b]& \frac{\xi\tau(t)}{2} \int_{\Gamma_{1}} \int_{0}^{1} z_{t} (x,\rho, t) k_{2} \bigl(z(x,\rho, t)\bigr) \,d\rho \,d\Gamma \\ &\quad =-\frac{\xi}{2} \int_{\Gamma_{1}} \int_{0}^{1} \bigl(1-\rho\tau' (t) \bigr)\frac{\partial}{\partial\rho} K_{2} \bigl(z(x, \rho, t)\bigr) \,d\rho \,d\Gamma,\end{aligned} $$

(3.9)

and it follows that

$$ \begin{aligned}[b]& \frac{d}{dt} \biggl(\frac{ \xi\tau(t)}{2} \int_{\Gamma_{1}} \int_{0}^{1} K_{2} \bigl(z(x,\rho, t) \bigr) \,d\rho \,d\Gamma \biggr) \\ & \quad = -\frac{\xi}{2} \int_{\Gamma_{1}} \int_{0}^{1} \frac{\partial }{\partial\rho} \bigl[\bigl(1-\rho \tau' (t)\bigr)K_{2} \bigl(z(x,\rho, t)\bigr) \bigr] \,d\rho \,d\Gamma \\ &\quad = \frac{\xi}{2} \int_{\Gamma_{1}} \bigl( K_{2} \bigl(z(x,0,t)\bigr)- K_{2} \bigl(z(x,1,t)\bigr) \bigr)\,d\Gamma +\frac{\xi}{2} \tau' (t) \int_{\Gamma_{1}} K_{2} \bigl(z(x,1,t)\bigr)\,d\Gamma.\end{aligned} $$

(3.10)

Thus from (3.7), (3.8), (3.10) and using (2.10) and (3.3), we deduce

$$ \begin{aligned}[b]\frac{d}{dt} E(t) \leq{}&{ -}\biggl(\mu_{1} - \frac{\xi C_{5}}{2} \biggr) \int _{\Gamma_{1}} k_{1} \bigl(u_{t} (t)\bigr) u_{t} (t) \,d\Gamma+|\mu_{2} | \int_{\Gamma_{1}} k_{2} \bigl(z(x,1,t)\bigr)u_{t} (t) \,d\Gamma \\ & -\frac{\xi}{2} \bigl(1-\tau' (t)\bigr) \int_{\Gamma_{1} } K_{2} \bigl(z(x,1,t)\bigr)\,d\Gamma- \int_{\Gamma_{1}} h(x) f(x) y_{t}^{2} (t) \,d\Gamma \\ & +\frac{1}{2} \bigl(g'\circ\nabla u\bigr) (t) - \frac{1}{2} g(t) \int_{\Omega}a(x) \big|\nabla u(t)\big|^{2} \,dx \\ & - \int_{\Omega}\bigl(\delta_{1} +b(x) \big|u_{t} (t)\big|^{m-2} \bigr) \big|u_{t} (t)\big|^{2} \,dx.\end{aligned} $$

(3.11)

Let us denote by $K_{2}^{*}$ the conjugate of the convex function $K_{2}$, i.e.,

$$K_{2}^{*} (s) =\sup_{t\in R^{+} } \bigl(st-K_{2} (t) \bigr), $$

then

$$ st\leq K_{2}^{*} (s)+K_{2} (t) ,\quad \forall s,t \geq0. $$

(3.12)

Moreover, $K_{2}$ is the Legendre transform of $K_{2}$ (thanks to the argument given in [34])

$$ K_{2}^{*} (s) =s\bigl(K_{2}' \bigr)^{-1} (s) -K_{2} \bigl( \bigl(K_{2}' \bigr)^{-1} (s) \bigr), \quad \forall s\geq0. $$

(3.13)

Then from the definition of $K_{2}$ and (3.13), we get

$$ K_{2}^{*} (s) =s(k_{2} )^{-1} (s) -K_{2} \bigl( (k_{2} )^{-1} (s) \bigr) ,\quad \forall s\geq0. $$

(3.14)

Let us recall the following relations (Eq. (3.7) in [35]) derived from (3.14):

$$\begin{aligned}& \begin{aligned}[b]& |\mu_{2} | \int_{\Gamma_{1}} k_{2} \bigl(z(x,1,t)\bigr)u_{t} (t) \,d\Gamma\\&\quad\leq|\mu_{2}| \int_{\Gamma_{1}} \bigl( k_{2} \bigl(z(x,1,t)\bigr) z(x,1,t)-K_{2} \bigl(z(x,1,t)\bigr)+K_{2} \bigl(u_{t} (t)\bigr) \bigr) \,d\Gamma,\end{aligned} \end{aligned}$$

(3.15)

$$\begin{aligned}& \begin{aligned}[b]K_{2}^{*} \bigl(k_{2} \bigl(z(x,1,t)\bigr) \bigr) &=z(x,1,t)k_{2} \bigl(z(x,1,t)\bigr)-K_{2} \bigl(z(x,1,t)\bigr) \\ & \leq(1-C_{4}) z(x,1,t)k_{2} \bigl(z(x,1,t)\bigr).\end{aligned} \end{aligned}$$

(3.16)

Using (3.11), (3.15) and (3.16), we obtain

$$\begin{aligned} \frac{d}{dt} E(t) \leq{}& {-}\biggl(\mu_{1} -\frac{\xi C_{5}}{2} \biggr) \int_{\Gamma_{1}} k_{1} \bigl(u_{t} (t)\bigr) u_{t} (t) \,d\Gamma\\ &+|\mu_{2} | \int_{\Gamma_{1}} \bigl[ K_{2} \bigl(u_{t} (t) \bigr)+K_{2}^{*} \bigl(k_{2} \bigl(z(x,1,t)\bigr)\bigr) \bigr]\,d \Gamma \\ & -\frac{\xi}{2} \bigl(1-\tau' (t)\bigr) \int_{\Gamma_{1}} K_{2} \bigl(z(x,1,t)\bigr) \,d\Gamma- \int_{\Gamma_{1}} h(x) f(x) y_{t}^{2} (t) \,d\Gamma+ \frac{1}{2} \bigl(g' \circ\nabla u\bigr) (t) \\ & -\frac{1}{2} g(t) \int_{\Omega} a(x) \big|\nabla u(t)\big|^{2} \,dx - \int _{\Omega} \bigl( \delta_{1} + b(x) \big|u_{t} (t)\big|^{m-2} \bigr) \big|u_{t} (t)\big|^{2} \,dx.\end{aligned} $$

Also using (2.10), (2.19), (3.16), this estimate becomes

$$\begin{aligned} \frac{d}{dt} E(t) \leq{}& {-}\biggl(\mu_{1} -|\mu_{2}| C_{5} -\frac{\xi C_{5}}{2} \biggr) \int_{\Gamma_{1}} k_{1} \bigl(u_{t} (t)\bigr) u_{t} (t) \,d\Gamma\\ &-\frac{\xi C_{4}}{2} \bigl(1-\tau' (t) \bigr) \int _{\Gamma_{1}} z(x,1,t) k_{2} (x,1,t) \,d\Gamma \\ & +|\mu_{2} |(1-C_{4}) \int_{\Gamma_{1}} z(x,1,t) k_{2} \bigl(z(x,1,t)\bigr) \,d\Gamma\\ &- \int_{\Gamma_{1}} h(x) f(x) y_{t}^{2} (t) \,d\Gamma+ \frac{1}{2} \bigl(g' \circ \nabla u\bigr) (t) \\ & -\frac{1}{2} g(t) \int_{\Omega} a(x)\big|\nabla u(t)\big|^{2} \,dx - \int _{\Omega} \bigl( \delta_{1} + b(x) |u_{t} (t)|^{m-2} \bigr) \big|u_{t} (t)\big|^{2} \,dx.\end{aligned} $$

Consequently, using (3.5), estimate (3.6) follows. Thus the proof of Lemma 3.2 is complete. □

Lemma 3.3

Let $(u, y, z)$ be the solution of (2.15)-(2.23). Assume that $I(0)>0$ and

$$ \gamma=C_{*}^{p} \biggl( \frac{2p}{l(p-2)} E(0) \biggr)^{(p-2) /2} < 1, $$

(3.17)

then $I(t)>0$ for all $t\geq0$.

Proof

Since $I(0)>0$, by continuity of $u(t)$ there exists $T_{*} < T$ such that $I(t)>0$ for all $t\in[0, T_{*} ]$. From (3.1), (3.2)

$$\begin{aligned} J(t) ={}&\frac{p-2}{2p} \biggl[ \biggl( \delta_{0}-a(x) \int_{0}^{t} g(s) \,ds \biggr)\big\| \nabla u(t) \big\| ^{2} +(g\circ\nabla u) (t) \\ & + \int_{\Gamma_{1} }m(x) h(x) y^{2} (t) \,d\Gamma+{\xi\tau (t)} \int_{\Gamma_{1}} \int_{0}^{1} K_{2} \bigl(z(x,\rho, t) \bigr) \,d\rho \,d\Gamma \biggr]+\frac{1}{p}I(t) \\ >{}& \frac{p-2}{2p} \biggl[ \biggl(\delta_{0}-a(x) \int_{0}^{t} g(s) \,ds \biggr) \big\| \nabla u(t) \big\| ^{2} +(g\circ\nabla u) (t) \\ & + \int_{\Gamma_{1} }h(x) m(x) y^{2} (t) \,d\Gamma+{\xi\tau (t)} \int_{\Gamma_{1}} \int_{0}^{1} K_{2} \bigl(z(x,\rho, t) \bigr) \,d\rho \,d\Gamma \biggr] .\end{aligned} $$

Hence from (2.3), (2.10) and the fact that $(g\circ \nabla u)(t)>0$, $\forall t \geq0$, we can deduce

$$ \begin{aligned}[b] & l\big\| \nabla u(t)\big\| ^{2} + \int_{\Gamma_{1} }h(x) m(x) y^{2} (t) \,d\Gamma +{\xi\tau(t)} \int_{\Gamma_{1}} \int_{0}^{1} K_{2} \bigl(z(x,\rho, t) \bigr) \,d\rho \,d\Gamma \\ &\quad \leq \biggl(\delta_{0}-a(x) \int_{0}^{t} g(s) \,ds \biggr)\big\| \nabla u(t)\big\| ^{2} + \int_{\Gamma_{1} }h(x) m(x) y^{2} (t) \,d\Gamma\\ &\qquad{}+{\xi\tau(t)} \int _{\Gamma_{1}} \int_{0}^{1} K_{2} \bigl(z(x,\rho, t) \bigr) \,d\rho \,d\Gamma \\ &\quad < \frac{2p}{p-2} J(t) ,\quad \forall t \in[0, T_{*} ),\end{aligned} $$

