General decay for a viscoelastic von Karman equation with delay and variable exponent nonlinearities

Abstract

In this paper, we consider a viscoelastic von Karman equation with damping, delay, and source effects of variable exponent type. Firstly, we show the global existence of solution applying the potential well method. Then, by making use of the perturbed energy method and properties of convex functions, we derive general decay results for the solution under more general conditions of a relaxation function. General decay results of solutions for viscoelastic von Karman equations with variable exponent nonlinearities have not been discussed before. Our results extend and complement many results for von Karman equations in the literature.

Introduction

In this work, we study the following viscoelastic von Karman equation with damping, delay, and source terms of variable exponents:

$$\begin{aligned}& u_{tt} + \Delta ^{2} u - \int ^{t}_{0} k (t-s) \Delta ^{2} u(s) \,ds + \alpha \vert u_{t} \vert ^{ m ( \cdot ) - 2 } u_{t} + \beta \bigl\vert u_{t} (t- \tau ) \bigr\vert ^{ m (\cdot ) -2} u_{t} ( t- \tau ) \\& \quad = \bigl[ u, \chi (u) \bigr] + \gamma \vert u \vert ^{ p (\cdot ) - 2 } u \quad \text{in } \Omega \times (0,\infty ), \end{aligned}$$

(1.1)

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

(1.2)

$$\begin{aligned}& u= \frac{\partial u}{\partial \nu } = 0 , \qquad \chi ( u ) = \frac{\partial \chi ( u ) }{\partial \nu } = 0 \quad \text{on } \partial \Omega \times (0,\infty ), \end{aligned}$$

(1.3)

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

(1.4)

$$\begin{aligned}& u_{t} (x, t - \tau ) = j_{0} ( x, t - \tau ) \quad \text{in } \Omega \times ( 0 , \tau ), \end{aligned}$$

(1.5)

where Ω is a bounded domain in ${\mathbb{R}}^{2}$ complementing a smooth boundary ∂Ω, ν is the unit normal vector outward to ∂Ω, $\alpha >0$, $\beta \in {\mathbb{R}}$, $\gamma >0$, $\tau >0$ is time delay, $u_{0}$, $u_{1}$, and $j_{0}$ are given initial data, and the von Karman bracket $[\cdot ,\cdot ]$ is defined as $[ v , \tilde{v} ] = v_{x_{1} x_{1}} \tilde{v}_{x_{2} x_{2}} + v_{x_{2} x_{2}} \tilde{v}_{x_{1} x_{1}} - 2 v_{x_{1} x_{2}} \tilde{v}_{x_{1} x_{2}}$, where $(x_{1}, x_{2}) = x \in \Omega$. The relaxation function k and the exponents $m (\cdot )$ and $p ( \cdot )$ will be specified later. This type of von Karman equations arises in many applications to the modeling of engineering and physical phenomena such as shells, nonlinear elastic plates, bifurcation theory, and so on.

In the absence of damping, delay, and source effects($\alpha = \beta = \gamma =0$) in (1.1), many authors [5, 22, 25, 26, 28, 30] have developed decay results by weakening the conditions of the relaxation function k. Munoz Rivera and Menzala [22] showed an exponential decay result when k satisfies

$$ k'(t) \leq - \zeta k(t), \quad \zeta >0 , $$

(1.6)

and a polynomial decay result when

$$ k'(t) \leq - \zeta k^{1+ \frac{1}{ q }}(t), \quad \zeta > 0 , q >2 . $$

(1.7)

The authors of [25, 30] improved those results under the condition

$$ k'(t) \leq - \zeta (t) k(t) , $$

(1.8)

where ζ fulfills $\zeta (t) >0$ and $\zeta '(t) \leq 0$ on $[0, \infty )$. Cavalcanti et al. [5] established a general decay result for a wider class of relaxation functions satisfying

$$ k'(t) \leq - K\bigl(k(t)\bigr), $$

(1.9)

where K is an increasing convex function on an interval $[ 0, \varepsilon )$ and satisfies $K(0) =0$. Recently, Park [26] established a more general decay result under the very general condition

$$ k'(t) \leq - \zeta (t) K\bigl(k(t)\bigr), $$

(1.10)

where K is an increasing convex function satisfying certain conditions, invoked by the pioneering work of Mustafa [23]. In [23], the author introduced condition (1.10) and established an explicit and general decay result for a viscoelastic wave equation when k satisfies (1.10).

