General decay for weak viscoelastic equation of Kirchhoff type containing Balakrishnan–Taylor damping with nonlinear delay and acoustic boundary conditions

Abstract

In this paper, we consider the general decay of solutions for the weak viscoelastic equation of Kirchhoff type containing Balakrishnan–Taylor damping with nonlinear delay and acoustic boundary conditions. By using suitable energy and Lyapunov functionals, we prove the general decay for the energy, which depends on the behavior of both σ and k.

1 Introduction

The objective of this work is to study the general decay of solutions for the weak viscoelastic equation of Kirchhoff type containing Balakrishnan–Taylor damping with nonlinear delay and acoustic boundary conditions

$$\begin{array}{c}|w_t{|}^{\rho }{w}_{tt}-(a_0+b_0{\parallel \mathrm{\nabla }w\parallel }^2+b_1(\mathrm{\nabla }w,\mathrm{\nabla }{w}_t))\mathrm{\Delta }w-\mathrm{\Delta }{w}_{tt}+\sigma (t){\int }_0^{t}k(t-s)\mathrm{\Delta }w(s)ds\hfill \\ =|w{|}^{p-2}w\text{in }\mathrm{\Omega }\times {\mathbb{R}}^{+},\hfill \end{array}$$

(1.1)

$$w=0\text{on }{\mathrm{\Gamma }}_0\times {\mathbb{R}}^{+},$$

(1.2)

$$\begin{array}{c}(a_0+b_0{\parallel \mathrm{\nabla }w\parallel }^2+b_1(\mathrm{\nabla }w,\mathrm{\nabla }{w}_t))\frac{\partial w}{\partial \nu }+\frac{\partial w_{tt}}{\partial \nu }-\sigma (t){\int }_0^{t}k(t-s)\frac{\partial w(s)}{\partial \nu }ds\hfill \\ +{\mu }_1|w_t(x,t){|}^{q-1}{w}_t(x,t)+{\mu }_2|w_t(x,t-\tau ){|}^{q-1}{w}_t(x,t-\tau )\hfill \\ =m(x)u_t\text{on }{\mathrm{\Gamma }}_1\times {\mathbb{R}}^{+},\hfill \end{array}$$

(1.3)

$$w_t+g(x)u_t+h(x)u=0\text{on }{\mathrm{\Gamma }}_1\times {\mathbb{R}}^{+},$$

(1.4)

$$\begin{aligned}& w(x,0)=w_{0} (x) ,\qquad w_{t} (x,0)=w_{1} (x) \quad \text{in } \Omega , \end{aligned}$$

(1.5)

$$\begin{aligned}& u(x,0)=u_{0} (x)\quad \text{on } \Gamma _{1}, \end{aligned}$$

(1.6)

$$\begin{aligned}& w_{t} (x,t-\tau ) =f_{0} (x,t-\tau )\quad \text{on } \Gamma _{1}, 0< t< \tau , \end{aligned}$$

(1.7)

where Ω is a bounded domain of ${\mathbb{R}}^n$ ($n\geq 1$) with a smooth boundary $\Gamma =\Gamma _{0} \cup \Gamma _{1}$. Here, $\Gamma _{0}$ and $\Gamma _{1}$ are closed and disjoint and ν is the unit outward normal to Γ. $w_{0}$, $w_{1}$, $u_{0}$, and $f_{0}$ are given functions. All the parameters $a_{0}$, $b_{0}$, $b_{1}$, ρ, p, q, $\mu _{1}$, and $\mu _{2}$ are positive constants, the functions $m,g,h:{\mathrm{\Gamma }}_1\to \mathbb{R}$ are essentially bounded. Moreover, k represents the kernel of the memory term and $\tau >0$ represents the time delay.

The equation (1.1) with $b_{0} = b_{1}=0$ and $a_{0}=\sigma (t)=1$,

$$\begin{array}{c}|w_t{|}^{\rho }{w}_{tt}-\mathrm{\Delta }w-\mathrm{\Delta }{w}_{tt}+{\int }_0^{t}k(t-s)\mathrm{\Delta }w(s)ds\hfill \\ =|w{|}^{p-2}w\text{in }\mathrm{\Omega }\times {\mathbb{R}}^{+},w=0\text{ on }\mathrm{\Gamma }\hfill \end{array}$$

(1.8)

has been studied by Messaoudi and Tatar [16]. The case of $\rho =1$ and $b_{1}=\sigma (t)= 0$ in the absence of the dispersion term, the equation (1.1) reduces to the well-known Kirchhoff equation that has been introduced in [8] in order to describe the nonlinear vibrations of an elastic string.

The model with Balakrishnan–Taylor damping ($b_{1} > 0$) and $k=0$, was initially proposed by Balakrishnan and Taylor in [2]. Several authors have studied the asymptotic behavior of the solution for the nonlinear viscoelastic Kirchhoff equations with Balakrishnan–Taylor damping (see [17, 22, 24] and references and therein). Recently, Al-Gharabli et al. [1] considered the following Balakrishnan–Taylor viscoelastic equation with a logarithmic source term

$$\begin{array}{c}|w_t{|}^{\rho }{w}_{tt}-(a_0+b_0{\parallel \mathrm{\nabla }w\parallel }^2+b_1(\mathrm{\nabla }w,\mathrm{\nabla }{w}_t))\mathrm{\Delta }w-\mathrm{\Delta }{w}_{tt}+{\int }_0^{t}k(t-s)\mathrm{\Delta }w(s)ds+h(w_t)\hfill \\ =kwln|w|\text{in }\mathrm{\Omega }\times {\mathbb{R}}^{+},w=0\text{ on }\mathrm{\Gamma }.\hfill \end{array}$$

(1.9)

They proved the general decay rates, using the multiplier method and some properties of the convex functions. Lian and Xu [11] investigated the problem (1.9) with weak and strong damping terms and $\rho =b_{0}=b_{1}= k=0$.

For $\sigma (t)>0$, Messaoudi [15] studied the following viscoelastic wave equation

$$w_{tt}-\mathrm{\Delta }w+\sigma (t){\int }_0^{t}k(t-s)\mathrm{\Delta }w(s)ds=0\text{in }\mathrm{\Omega }\times {\mathbb{R}}^{+}.$$

The author obtained the general decay result that depends on the behavior of both σ and k. For other related works, we refer the readers to [3, 13, 14].

Since most phenomena naturally depend not only on the present state but also on some past occurrences, in recent years, there has been published much work concerning the wave equation with delay effects that often appear in many practical problems [18–21]. Feng and Li [7] proved the general energy decay for a viscoelastic Kirchhoff plate equation with a time delay. Lee et al. [9] showed the general energy decay of solutions for system (1.1)–(1.7) with $\sigma (t)=1$ and $q=1$.

Motivated by previous work, we study the general energy decay of solutions for the system (1.1)–(1.7) that depends on the behavior of the potential σ and the relaxation function k satisfying the suitable conditions. The acoustic boundary condition (1.4) and the coupled impenetrability boundary condition (1.3) were proposed by Beale and Rosencrans [5]. For physical application of acoustic boundary conditions, we refer to [4, 6]. The stability of various models with acoustic boundary conditions has been discussed by many researchers [10, 12, 14, 23]. The outline of this paper is as follows. In Sect. 2, we present some preparations and hypotheses for our main result. In Sect. 3, we obtain the general energy decay of the system (1.1)–(1.7) by using the energy-perturbation method.

2 Preliminary

In this section, we present some material that we shall use in order to prove our result. We denote by

$$ V= \bigl\{ w\in H^{1} (\Omega ) : w=0 \text{ on } \Gamma _{0} \bigr\} .$$

The Poincaré inequality holds in V, i.e., there exists a constant $C_{*}$ such that

$$ \Vert w \Vert _{r} \leq C_{*} \Vert \nabla w \Vert , \quad 2\leq r \leq \frac{2n}{n-2}, \forall w \in V, $$

(2.1)

and there exists a constant $\tilde{C}_{*}$ such that

$$ \Vert w \Vert _{r,\Gamma _{1}} \leq \tilde{C}_{*} \Vert \nabla w \Vert , \quad \forall w \in V. $$

(2.2)

For our study of problem (1.1)–(1.7), we will need the following assumptions.