(3.18)

it follows that

$$ \big\| \nabla u(t)\big\| ^{2} \leq\frac{2p}{l(p-2)} J(t) \leq \frac{2p}{l(p-2)} E(t) \leq\frac{2p}{l(p-2)} E(0), \quad \forall t\in[0, T_{*}). $$

(3.19)

Thus from (2.1), (3.17) and (3.19), we arrive at

$$\begin{aligned} \big\| u(t)\big\| _{p}^{p} &\leq C_{*}^{p} \big\| \nabla u(t) \big\| ^{p} \leq C_{*}^{p} \big\| \nabla u(t)\big\| ^{p-2} \big\| \nabla u(t)\big\| ^{2} \\ & \leq C_{*}^{p} \biggl( \frac{2p}{l(p-2)} E(0) \biggr)^{\frac {p-2}{2}}\big\| \nabla u(t)\big\| ^{2} \leq\big\| \nabla u(t) \big\| ^{2} , \quad \forall t \in [0, T_{*} ).\end{aligned} $$

(3.20)

Hence $\|u(t)\|_{p}^{p} \leq C\|\nabla u(t)\|^{2}$, $\forall t\in[0, T_{*})$, which implies that $I(t)>0$, $\forall\in[0, T_{*})$. Note that

$$C_{*}^{p} \biggl( \frac{2p}{l(p-2)} E(T_{*}) \biggr)^{\frac{p-2}{p}} \leq C_{*}^{p} \biggl( \frac{2p}{l(p-2)} E(0) \biggr)^{\frac{p-2}{p}}< 1. $$

We repeat the procedure with $T_{*}$ extended to T. □

Theorem 3.1

Let $(u, y, z)$ be the solutions of problem (2.15)-(2.23). Suppose that (3.17) holds and $I(0)>0$, then the solution $(u,y,z)$ is a global time.

Proof

It suffices to show that

$$\big\| u_{t} (t)\big\| ^{2} +\big\| \nabla u(t)\big\| ^{2} +{\xi \tau(t)} \int_{\Gamma_{1}} \int_{0}^{1} K_{2} \bigl(z(x,\rho, t) \bigr) \,d\rho \,d\Gamma+ \int_{\Gamma_{1} } h(x) m(x) y^{2} (t)\, d \Gamma $$

is bounded independent of t. Under the hypotheses in Theorem 3.1, we see from Lemma 3.3 that $I(t)>0$ on $[0,T]$. Using Lemma 3.2, from (3.18) it follows that

$$\begin{gathered} \frac{1}{2}\big\| u_{t} (t)\big\| ^{2}+\frac{p-2}{2p} \biggl(l \big\| \nabla u (t)\big\| ^{2} +{\xi\tau(t)} \int_{\Gamma_{1}} \int_{0}^{1} K_{2} \bigl(z(x,\rho, t) \bigr) \,d\rho \,d\Gamma \\ \qquad{}+ \int_{\Gamma_{1} } h(x) m(x) y^{2} (t) \,d\Gamma \biggr) \\ \quad \leq\frac{1}{2} \big\| u_{t} (t)\big\| ^{2} +J(t)+ \frac{1}{2} \big\| u_{t} (t)\big\| _{\Gamma_{1}}^{2}=E(t)< E(0) .\end{gathered} $$

Thus, there exists a constant $C>0$ depending p and l such that

$$\begin{gathered} \big\| u_{t} (t)\big\| ^{2} +\big\| \nabla u(t)\big\| ^{2} +{\xi \tau(t)} \int_{\Gamma_{1}} \int_{0}^{1} K_{2} \bigl(z(x,\rho, t) \bigr) \,d\rho \,d\Gamma+ \int_{\Gamma_{1}} h(x) m(x) y^{2} (t) \,d\Gamma \\ \quad\leq CE(0)< + \infty. \end{gathered}$$

Thus the proof of Theorem 3.1 is finished. □

4 General energy decay rate

In this section, we shall investigate the asymptotic behavior of the energy function $E(t)$. For this purpose we construct a Lyapunov function $\mathcal {L} (t)$ equivalent to $E(t)$, which we can show to lead to the desired result. First, we define some functional and establish several lemmas. Let

$$ {\mathcal {L}}(t)=ME(t) +\varepsilon\Psi(t) +\Phi(t)+\varepsilon \Lambda(t), $$

(4.1)

where M and ε are positive constants to be chosen later and

$$\begin{aligned}& \begin{aligned}[b] \Psi(t)={}& \int_{\Omega}u_{t} (t) u(t) \,dx + \int_{\Gamma_{1} } h(x) u(t) y(t) \,d\Gamma+\frac{1}{2} \int_{\Gamma _{1} } h(x) f(x) y^{2} (t) \,d\Gamma \\ & + \int_{\Gamma_{1}} u(t) u_{t} (t) \,d\Gamma, \end{aligned} \end{aligned}$$

(4.2)

$$\begin{aligned}& \begin{aligned}[b]\Phi(t) ={}&{-} \int_{\Omega}a(x) u_{t} (t) \int_{0}^{t} g(t-s) \bigl(u(t)-u(s)\bigr) \,ds \,dx \\ & - \int_{\Gamma_{1}} a(x) u_{t} (t) \int_{0}^{t} g(t-s) \bigl( u(t)- u(s)\bigr) \,ds\,d \Gamma,\end{aligned} \end{aligned}$$

(4.3)

and

$$ \Lambda(t) =\tau(t) \int_{\Gamma_{1}} \int_{0}^{1} e^{-\rho\tau(t)} K_{2} \bigl(z(x,\rho,t)\bigr) \,d\rho \,d\Gamma. $$

(4.4)

The functional $\mathcal {L} (t)$ is equivalent to the energy function $E(t)$ by the following lemma.

Lemma 4.1

For $\varepsilon>0$ small enough while M is large enough, there exist two positive constants $\beta_{1}$ and $\beta_{2}$ such that

$$ \beta_{1} E(t) \leq\mathcal{L} (t)\leq \beta_{2} E(t), \quad\forall t \geq0. $$

(4.5)

Proof

From Hölder’s and Young’s inequality, (2.1), (2.2) and (4.2)-(4.4) we have

$$\begin{aligned}& \begin{aligned}[b] \big|\Psi(t)\big| \leq{}& \bigg| \int_{\Omega}u(t) u_{t} (t) \,dx \bigg| + \bigg| \int_{\Gamma_{1}} h(x) u(t) y(t) \,d\Gamma \bigg|+\frac{1}{2} \bigg| \int_{\Gamma_{1}} h(x) f(x) y^{2} (t) \,d\Gamma \bigg| \\ & + \bigg| \int_{\Gamma_{1}} u(t) u_{t} (t) \,d\Gamma \bigg| \\ \leq{}&\frac{1}{2} \big\| u_{t} (t)\big\| ^{2} + \frac{C_{*}^{2} }{2} \big\| \nabla u(t)\big\| ^{2} \\ &+\frac{\|h\|_{\infty}^{1/2} \|m\|_{\infty}^{1/2}}{m_{0}} \biggl( \int_{\Gamma_{1}} h(x) m(x) y^{2} (t) \,d\Gamma \biggr)^{1/2} \biggl( \int_{\Gamma_{1}} \big|u(t)\big|^{2} \,d\Gamma \biggr)^{1/2} \\ & +\frac{1}{2} \int_{\Gamma_{1}} h(x) f(x) y^{2} (y) \,d\Gamma + \frac{1}{2} \big\| u_{t} (t)\big\| _{\Gamma_{1}}^{2} + \frac{{\tilde{C}}_{*}^{2} }{2} \big\| \nabla u(t)\big\| ^{2} \\ \leq{}&\frac{1}{2} \big\| u_{t} (t)\big\| ^{2} + \frac{C_{*}^{2} }{2} \big\| \nabla u(t)\big\| ^{2} +\frac{\|h\|_{\infty} \|m\|_{\infty}}{2 m^{2}_{0}} \int_{\Gamma_{1}} h(x) m(x) y^{2} (t) \,d\Gamma \\ & + \frac{{\tilde{C}}_{*}^{2}}{2} \big\| \nabla u(t)\big\| ^{2} +\frac{\|f\| _{\infty}}{2m_{0}} \int_{\Gamma_{1}} h(x) m(x) y^{2} (t) \,d\Gamma+ \frac {1}{2} \big\| u_{t} (t)\big\| _{\Gamma_{1}}^{2} + \frac{{\tilde{C}}_{*}^{2}}{2} \big\| \nabla u(t)\big\| ^{2} \\ ={}&\frac{1}{2} \big\| u_{t} (t)\big\| ^{2} + \biggl( \frac{ C_{*}^{2}}{2}+\frac {{\tilde{C}}_{*}^{2}}{2} \biggr) \big\| \nabla u(t)\big\| ^{2}\\ & + \biggl(\frac{\|f\| _{\infty}}{2m_{0}} +\frac{\|h\|_{\infty} \|m\|_{\infty}}{ 2 m_{0}^{2}} \biggr) \int_{\Gamma_{1}} h(x) m(x) y^{2} (t) \,d\Gamma +\frac{1}{2} \big\| u_{t} (t)\big\| _{\Gamma_{1}}^{2},\end{aligned} \end{aligned}$$

(4.6)

$$\begin{aligned}& \begin{aligned}[b]\big|\Phi(t)\big| \leq{}& \bigg| - \int_{\Omega}a(x) u_{t} (t) \int_{0}^{t} g(t-s) \bigl(u(t)-u(s)\bigr) \,ds \,dx \bigg| \\ & + \bigg|- \int_{\Gamma_{1}} a(x) u_{t} (t) \int_{0}^{t} g(t-s) \bigl( u(t)- u(s)\bigr) \,ds\,d \Gamma \bigg| \\ \leq{}&\frac{1}{2}\big\| u_{t} (t)\big\| ^{2} \\ &+ \frac{1}{2} \int_{\Omega}\biggl( a(x) u_{t} (t) \int_{0}^{t} g(t-s) \bigl(u(t)-u(s)\bigr) \,ds \biggr)^{2} \,dx \\ & +\frac{1}{2}\big\| u_{t} (t)\big\| _{\Gamma_{1}}^{2} +\frac{1}{2} \int _{\Gamma_{1}} \biggl( a(x) u_{t} (t) \int_{0}^{t} g(t-s) \bigl( u(t)- u(s)\bigr) \,ds \biggr)^{2} \,d\Gamma \\ \leq{}&\frac{1}{2} \big\| u_{t} (t)\big\| ^{2}\\ & + \frac{\|a\|_{\infty}}{2} \int_{0}^{t} g(s) \,ds \int_{\Omega}\int_{0}^{t} g(t-s) \big|u(t)-u(s)\big|^{2} \,ds \,dx +\frac{1}{2} \big\| u_{t} (t)\big\| _{\Gamma_{1}}^{2} \\ & +\frac{\|a\|_{\infty}}{2} \int_{0}^{t} g(s) \,ds \int_{\Gamma _{1}} \int_{0}^{t} g(t-s) \big|u(t)-u(s)\big|^{2} \,ds \,d \Gamma \\ \leq{}&\frac{1}{2} \big\| u_{t} (t)\big\| ^{2} + \frac{1}{2}\big\| u_{t} (t)\big\| _{\Gamma _{1}}^{2} + \frac{(\delta_{0} -l)}{2} \bigl(C_{*}^{2} +{\tilde{C}}_{*}^{2}\bigr) (g \circ \nabla u) (t),\end{aligned} \end{aligned}$$