On one hand, the study of elliptic, parabolic, and hyperbolic problems with nonlinearities of variable exponent type has been attracting much interest [1, 2, 19, 21, 31]. The nonlinearities of such type describe various physical applications, for example, electrorheological fluids [29], nonlinear elastics [33], non-Newtonian fluids [3], and image precessing [1]. In recent years, some authors studied the following wave equation with such nonlinearities:

$$\begin{aligned} u_{tt} - \Delta u + \int ^{t}_{0} k (t-s) \Delta u(s) \,ds + \alpha \vert u_{t} \vert ^{ m ( x ) - 2 } u_{t} = \gamma \vert u \vert ^{ p ( x ) - 2 } u . \end{aligned}$$

(1.11)

In case $k =0$ in (1.11), Messaoudi et al. [21] proved the local existence of solution and showed a blow-up result of the solution with negative initial energy when the exponents $m ( x ) \geq 2$ and $p( x ) \geq 2$ satisfy some hypotheses. Later, in [12], the authors proved the global existence of solution for the same equation by giving some conditions on initial data. Moreover, they showed that the solution decays exponentially when $m(x) = 2$ and polynomially when $m(x) \geq 2$ and $m_{2} >2$, where $m_{2} = \operatorname{ess} \sup_{x \in \Omega } m (x)$, by using an integral inequality introduced by Komornik [17]. In case $k \neq 0$ in (1.11), Park and Kang [27] obtained similar results of [21] for the solution with certain positive initial energy. Most recent, Messaoudi et al. [20] established very general decay results when $m (x) > 1$ and the relaxation function k satisfies (1.10). Their results generalize and extend the previous results for problem (1.11).

Inspired by these works, in this article, we consider the viscoelastic von Karman system (1.1)–(1.5) with damping, source, and time delay effects of variable-exponent type. Time delay appears in the phenomena depending on some past occurrences as well as on the present state, and may cause instability. We refer to [7, 32] for more applications of time delay and [11, 16, 24] for various decay results of delayed equations. In the absence of memory and time delay($k = \beta = 0$) in (1.1), Ha and Park [13] proved the global existence of solution and showed exponential or polynomial decay results depending on $m_{2} \geq 2$. At this point, it is worth to say that there are no works on the global existence of solution and general decay of the solution for viscoelastic von Karman equations with variable exponent damping and source terms. Due to the presence of source effect, we have some difficulty in deriving desired general decay results. We overcome this by giving some conditions on initial data. Moreover, as far as we know, the global existence and decay of solutions for viscoelastic von Karman equations with delay of variable exponent type have not been considered before. Thus, we intend to discuss the issues for problem (1.1)–(1.5).

Here are the contents of this paper. We give preliminaries in Sect. 2. We show a global existence result in Sect. 3. We establish general decay results for both cases $1 < m_{1} <2$ and $m_{1} > 2$, where $m _{1} = \operatorname{ess} \inf_{x \in \Omega } m (x)$.

Preliminaries

In this section, we present notations, review necessary materials, give assumptions, and state a local existence result.