(H1) The constants ρ and q satisfy

$$ 0< \rho , q \leq \frac{2}{n-2} \quad \text{if } n\geq 3, \qquad \rho , q >0 \quad \text{if } n=1,2, $$

(2.3)

and p satisfies

$$ 0< p \leq \frac{4}{n-2}\quad \text{if } n\geq 3 ,\qquad p>2 \quad \text{if } n=1,2. $$

(2.4)

For the relaxation function k and potential σ, as in [15], we assume that

(H2) $k,\sigma :{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ are nonincreasing differentiable functions such that k is a $C^{2}$ function and σ is a $C^{1}$ function satisfying

$$\begin{aligned}& \begin{aligned} & k(0)>0 ,\quad \int _{0}^{\infty }k(s)\,ds =k_{0} < \infty , \qquad \sigma (t) >0, \\ & a_{0}- \sigma (t) \int _{0}^{t} k(s)\,ds \geq l >0 ,\quad \forall t \geq 0, \end{aligned} \end{aligned}$$

(2.5)

$$\begin{aligned}& { \biggl( \sigma (t) \int _{0}^{t} k(s)\,ds \biggr)' \geq 0 ,\quad \forall t\in [0, t_{0}],} \end{aligned}$$

(2.6)

where l and $t_{0}$ are suitable positive constants. There exists a nonincreasing differentiable function $\zeta :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ with

$$\begin{aligned}& \zeta (t)>0,\quad k'(t) \leq -\zeta (t) k(t), \qquad \forall t \geq 0,\quad \lim_{t\to \infty} \frac{-\sigma ' (t)}{\zeta (t)\sigma (t) }=0 . \end{aligned}$$

(2.7)

(H3) There exist three positive constants $m_{1}$, $g_{1}$, and $h_{1}$ such that

$$ m_{1} \leq m(x),\qquad g_{1} \leq g(x) , \qquad h_{1} \leq h(x) ,\quad x\in \Gamma _{1}. $$

(2.8)

(H4) We assume that the constants $\mu _{1}$ and $\mu _{2}$ satisfy $\mu _{2} < \mu _{1}$.

Remark 2.1

([15])

1. Note that (2.7) implies that ${ \lim_{t\to \infty} \frac{-\sigma ' (t)}{\sigma (t) }=0.}$

2. Examples of functions k and σ satisfying (H2) are

$$ \sigma (t) = \frac{1}{1+t} , \qquad k(t)=ae^{-b(1+t)^{c}},\quad 0< c \leq 1, $$

for $a, b>0$, to be chosen properly.

As in [19], let us introduce the function

$$ z(x, \delta , t)=w_{t} (x, t-\tau \delta ),\quad x\in \Omega , \delta \in (0,1), \forall t>0. $$

(2.9)

Then, problem (1.1)–(1.7) is equivalent to

$$\{\begin{array}{c}|w_t{|}^{\rho }{w}_{tt}-(a_0+b_0{\parallel \mathrm{\nabla }w\parallel }^2+b_1(\mathrm{\nabla }w,\mathrm{\nabla }{w}_t))\mathrm{\Delta }w-\mathrm{\Delta }{w}_{tt}+\sigma (t){\int }_0^{t}k(t-s)\mathrm{\Delta }w(s)ds\hfill \\ =|w{|}^{p-2}w\text{in }\mathrm{\Omega }\times {\mathbb{R}}^{+},\hfill \\ w=0\text{on }{\mathrm{\Gamma }}_0\times {\mathbb{R}}^{+},\hfill \\ (a_0+b_0{\parallel \mathrm{\nabla }w\parallel }^2+b_1(\mathrm{\nabla }w,\mathrm{\nabla }{w}_t))\frac{\partial w}{\partial \nu }+\frac{\partial w_{tt}}{\partial \nu }-\sigma (t){\int }_0^{t}k(t-s)\frac{\partial w(s)}{\partial \nu }ds\hfill \\ +{\mu }_1|w_t(x,t){|}^{q-1}{w}_t(x,t)+{\mu }_2|z(x,1,t){|}^{q-1}z(x,1,t)\hfill \\ =m(x)u_t\text{on }{\mathrm{\Gamma }}_1\times (0,1)\times {\mathbb{R}}^{+},\hfill \\ \tau z_t(x,\delta ,t)+z_{\delta }(x,\delta ,t)=0\text{on }{\mathrm{\Gamma }}_1\times (0,1)\times {\mathbb{R}}^{+},\hfill \\ w_t+g(x)u_t+h(x)u=0\text{on }{\mathrm{\Gamma }}_1\times {\mathbb{R}}^{+},\hfill \\ w(x,0)=w_0(x),w_t(x,0)=w_1(x)\text{in }\mathrm{\Omega },\hfill \\ u(x,0)=u_0(x)\text{on }{\mathrm{\Gamma }}_1,\hfill \\ z(x,\delta ,0)=f_0(x,-\tau \delta )\text{on }{\mathrm{\Gamma }}_1\times (0,1).\hfill \end{array}$$

(2.10)

By combining with the argument of [5], we now state the local existence result of problem (2.10), which can be obtained.

Theorem 2.1

Suppose that (H1)–(H4) hold and that $(w_{0} , w_{1}) \in (H^{2} (\Omega )\cap V)\times V$, $u_{0} \in L^{2} (\Gamma _{1})$ and $f_{0} \in L^{2} (\Gamma _{1} \times (0,1))$. Then, for any $T>0$, there exists a unique solution $(w,u,z)$ of problem (2.10) on $[0,T]$ such that

$$\begin{aligned}& w\in L^{\infty } \bigl(0,T; H^{2} (\Omega )\cap V \bigr),\qquad w_{t} \in L^{\infty }(0,T; V) \cap L^{q+1} \bigl(\Gamma _{1} \times (0,T) \bigr), \\& m^{1/2} u\in L^{\infty } \bigl(0,T; L^{2} (\Gamma _{1}) \bigr),\qquad m^{1/2} u_{t} \in L^{2} \bigl(0,T; L^{2} (\Gamma _{1}) \bigr). \end{aligned}$$

3 Main result

In this section, we state and show our main result. For this purpose, we define

$$\begin{aligned} J(t) =& \frac{1}{2} \biggl( a_{0}-\sigma (t) \int _{0}^{t} k(s)\,ds \biggr) \bigl\Vert \nabla w(t) \bigr\Vert ^{2} +\frac{b_{0}}{4} \bigl\Vert \nabla w(t) \bigr\Vert ^{4} \\ &{} + \frac{1}{2} \bigl\Vert \nabla w_{t} (t) \bigr\Vert ^{2} +\frac{1}{2} \sigma (t) (k \circ \nabla w) (t) \\ & {}+\frac{\xi}{2} \int _{\Gamma _{1}} \int _{0}^{1} \bigl\vert z (x, \delta , t) \bigr\vert ^{q+1}\,d\delta \,d\Gamma +\frac{1}{2} \int _{\Gamma _{1}} h(x) m(x) u^{2} (t)\,d\Gamma - \frac{1}{p} \bigl\Vert w(t) \bigr\Vert _{p}^{p} , \end{aligned}$$

(3.1)

and

$$\begin{aligned} I(t) =& \biggl( a_{0}-\sigma (t) \int _{0}^{t} k(s)\,ds \biggr) \bigl\Vert \nabla w(t) \bigr\Vert ^{2} +\frac{b_{0}}{2} \bigl\Vert \nabla w(t) \bigr\Vert ^{4} \\ &{}+ \bigl\Vert \nabla w_{t} (t) \bigr\Vert ^{2} + \sigma (t) ( k\circ \nabla w) (t) \\ & {}+\xi \int _{\Gamma _{1}} \int _{0}^{1} \bigl\vert z(x, \delta , t) \bigr\vert ^{q+1}\,d\delta \,d\Gamma + \int _{\Gamma _{1}} h(x) m(x) u^{2} (t)\,d\Gamma - \bigl\Vert w(t) \bigr\Vert _{p}^{p} , \end{aligned}$$

(3.2)

where ${(k\circ w)(t) = \int _{0}^{t} k(t-s) \|w(t)-w(s) \|^{2}\,ds}$. From direct calculation, we find that

$$\begin{aligned}& \sigma (t) (k\ast w, w_{t} ) \\& \quad = -\frac{\sigma (t)}{2} k(t) \bigl\Vert w(t) \bigr\Vert ^{2} - \frac{d}{dt} \biggl[ \frac{\sigma (t)}{2} (k\circ w) (t) - \frac{\sigma (t)}{2} \biggl( \int _{0}^{t} k(s)\,ds \biggr) \bigl\Vert w(t) \bigr\Vert ^{2} \biggr] \\& \qquad {}+\frac{\sigma (t)}{2} \bigl(k'\circ w \bigr) (t) + \frac{\sigma '(t)}{2} (k\circ w) (t) -\frac{\sigma '(t)}{2} \biggl( \int _{0}^{t} k(s)\,ds \biggr) \bigl\Vert w(t) \bigr\Vert ^{2} , \end{aligned}$$

(3.3)

and

$$ (k \ast w, w) \leq \biggl( \int _{0}^{t} k(s)\,ds \biggr) \bigl\Vert w(t) \bigr\Vert ^{2} + \frac{1}{4} (k\circ w) (t), $$

(3.4)

where ${(k\ast w)(t) =\int _{0}^{t} k(t-s) w(s)\,ds }$.