(4.7)

where we used

$$\delta_{0} -\|a\|_{\infty} \int_{0}^{\infty}g(s) \,ds =l>0 $$

and

$$\begin{aligned}[b] \big|\Lambda(t)\big|& = \bigg| \tau(t) \int_{\Gamma_{1}} \int_{0}^{1} e^{-\rho \tau(t)} K_{2} \bigl(z(x,\rho,t)\bigr) \,d\rho \,d\Gamma \bigg| \\ & \leq C\tau(t) \int_{\Gamma_{1}} \int_{0}^{1} K_{2} \bigl(z(x,\rho,t)\bigr) \,d\rho \,d\Gamma,\end{aligned} $$

(4.8)

where C is a positive constant. Combining (4.1) and (4.6)-(4.8), then we arrive at

$$\begin{aligned} \big| \mathcal {L}(t) -ME(t) \big| \leq{}&\frac{1}{2} (\varepsilon+1) \big\| u_{t} (t)\big\| ^{2} +\frac {\varepsilon}{2}\bigl(C_{*}^{2} +{\tilde{C}}_{*}^{2}\bigr)\big\| \nabla u(t)\big\| ^{2} \\ & +\frac{(\delta_{0} -l)}{2}\bigl(C_{*}^{2} +{\tilde{C}}_{*}^{2} \bigr) (g\circ \nabla u) (t) \\ & + \frac{\varepsilon}{2m_{0}} \biggl( \|f\|_{\infty} + \frac {\|h\|_{\infty} \|m\|_{\infty}}{ m_{0}} \biggr) \int_{\Gamma_{1}} h(x) m(x) y^{2} (t) \,d\Gamma \\ & +\frac{1}{2} (\varepsilon+1) \big\| u_{t} (t) \big\| _{\Gamma_{1}}^{2} +\varepsilon C \tau(t) \int_{\Gamma_{1}} \int_{0}^{1} K_{2} \bigl(z(x,\rho,t)\bigr) \,d\rho \,d\Gamma \\ \leq{}& C E(t),\end{aligned} $$

where C is a positive constant. Choosing $M>0$ large, we complete the proof of Lemma 4.1. □

Lemma 4.2

Let $(u,y,z)$ be the solution of (2.15)-(2.23). Then the functional Ψ defined in (4.2) satisfies

$$\begin{aligned}[b] \frac{d}{dt}\Psi(t) \leq{}& \biggl( 1+ \frac{\delta_{1}}{4 \eta } \biggr) \big\| u_{t} (t)\big\| ^{2} \\ &- \biggl[ \frac{\delta_{0}}{2} - \delta_{1} C_{*}^{2} \eta-\eta\tilde {C}_{*}^{2} \bigl(1+ \mu_{1} +|\mu_{2}|\bigr) -\frac{1}{2\delta_{0}} (1+\eta) ( \delta_{0} -l)^{2} \\ & -\frac{\|b\|_{\infty}\eta^{ - m} C_{*}^{m}}{m} \biggl( \frac {2p}{l(p-2)} E(0) \biggr)^{\frac{m-2}{2}} \biggr]\big\| \nabla u(t)\big\| ^{2} \\ & +\frac{1}{2\delta_{0}} \biggl( 1+\frac{1}{\eta} \biggr) ( \delta_{0} -l) (g\circ\nabla u) (t) +\big\| u(t)\big\| _{p}^{p} +\big\| u_{t} (t) \big\| _{\Gamma_{1}}^{2} \\ & + \frac{\|b\|_{\infty} (m-1)}{m} \eta^{\frac{m}{m-1}} \big\| u_{t} (t) \big\| _{m}^{m} - \int_{\Gamma_{1}} h(x) m(x) y^{2} (t) \,d\Gamma\\ &+ \frac{\|h\| _{\infty}\|f\|_{\infty}}{\eta f_{0}^{2}} \int_{\Gamma_{1}} h(x) f(x) y_{t}^{2} (t) \,d\Gamma \\ & +\frac{\mu_{1} }{4\eta} \int_{\Gamma_{1} } k_{1}^{2} \bigl(u_{t} (t)\bigr) \,d\Gamma+\frac{|\mu_{2}|}{4\eta} \int_{\Gamma_{1}} k_{2}^{2} \bigl(z(x,1,t)\bigr)\,d \Gamma. \end{aligned} $$

(4.9)

Proof

Taking the derivatives of $\Psi(t)$ defined in (4.2) and using (2.15)-(2.18) we have

$$\begin{aligned} \frac{d}{dt} \Psi(t) =& \int_{\Omega}u(t) u_{tt}(t) \,dx+ \int_{\Gamma_{1}} u(t) u_{tt}(t) \,d\Gamma+ \int_{\Omega}\big|u_{t} (t)\big|^{2} \,dx + \int_{\Gamma_{1}} \big|u_{t} (t)\big|^{2} \,d\Gamma \\ & {} + \int_{\Gamma_{1}} h(x) u_{t} (t) y(t) \,d\Gamma+ \int_{\Gamma _{1}} h(x) u(t) y_{t} (t) \,d\Gamma+ \int_{\Gamma_{1}} h(x) f(x) y(t) y_{t} (t) \,d\Gamma \\ =&-\delta_{0} \int_{\Omega}\big|\nabla u(t)\big|^{2} \,dx + \int_{\Omega}\nabla u(t) \int_{0}^{t} g(t-s) \bigl(a(x) \nabla u(s)\bigr) \,ds\,dx \\ &{} -\delta_{1} \int_{\Omega}u(t) u_{t} (t) \,dx - \int_{\Omega}b(x) \big|u_{t} (t)\big|^{m-2} u_{t} (t) u(t) \,dx \\ & {} + \int_{\Omega}\big|u(t)\big|^{p} \,dx + \int_{\Omega}\big|u_{t} (t)\big|^{2} \,dx + \int _{\Gamma_{1}} \big|u_{t} (t)\big|^{2} \,d\Gamma \\ & {} + 2 \int_{\Gamma_{1}} h(x) u(t) y_{t} (t) \,d\Gamma- \int _{\Gamma_{1}} h(x) m(x) y^{2}(t) \,d\Gamma \\ & {} -\mu_{1} \int_{\Gamma_{1}} k_{1} \bigl(u_{t} (t)\bigr) u(t) \,d\Gamma-\mu_{2} \int_{\Gamma_{1}} k_{2} \bigl(z(x,1,t)\bigr) u(t) \,d \Gamma. \end{aligned}$$

(4.10)

Now, by using Hölder’s and Young’s inequality, $(H1)$, (2.1) and (2.2) we estimate the right hand side of (2.10) as follows, for any $\eta>0$:

$$\begin{aligned}& \begin{aligned}[b] & \bigg| \int_{\Omega}\nabla u(t) \int_{0}^{t} g(t-s) \bigl(a(x) \nabla u(s)\bigr) \,ds \,dx \bigg| \\ & \quad \leq \biggl[ \frac{\delta_{0} }{2} +\frac{1}{2\delta_{0}}(1+\eta) ( \delta_{0} -l )^{2} \biggr] \big\| \nabla u(t)\big\| ^{2} \\ &\qquad{} + \frac{1}{2\delta_{0}} \biggl(1+\frac{1}{\eta}\biggr) ( \delta_{0} -l) (g\circ\nabla u) (t), \end{aligned} \end{aligned}$$

(4.11)

$$\begin{aligned}& \bigg|-\delta_{1} \int_{\Omega}u (t) u_{t} (t) \,dx \bigg| \leq \delta_{1} \eta C_{*}^{2} \big\| \nabla u(t)\big\| ^{2} + \frac{\delta _{1}}{4\eta} \big\| u_{t} (t)\big\| ^{2} , \end{aligned}$$

(4.12)

$$\begin{aligned}& \begin{aligned}[b]& \bigg|- \int_{\Omega}b(x) \big|u_{t} (t)\big|^{m-2} u_{t} (t) u(t) \,dx \bigg| \\ & \quad \leq\frac{\|b\|_{\infty}\eta^{-m}}{m} C_{*}^{m} \biggl( \frac {2p}{l(p-2)} E(0) \biggr)^{\frac{m-2}{2}}\big\| u(t)\big\| _{m}^{m}+\|b\|_{\infty} \frac{m-1}{m} \eta^{m/{m-1}}\big\| u_{t} (t)\big\| _{m}^{m} , \end{aligned} \end{aligned}$$

(4.13)

$$\begin{aligned}& \begin{aligned}[b] 2 \bigg| \int_{\Gamma_{1}} h(x) u(t) y_{t} (t) \,d\Gamma \bigg| &=2 \bigg| \int_{\Gamma_{1}} \frac{h(x) f(x) u(t) y_{t} (t)}{f(x)} \,d\Gamma \bigg| \\ & \leq\eta\tilde{C}_{*}^{2} \big\| \nabla u(t)\big\| ^{2} + \frac{\|h\|_{\infty}\|f\|_{\infty}}{\eta f^{2}_{0}} \int_{\Gamma_{1}} h(x) f(x) y_{t}^{2} (t) \,d\Gamma,\end{aligned} \end{aligned}$$

(4.14)

$$\begin{aligned}& \bigg| \mu_{1} \int_{\Gamma_{1}} k_{1}\bigl(u_{t} (x,t)\bigr) u(t) \,d\Gamma \bigg| \leq\mu_{1} \eta\tilde{C}_{*}^{2} \big\| \nabla u(t) \big\| ^{2} +\frac{\mu _{1}}{4\eta} \int_{\Gamma_{1}} k_{1}^{2} \bigl(u_{t} (t)\bigr) \,d\Gamma, \end{aligned}$$

(4.15)

and

$$ \bigg|\mu_{2} \int_{\Gamma_{1}} k_{2} \bigl(z(x,1,t)\bigr) u(t) \,d\Gamma \bigg| \leq|\mu_{2} |\eta\tilde{C}_{*}^{2} \big\| \nabla u(t) \big\| ^{2} +\frac{|\mu_{2} |}{4\eta} \int_{\Gamma_{1}} k_{2}^{2} \bigl(z(x,1,t)\bigr) \,d \Gamma. $$

(4.16)