We denote by $\| \cdot \|_{Y}$ the norm of a norm space Y. To simplify notations, we denote $\| \cdot \|_{L^{s} (\Omega )}$ as $\| \cdot \|_{s}$ for $1 \leq s \leq \infty$. We use the letter $B_{s}$ to denote the embedding constant satisfying

$$ \Vert v \Vert _{ s } \leq B_{ s } \Vert \Delta v \Vert _{2} \quad \text{for } s \in [2, \infty ], v \in H_{0}^{2}( \Omega ) . $$

(2.1)

Here, we recall Lebesgue and Sobolev spaces of variable exponents (see e.g. [8, 9, 18]). Let D be a bounded domain of ${\mathbb{R}}^{n}$, $n \geq 1$, and $r : \Omega \to [1, \infty ]$ be a measurable function. The Lebesque space

$$ L^{ r (\cdot )}( D ) = \biggl\{ v : D \to {\mathbb{R}} | v \text{ is measurable in } D, \int _{ D } \bigl\vert \mu v (x) \bigr\vert ^{ r (x)} \,dx < \infty \text{ for some } \mu >0 \biggr\} $$

is a Banach space with respect to the Luxembourg-type norm

$$ \Vert v \Vert _{ r (\cdot )} = \inf \biggl\{ \mu > 0 | \int _{ D } \biggl\vert \frac{ v (x)}{\mu } \biggr\vert ^{ r (x)} \,dx \leq 1 \biggr\} . $$

It is said that $r ( \cdot )$ satisfies the log-Hölder continuity condition if

$$\begin{aligned} \bigl\vert r (x) - r ( \tilde{ x } ) \bigr\vert \leq - \frac{ b_{1} }{ \log \vert x - \tilde{ x } \vert } \end{aligned}$$

(2.2)

for all $x,\tilde{ x } \in D \text{ with } | x - \tilde{ x } | < b_{2}$, where $b_{1} > 0$ and $0 < b_{2} < 1$.

Throughout this paper, we let

$$ r_{1} : = \operatorname{ess} \inf_{x\in D } r (x) \quad \text{and} \quad r_{2} : = \operatorname{ess}\sup _{x\in D } r (x) . $$

We remind the following property of von Karman bracket.

Lemma 2.1

([6])

Let $v_{1}, v_{2}, v_{3} \in H^{2}(\Omega )$. If at least one of them is an element of $H^{2}_{0}(\Omega )$, then

$$ \int _{\Omega } [ v_{1}, v_{2}] v_{3} \,dx = \int _{\Omega } [ v_{1}, v_{3}] v_{2} \,dx. $$

We give the following assumptions.

$(A_{1})$ The exponents $p (\cdot )$ and $m (\cdot )$ are continuous functions on Ω̅ satisfying (2.2) and

$$\begin{aligned}& 2 < p_{1} \leq p (x) \leq p_{2} < \infty , \end{aligned}$$

(2.3)

$$\begin{aligned}& 1 < m_{1} \leq m (x) \leq m_{2} < \infty . \end{aligned}$$

(2.4)

$(A_{2})$ The coefficients α and β have the relation

$$ \vert \beta \vert < \frac{ \alpha }{ 1 + \frac{1}{ m_{1} } - \frac{1}{ m_{2} } } . $$

(2.5)

$(A_{3})$ $k : {\mathbb{R}}^{+} \to {\mathbb{R}}^{+}$ is a continuously differentiable function and satisfies

$$ k (0) >0, \quad 1 - \int ^{\infty }_{0} k (s) \,ds = k_{l} > 0 , $$

and

$$ k'(t) \leq - \zeta (t) K\bigl(k(t)\bigr) \quad \text{for all } t \geq 0, $$

(2.6)

where $K : (0, \infty ) \to (0, \infty )$ is a continuously differentiable function, which is either a linear function or a strictly increasing and strictly convex $C^{2}$-function on $(0, \varepsilon ]$, $\varepsilon \leq k(0)$, $K(0) = K '(0)=0$, and ζ is positive, differentiable, and nonincreasing.

Remark 2.1

1. For examples of the function k satisfying $(A_{3})$, we refer to [23].

2. Since K is a strictly increasing and strictly convex $C^{2}$-function on $(0, \varepsilon ]$ satisfying $K(0) = K '(0)=0$, there exists an extension K̅ of K, which is a strictly increasing and strictly convex $C^{2}$ function on $(0, \infty )$. We mention [23] for details.