Now, we denote the modified energy functional $E(t)$ associated with problem (2.10) by

$$\begin{aligned} E(t) =&\frac{1}{\rho +2} \bigl\Vert w_{t} (t) \bigr\Vert _{\rho +2}^{\rho +2}+ \frac{1}{2} \biggl( a_{0}- \sigma (t) \int _{0}^{t} k(s)\,ds \biggr) \bigl\Vert \nabla w(t) \bigr\Vert ^{2} \\ &{} +\frac{b_{0}}{4} \bigl\Vert \nabla w(t) \bigr\Vert ^{4}+ \frac{1}{2} \bigl\Vert \nabla w_{t} (t) \bigr\Vert ^{2} \\ & {}+\frac{1}{2}\sigma (t) ( k\circ \nabla w) (t) +\frac{\xi}{2} \int _{\Gamma _{1}} \int _{0}^{1} \bigl\vert z (x, \delta , t) \bigr\vert ^{q+1}\,d\delta \,d\Gamma \\ &{}+\frac{1}{2} \int _{\Gamma _{1}} h(x) m(x) u^{2} (t)\,d\Gamma - \frac{1}{p} \bigl\Vert w(t) \bigr\Vert _{p}^{p} \\ =&\frac{1}{\rho +2} \bigl\Vert w_{t} (t) \bigr\Vert _{\rho +2}^{\rho +2} +J(t), \end{aligned}$$

(3.5)

where ξ is a positive constant such that

$$ \frac{2\tau {\mu _{2} }q}{q+1}< \xi < \frac{2\tau {\mu _{1} }(q+1)-2\tau \mu _{2}}{q+1} . $$

(3.6)

Note that this choice of ξ is possible from assumption (H4).

Lemma 3.1

Assume that (H2) and (H4) hold. Then, for the solution of problem (2.10), the energy functional $E(t)$ satisfies

$$\begin{aligned} E'(t) \leq& -C_{1} \bigl\Vert w_{t} (t) \bigr\Vert _{q+1, \Gamma _{1} }^{q+1}-C_{2} \int _{\Gamma _{1}} \bigl\vert z(x,1,t) \bigr\vert ^{q+1} \,d\Gamma -b_{1} \biggl( \frac{1}{2} \frac{d}{dt} \bigl\Vert \nabla w(t) \bigr\Vert ^{2} \biggr)^{2} \\ & {}-\frac{\sigma (t)k(t)}{2} \bigl\Vert \nabla w(t) \bigr\Vert ^{2} - \frac{\sigma '(t)}{2} \biggl( \int _{0}^{t} k(s)\,ds \biggr) \bigl\Vert \nabla w(t) \bigr\Vert ^{2} +\frac{\sigma '(t)}{2} (k\circ \nabla w) (t) \\ & {}+\frac{\sigma (t)}{2} \bigl(k'\circ \nabla w \bigr) (t)- \int _{\Gamma _{1}} m(x) g(x) u_{t}^{2} (t)\,d\Gamma \leq 0, \quad \forall t \in [0, t_{0}], \end{aligned}$$

(3.7)

where $C_{1}$ and $C_{2}$ are some positive constants.

Proof

Multiplying in the first equation of (2.10) by $w_{t}$ and integrating over Ω, using (3.3), we have

$$\begin{aligned}& \frac{d}{dt} \biggl[ \frac{1}{\rho +2} \bigl\Vert w_{t} (t) \bigr\Vert _{\rho +2}^{ \rho +2} +\frac{1}{2} \biggl( a_{0}-\sigma (t) \int _{0}^{t} k(s)\,ds \biggr) \bigl\Vert \nabla w(t) \bigr\Vert ^{2} \\& \qquad {} +\frac{b_{0}}{4} \bigl\Vert \nabla w(t) \bigr\Vert ^{4} + \frac{1}{2} \bigl\Vert \nabla w_{t} (t) \bigr\Vert ^{2} \\& \qquad {}+\frac{1}{2} \sigma (t) (k\circ \nabla w) (t) - \frac{1}{p} \bigl\Vert w(t) \bigr\Vert _{p}^{p} + \frac{1}{2} \int _{\Gamma _{1}} h(x) m(x) u^{2} (t)\,d\Gamma \biggr] \\& \quad =-\mu _{1} \bigl\Vert w_{t} (t) \bigr\Vert _{q+1, \Gamma _{1}}^{q+1} -\mu _{2} \int _{ \Gamma _{1}} \bigl\vert z(x,1,t) \bigr\vert ^{q-1} z(x,1,t) w_{t} (t)\,d\Gamma -b_{1} \biggl( \frac{1}{2} \frac{d}{dt} \bigl\Vert \nabla w(t) \bigr\Vert ^{2} \biggr)^{2} \\& \qquad {}-\frac{\sigma (t)}{2} k(t) \bigl\Vert \nabla w(t) \bigr\Vert ^{2} - \frac{\sigma '(t)}{2} \biggl( \int _{0}^{t} k(s)\,ds \biggr) \bigl\Vert \nabla w(t) \bigr\Vert ^{2} +\frac{\sigma '(t)}{2} (k\circ \nabla w) (t) \\& \qquad {} +\frac{\sigma (t)}{2} \bigl(k'\circ \nabla w \bigr) (t) - \int _{\Gamma _{1}} m(x) g(x) u_{t}^{2} (t)\,d\Gamma . \end{aligned}$$

(3.8)

Multiplying the equation in the fourth equation of (2.10) by $\xi |z|^{q-1} z$ and integrating the result over $\Gamma _{1} \times (0,1)$, we obtain

$$\begin{aligned}& \frac{\xi}{2}\frac{d}{dt} \int _{\Gamma _{1}} \int _{0}^{1} \bigl\vert z(x, \delta , t) \bigr\vert ^{q+1}\,d\delta \,d\Gamma \\& \quad =-\frac{\xi}{2\tau} \int _{\Gamma _{1}} \int _{0}^{1} \frac{\partial}{\partial \delta} \bigl\vert z(x, \delta , t) \bigr\vert ^{q+1}\,d\delta \,d\Gamma \\& \quad =-\frac{\xi}{2\tau } \int _{\Gamma _{1}} \bigl\vert z(x,1,t) \bigr\vert ^{q+1} \,d\Gamma +\frac{\xi}{2\tau} \int _{\Gamma _{1}} \bigl\vert w_{t} (t) \bigr\vert ^{q+1}\,d\Gamma . \end{aligned}$$

(3.9)

By using Young’s inequality, we obtain

$$\begin{aligned}& \biggl\vert \mu _{2} \int _{\Gamma _{1}} \bigl\vert z(x,1,t) \bigr\vert ^{q-1} z(x,1,t) w_{t} (t) \Gamma \biggr\vert \\& \quad \leq \frac{\mu _{2} q}{q+1} \int _{\Gamma _{1}} \bigl\vert z(x,1,t) \bigr\vert ^{q+1} \,d\Gamma +\frac{\mu _{2} }{q+1} \int _{\Gamma _{1}} \bigl\vert w_{t} (t) \bigr\vert ^{q+1}\,d\Gamma . \end{aligned}$$

(3.10)

Thus, from (3.8)–(3.10) and the definition of $E(t)$, we have

$$\begin{aligned} E'(t) \leq& - \biggl( \mu _{1} -\frac{\xi}{2\tau} - \frac{\mu _{2}}{q+1} \biggr) \bigl\Vert w_{t} (t) \bigr\Vert _{q+1, \Gamma _{1}}^{q+1}- \biggl( \frac{\xi}{2\tau}-\frac{\mu _{2} q}{q+1} \biggr) \int _{\Gamma _{1}} \bigl\vert z(x,1,t) \bigr\vert ^{q+1} \,d\Gamma \\ & {}-b_{1} \biggl( \frac{1}{2} \frac{d}{dt} \bigl\Vert \nabla w(t) \bigr\Vert ^{2} \biggr)^{2} - \frac{\sigma (t)}{2} k(t) \bigl\Vert \nabla w(t) \bigr\Vert ^{2} - \frac{\sigma '(t)}{2} \biggl( \int _{0}^{t} k(s)\,ds \biggr) \bigl\Vert \nabla w(t) \bigr\Vert ^{2} \\ & {}+\frac{\sigma '(t)}{2} (k\circ \nabla w) (t)+ \frac{\sigma (t)}{2} \bigl(k'\circ \nabla w \bigr) (t) - \int _{\Gamma _{1}} m(x) g(x) u_{t}^{2} (t)\,d\Gamma . \end{aligned}$$