Thus from (4.10)-(4.16) we conclude that

$$\begin{aligned} \frac{d}{dt}\Psi(t) \leq &\biggl( 1+\frac{\delta_{1}}{4\eta} \biggr) \big\| u_{t} (t)\big\| ^{2}\\ &{}- \biggl[ \frac{\delta_{0}}{2}- \delta_{1} C_{*}^{2} \eta-\eta\tilde{C}_{*}^{2} \bigl(1+\mu _{1} +|\mu_{2}|\bigr) -\frac{1}{2\delta_{0}}(1+\eta) ( \delta_{0} -l)^{2} \\ & {} -\frac{\|b\|_{\infty}\eta^{-m} C_{*}^{m}}{m} \biggl( \frac {2p}{l(p-2)} E(0) \biggr)^{\frac{m-2}{2}} \biggr] \big\| \nabla u(t)\big\| ^{2} \\ &{} +\frac{1}{2\delta_{0}} \biggl(1+\frac{1}{\eta}\biggr) ( \delta_{0} -l ) (g\circ\nabla u) (t) +\big\| u(t)\big\| _{p}^{p} +\big\| u_{t} (t) \big\| _{\Gamma}^{2}\\ &{} +\frac{\| b\|_{\infty}(m-1) \eta^{m/m-1}}{m} \big\| u_{t} (t)\big\| _{m}^{m} \\ & {} - \int_{\Gamma_{1}} h(x) m(x) y^{2} (t) \,d\Gamma+ \frac{\|h\| _{\infty}\|f\|_{\infty}}{\eta f_{0}^{2} } \int_{\Gamma_{1}} h(x) f(x) y_{t}^{2} (t) \,d\Gamma \\ & {} + \frac{\mu_{1}}{4\eta} \int_{\Gamma_{1}} k_{1}^{2} \bigl(u_{t} (t)\bigr) \,d\Gamma+\frac{|\mu_{2}|}{4\eta} \int_{\Gamma_{1}} k_{2}^{2} \bigl(z(x,1,t)\bigr) \,d \Gamma. \end{aligned}$$

Thus we finished the proof of Lemma 4.2. □

Lemma 4.3

Let $(u,y,z)$ be the solution of problem (2.15)-(2.23). Then the functional $\Phi(t)$ defined in (4.3) satisfies

$$\begin{aligned}[b] \frac{d}{dt} \Phi(t) \leq{}&{-} \biggl( a_{0} \int_{0}^{t} g(s) \,ds -\delta_{1} \eta-\eta \biggr) \big\| u_{t} (t)\big\| ^{2} \\ & +\eta \biggl[ \delta_{0}^{2} + \delta_{0}^{2} \alpha_{1}^{2} +2\alpha _{1}^{2} (\delta_{0} -l)^{2} +2( \delta_{0} -l)^{2} +C_{*}^{2(p-1)} \biggl( \frac {2p E(0)}{l(p-2)} \biggr)^{p-2} \biggr] \big\| \nabla u(t)\big\| ^{2} \\ & + \biggl[ \frac{\delta_{0}-l}{4\eta} \bigl(1+2C_{*}^{2} +8\eta+\|a\| _{\infty}+2\|a\|_{\infty}C_{*}^{2} \\ &+\mu_{1} \|a \|_{\infty}\tilde{C}_{*}^{2} +|\mu _{2}| \|a \|_{\infty}\tilde{C}_{*}^{2} +\|a\|_{\infty}\tilde{C}_{*}^{2} \bigr) \\ & +\frac{2\eta^{m} \|b\|_{\infty}}{m} (\delta_{0} -l)^{m-1} C_{*}^{m} \biggl( \frac{2p}{l(p-2)}E(0) \biggr)^{\frac{m-2}{2}} \biggr] (g\circ \nabla u) (t) \\ & -\frac{g(0)\|a\|_{\infty}}{4\eta} \bigl(C_{*}^{2} +\tilde{C}_{*}^{2} \bigr) \bigl(g'\circ\nabla u\bigr) (t) - \biggl( a_{0} \int_{0}^{t} g(s) \,ds -\eta \biggr) \big\| u_{t} (t)\big\| _{\Gamma_{1}}^{2} \\ & +\mu_{1} \eta \int_{\Gamma_{1}} k_{1}^{2} \bigl(u_{t} (t)\bigr)\,d\Gamma+|\mu _{2}| \eta \int_{\Gamma_{1}} k_{2}^{2} \bigl(z(x,1,t)\bigr)\,d \Gamma \\ &+\frac{\eta\|h\|_{\infty}}{f_{0}} \int_{\Gamma_{1}} h(x) f(x) y_{t}^{2} (t) \,d\Gamma.\end{aligned} $$

(4.17)

Proof

Taking the derivative of $\Phi(t)$ defined in (4.3) and using (2.15)-(2.18), we have

$$\begin{aligned} \frac{d}{dt} \Phi(t) =& - \int_{\Omega}a(x) u_{tt}(t) \int_{0}^{t} g(t-s) \bigl(u(t) -u(s)\bigr) \,ds \,dx \\ & {}- \int_{\Omega}a(x) u_{t} (t) \int_{0}^{t} g'(t-s) \bigl(u(t)-u(s) \bigr) \,ds \,dx \\ &{}- \biggl( \int_{0}^{t} g(s) \,ds \biggr) \int_{\Omega}a(x) u_{t}^{2} (t) \,dx \\ & {} - \int_{\Gamma_{1}} a(x) u_{tt}(t) \int_{0}^{t} g(t-s) \bigl(u(t)-u(s)\bigr) \,ds \,dx \end{aligned}$$

(4.18)

$$\begin{aligned} & {}- \int_{\Gamma_{1}} a(x) u_{t} (t) \int_{0}^{t} g'(t-s) \bigl(u(t)-u(s) \bigr) \,ds \,dx \\ &{} - \biggl( \int_{0}^{t} g(s) \,ds \biggr) \int_{\Gamma_{1}} a(x) u_{t}^{2} (t) \,d\Gamma \\ =& \int_{\Omega}\delta_{0} a(x) \nabla u(t) \int_{0}^{t} g(t-s) \bigl(\nabla u(t)-\nabla u(s) \bigr) \,ds \,dx \\ &{} + \int_{\Omega}\delta_{0} \nabla u(t)\cdot\nabla a(x) \int _{0}^{t} g(t-s) \bigl(u(t)-u(s)\bigr) \,ds \,dx \\ & {} - \int_{\Omega}\biggl( \int_{0}^{t} g(t-s) a(x) \nabla u(s)\cdot \nabla a(x) \,ds \biggr) \biggl( \int_{0}^{t} g(t-s) \bigl(u(t)-u(s)\bigr) \,ds \biggr) \,dx \\ &{} - \int_{\Omega}a(x) \biggl( \int_{0}^{t} g(t-s) a(x) \nabla u(s) \,ds \biggr) \biggl( \int_{0}^{t} g(t-s) \bigl(\nabla u(t)-\nabla u(s) \bigr) \,ds \biggr) \,dx \\ & {} + \int_{\Omega}a(x) \bigl(\delta_{1} +b(x) \big|u_{t} (t)\big|^{m-2} \bigr)u_{t} (t) \biggl( \int_{0}^{t} g(t-s) \bigl(u(t)-u(s)\bigr)\,ds \biggr) \,dx \\ & {} - \int_{\Omega}a(x) \big|u(t)\big|^{p-2} u(t) \biggl( \int_{0}^{t} g(t-s) \bigl(u(t)-u(s)\bigr)\,ds \biggr) \,dx \\ &{} - \int_{\Omega}a(x) u_{t} (t) \int_{0}^{t} g'(t-s) \bigl(u(t) -u(s) \bigr) \,ds \,dx \\ &{}- \biggl( \int_{0}^{t} g(s) \,ds \biggr) \int_{\Omega}a(x) u_{t}^{2} (t) \,dx \\ &{} + \int_{\Gamma_{1}} a(x) \bigl[ \mu_{1} k_{1} \bigl(u_{t} (t)\bigr)+\mu_{2} k_{2} \bigl(z(x,1,t) \bigr)-h(x) y_{t} (t) \bigr] \\ &{}\times \biggl( \int_{0}^{t} g(t-s) \bigl(u(t)-u(s)\bigr) \,ds \biggr) \,d\Gamma \\ & {} - \int_{\Gamma_{1}} a(x) u_{t} (t) \biggl( \int_{0}^{t} g'(t-s) \bigl(u(t)-u(s) \bigr) \,ds \biggr) \\ &{}- \biggl( \int_{0}^{t} g(s) \,ds \biggr) \int_{\Gamma _{1}} a(x) u_{t}^{2} (t) \,d \Gamma . \end{aligned}$$

(4.19)

Similarly to (4.9), we estimate each terms in the right hand side of (4.19). Using Hölder’s and Young’s inequality, $(H1)$, $(H2)$, (2.1), (2.2), (2.3), (2.5), (2.6) and (3.19), for any $\eta>0$, we have

$$\begin{aligned}& \bigg| \int_{\Omega}\delta_{0} a(x) \nabla u(t) \int_{0}^{t} g(t-s) \bigl(\nabla u(t)-\nabla u(s) \bigr) \,ds \,dx \bigg| \\& \quad \leq\delta_{0}^{2} \eta\big\| \nabla u(t)\big\| ^{2} + \frac{1}{4\eta} \int _{\Omega}\biggl( a(x) \int_{0}^{t} g(t-s) \bigl(\nabla u(t)-\nabla u(s) \bigr) \,ds \biggr)^{2} \,dx \\& \quad\leq\delta_{0}^{2} \eta\big\| \nabla u(t)\big\| ^{2} + \frac{\delta_{0} -l}{4\eta } (g\circ\nabla u) (t), \end{aligned}$$

(4.20)

$$\begin{aligned}& \bigg| \int_{\Omega}\delta_{0} \nabla u(t) \cdot\nabla a(x) \int _{0}^{t} g(t-s) \bigl(u(t)-u(s)\bigr) \,ds \,dx\bigg| \\& \quad \leq\delta_{0} \alpha_{1} \int_{\Omega}\big|\nabla u(t)\big| \sqrt{a(x)} \biggl( \int_{0}^{t} g(s) \,ds \biggr)^{1/2} \biggl( \int_{0}^{t} g(t-s) \bigl(u(t)-u(s) \bigr)^{2} \,ds \biggr)^{1/2} \,dx \\& \quad \leq\delta_{0}^{2} \alpha_{1}^{2} \eta\big\| \nabla u(t)\big\| ^{2} +\frac{(\delta _{0}-l)C_{*}^{2}}{4\eta} (g\circ\nabla u) (t) , \end{aligned}$$