As in [24], we introduce a function y as

$$\begin{aligned} y (x, \rho , t) = u_{t} (x, t - \rho \tau ) \quad \text{for } (x, \rho , t) \in \Omega \times (0,1) \times (0,T). \end{aligned}$$

Then, problem (1.1)–(1.5) reads as

$$\begin{aligned}& u_{tt} + \Delta ^{2} u - \int ^{t}_{0} k (t-s) \Delta ^{2} u(s) \,ds + \alpha \vert u_{t} \vert ^{ m ( \cdot ) - 2 } u_{t} + \beta \bigl\vert y ( x, 1, t ) \bigr\vert ^{ m (\cdot ) -2} y ( x, 1, t ) \\& \quad = \bigl[ u, \chi (u) \bigr] + \gamma \vert u \vert ^{ p (\cdot ) - 2 } u \quad \text{in } \Omega \times (0,\infty ), \end{aligned}$$

(2.7)

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

(2.8)

$$\begin{aligned}& \tau y_{t} ( x, \rho , t) + y_{\rho } (x, \rho , t) =0 \quad \text{for } (x, \rho , t ) \in \Omega \times (0, 1) \times (0, T), \end{aligned}$$

(2.9)

$$\begin{aligned}& u= \frac{\partial u}{\partial \nu } = 0 , \qquad \chi ( u ) = \frac{\partial \chi ( u ) }{\partial \nu } = 0 \quad \text{on } \partial \Omega \times (0,\infty ), \end{aligned}$$

(2.10)

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

(2.11)

$$\begin{aligned}& y( x, \rho , 0 ) = j_{0} ( x, - \rho \tau ) := y_{0} \quad \text{in } \Omega \times ( 0, 1 ), \end{aligned}$$

(2.12)

By virtue of the arguments of [4, 15, 27], we have the following local existence result.

Theorem 2.1

(Local existence)

Under $(A_{1})$, $(A_{2})$, and $(A_{3})$, problem (2.7)–(2.12) has a unique local solution $( u, y )$ satisfying

$$\begin{aligned}& u \in L^{\infty } \bigl(0,T; H^{2}_{0}(\Omega ) \bigr) , \qquad u_{t} \in L^{\infty } \bigl( 0,T; L^{2}(\Omega )\bigr) \cap L^{ m (\cdot )} \bigl( \Omega \times ( 0, T)\bigr) ,\\& y \in L^{\infty } \bigl( 0, T ; L^{ m (\cdot )} \bigl( \Omega \times (0,1) \bigr) \bigr) \end{aligned}$$

for every $( u_{0}, u_{1} , y_{0} ) \in H^{2}_{0}(\Omega ) \times L^{2}(\Omega ) \times L^{ m (\cdot ) } ( \Omega \times ( 0, 1) )$.

Global existence

In this section, we derive the global existence of solution to problem (2.7)–(2.12). In the proof of global existence and decay results, we will use the following lemma several times.

Lemma 3.1

Let r be a continuous function on Ω̅ with $2 \leq r_{1} \leq r(x) \leq r_{2} < \infty$. Then, for $v \in H^{2}_{0} (\Omega )$, it holds

$$\begin{aligned} \int _{\Omega } \vert v \vert ^{ r (x)} \,dx \leq \bigl( B_{r_{1}}^{r_{1}} \Vert \Delta v \Vert _{2}^{ r_{1} -2 } + B_{r_{2}}^{ r_{2} } \Vert \Delta v \Vert _{2}^{ r_{2} -2} \bigr) \Vert \Delta v \Vert _{2}^{2} . \end{aligned}$$

(3.1)