Using (3.6), we take $C_{1} =\mu _{1} -\frac{\xi}{2\tau} -\frac{\mu _{2}}{q+1}>0$ and $C_{2} =\frac{\xi}{2\tau}-\frac{\mu _{2} q}{q+1}>0$. From (2.6), we obtain the desired inequality (3.7). □

Lemma 3.2

Suppose that (H1) and (H2) hold. Let $(w,u,z)$ be the solution of problem (2.10). Assume that $I(0)>0$ and

$$ \alpha =\frac{C_{*}^{p}}{l} \biggl( \frac{2pE(0)}{l(p-2)} \biggr)^{ \frac{p-2}{2} }< 1 . $$

(3.11)

Then, $I(t)>0$ for $t\in [0,T]$, where $I(t)$ is defined in (3.2).

Proof

Since $I(0)>0$ and continuity of $w(t)$, then there exists $t_{1} < T$ such that

$$ I(t) \geq 0,\quad \forall t\in [0, t_{1}]. $$

(3.12)

From (2.5), (3.1), (3.2), and (3.12), we obtain

$$\begin{aligned} J(t) =& \frac{p-2}{2p} \biggl[ \biggl( a_{0}-\sigma (t) \int _{0}^{t} k(s)\,ds \biggr) \bigl\Vert \nabla w(t) \bigr\Vert ^{2} +\frac{b_{0}}{2} \bigl\Vert \nabla w(t) \bigr\Vert ^{4} \\ &{}+ \bigl\Vert \nabla w_{t} (t) \bigr\Vert ^{2} + \sigma (t) ( k\circ \nabla w) (t) \\ & {}+\xi \int _{\Gamma _{1}} \int _{0}^{1} \bigl\vert z (x, \delta , t) \bigr\vert ^{q+1}\,d\delta \,d\Gamma + \int _{\Gamma _{1}} h(x) m(x) u^{2} (t)\,d\Gamma \biggr]+ \frac{1}{p} I(t) \\ \geq& \frac{p-2}{2p} l \bigl\Vert \nabla w(t) \bigr\Vert ^{2} , \quad \forall t \in [0,t_{1}]. \end{aligned}$$

(3.13)

Using (3.5), (3.7), and (3.13), we obtain

$$\begin{aligned} l \bigl\Vert \nabla w(t) \bigr\Vert ^{2} \leq \frac{2p}{p-2} J(t) \leq \frac{2p}{p-2} E(t) \leq \frac{2p}{p-2} E(0),\quad \forall t\in \bigl[0, T^{*} \bigr], \end{aligned}$$

(3.14)

where $T^{*} =\min \{t_{0}, t_{1}\}$. Applying (2.1), (2.5), (3.11), and (3.14), we have

$$\begin{aligned} \bigl\Vert w(t) \bigr\Vert _{p}^{p} \leq& C_{*}^{p} \bigl\Vert \nabla w(t) \bigr\Vert ^{p} \\ \leq&\alpha l \bigl\Vert \nabla w(t) \bigr\Vert ^{2} \leq \biggl( a_{0}-\sigma (t) \int _{0}^{t} k(s)\,ds \biggr) \bigl\Vert \nabla w(t) \bigr\Vert ^{2}, \quad \forall t\in \bigl[0,T^{*} \bigr]. \end{aligned}$$

Consequently, we arrive at

$$\begin{aligned} I(t) =& \biggl( a_{0}-\sigma (t) \int _{0}^{t} k(s)\,ds \biggr) \bigl\Vert \nabla w(t) \bigr\Vert ^{2} +\frac{b_{0}}{2} \bigl\Vert \nabla w(t) \bigr\Vert ^{4} + \bigl\Vert \nabla w_{t} (t) \bigr\Vert ^{2} +\sigma (t) (k\circ \nabla w) (t) \\ & {}+\xi \int _{\Gamma _{1}} \int _{0}^{1} \bigl\vert z (x, \delta , t) \bigr\vert ^{q+1}\,d\delta \,d\Gamma + \int _{\Gamma _{1}} h(x) m(x) u^{2} (t)\,d\Gamma \\ &{}- \bigl\Vert w(t) \bigr\Vert _{p}^{p}>0,\quad \forall t\in \bigl[0,T^{*} \bigr]. \end{aligned}$$

By repeating this procedure, and using the fact that

$$ \lim_{t\to T^{*}} \frac{C_{*}^{p}}{l} \biggl( \frac{2pE(t)}{l(p-2)} \biggr)^{\frac{p-2}{2}} \leq \alpha < 1, $$

$T^{*}$ is extended to T. Thus, the proof is complete. □

We state the global existence result, which can be obtained by the arguments of [9, 22, 24].

Theorem 3.1

Suppose that (H1)–(H4) hold. Let $(w_{0} , w_{1})\in (H^{2} (\Omega ) \cap V)\times V$, $u_{0} \in L^{2} (\Gamma _{1})$, $f_{0} \in L^{2} (\Gamma _{1} \times (0,1))$. If $I(0)>0$ and satisfy (3.11), then the solution $(w,u,z)$ of (2.10) is bounded and global in time.

Now, we will establish the general decay property of the solution for problem (2.10) in the case $\mu _{2} < \mu _{1}$. For this purpose, we define the functional

$$ \Xi (t) =M E(t) +\varepsilon \sigma (t) \Phi _{1} (t)+ \sigma (t) \Phi _{2} (t), $$

(3.15)

where M and ε are positive constants that will be specified later and

$$\begin{aligned} \Phi _{1} (t) = &\frac{1}{\rho +1} \int _{\Omega } \bigl\vert w_{t} (t) \bigr\vert ^{\rho} w_{t} (t) w(t)\,dx +\frac{b_{1}}{4} \bigl\Vert \nabla w(t) \bigr\Vert ^{4} + \int _{ \Omega }\nabla w_{t} (t) \nabla w(t)\,dx \\ & {}+ \int _{\Gamma _{1}} m(x) w(t) u(t)\,d\Gamma +\frac{1}{2} \int _{\Gamma _{1}} m(x) g(x) u^{2} (t)\,d\Gamma , \end{aligned}$$

(3.16)

and

$$\begin{aligned} \Phi _{2}(t) =& -\frac{1}{\rho +1} \int _{\Omega } \bigl\vert w_{t} (t) \bigr\vert ^{\rho }w_{t} (t) \int _{0}^{t} k(t-s) \bigl(w(t)-w(s) \bigr)\,ds\,dx \\ & {}- \int _{\Omega }\nabla w_{t} (t) \int _{0}^{t} k(t-s) \bigl( \nabla w(t)-\nabla w(s) \bigr)\,ds\,dx . \end{aligned}$$

(3.17)

Before we show our main result, we need the following lemmas.

Lemma 3.3

Let $w\in L^{\infty }([0,T]; H_{0}^{1} (\Omega ))$, then we have

$$\begin{aligned} \int _{\Omega } \biggl( \sigma (t) \int _{0}^{t} k(t-s) \bigl(w(t)-w(s) \bigr)\,ds \biggr)^{\rho +2}\,dx \leq (a_{0}-l)^{\rho +1} \alpha _{1} \sigma (t) (k \circ \nabla w) (t), \end{aligned}$$

(3.18)

where $\alpha _{1} =C_{*}^{\rho +2} ( \frac{2p E(0)}{l(p-2)} )^{ \frac{\rho}{2}}$.