(4.21)

$$\begin{aligned}& \bigg|- \int_{\Omega}\biggl( \int_{0}^{t} g(t-s) a(x) \nabla u(s) \cdot\nabla a(x) \,ds \biggr) \biggl( \int_{0}^{t} g(t-s) \bigl(u(t)-u(s)\bigr) \,ds \biggr) \,dx \bigg| \\& \quad \leq\alpha_{1}^{2} \eta \int_{\Omega}a^{2} (x) \biggl( \int_{0}^{t} g(t-s) \big|\nabla u(s)\big|\,ds \biggr)^{2} \,dx \\& \qquad{}+\frac{1}{4\eta} \int_{\Omega}a(x) \biggl( \int_{0}^{t} g(t-s) \bigl(u(t)-u(s)\bigr) \,ds \biggr)^{2} \,dx \\& \quad \leq2\alpha_{1}^{2} \eta(\delta_{0} -l)^{2} \big\| \nabla u(t)\big\| ^{2} + \biggl(2\alpha_{1}^{2} \eta(\delta_{0} -l)+\frac{(\delta_{0} -l)C_{*}^{2}}{4\eta } \biggr) (g\circ\nabla u) (t), \end{aligned}$$

(4.22)

$$\begin{aligned}& \bigg|- \int_{\Omega}a(x) \biggl( \int_{0}^{t} g(t-s) a(x) \nabla u(s) \,ds \biggr) \biggl( \int_{0}^{t} g(t-s) \bigl(\nabla u(t)-\nabla u(s) \bigr) \,ds \biggr)\,dx \bigg| \\& \quad\leq\eta \int_{\Omega}a^{2} (x) \biggl( \int_{0}^{t} g(t-s) \big|\nabla u(s)\big| \,ds \biggr)^{2} \,dx \\& \qquad{}+\frac{1}{4\eta} \int_{\Omega}a^{2} (x) \biggl( \int_{0}^{t} g(t-s) \bigl(\nabla u(t)-\nabla u(s) \bigr) \,ds \biggr)^{2} \,dx \\& \quad\leq 2\eta(\delta_{0} -l)^{2}\big\| \nabla u(t) \big\| ^{2} + \biggl( 2\eta+\frac {\|a\|_{\infty}}{4\eta} \biggr) (\delta_{0} -l) (g\circ\nabla u) (t), \end{aligned}$$

(4.23)

$$\begin{aligned}& \bigg| \int_{\Omega}a(x) \bigl(\delta_{1} +b(x) \big|u_{t} (t)\big|^{m-2} \bigr) u_{t} (t) \biggl( \int_{0}^{t} g(t-s) \bigl(u(t)-u(s)\bigr) \,ds \biggr) \,dx \bigg| \\& \quad \leq\delta_{1} \bigg| \int_{\Omega}a(x) u_{t} (t) \int_{0}^{t} g(t-s) \bigl(u(t)-u(s)\bigr) \,ds \,dx\bigg| \\& \qquad{} +\bigg| \int_{\Omega}a(x) b(x) \big|u_{t} (t)\big|^{m-1} \int_{0}^{t} g(t-s) \bigl(u(t)-u(s)\bigr) \,ds \,dx \bigg| \\& \quad \leq\delta_{1} \eta\big\| u_{t} (t)\big\| ^{2} + \frac{(\delta_{0} -l)\|a\| _{\infty}C_{*}^{2} }{4\eta} (g\circ\nabla u) (t) +\frac{m-1}{m} \|b\| _{\infty}\eta^{-\frac{m}{m-1}}\big\| u_{t} (t)\big\| _{m}^{m} \\& \qquad{}+\frac{\eta^{m}}{m}\|b\|_{\infty}(\delta_{0} -l)^{m-1} C_{*}^{m} \int_{\Omega}\int_{0}^{t} g(t-s) \big|\nabla u(t)-\nabla u(s)\big|^{2} \big|\nabla u(t)-\nabla u(s)\big|^{m-2} \,ds \,dx \\& \quad=\delta_{1} \eta\big\| u_{t} (t)\big\| ^{2} + \frac{m-1}{m} \|b\|_{\infty}\eta ^{-\frac{m}{m-1}}\big\| u_{t} (t) \big\| _{m}^{m} + \biggl[ \frac{(\delta_{0} -l)\|a\| _{\infty}C_{*}^{2}}{4\eta} \\& \qquad{} +\frac{2\eta^{m}}{m}\|b\|_{\infty}(\delta_{0} -l)^{m-1} C_{*}^{m} \biggl( \frac{2p}{l(p-2)} E(0) \biggr)^{\frac{m-2}{2}} \biggr] (g\circ \nabla u) (t) , \end{aligned}$$

(4.24)

$$\begin{aligned}& \bigg|- \int_{\Omega}a(x) \big|u(t)\big|^{p-2} u(t) \int_{0}^{t} g(t-s) \bigl(u(t)-u(s)\bigr) \,ds \,dx\bigg| \\& \quad \leq\eta\big\| u(t)\big\| _{2(p-1)}^{2(p-1)}+\frac{(\delta_{0} -l)\|a\| _{\infty}C_{*}^{2} }{4\eta} (g\circ \nabla u) (t) \\& \quad \leq\eta C_{*}^{2(p-1)} \biggl( \frac{2pE(0)}{l(p-2)} \biggr)^{p-2} \big\| \nabla u(t)\big\| ^{2} +\frac{(\delta_{0} -l)\|a\|_{\infty}C_{*}^{2} }{4\eta} (g\circ\nabla u) (t), \end{aligned}$$

(4.25)

$$\begin{aligned}& \bigg|- \int_{\Omega}a(x) u_{t} (t) \int_{0}^{t} g'(t-s) \bigl(u(t)-u(s) \bigr) \,ds \,dx \bigg| \\& \quad \leq\eta\big\| u_{t} (t)\big\| ^{2} -\frac{g(0) \|a\|_{\infty}C_{*}^{2} }{4\eta} \bigl(g'\circ\nabla u\bigr) (t), \end{aligned}$$

(4.26)

$$\begin{aligned}& \bigg| \mu_{1} \int_{\Gamma_{1}} a(x) k_{1} \bigl(u_{t} (t)\bigr) \biggl( \int _{0}^{t} g(t-s) \bigl(u(t)-u(s)\bigr) \,ds \biggr) \,dx \bigg| \\& \quad \leq\mu_{1} \eta \int_{\Gamma_{1}} k_{1}^{2} \bigl(u_{t} (t)\bigr)\,d\Gamma+\frac {\mu_{1} (\delta_{0} -l) \|a\|_{\infty}\tilde{C}_{*}^{2} }{4\eta} (g\circ \nabla u) (t), \end{aligned}$$

(4.27)

$$\begin{aligned}& \bigg| \mu_{2} \int_{\Gamma_{1}} a(x) k_{2} \bigl(z(x,1,t)\bigr) \biggl( \int _{0}^{t} g(t-s) \bigl(u(t)-u(s)\bigr) \,ds \biggr) \,dx \bigg| \\& \quad \leq|\mu_{2}|\eta \int_{\Gamma_{1}} k_{2}^{2} \bigl(z(x,1,t)\bigr)\,d \Gamma+\frac {|\mu_{2} | (\delta_{0} -l) \|a\|_{\infty}\tilde{C}_{*}^{2} }{4\eta} (g\circ \nabla u) (t), \end{aligned}$$

(4.28)

$$\begin{aligned}& \bigg|- \int_{\Gamma_{1}} a(x) h(x) y_{t} (t) \int_{0}^{t} g(t-s) \bigl(u(t)-u(s)\bigr) \,ds \,dx \bigg| \\& \quad \leq\frac{\eta\|h\|_{\infty}}{f_{0}} \int_{\Gamma_{1}} h(x) f(x) y_{t}^{2} (t) \,d\Gamma+ \frac{(\delta_{0} -l) \|a\|_{\infty}\tilde{C}_{*}^{2} }{4\eta} (g\circ\nabla u) (t), \end{aligned}$$

(4.29)

$$\begin{aligned}& \bigg| - \int_{\Gamma_{1}} a(x) u_{t} (t) \biggl( \int_{0}^{t} g'(t-s) \bigl(u(t)-u(s) \bigr) \,ds \biggr) \,d\Gamma \bigg| \\& \quad \leq\eta\big\| u_{t} (t)\big\| _{\Gamma_{1}}^{2}- \frac{g(0) \|a\|_{\infty}\tilde{C}_{*}^{2} }{4\eta} \bigl(g'\circ\nabla u\bigr) (t). \end{aligned}$$

(4.30)

Combining the estimates (4.20)-(4.30), then (4.19) becomes

$$\begin{aligned} \frac{d}{dt} \Phi(t) \leq{}&{-} \biggl( a_{0} \int_{0}^{t} g(s) \,ds -\delta_{1} \eta-\eta \biggr) \big\| u_{t} (t)\big\| ^{2} \\ & +\eta \biggl[ \delta_{0}^{2} + \delta_{0}^{2} \alpha_{1}^{2} +2\alpha _{1}^{2} (\delta_{0} -l)^{2} +2( \delta_{0} -l)^{2} +C_{*}^{2(p-1)} \biggl( \frac {2p E(0)}{l(p-2)} \biggr)^{p-2} \biggr] \big\| \nabla u(t)\big\| ^{2} \\ & + \biggl[ \frac{\delta_{0}-l}{4\eta} \bigl(1+2C_{*}^{2} +8\eta+\|a\| _{\infty}+2\|a\|_{\infty}C_{*}^{2}\\ & +\mu_{1} \|a \|_{\infty}\tilde{C}_{*}^{2} +|\mu _{2}| \|a \|_{\infty}\tilde{C}_{*}^{2} +\|a\|_{\infty}\tilde{C}_{*}^{2} \bigr) \\ & +\frac{2\eta^{m} \|b\|_{\infty}}{m} (\delta_{0} -l)^{m-1} C_{*}^{m} \biggl( \frac{2p}{l(p-2)}E(0) \biggr)^{\frac{m-2}{2}} \biggr] (g\circ \nabla u) (t) \\ & -\frac{g(0)\|a\|_{\infty}}{4\eta} \bigl(C_{*}^{2} +\tilde{C}_{*}^{2} \bigr) \bigl(g'\circ\nabla u\bigr) (t) - \biggl( a_{0} \int_{0}^{t} g(s) \,ds -\eta \biggr) \big\| u_{t} (t)\big\| _{\Gamma_{1}}^{2} \\ & +\mu_{1} \eta \int_{\Gamma_{1}} k_{1}^{2} \bigl(u_{t} (t)\bigr)\,d\Gamma+|\mu _{2}| \eta \int_{\Gamma_{1}} k_{2}^{2} \bigl(z(x,1,t)\bigr)\,d \Gamma\\ & +\frac{\eta\|h\|_{\infty}}{f_{0}} \int_{\Gamma_{1}} h(x) f(x) y_{t}^{2} (t) \,d\Gamma.\end{aligned} $$