Proof

Using (2.1), we have

$$\begin{aligned} \int _{\Omega } \vert v \vert ^{ r (x)} \,dx = & \int _{ \{ x \in \Omega : \vert v (x) \vert < 1 \} } \vert v \vert ^{ r (x)} \,dx + \int _{ \{ x \in \Omega : \vert v (x) \vert \geq 1 \} } \vert v \vert ^{ r (x)} \,dx \\ \leq & \int _{ \{ x \in \Omega : \vert v (x) \vert < 1 \} } \vert v \vert ^{ r_{1} } \,dx + \int _{ \{ x \in \Omega : \vert v (x) \vert \geq 1 \} } \vert v \vert ^{ r_{2} } \,dx \\ \leq & \Vert v \Vert _{ r_{1} }^{ r_{1} } + \Vert v \Vert _{ r_{2} }^{ r_{2} } \leq B_{ r_{1} }^{ r_{1} } \Vert \Delta v \Vert _{2}^{ r_{1} } + B_{ r_{2} }^{ r_{2} } \Vert \Delta v \Vert _{2}^{ r_{2} }. \end{aligned}$$

 □

We define the energy E of the solution to (2.7)–(2.12) as

$$\begin{aligned} E(t) = & \frac{1}{ 2 } \Vert u_{t} \Vert _{2}^{2} + \frac{1}{ 2 } \biggl( 1- \int ^{t}_{0} k(s) \,ds \biggr) \Vert \Delta u \Vert _{2}^{2} + \frac{1}{4} \bigl\Vert \Delta \chi ( u ) \bigr\Vert _{2}^{2} + \frac{1}{ 2 } ( k \Box \Delta u ) \\ &{} + \frac{ \xi \tau }{ 2 } \int _{\Omega } \int ^{1}_{0} \bigl\vert y (x, \rho , t ) \bigr\vert ^{ m (x) } \,d\rho \,dx - \gamma \int _{\Omega } \frac{ \vert u \vert ^{ p (x) }}{ p (x) } \,dx , \end{aligned}$$

(3.2)

where

$$ (k \Box \Delta u ) (t) = \int ^{t}_{0} k(t-s) \bigl\Vert \Delta u (t) - \Delta u (s) \bigr\Vert _{2}^{2} \,ds $$

and

$$ \frac{ 2 \vert \beta \vert ( m_{2} -1 ) }{ m_{2} } < \xi < \frac{ 2 ( \alpha m_{1} - \vert \beta \vert ) }{ m_{1} } . $$

(3.3)

Let

$$\begin{aligned} J( t) = \frac{1}{ 2 } \biggl( 1- \int ^{t}_{0} k(s) \,ds \biggr) \Vert \Delta u \Vert _{2}^{2} + \frac{1}{ 2 } ( k \Box \Delta u ) - \frac{ \gamma }{ p_{1} } \int _{\Omega } \vert u \vert ^{ p (x) } \,dx \end{aligned}$$

(3.4)

and

$$\begin{aligned} I( t) = \biggl( 1- \int ^{t}_{0} k(s) \,ds \biggr) \Vert \Delta u \Vert _{2}^{2} + ( k \Box \Delta u ) - \gamma \int _{\Omega } \vert u \vert ^{ p (x) } \,dx . \end{aligned}$$

(3.5)

Then we see

$$\begin{aligned} J(t) = \biggl( \frac{1}{2} - \frac{1}{ p_{1} } \biggr) \biggl\{ \biggl( 1- \int ^{t}_{0} k(s) \,ds \biggr) \Vert \Delta u \Vert _{2}^{2} + ( k \Box \Delta u ) \biggr\} + \frac{1}{ p_{1} } I(t) \end{aligned}$$

(3.6)

and

$$\begin{aligned} E(t) \geq J(t) + \frac{1}{ 2 } \Vert u_{t} \Vert _{2}^{2} + \frac{1}{4} \bigl\Vert \Delta \chi ( u ) \bigr\Vert _{2}^{2} + \frac{ \xi \tau }{ 2 } \int _{\Omega } \int ^{1}_{0} \bigl\vert y (x, \rho , t ) \bigr\vert ^{ m (x) } \,d\rho \,dx . \end{aligned}$$

(3.7)