Proof

From (2.1), (2.5), (3.14), and Hölder’s inequality, we obtain

$$\begin{aligned}& \int _{\Omega } \biggl( \sigma (t) \int _{0}^{t} k(t-s) \bigl(w(t)-w(s) \bigr)\,ds \biggr)^{\rho +2}\,dx \\& \quad \leq \int _{\Omega } \biggl( \sigma (t) \int _{0}^{t} k(t-s)\,ds \biggr)^{\rho +1} \biggl( \sigma (t) \int _{0}^{t} k(t-s) \bigl\vert w(t)-w(s) \bigr\vert ^{ \rho +2}\,ds \biggr)\,dx \\& \quad \leq (a_{0}-l)^{\rho +1} C_{*}^{\rho +2} \biggl( \frac{2p E(0)}{l(p-2)} \biggr)^{\frac{\rho}{2}} \sigma (t) (k\circ \nabla w ) (t). \end{aligned}$$

 □

Lemma 3.4

Let $(w, u, z)$ be the solution of (2.10) and suppose that (H1)–(H3) hold, then there exist two positive constants $\beta _{1}$ and $\beta _{2}$ such that

$$ \beta _{1} E(t) \leq \Xi (t) \leq \beta _{2} E(t), \quad \forall t \geq 0. $$

(3.19)

Proof

Using (2.1), (2.2), (2.8), (3.14), and Young’s inequality, we obtain

$$\begin{aligned}& \biggl\vert \frac{1}{\rho +1} \int _{\Omega } \bigl\vert w_{t} (t) \bigr\vert ^{\rho }w_{t} (t) w(t)\,dx \biggr\vert \\& \quad \leq \frac{1}{\rho +2} \bigl\Vert w_{t} (t) \bigr\Vert _{\rho +2}^{\rho +2} + \frac{\alpha _{1}}{(\rho +2)(\rho +1)} \bigl\Vert \nabla w(t) \bigr\Vert ^{2} , \end{aligned}$$

(3.20)

$$\begin{aligned}& \biggl\vert \int _{\Omega }\nabla w_{t} (t) \nabla w(t)\,dx \biggr\vert \leq \frac{1}{2} \bigl\Vert \nabla w_{t} (t) \bigr\Vert ^{2} +\frac{1}{2} \bigl\Vert \nabla w(t) \bigr\Vert ^{2}, \end{aligned}$$

(3.21)

$$\begin{aligned}& \biggl\vert \int _{\Gamma _{1}} m(x) w(t) u(t)\,d\Gamma \biggr\vert \leq \frac{ \Vert m \Vert _{\infty}}{2h_{1} } \int _{\Gamma _{1}} h(x) m(x) u^{2} (t)\,d\Gamma + \frac{\tilde{C}^{2}_{*}}{2} \bigl\Vert \nabla w(t) \bigr\Vert ^{2}. \end{aligned}$$

(3.22)

Similarly, using (2.1), (2.5), (3.18), and Young’s inequality, we see that

$$\begin{aligned}& \biggl\vert \frac{1}{\rho +1} \int _{\Omega }\sigma (t) \bigl\vert w_{t} (t) \bigr\vert ^{ \rho }w_{t} (t) \int _{0}^{t} k(t-s) \bigl(w(t)-w(s) \bigr)\,ds \,dx \biggr\vert \\& \quad \leq \frac{\sigma (t)}{\rho +2} \bigl\Vert w_{t} (t) \bigr\Vert _{\rho +2}^{\rho +2}+ \frac{(a_{0}-l)^{\rho +1} \alpha _{1}}{(\rho +2)(\rho +1)} \sigma (t) (k \circ \nabla w) (t), \end{aligned}$$

(3.23)

and

$$\begin{aligned}& \biggl\vert - \int _{\Omega }\sigma (t)\nabla w_{t} (t) \int _{0}^{t} k(t-s) \bigl(\nabla w(t)-\nabla w(s) \bigr)\,ds \,dx \biggr\vert \\& \quad \leq \frac{\sigma (t)}{2} \bigl\Vert \nabla w_{t} (t) \bigr\Vert ^{2} +\frac{k_{0}}{2} \sigma (t) (k \circ \nabla w) (t). \end{aligned}$$

(3.24)

Combining (3.15)–(3.17), (3.20)–(3.24), and using (H2), we obtain

$$\begin{aligned}& \bigl\vert \Xi (t) -ME(t) \bigr\vert \\& \quad \leq \varepsilon \sigma (t) \bigl\vert \Phi _{1} (t) \bigr\vert + \sigma (t) \bigl\vert \Phi _{2} (t) \bigr\vert \\& \quad \leq \frac{\sigma (t)(\varepsilon +1)}{\rho +2} \bigl\Vert w_{t} (t) \bigr\Vert _{ \rho +2}^{\rho +2} +\frac{\sigma (t)(\varepsilon +1)}{2} \bigl\Vert \nabla w_{t} (t) \bigr\Vert ^{2} \\& \qquad {}+\varepsilon \sigma (t) \biggl( \frac{\alpha _{1}}{(\rho +2)(\rho +1)}+ \frac{\tilde{C}_{*}^{2} }{2}+ \frac{1}{2} \biggr) \bigl\Vert \nabla w(t) \bigr\Vert ^{2} \\& \qquad {}+ \biggl( \frac{(a_{0}-l)^{\rho +1} \alpha _{1}}{(\rho +2)(\rho +1)} + \frac{k_{0}}{2} \biggr) \alpha (t) (k \circ \nabla w) (t) \\& \qquad {}+\varepsilon \sigma (t) \biggl(\frac{b_{1}}{4} \bigl\Vert \nabla w(t) \bigr\Vert ^{4} + \frac{ \Vert m \Vert _{\infty} + \Vert g \Vert _{\infty}}{2h_{1}} \int _{\Gamma _{1}} h(x) m(x) u^{2} (t)\,d\Gamma \biggr) \\& \quad \leq CE(t), \end{aligned}$$

where C is some positive constant. Choosing $M>0$ sufficiently large and ε small, we obtain (3.19). □

The following theorem is our main result.

Theorem 3.2

Suppose that (H1)–(H4) and (3.6) hold. If $(w_{0}, w_{1}) \in (H^{2} (\Omega )\cap V) \times V$, $u_{0} \in L^{2} (\Gamma _{1})$, $f_{0} \in L^{2} (\Gamma _{1} \times (0,1))$ and satisfying (3.11). Then, for any $t>t_{0}^{*}$, there exist positive constants K and κ such that the energy of the solution for problem (2.10) satisfies

$$\begin{aligned} E(t) =K e^{-\kappa \int _{t_{0}^{*}}^{t} \sigma (s) \zeta (s)\,ds} , \quad \forall t\geq t_{0}^{*} . \end{aligned}$$

(3.25)

Proof

From Lemma 3.4, it suffices to prove that we obtain the estimate of $\Xi (t)$. For this purpose, first we estimate $\Phi _{1}'(t)$. It follows from (2.10) and (3.16) that

$$\begin{aligned} \Phi _{1}'(t) =&- \bigl( a_{0}+b_{0} \bigl\Vert \nabla w(t) \bigr\Vert ^{2} \bigr) \int _{\Omega} \bigl\vert \nabla w(t) \bigr\vert ^{2} \,dx+ \int _{\Omega }\nabla w(t) \sigma (t) \int _{0}^{t} k(t-s) \nabla w(s)\,ds \,dx \\ & {}-\mu _{1} \int _{\Gamma _{1}} \bigl\vert w_{t} (t) \bigr\vert ^{q-1} w_{t} (t) w(t)\,d\Gamma -\mu _{2} \int _{\Gamma _{1}} \bigl\vert z(x,1,t) \bigr\vert ^{q-1} z(x,1,t) w(t)\,d\Gamma \\ &{}+\frac{1}{\rho +1} \bigl\Vert w_{t} (t) \bigr\Vert _{\rho +2}^{\rho +2} \\ & {}+2 \int _{\Gamma _{1}} m(x) w(t) u_{t} (t)\,d\Gamma - \int _{\Gamma _{1}} h(x) m(x) u^{2} (t)\,d\Gamma + \int _{\Omega} \bigl\vert \nabla w_{t} (t) \bigr\vert ^{2}\,dx \\ &{}+ \int _{\Omega} \bigl\vert w(t) \bigr\vert ^{p}\,dx . \end{aligned}$$

(3.26)

We will estimate the right-hand side of (3.26). By using (2.2), (2.5), (2.8), (3.14), and Young’s inequality, for any $\eta >0$, we have

$$\begin{aligned}& \biggl\vert \int _{\Omega} \nabla w(t) \sigma (t) \int _{0}^{t} k(t-s) \nabla w(s)\,ds \,dx \biggr\vert \\& \quad \leq \biggl\vert \int _{\Omega} \nabla w(t) \sigma (t) \int _{0}^{t} k(t-s) \bigl(\nabla w(s)-\nabla w(t) \bigr)\,ds \,dx \biggr\vert \\& \qquad {} +\sigma (t) \int _{0}^{t} k(s)\,ds \int _{\Omega } \bigl\vert \nabla w(t) \bigr\vert ^{2} \,dx \\& \quad \leq (1+\eta ) (a_{0}-l) \bigl\Vert \nabla w(t) \bigr\Vert ^{2} + \frac{\sigma (t)}{4\eta} (k\circ \nabla w) (t) , \end{aligned}$$