 □

Lemma 4.4

Let $(u,y,z)$ be the solution of problem (2.15)-(2.23). Then the functional $\Lambda(t)$ defined in (4.4) satisfies

$$\begin{aligned}[b] \frac{d}{dt}\Lambda(t) \leq{}&{-}\rho\Lambda(t) +C_{5} \int_{\Gamma _{1}} k_{1} \bigl(u_{t} (t)\bigr) u_{t} (t) \,d\Gamma\\ &-C_{4} (1-d)e^{-\tau(t)} \int_{\Gamma _{1}} k_{2} \bigl(z(x,1,t)\bigr)z(x,1,t) \,d\Gamma.\end{aligned} $$

(4.31)

Proof

Multiplying (2.19) by $e^{-\rho\tau(t)}k_{2} (z(x,\rho, t))$ and integrating over $\Gamma_{1} \times(0,1)$, we obtain

$$\begin{aligned}[b] & \tau(t) \int_{\Gamma_{1}} e^{-\rho\tau(t)} \int_{0}^{1} z_{t} (x, \rho, t) k_{2} \bigl(z(x,\rho,t)\bigr) \,d\rho \,d\Gamma \\ &\quad =- \int_{\Gamma_{1}} \int_{0}^{1} \bigl(1-\rho\tau' (t) \bigr)e^{-\rho\tau (t)}\frac{\partial}{\partial\rho} K_{2} \bigl(z(x,\rho,t)\bigr)\,d \rho \,d\Gamma .\end{aligned} $$

(4.32)

Differentiating (4.4) with respect to t and using (4.32), we get

$$\begin{aligned}[b] & \frac{d}{dt} \biggl( \tau(t) \int_{\Gamma_{1}} \int_{0}^{1} e^{-\rho \tau(t)} K_{2} \bigl(z(x,\rho, t)\bigr)\,d\rho \,d\Gamma \biggr) \\ &\quad =\tau'(t) \bigl(1-\tau(t)\bigr) \int_{\Gamma_{1}} \int_{0}^{1} e^{-\rho\tau (t)}\frac{\partial}{\partial\rho} K_{2} \bigl(z(x,\rho,t)\bigr)\,d\rho \,d\Gamma \\ & \qquad{} - \int_{\Gamma_{1}} \int_{0}^{1} \bigl(1-\rho\tau'(t) \bigr)e^{-\rho\tau (t)}\frac{\partial}{\partial\rho} K_{2} \bigl(z(x,\rho, t)\bigr) \,d\rho \,d\Gamma .\end{aligned} $$

(4.33)

Then, by integration by parts and using (2.12), (4.33) lead to

$$\begin{aligned} \frac{d}{dt} \Lambda(t) \leq{}&{-}\bigl(1-\rho\tau'(t) +\tau'(t)\bigr) \Lambda(t)+ \int_{\Gamma_{1}} \bigl[ K_{2} \bigl(z(x,0,t)\bigr)-e^{-\tau(t)} K_{2} \bigl(z(x,1,t)\bigr) \bigr]\,d\Gamma \\ & +\tau'(t) e^{-\tau(t)} \int_{\Gamma_{1}} K_{2} \bigl(z(x,1,t)\bigr)\,d\Gamma \\ ={}&{-}\rho\Lambda(t) + \int_{\Gamma_{1}} K_{2} \bigl(z(x,0,t)\bigr) \,d\Gamma +(d-1)e^{-\tau(t)} \int_{\Gamma_{1}} K_{2} \bigl(z(x,1,t)\bigr)\,d\Gamma.\end{aligned} $$

Using (2.10), then (4.31) holds. □

Now we are in a position to state our main result.

Theorem 4.1

Assume that (H1)-(H5) and (3.5) hold. Then, for each $t_{0} >0$, there exist positive constants θ, $\theta_{1}$, $\theta_{2}$ and $\varepsilon_{0}$ such that the solution energy of (2.15)-(2.23) satisfies

$$E(t)\leq\theta O^{-1} \biggl( \theta_{1} \int_{t_{0}}^{t} \zeta(s) \,ds + \theta_{2} \biggr) \quad\textit{for } t\geq t_{0} , $$

where

$$ O(t) = \int_{t}^{1} \frac{1}{O_{1} (s) }\,ds $$

and

$$ O_{1}(t)=\left \{ \textstyle\begin{array}{l@{\quad}l} t &\textit{if } K_{1} \textit{ is linear on } [0,\varepsilon _{1}], \\ tK_{1}' (\varepsilon_{0} t) & \textit{if } K_{1}'(0)=0 \textit{ and } K_{1}''(t)>0 \textit{ on } (0,\varepsilon_{1}]. \end{array}\displaystyle \right . $$

Proof

Since the function $g(t)$ is positive, there exists $t_{0} >0$ such that

$$\int_{0}^{t} g(s) \,ds \geq \int_{0}^{t_{0}} g(s) \,ds : = g_{0} \quad \hbox{for } t\geq t_{0}. $$

Using (3.6), (4.9), (4.17) and (4.31), we arrive at

$$\begin{aligned} \frac{d}{dt} \mathcal {L}(t) \leq&-M \biggl(\mu_{1} -|\mu_{2}|C_{5} - \frac{\xi C_{5}}{2} \biggr) \int_{\Gamma_{1}} k_{1} \bigl(u_{t} (t) \bigr) u_{t} (t) \,d\Gamma \\ & {} -M \biggl[ \frac{ \xi C_{4}}{2} \bigl(1-\tau' (t)\bigr) -| \mu_{2}|(1-C_{4} ) \biggr] \int_{\Gamma_{1}} k_{2} \bigl(z(x,1,t)\bigr) z(x,1,t) \,d\Gamma \\ & {} -M \int_{\Gamma_{1}} h(x) f(x) y_{t}^{2} (t) \,d\Gamma+ \frac {M}{2} \bigl(g' \circ\nabla u\bigr) (t) - \frac{a_{0} M}{2}g(t) \big\| \nabla u(t)\big\| ^{2} \\ & {} -\delta_{1} M \big\| u_{t} (t)\big\| ^{2} -b_{0} M \big\| u_{t} (t) \big\| _{m}^{m} + \varepsilon \biggl( 1+\frac{\delta_{1}}{4\eta} \biggr) \big\| u_{t} (t) \big\| ^{2} \\ & {} -\varepsilon \biggl[\frac{\delta_{0}}{2}-\delta_{1} C_{*}^{2} \eta -\eta\tilde{C}_{*}^{2} \bigl(1+\mu_{1}+| \mu_{2}|\bigr)-\frac{1}{2\delta_{0}}(1+\eta ) (\delta_{0} -l)^{2} \\ & {} -\frac{\|b\|_{\infty} \eta^{-m} C_{*}^{m}}{m} \biggl(\frac {2p}{l(p-2)}E(0) \biggr)^{\frac{m-2}{2}} \biggr]\big\| \nabla u(t)\big\| ^{2} \\ & {}+\frac{\varepsilon}{2\delta_{0}} \biggl( 1+\frac{1}{\eta } \biggr) ( \delta_{0} -l) (g\circ\nabla u) (t) +\varepsilon\big\| u(t) \big\| _{p}^{p} +\varepsilon\big\| u_{t} (t) \big\| _{\Gamma_{1}}^{2} \\ &{} +\frac{\varepsilon\|b\|_{\infty}(m-1)}{m} \eta^{\frac {m}{m-1}}\big\| u_{t} (t) \big\| _{m}^{m} -\varepsilon \int_{\Gamma_{1}} h(x) m(x) y^{2} (t) \,d\Gamma \\ & {}+\frac{\varepsilon\|h\|_{\infty}\|f\|_{\infty}}{\eta f_{0}^{2}} \int_{\Gamma_{1}} h(x) f(x) y_{t}^{2} (t) \,d\Gamma+ \frac{\mu_{1} \varepsilon}{4\eta} \int_{\Gamma_{1}} k_{1}^{2} \bigl(u_{t} (t)\bigr) \,d\Gamma \\ & {} + \frac{|\mu_{2}|\varepsilon}{4\eta} \int_{\Gamma_{1}} k_{2}^{2} \bigl(z(x,1,t)\bigr)\,d \Gamma-(a_{0} g_{0}-\delta_{1} \eta-\eta) \big\| u_{t} (t)\big\| ^{2} \\ &{} +\eta \biggl[\delta_{0}^{2} +\delta_{0}^{2} \alpha_{1}^{2} +2\alpha _{1}^{2} ( \delta_{0} -l)^{2} +2(\delta_{0} -l)^{2} +C_{*}^{2(p-1)} \biggl( \frac {2p E(0)}{l(p-2)} \biggr)^{p-2} \biggr] \big\| \nabla u(t)\big\| ^{2} \\ & {} + \biggl[ \frac{(\delta_{0} -l)}{4\eta} \bigl(1+2C_{*}^{2} +8\eta+\| a \|_{\infty}+2\|a\|_{\infty}C_{*}^{2}\\ &{} +\mu_{1} \|a \|_{\infty}\tilde{C}_{*}^{2} +|\mu _{2}| \|a \|_{\infty}\tilde{C}_{*}^{2} +\|a\|_{\infty}\tilde{C}_{*}^{2} \bigr) \\ & {} +\frac{2\eta^{m} \|b\|_{\infty}}{m}(\delta_{0} -l)^{m-1} C_{*}^{m} \biggl(\frac{2p}{l(p-2)} E(0) \biggr)^{\frac{m-2}{2}} \biggr] (g\circ \nabla u) (t) \end{aligned}$$

$$\begin{aligned} & {}-\eta C_{*}^{2(p-2)} \bigl(C_{*}^{2} +\tilde{C}_{*}^{2} \bigr) \bigl(g'\circ\nabla u\bigr) (t) -(a_{0} g_{0} -\eta)\big\| u_{t} (t) \big\| _{\Gamma_{1}}^{2}\\ &{} +\mu_{1} \eta \int _{\Gamma_{1}} k_{1}^{2} \bigl(u_{t} (t)\bigr)\,d\Gamma+|\mu_{2} |\eta \int_{\Gamma_{1}} k_{2}^{2} \bigl(z(x,1,t)\bigr) \,d \Gamma \\ &{} +\frac{\eta\|h\|_{\infty}}{f_{0}} \int_{\Gamma_{1}} h(x) f(x) y_{t}^{2} (t) \,d\Gamma- \varepsilon\rho\Lambda(t)+\varepsilon C_{5} \int_{\Gamma_{1}} k_{1} \bigl(u_{t} (t)\bigr) u_{t} (t) \,d\Gamma \\ &{} -\varepsilon C_{4} (1-d) e^{-\tau(t)} \int_{\Gamma_{1}} k_{2} \bigl(z(x,1,t)\bigr) z(x,1,t) \,d\Gamma \\ =&- \biggl[\delta_{1} M -\varepsilon \biggl( 1+\frac{\delta_{1}}{4\eta } \biggr) +a_{0} g_{0} -\eta(\delta_{1} +1) \biggr] \big\| u_{t} (t)\big\| ^{2} \\ & {}- \biggl[\frac{M}{2} +\eta C_{*}^{2(p-2)} \bigl(C_{*}^{2} +\tilde{C}_{*}^{2} \bigr) \biggr] \bigl(g' \circ\nabla u\bigr) (t) +\varepsilon\big\| u(t) \big\| _{p}^{p} \\ & {} - \biggl( M-\frac{\varepsilon\|h\|_{\infty}\|f\|_{\infty}}{\eta f_{0}^{2}} -\frac{\eta\|h\|_{\infty}}{f_{0}} \biggr) \int_{\Gamma _{1}} h(x) f(x) y_{t}^{2} (t) \,d\Gamma \\ &{}- \varepsilon \int_{\Gamma_{1}} h(x) m(x) y^{2} (t) \,d\Gamma \\ & {} - \biggl\{ \frac{a_{0} M}{2} g(t) +\varepsilon \biggl[ \frac {\delta_{0}}{2} -\delta_{1} C_{*}^{2} \eta-\eta\tilde{C}_{*}^{2} \bigl(1+\mu_{1} +|\mu _{2}|\bigr) - \frac{(1+\eta)(\delta_{0} -l)^{2}}{2\delta_{0}} \\ & {} -\frac{\|b\|_{\infty}\eta^{-m}C_{*}^{m}}{m} \biggl( \frac {2p}{l(p-2)} E(0) \biggr)^{\frac{m-2}{2}} \biggr] \\ & {}-\eta \biggl[ \delta _{0}^{2} + \delta_{0}^{2} \alpha_{1}^{2} +2 \alpha_{1}^{2}(\delta_{0} -l)^{2} +2( \delta _{0} -l)^{2} + C_{*}^{2(p-1)} \biggl( \frac{2pE(0)}{l(p-2)} \biggr)^{p-2} \biggr] \biggr\} \big\| \nabla u(t)\big\| ^{2} \\ & {}+ \biggl[ \frac{\varepsilon}{2\delta_{0}} \biggl( 1+\frac {1}{\eta} \biggr) ( \delta_{0} -l) +\frac{(\delta_{0} -l)}{4\eta} \bigl(1+2C_{*}^{2} +8\eta+ \|a\|_{\infty}+2\|a\|_{\infty}C_{*}^{2} \\ & {} +\mu_{1} \|a\| _{\infty}\tilde{C}_{*}^{2}+|\mu_{2}| \|a\|_{\infty}\tilde{C}_{*}^{2} +\|a \|_{\infty}\tilde{C}_{*}^{2} \bigr)\\ &{}+\frac{2\eta^{m} \|b\|_{\infty}}{m}( \delta_{0} -l)^{m-1} C_{*}^{m} \biggl( \frac{2p}{l(p-2)} E(0) \biggr)^{\frac{m-2}{2}} \biggr] (g\circ \nabla u) (t) \\ & {} - \biggl( b_{0} M -\frac{\varepsilon\|b\|_{\infty}(m-1)}{m} \eta^{\frac{m}{m-1}} \biggr) \big\| u_{t} (t)\big\| _{m}^{m} -(a_{0} g_{0} -\eta -\varepsilon)\big\| u_{t} (t) \big\| _{\Gamma_{1}}^{2} -\varepsilon\rho\Lambda(t) \\ & {}- \biggl[ M\biggl(\mu_{1} -|\mu_{2}| C_{5} - \frac{\xi C_{5}}{2}\biggr) -\varepsilon C_{5} \biggr] \int_{\Gamma_{1}} k_{1} \bigl(u_{t} (t) \bigr)u_{t} (t) \,d\Gamma\\ &{}+\mu_{1} \biggl(\frac{ \varepsilon}{4\eta}+ \eta \biggr) \int_{\Gamma_{1}} k_{1}^{2} \bigl(u_{t} (t)\bigr) \,d\Gamma \\ & {}- \biggl\{ M \biggl[ \frac{\xi C_{4}}{2} \bigl(1-\tau' (t) \bigr)-|\mu _{2}|(1-C_{4}) \biggr] +\varepsilon C_{4} (1-d)e^{-\tau(t)} \biggr\} \\ &{}\times \int _{\Gamma_{1}} k_{2} \bigl(z(x,1,t)\bigr)z(x,1,t)\,d\Gamma \\ & {}+ |\mu_{2} | \biggl(\frac{ \varepsilon}{4\eta}+\eta \biggr) \int_{\Gamma_{1}} k_{2}^{2} \bigl(z(x,1,t)\bigr)\,d \Gamma. \end{aligned}$$

At this point, we choose $\varepsilon>0$ small enough and we pick $\eta>0$ sufficiently small and M is so large such that

$$\begin{gathered} M_{1} = \delta_{1} M -\varepsilon \biggl( 1+ \frac{\delta_{1}}{4\eta } \biggr) +a_{0} g_{0} -\eta( \delta_{1} +1)>0, \\ M_{2} = \frac{M}{2} +\eta C_{*}^{2(p-2)} \bigl(C_{*}^{2} +\tilde{C}_{*}^{2}\bigr) >0, \\ M_{3} = M -\frac{\varepsilon\|h\|_{\infty}\|f\|_{\infty}}{\eta f_{0}^{2}}-\frac{\eta\|h\|_{\infty}}{f_{0}} >0, \\ \begin{aligned} M_{4} ={}& \frac{a_{0} M}{2} g(t) \\ &+\varepsilon \biggl[ \frac{\delta _{0}}{2} -\delta_{1} C_{*}^{2} \eta-\eta\tilde{C}_{*}^{2} \bigl(1+\mu_{1} +|\mu_{2}|\bigr) -\frac{(1+\eta)(\delta_{0} -l)^{2}}{2\delta_{0}} \\ & +\frac{\|b\|_{\infty}\eta^{-m}C_{*}^{m}}{m} \biggl( \frac {2p}{l(p-2)} E(0) \biggr)^{\frac{m-2}{2}} \biggr] \\ & -\eta \biggl[ \delta _{0}^{2} + \delta_{0}^{2} \alpha_{1}^{2} +2 \alpha_{1}^{2}(\delta_{0} -l)^{2} +2( \delta _{0} -l)^{2} + C_{*}^{2(p-1)} \biggl( \frac{2pE(0)}{l(p-2)} \biggr)^{p-2} \biggr] \\ >{}&0,\end{aligned} \\ M_{5} = b_{0} M -\frac{\varepsilon\|b\|_{\infty}(m-1)}{m} \eta^{\frac {m}{m-1}} >0 , \\ M_{6} = a_{0} g_{0} -\eta-\varepsilon>0, \\ M_{7} = M\biggl( \mu_{1} -|\mu_{2}|C_{5} -\frac{\xi C_{5}}{2} \biggr)-\varepsilon C_{5} > 0 .\end{gathered} $$

Therefore, for all $t\geq t_{0}$, we deduce

$$\begin{aligned} \frac{d}{dt}\mathcal{L}(t) \leq{}&{-}M_{1} \big\| u_{t} (t)\big\| ^{2} -M_{2} (g\circ\nabla u) (t)\\ & -M_{3} \int _{\Gamma_{1}} h(x) f(x) y_{t}^{2} (t) \,d\Gamma- \varepsilon \int_{\Gamma_{1}} h(x) m(x) y^{2} (t) \,d\Gamma \\ & -M_{4} \big\| \nabla u(t)\big\| ^{2} -M_{5} \big\| u_{t} (t)\big\| _{m}^{m} -M_{6} \big\| u_{t} (t)\big\| _{\Gamma_{1}}^{2} -\varepsilon\rho \Lambda(t)\\ &-M_{7} \int_{\Gamma_{1}} k_{1} \bigl(u_{t} (t) \bigr)u_{t} (t) \,d\Gamma + \mu_{1} \biggl(\frac{ \varepsilon}{4\eta}+\eta \biggr) \int_{\Gamma_{1}}k_{1}^{2} \bigl(u_{t} (t)\bigr)\,d\Gamma\\ &-M_{8} \int_{\Gamma_{1}} k_{2} \bigl(z(x,1,t)\bigr)z(x,1,t)\,d\Gamma + |\mu_{2} | \biggl(\frac{ \varepsilon}{4\eta}+\eta \biggr) \int_{\Gamma_{1}} k_{2}^{2} \bigl(z(x,1,t)\bigr)\,d \Gamma\\ & +M_{9} (g\circ\nabla u) (t),\end{aligned} $$

where

$$\begin{gathered} M_{8} = M \biggl[ \frac{\xi C_{4}}{2} \bigl(1-\tau' (t)\bigr)-|\mu _{2}|(1-C_{4}) \biggr] +\varepsilon C_{4} (1-d)e^{-\tau(t)} >0 , \\ \begin{aligned} M_{9} ={}& \frac{\varepsilon}{2\delta_{0}} \biggl( 1+\frac{1}{\eta } \biggr) ( \delta_{0} -l) +\frac{(\delta_{0} -l)}{4\eta} \bigl(1+2C_{*}^{2} +8\eta+ \|a\|_{\infty}+2\|a\|_{\infty}C_{*}^{2} \\ & +\mu_{1} \|a\|_{\infty}\tilde{C}_{*}^{2}+|\mu_{2}| \|a\|_{\infty}\tilde{C}_{*}^{2} +\|a \|_{\infty}\tilde{C}_{*}^{2} \bigr)\\ &+\frac{2\eta^{m} \|b\|_{\infty}}{m}( \delta_{0} -l)^{m-1} C_{*}^{m} \biggl( \frac{2p}{l(p-2)} E(0) \biggr)^{\frac{m-2}{2}}\\ >{}&0 ,\end{aligned}\end{gathered} $$

which implies that

$$\begin{aligned} \frac{d}{dt}\mathcal{L}(t) \leq{}&{-}M_{10} E(t) +M_{11} (g\circ\nabla u ) (t) + \mu_{1} \biggl(\frac{ \varepsilon}{4\eta}+\eta \biggr) \int_{\Gamma_{1}} k_{1}^{2} \bigl(u_{t}(t) \bigr)\,d\Gamma \\ &+ |\mu_{2} | \biggl(\frac{ \varepsilon}{4\eta}+\eta \biggr) \int _{\Gamma_{1}} k_{2}^{2} \bigl(z(x,1,t)\bigr)\,d \Gamma,\end{aligned} $$

where $M_{10}$ and $M_{11}$ are some positive constants.