(3.27)

$$\begin{aligned}& \biggl\vert \mu _{1} \int _{\Gamma _{1}} \bigl\vert w_{t} (t) \bigr\vert ^{q-1} w_{t} (t) w(t)\,d\Gamma \biggr\vert \leq \mu _{1} \eta \alpha _{2} \bigl\Vert \nabla w(t) \bigr\Vert ^{2} + \mu _{1} C_{\eta} \bigl\Vert w_{t} (t) \bigr\Vert _{q+1, \Gamma _{1}}^{q+1} \end{aligned}$$

(3.28)

$$\begin{aligned}& \biggl\vert \mu _{2} \int _{\Gamma _{1}} \bigl\vert z(x,1,t) \bigr\vert ^{q-1} z(x,1,t) u(t)\,d\Gamma \biggr\vert \\& \quad \leq \mu _{2} \eta \alpha _{2} \bigl\Vert \nabla w(t) \bigr\Vert ^{2}+ \mu _{2} C_{\eta} \bigl\Vert z(x,1,t) \bigr\Vert _{q+1, \Gamma _{1}}^{q+1} \end{aligned}$$

(3.29)

and

$$\begin{aligned} & \biggl\vert 2 \int _{\Gamma _{1}} m(x) w(t) u_{t} (t)\,d\Gamma \biggr\vert \leq \eta \tilde{C}_{*}^{2} \bigl\Vert \nabla w(t) \bigr\Vert ^{2} + \frac{ \Vert m \Vert _{\infty}}{\eta g_{1}} \int _{\Gamma _{1}} m(x) g(x) u_{t}^{2} (t)\,d\Gamma , \end{aligned}$$

(3.30)

where $\alpha _{2}=\tilde{C}_{*}^{{q+1}} ( \frac{2pE(0) }{l(p-2)} )^{ \frac{q-1}{2}}$. Choosing η small enough such that

$$ \eta \bigl(a_{0}-l+\tilde{C}_{*}^{2}+\mu _{1} \alpha _{2}+\mu _{2} \alpha _{2} \bigr) \leq \frac{l}{2}$$

and substituting of (3.27)–(3.30) into (3.26), we obtain

$$\begin{aligned} \Phi _{1}'(t) \leq& -\frac{l}{2} \bigl\Vert \nabla w(t) \bigr\Vert ^{2} -b_{0} \bigl\Vert \nabla w(t) \bigr\Vert ^{4} +\frac{1}{\rho +1} \bigl\Vert w_{t} (t) \bigr\Vert _{\rho +2}^{\rho +2}+ \bigl\Vert \nabla w_{t} (t) \bigr\Vert ^{2} + \bigl\Vert w(t) \bigr\Vert _{p}^{p} \\ & {}+\frac{\sigma (t)}{4\eta} (k\circ \nabla w) (t) +\mu _{1} C_{ \eta} \bigl\Vert w_{t} (t) \bigr\Vert _{q+1, \Gamma _{1}}^{q+1} +\mu _{2} C_{\eta} \bigl\Vert z(x,1,t) \bigr\Vert _{q+1, \Gamma _{1}}^{q+1} \\ & {}+\frac{ \Vert m \Vert _{\infty}}{\eta g_{1}} \int _{\Gamma _{1}} m(x) g(x) u_{t}^{2} (t)\,d\Gamma - \int _{\Gamma _{1}} h(x) m(x) u^{2} (t)\,d\Gamma . \end{aligned}$$

(3.31)

Next, we would like to estimate $\Phi _{2}'(t)$. Taking the derivative of $\Phi _{2} (t)$ in (3.17) and using (2.10), we obtain

$$\begin{aligned} \Phi _{2}'(t) =& \bigl( a_{0}+b_{0} \bigl\Vert \nabla w(t) \bigr\Vert ^{2} \bigr) \int _{ \Omega }\nabla w(t) \int _{0}^{t} k(t-s) \bigl(\nabla w(t)-\nabla w(s) \bigr)\,ds \,dx \\ & {}+b_{1} \int _{\Omega }\nabla w(t) \nabla w_{t} (t)\,dx \int _{ \Omega }\nabla w(t) \int _{0}^{t} k(t-s) \bigl(\nabla w(t)-\nabla w(s) \bigr)\,ds \,dx \\ & {}- \int _{\Omega }\sigma (t) \int _{0}^{t} k(t-s) \nabla w(s)\,ds \int _{0}^{t} k(t-s) \bigl(\nabla w(t)-\nabla w(s) \bigr)\,ds \,dx \\ & {}- \int _{\Omega } \bigl\vert w(t) \bigr\vert ^{p-2} w(t) \int _{0}^{t} k(t-s) \bigl(w(t)-w(s) \bigr)\,ds \,dx \\ & {}- \int _{\Omega }\nabla w_{t} (t) \int _{0}^{t} k'(t-s) \bigl( \nabla w(t)-\nabla w(s) \bigr)\,ds \,dx \\ & {}-\frac{1}{\rho +1} \int _{\Omega } \bigl\vert w_{t} (t) \bigr\vert ^{\rho }w_{t} (t) \int _{0}^{t} k'(t-s) \bigl(w(t)-w(s) \bigr)\,ds \,dx \\ & {}- \int _{\Gamma _{1}} m(x) u_{t} (t) \int _{0}^{t} k(t-s) \bigl(w(t)-w(s) \bigr)\,ds \,d \Gamma \\ & {}+\mu _{2} \int _{\Gamma _{1}} \bigl\vert z(x,1,t) \bigr\vert ^{q-1} z( x,1,t) \int _{0}^{t} k(t-s) \bigl(w(t)-w(s) \bigr)\,ds \,d \Gamma \\ & {}+\mu _{1} \int _{\Gamma _{1}} \bigl\vert w_{t} (x,t) \bigr\vert ^{q-1} w_{t} (x,t) \int _{0}^{t} k(t-s) \bigl(w(t)-w(s) \bigr)\,ds \,d \Gamma \\ & {}- \biggl( \int _{0}^{t} k(s)\,ds \biggr) \bigl\Vert \nabla w_{t} (t) \bigr\Vert ^{2} - \frac{1}{\rho +1} \biggl( \int _{0}^{t} k(s)\,ds \biggr) \bigl\Vert w_{t} (t) \bigr\Vert _{\rho +2}^{ \rho +2} \\ :=& E_{1} + E_{2} +\cdots + E_{9} - \biggl( \int _{0}^{t} k(s)\,ds \biggr) \bigl\Vert \nabla w_{t} (t) \bigr\Vert ^{2} \\ &{}-\frac{1}{\rho +1} \biggl( \int _{0}^{t} k(s)\,ds \biggr) \bigl\Vert w_{t} (t) \bigr\Vert _{\rho +2}^{\rho +2} . \end{aligned}$$

(3.32)

Now, we will estimate the right-hand side of (3.32). By (2.1), (2.2), (2.5), (2.8), (3.7), (3.13), (3.14), and Young’s inequality, for any $\gamma >0$, we derive the following inequalities

$$\begin{aligned}& \vert E_{1} \vert \leq \biggl\vert \int _{\Omega } \biggl( a_{0}+\frac{2b_{0} pE(0)}{l(p-2)} \biggr) \nabla w(t) \int _{0}^{t} k(t-s) \bigl(\nabla w(t)-\nabla w(s) \bigr)\,ds \,dx \biggr\vert \\& \hphantom{\vert E_{1} \vert }\leq \gamma \bigl\Vert \nabla w(t) \bigr\Vert ^{2} + \frac{k_{0}}{4\gamma} \biggl( a_{0} + \frac{2b_{0} pE(0)}{l(p-2)} \biggr)^{2} (k\circ \nabla w) (t), \end{aligned}$$

(3.33)

$$\begin{aligned}& \vert E_{2} \vert \leq \gamma b_{1}^{2} \biggl( \int _{\Omega }\nabla w(t) \nabla w_{t} (t)\,dx \biggr)^{2} \bigl\Vert \nabla w(t) \bigr\Vert ^{2} \\& \hphantom{\vert E_{2} \vert \leq}{} + \frac{1}{4\gamma} \int _{\Omega } \biggl( \int _{0}^{t} k(t-s) \bigl(\nabla w(t)-\nabla w(s) \bigr)\,ds \biggr)^{2}\,dx \\& \hphantom{\vert E_{2} \vert } \leq -\frac{2\gamma b_{1}^{2} pE(0)}{l(p-2)} E' (t) + \frac{k_{0}}{4\gamma}(k\circ \nabla w) (t), \end{aligned}$$