Multiplying the above inequality by $\zeta(t)$, we obtain, for any $t\geq t_{0}$,

$$\begin{aligned} \zeta(t)\mathcal{L}'(t) \leq{}&{-}M_{10} \zeta(t) E(t)+ M_{11} \zeta (t) (g\circ\nabla u) (t) + \mu_{1} \biggl( \frac{ \varepsilon}{4\eta }+\eta \biggr) \zeta(t) \int_{\Gamma_{1}} k_{1}^{2} \bigl(u_{t} (t)\bigr) \,d\Gamma\\ & + |\mu_{2} | \biggl(\frac{ \varepsilon}{4\eta}+\eta \biggr) \zeta (t) \int_{\Gamma_{1}} k_{2}^{2} \bigl(z(x,1,t)\bigr) \,d \Gamma.\end{aligned} $$

From (2.9), we obtain

$$k_{2}^{2} (s) \leq csk_{2} (s) \quad\hbox{for all } s\in R, $$

where c is some positive constant. Recalling (2.4) and (3.6), we get for any $t\geq t_{0}$

$$ \zeta(t)\mathcal{L}'(t) \leq-M_{10} \zeta(t) E(t)+ M_{12} \zeta (t) \int_{\Gamma_{1}} k_{1}^{2} \bigl(u_{t} (t)\bigr) \,d\Gamma- M_{13} E'(t) . $$

(4.34)

Now, we define

$$G(t)=\zeta(t) \mathcal{L}(t) +M_{13} E(t). $$

As ζ is a non-increasing positive function, by using Lemma 4.1, the function $G(t)$ is equivalent to $E(t)$. Using the fact that $\zeta'(t)\leq0$, (4.34) implies that

$$ G'(t)\leq-M_{10} \zeta(t) E(t) + M_{12} \zeta(t) \int_{\Gamma_{1}} k_{1}^{2} \bigl(u_{t} (t)\bigr) \,d\Gamma\quad\hbox{for } t \geq t_{0}. $$

(4.35)

In the following, we shall estimate the term $\int_{\Gamma_{1}} k_{1}^{2} (u_{t} (t)) \,d\Gamma$ in (4.35). To do this, let $\Gamma_{11} =\{ x\in\Gamma_{1} : |u_{t} |>\varepsilon _{1}\}$, $\Gamma_{12} =\{ x\in\Gamma_{1} : |u_{t} |\leq\varepsilon_{1} \}$.

Case (I): $K_{1}$ is linear on $[0, \varepsilon_{1}]$.

There exist positive constants $C_{1}$ and $C_{2}$ such that

$$\begin{aligned} C_{1} |s| \leq\big|k_{1}(s)\big| \leq C_{2} |s| \quad\hbox{for } s\in R. \end{aligned}$$

This and (4.35) yield

$$ G'(t)\leq-M_{10} \zeta(t) E(t) - M_{13} E'(t)\quad \hbox{for } t \geq t_{0}, $$

where $M_{13}$ is a positive constant. This gives

$$ \bigl( G(t) + M_{13} E(t) \bigr) ' \leq-M_{10} \zeta(t) E(t) \quad\hbox{for } t \geq t_{0} . $$

Employing that G is equivalent to E, we get

$$ E(t)\leq C e^{- \nu\int^{t}_{t_{0}} \zeta(s) \,ds} \quad\hbox{for } t \geq t_{0}, $$

where C and ν are positive constants. Owing to $K_{1} (s) = \sqrt {s} k_{1} (\sqrt{s}) =cs$,

$$ E(t)\leq C e^{- \nu\int^{t}_{t_{0}} \zeta(s) \,ds} = K_{1}^{-1} \biggl( \int ^{t}_{t_{0}} \zeta(s) \,ds \biggr)\quad \hbox{for } t \geq t_{0} . $$

Case (II): $K_{1}'(0) =0$ and $K_{1} '' (t) >0$ on $[0, \varepsilon _{1}]$.

Since $K_{1}$ is convex and increasing, $K_{1}^{-1}$ is concave and increasing. By (2.7), (2.8), (3.6) and the reversed Jensen inequality

$$\begin{aligned} \zeta(t) \int_{\Gamma_{1}} k_{1}^{2} \bigl(u_{t} (t)\bigr) \,d\Gamma & = \zeta(t) \int_{\Gamma_{11}}k_{1}^{2} \bigl(u_{t} (t)\bigr)\,d\Gamma + \zeta(t) \int_{\Gamma_{12}}k_{1}^{2} \bigl(u_{t} (t)\bigr)\,d\Gamma \\ & \leq C_{2} \zeta(0) \int_{\Gamma_{11}}u_{t} (t) k_{1} \bigl(u_{t} (t)\bigr)\,d\Gamma + \zeta(t) \int_{\Gamma_{12}}K_{1}^{-1} \bigl(u_{t} k_{1} (u_{t})\bigr)\,d\Gamma \\ & \leq - \mu_{3} E' (t) + \zeta(t) | \Gamma_{12} |K_{1}^{-1} \biggl( \frac{1}{|\Gamma_{12}|} \int_{\Gamma_{12}}u_{t} k_{1} (u_{t} ) \,d \Gamma \biggr) \\ & \leq - \mu_{3} E' (t) + \zeta(t) | \Gamma_{12}| K_{1}^{-1} \bigl( - \mu_{4} E'(t) \bigr) ,\end{aligned} $$

where $\mu_{3}$ and $\mu_{4}$ are positive constants. Thus, we get from (4.35)

$$\begin{aligned} G_{1}'(t) \leq- M_{10} \zeta(t) E(t)+ | \Gamma_{12}| M_{12} \zeta(t) K_{1}^{-1} \bigl(- \mu_{4} E'(t) \bigr), \end{aligned}$$

where $G_{1}(t) = G(t) + \mu_{3} M_{12} E(t)$. Now, for $0 < \varepsilon _{0} < \varepsilon_{1}$ and $\mu>0$, we define

$$ H(t)=K_{1}' \biggl( \varepsilon_{0} \frac{E(t)}{E(0)} \biggr) G_{1} (t) +\mu E(t), \quad \forall t\geq t_{0} . $$

(4.36)

It is easily noted that

$$ \gamma_{1} H(t)\leq E(t) \leq\gamma_{2} H(t), $$

(4.37)

where $\gamma_{1}$, $\gamma_{2}$ are positive constant. Thanks to the similar argument in (3.13) and (3.12), we have

$$\begin{aligned} H'(t) ={} & \varepsilon_{0} \frac{E'(t)}{E(0)} K_{1}'' \biggl( \varepsilon _{0} \frac{E(t)}{E(0)} \biggr) G_{1}(t)+K_{1}' \biggl(\varepsilon_{0} \frac {E'(t)}{E(0)} \biggr)G_{1}'(t) +\mu E'(t) \\ \leq{}& {-} M_{10} \zeta(t) E(t) K_{1}' \biggl( \varepsilon_{0} \frac {E(t)}{E(0)} \biggr) \\ &+ M_{12} | \Gamma_{12}| \zeta(t) K_{1}' \biggl( \varepsilon_{0} \frac {E(t)}{E(0)} \biggr) K_{1}^{-1} \bigl(- \mu_{4} E'(t) \bigr) +\mu E'(t) \\ \leq{}&{ -}M_{10} \zeta(t) E(t) K_{1}' \biggl( \varepsilon_{0} \frac {E(t)}{E(0)} \biggr) +\mu E'(t) \\ & + M_{12}|\Gamma_{12}| \zeta(t) K_{1}^{*} \biggl\{ K_{1}' \biggl( \varepsilon_{0} \frac{E(t)}{E(0)} \biggr) \biggr\} - \mu_{4} |\Gamma _{12}| M_{12} \zeta(t) E'(t) \\ \leq{}& {-}M_{10} \zeta(t) E(t) K_{1}' \biggl( \varepsilon_{0} \frac {E(t)}{E(0)} \biggr) + (\mu- M_{13} ) E'(t) \\ & + M_{14} \zeta(t) K_{1}' \biggl( \varepsilon_{0} \frac {E(t)}{E(0)} \biggr) \varepsilon_{0} \frac{E(t)}{E(0)} - M_{14} \zeta(t) K_{1} \biggl( \varepsilon_{0} \frac{E(t)}{E(0)} \biggr) \\ \leq{}& {-} \bigl( M_{10} E(0) - \varepsilon_{0} M_{14} \bigr) \zeta (t) \frac{E(t)}{E(0)} K_{1}' \biggl(\varepsilon_{0} \frac {E(t)}{E(0)} \biggr) + (\mu- M_{13} ) E'(t) ,\end{aligned} $$

where $M_{13} =|\Gamma_{12} | M_{12} \mu_{4} \zeta(0)$ and $M_{14} = |\Gamma_{12}| M_{12}$. Taking $\varepsilon_{0}$ sufficiently small such that $M_{10} E(0) - \varepsilon_{0} M_{14} >0$ and $\mu>0$ suitably so that $\mu- M_{13} >0$, we arrive at

$$\begin{aligned} H'(t) \leq- \mu_{5} \zeta(t) \frac{E(t)}{E(0)} K_{1}' \biggl(\varepsilon_{0} \frac{E(t)}{E(0)} \biggr) = - \mu_{5} \zeta(t) O_{1} \biggl( \frac{E(t)}{E(0)} \biggr) \quad\hbox{for } t \geq t_{0}, \end{aligned}$$

(4.38)

where $O_{1} (t)=tK_{1}' (\varepsilon_{0} t)$ and $\mu_{5}$ is a positive constant. Finally, we define

$${\mathcal {E} }(t)=\gamma_{1} \frac{H(t)}{E(0)}. $$

Using (4.37), we see that ${\mathcal {E} }$ is equivalent to E. Therefore,

$${\mathcal {E} }'(t)\leq-\theta_{1} \zeta(t) O_{1} \bigl({\mathcal {E} }(t)\bigr)\quad \hbox{for } t\geq t_{0}, $$

for some $\theta_{1} >0$, and

$${\mathcal {E} }(t) \leq O^{-1} \biggl( \theta_{1} \int_{t_{0}}^{t} \zeta(s) \,ds +\theta_{2} \biggr)\quad\hbox{for } t\geq t_{0} . $$

Thus the proof is completed. □

5 Conclusion

We have investigated the energy decay rate of the nonlinear viscoelastic problem with dynamic and acoustic boundary conditions.

It is well known that viscoelastic materials have memory effects, which is due to the mechanical response influenced by the history of the materials themselves. As these materials have a wide application in the natural sciences, their dynamics is interesting and of great importance. Also, the dynamic boundary conditions are not only important from the theoretical point of view but also arise in numerous practical problems and the acoustic boundary conditions are related to noise control and suppression in practical applications. Moreover, time delay so often arises in many physical, chemical, biological and economical phenomena because these phenomena depend not only on the present state but also on the history of the system in a more complicated way. We established a decay rate estimate for the energy by introducing suitable Lyapunov functionals.