(3.34)

$$\begin{aligned}& \vert E_{3} \vert \leq \gamma \int _{\Omega }\sigma (t) \biggl( \int _{0}^{t} k(t-s) \bigl( \bigl\vert \nabla w(t)-\nabla w(s) \bigr\vert + \bigl\vert \nabla w(t) \bigr\vert \bigr)\,ds \biggr)^{2}\,dx \\& \hphantom{\vert E_{3} \vert \leq}{}+\frac{1}{4\gamma} \int _{\Omega }\sigma (t) \biggl( \int _{0}^{t} k(t-s) \bigl\vert \nabla w(t)-\nabla w(s) \bigr\vert \,ds \biggr)^{2}\,dx \\& \hphantom{\vert E_{3} \vert } \leq \biggl( 2\gamma +\frac{1}{4\gamma} \biggr) (a_{0}-l) (k \circ \nabla w) (t) +2\gamma (a_{0}-l)k_{0} \bigl\Vert \nabla w(t) \bigr\Vert ^{2}, \end{aligned}$$

(3.35)

$$\begin{aligned}& \vert E_{4} \vert \leq \gamma \int _{\Omega } \bigl\vert w(t) \bigr\vert ^{2(p-1)}\,dx + \frac{C_{*}^{2} k_{0}}{4\gamma}(k \circ \nabla w) (t) \\& \hphantom{\vert E_{4} \vert }\leq \gamma \alpha _{3} \bigl\Vert \nabla w(t) \bigr\Vert ^{2} +\frac{C_{*}^{2} k_{0}}{4\gamma} (k \circ \nabla w) (t), \end{aligned}$$

(3.36)

$$\begin{aligned}& \vert E_{5} \vert \leq \gamma \bigl\Vert \nabla w_{t} (t) \bigr\Vert ^{2} - \frac{k(0)}{4\gamma} \bigl(k'\circ \nabla w \bigr) (t), \end{aligned}$$

(3.37)

$$\begin{aligned}& \vert E_{6} \vert \leq \frac{\gamma \alpha _{4}}{\rho +1} \bigl\Vert \nabla w_{t} (t) \bigr\Vert ^{2} -\frac{k(0) C_{*}^{2} }{4\gamma (\rho +1)} \bigl(k'\circ \nabla w \bigr) (t), \end{aligned}$$

(3.38)

$$\begin{aligned}& \vert E_{7} \vert \leq \frac{\gamma \Vert m \Vert _{\infty}}{g_{1}} \int _{\Gamma _{1}} m(x) g(x) u_{t}^{2} (t)\,d\Gamma +\frac{ \tilde{C}_{*}^{2}k_{0}}{4\gamma} (k \circ \nabla w) (t), \end{aligned}$$

(3.39)

$$\begin{aligned}& \vert E_{8} \vert \leq \gamma \mu _{2} \bigl\Vert z(x,1,t) \bigr\Vert _{q+1, \Gamma _{1}}^{q+1} +\mu _{2} C_{ \gamma }k_{0}^{q} \alpha _{2} (k\circ \nabla w) (t) , \end{aligned}$$

(3.40)

and

$$\begin{aligned} & \vert E_{9} \vert \leq \gamma \mu _{1} \alpha _{5} \bigl\Vert \nabla w_{t} (t) \bigr\Vert ^{2} + \frac{\mu _{1} \tilde{C}_{*}^{2} k_{0}}{4\gamma} (k\circ \nabla w) (t), \end{aligned}$$

(3.41)

where $\alpha _{3}=C_{*}^{2(p-1)} ( \frac{2pE(0)}{l(p-2)} )^{p-2}$, $\alpha _{4} =C_{*}^{2(\rho +1)} ( \frac{2pE(0)}{p-2} )^{\rho}$, and $\alpha _{5} = \tilde{C}_{*}^{2q} ( \frac{2pE(0)}{p-2} )^{q-1}$. Thus, from (3.32)–(3.41), we conclude that

$$\begin{aligned} \Phi '_{2}(t) \leq& - \frac{1}{\rho +1} \biggl( \int _{0}^{t} k(s)\,ds \biggr) \bigl\Vert w_{t} (t) \bigr\Vert _{\rho +2}^{\rho +2} \\ &{}- \biggl( \int _{0}^{t} k(s)\,ds - \gamma \biggl(1+ \mu _{1} \alpha _{5} +\frac{\alpha _{4}}{\rho +1} \biggr) \biggr) \bigl\Vert \nabla w_{t} (t) \bigr\Vert ^{2} \\ & {}+ C_{3} (k\circ \nabla w) (t)-\frac{k(0)}{4\gamma} \biggl( 1+ \frac{C_{*}^{2}}{\rho +1} \biggr) \bigl(k'\circ \nabla w \bigr) (t) \\ &{} - \frac{2\gamma b_{1}^{2} p E(0)}{l(p-2)} E'(t)+\gamma \mu _{2} \bigl\Vert z(x,1,t) \bigr\Vert _{q+1, \Gamma _{1}}^{q+1} \\ & {}+\gamma \bigl( 1+2(a_{0}-l)k_{0} +\alpha _{3} \bigr) \bigl\Vert \nabla w(t) \bigr\Vert ^{2} + \frac{\gamma \Vert m \Vert _{\infty}}{g_{1}} \int _{\Gamma _{1}} m(x) g(x) u_{t}^{2} (t)\,d\Gamma , \end{aligned}$$

(3.42)

where $C_{3} = \frac{1}{4\gamma} \{ k_{0} ( a_{0}+ \frac{2 b_{0} p E(0)}{l(p-2)} )^{2} +( 8 \gamma ^{2} +1) (a_{0}-l) +k_{0}(1+{C}_{*}^{2}+\tilde{C}_{*}^{2} +\mu _{1} \tilde{C}_{*}^{2})+4 \gamma \mu _{2} C_{\gamma }k_{0}^{q} \alpha _{2} \}$. Similarly to Lemma 3.4, for any $\lambda >0$, we obtain

$$\begin{aligned} \sigma '(t) \Phi _{1} (t) \leq& -\frac{ \sigma '(t)}{\rho +2} \bigl\Vert w_{t} (t) \bigr\Vert _{\rho +2}^{\rho +2} - C_{4}\sigma '(t) \bigl\Vert \nabla w(t) \bigr\Vert ^{2} - \frac{ \sigma '(t) }{2} \bigl\Vert \nabla w_{t} (t) \bigr\Vert ^{2} \\ & {}-\frac{ \sigma '(t)b_{1}}{4} \bigl\Vert \nabla w(t) \bigr\Vert ^{4} \\ &{} - \frac{ \sigma '(t) ( \Vert m \Vert _{\infty}+ \Vert g \Vert _{\infty})}{2h_{1} } \int _{ \Gamma _{1}} h(x) m(x) u^{2} (t)\,d\Gamma \end{aligned}$$

(3.43)

and

$$\begin{aligned} \sigma '(t) \Phi _{2}(t) \leq -\frac{\lambda \sigma '(t)}{\rho +2} \bigl\Vert w_{t} (t) \bigr\Vert _{\rho +2}^{\rho +2}-{\lambda \sigma '(t)} \bigl\Vert \nabla w_{t} (t) \bigr\Vert ^{2} -C_{5}\sigma '(t) (k \circ \nabla w) (t) , \end{aligned}$$

(3.44)

where $C_{4} = \frac{1}{2} +\frac{\tilde{C}_{*}^{2}}{2}+ \frac{\alpha _{1}}{(\rho +2)(\rho +1)}$ and $C_{5} = \frac{C_{\lambda}{k_{0}}^{\rho +1} \alpha _{1}}{(\rho +2)(\rho +1)} + \frac{k_{0}}{4\lambda}$. Since k is positive, we have, for any $t_{0}^{*} >0$, $\int _{0}^{t} k(s)\,ds \geq \int _{0}^{t_{0}^{*}} k(s)\,ds := k_{1}>0$, for all $t\geq t_{0}^{*}$. Applying (3.7), (3.31), and (3.42)–(3.44), we find that for any $t\geq t_{0}^{*}$,

$$\begin{aligned} \Xi '(t) =&M E'(t) +\varepsilon \sigma ' (t) \Phi _{1}(t) + \varepsilon \sigma (t) \Phi _{1}'(t) +\sigma '(t) \Phi _{2} (t) + \sigma (t) \Phi _{2}'(t) \\ \leq& -\sigma (t) \biggl( \frac{k_{1}-\varepsilon}{\rho +1}+ \frac{(\varepsilon +\lambda ) \sigma '(t)}{(\rho +2)\sigma (t)} \biggr) \bigl\Vert w_{t} (t) \bigr\Vert _{\rho +2}^{\rho +2} -\sigma (t) \biggl( b_{0} \varepsilon + \frac{\varepsilon b_{1} \sigma '(t)}{4\sigma (t)} \biggr) \bigl\Vert \nabla w(t) \bigr\Vert ^{4} \\ & {}-\sigma (t) \biggl( \biggl(\frac{k(t)}{2} + \frac{ \sigma '(t)}{2\sigma (t)} \int _{0}^{t} k(s)\,ds \biggr)M \\ &{}+ \frac{\varepsilon C_{4} \sigma '(t)}{\sigma (t)} + \frac{\varepsilon l}{2} -\gamma \bigl( 1+2 (a_{0}-l)k_{0} + \alpha _{3} \bigr) \biggr) \bigl\Vert \nabla w(t) \bigr\Vert ^{2} \\ & {}-\sigma (t) \biggl( k_{1}-\gamma \biggl(1+\mu _{1} \alpha _{5} + \frac{\alpha _{4} }{q+1} \biggr)-\varepsilon + \frac{(\varepsilon +2\lambda )\sigma '(t)}{2\sigma (t)} \biggr) \bigl\Vert \nabla w_{t} (t) \bigr\Vert ^{2} \\ &{} +\varepsilon \sigma (t) \bigl\Vert w(t) \bigr\Vert _{p}^{p} \\ & {}+\sigma (t) \biggl( \frac{M\sigma '(t)}{2\sigma (t)}+ \frac{\varepsilon \sigma (t)}{4\eta} - \frac{C_{5}\sigma '(t)}{\sigma (t)} +C_{3} \biggr) (k\circ \nabla w) (t) \\ &{} -\sigma (t) \biggl( \frac{C_{1} M}{\sigma (t)}-\varepsilon \mu _{1} C_{ \eta } \biggr) \bigl\Vert w_{t} (t) \bigr\Vert _{q+1, \Gamma _{1}}^{q+1} \\ & {}-\sigma (t) \biggl( \frac{C_{2} M}{\sigma (t)}-\varepsilon \mu _{2} C_{\eta}-\gamma \mu _{2} \biggr) \bigl\Vert z(x,1,t) \bigr\Vert _{q+1, \Gamma _{1}}^{q+1} \\ &{} + \sigma (t) \biggl( \frac{M}{2}- \frac{k(0)}{4\gamma} \biggl( 1+ \frac{C_{*}^{2}}{\rho +1} \biggr) \biggr) \bigl(k'\circ \nabla w \bigr) (t) \\ & {}-\sigma (t) \biggl( \varepsilon + \frac{\varepsilon \sigma '(t) ( \Vert m \Vert _{\infty }+ \Vert g \Vert _{\infty}) }{2h_{1}\sigma (t)} \biggr) \int _{\Gamma _{1}} h(x) m(x) u^{2} (t)\,d\Gamma \\ &{}-\sigma (t) \frac{2\gamma b_{1}^{2} pE(0)}{l(p-2)} E'(t) \\ & {}-\sigma (t) \biggl( \frac{M}{\sigma (t)} - \frac{\varepsilon \Vert m \Vert _{\infty}}{\eta g_{1}} - \frac{\gamma \Vert m \Vert _{\infty }}{g_{1}} \biggr) \int _{\Gamma _{1}} m(x) g(x) u_{t}^{2} (t)\,d\Gamma . \end{aligned}$$

(3.45)

Since $\lim_{t \to \infty} \frac{\sigma '(t)}{\sigma (t)}=0$, we choose $t_{0}^{*} >0$ sufficiently large. At this point, we pick $\varepsilon >0$ and $\gamma >0$ sufficiently small and we take M sufficiently large such that for $t\geq t_{0}^{*}$,

$$\begin{aligned} & M_{1} =\frac{k_{1}-\varepsilon}{\rho +1}+ \frac{(\varepsilon +\lambda ) \sigma '(t)}{(\rho +2)\sigma (t)} >0, \\ & M_{2} = \biggl(\frac{k(t)}{2} +\frac{ \sigma '(t)}{2\sigma (t)} \int _{0}^{t} k(s)\,ds \biggr)M + \frac{\varepsilon C_{4} \sigma '(t)}{\sigma (t)} + \frac{\varepsilon l}{2} -\gamma \bigl( 1+2 (a_{0}-l)k_{0} + \alpha _{3} \bigr) >0, \\ & M_{3} =k_{1}-\gamma \biggl(1+\mu _{1} \alpha _{5} + \frac{\alpha _{4} }{q+1} \biggr)-\varepsilon + \frac{(\varepsilon +2\lambda )\sigma '(t)}{2\sigma (t)}>0, \\ & M_{4}= \frac{M\sigma '(t)}{2\sigma (t)}+ \frac{\varepsilon \sigma (t)}{4\eta} - \frac{C_{5}\sigma '(t)}{\sigma (t)} +C_{3} >0, \qquad M_{5}= \frac{C_{1} M}{\sigma (t)}- \varepsilon \mu _{1} C_{\eta }>0, \\ & M_{6}=\frac{C_{2} M}{\sigma (t)}-\varepsilon \mu _{2} C_{\eta}- \gamma \mu _{2}>0,\qquad M_{7}= \frac{M}{2}-\frac{g(0)}{4\gamma} \biggl( 1+ \frac{C_{*}^{2}}{\rho +1} \biggr) >0 , \end{aligned}$$

and

$$\begin{aligned} & M_{8}=\frac{M}{\sigma (t)} - \frac{\varepsilon \Vert m \Vert _{\infty}}{\eta g_{1}} - \frac{\gamma \Vert m \Vert _{\infty }}{g_{1}} >0. \end{aligned}$$

Then, for any $t\geq t_{0}^{*}$, using (3.5) and (3.45), we deduce that

$$\begin{aligned} \Xi '(t) \leq -M_{9} \sigma (t) E(t) +M_{10} \sigma (t) (k\circ \nabla w) (t) -M_{11}\sigma (t) E'(t), \end{aligned}$$

(3.46)

where $M_{9}$ and $M_{10}$ are some positive constants and $M_{11} =\frac{2\gamma b_{1}^{2} p E(0)}{l(p-2)}$. Multiplying (3.46) by $\zeta (t)$ and using (2.7) and (3.7), we obtain for any $t\geq t_{0}^{*}$,

$$\begin{aligned} \zeta (t) \Xi '(t) \leq& -M_{9} \sigma (t)\zeta (t) E(t) -M_{10} \sigma (t) \bigl(k'\circ \nabla w \bigr) (t) -M_{11} \sigma (t) \zeta (t) E'(t) \\ \leq& -M_{9}\sigma (t) \zeta (t) E(t)- \bigl(2M_{10}+M_{11} \sigma (t)\zeta (t) \bigr)E'(t). \end{aligned}$$

(3.47)

Now, we define

$$ G(t)=\zeta (t) \Xi (t) + \bigl(2M_{10} +M_{11} \sigma (t) \zeta (t) \bigr)E(t). $$

Using the fact that ζ and σ are nonincreasing positive functions and $\zeta '(t) \leq 0$ and $\sigma '(t) \leq 0$, (3.47) implies that

$$\begin{aligned} G'(t) \leq -M_{9} \sigma (t) \zeta (t) E(t) \leq -\kappa \sigma (t) \zeta (t) G(t), \end{aligned}$$

(3.48)

where κ is a positive constant. Integrating (3.48) between $t_{0}^{*}$ and t gives the following estimation for the function $G(t)$

$$ G(t) \leq G \bigl(t_{0}^{*} \bigr)e^{-\kappa \int _{t_{0}^{*}}^{t} \sigma (s) \zeta (s)\,ds},\quad \forall t\geq t_{0}^{*}. $$

Again, employing that $G(t)$ is equivalent to $E(t)$, we deduce

$$ E(t) \leq K e^{-\kappa \int _{t_{0}^{*}}^{t} \sigma (s) \zeta (s)\,ds},\quad \forall t\geq t_{0}^{*}, $$

where K is a positive constant. Thus, the proof of Theorem 3.2 is completed. □