Lemma 3.2

Let $(A_{1})$, $(A_{2})$, and $(A_{3})$ hold. Then there exists $C_{0} >0$ satisfying

$$\begin{aligned} E '(t) \leq& - C_{0} \biggl( \int _{\Omega } \vert u_{t} \vert ^{ m (x)} \,dx + \int _{\Omega } \bigl\vert y ( x, 1, t) \bigr\vert ^{ m (x)} \,dx \biggr) + \frac{1}{2} \bigl( k ' \Box \Delta u \bigr) - \frac{ k (t) }{2} \Vert \Delta u \Vert _{2}^{2} \\ \leq& 0. \end{aligned}$$

(3.8)

Proof

From (2.7), (2.8), (2.10), we have

$$\begin{aligned}& \frac{d}{dt} \biggl\{ \frac{1}{2} \Vert u_{t} \Vert _{2}^{2} + \frac{1}{2} \biggl( 1 - \int ^{t}_{0} k ( s ) \,ds \biggr) \Vert \Delta u \Vert _{2}^{2} + \frac{1}{4} \bigl\Vert \Delta \chi ( u ) \bigr\Vert _{2}^{2} \\& \qquad {} + \frac{1}{2} ( k \Box \Delta u ) - \gamma \int _{\Omega } \frac{ \vert u \vert ^{ p (x) } }{ p (x) } \,dx \biggr\} \\& \quad = - \alpha \int _{\Omega } \vert u_{t} \vert ^{ m (x)} \,dx - \beta \int _{ \Omega } u_{t} \bigl\vert y (x, 1 , t) \bigr\vert ^{ m (x) -2} y (x,1, t) \,dx \\& \qquad {}+ \frac{1}{2}\bigl( k ' \Box \Delta u \bigr) - \frac{ k (t) }{2} \Vert \Delta u \Vert _{2}^{2} . \end{aligned}$$

(3.9)

Using (2.9) and the equality $y (x, 0 , t) = u_{t} (x,t )$, we get

$$\begin{aligned}& \frac{\partial }{\partial t} \biggl( \int _{\Omega } \int ^{1}_{0} \bigl\vert y (x, \rho , t) \bigr\vert ^{ m (x) } \,d\rho \,dx \biggr) \\& \quad = \int _{\Omega } \int ^{1}_{0} m (x) \bigl\vert y (x, \rho , t) \bigr\vert ^{ m (x) -2} y (x, \rho , t) y_{t} (x, \rho , t) \,d\rho \,dx \\& \quad = - \frac{1}{ \tau } \int _{\Omega } \int ^{1}_{0} m (x) \bigl\vert y (x, \rho , t) \bigr\vert ^{ m (x) -2} y (x, \rho , t) y_{\rho } (x,\rho , t) \,d\rho \,dx \\& \quad = - \frac{1}{ \tau } \int _{\Omega } \int ^{1}_{0} \frac{\partial }{\partial \rho } \bigl( \bigl\vert y (x,\rho , t) \bigr\vert ^{ m (x) } \bigr) \,d\rho \,dx \\& \quad = - \frac{1}{ \tau } \int _{\Omega } \bigl\vert y (x, 1 , t) \bigr\vert ^{ m (x) } \,dx + \frac{1}{ \tau } \int _{\Omega } \vert u_{t} \vert ^{ m (x) } \,dx . \end{aligned}$$

(3.10)

Using Young’s inequality with $\frac{ m (x) -1 }{ m (x)} + \frac{ 1 }{ m (x)} =1$ and the fact that the function $f (s) = \frac{s-1}{s}$ is increasing for $s > 0$, we find

$$\begin{aligned}& - \beta \int _{\Omega } u_{t} \bigl\vert y (x, 1 , t) \bigr\vert ^{ m (x) -2} y (x,1, t) \,dx \\& \quad \leq \vert \beta \vert \int _{\Omega } \frac{ \vert u_{t} \vert ^{ m (x)} }{ m (x)} \,dx + \vert \beta \vert \int _{\Omega } \frac{ m (x) -1 }{ m (x)} \bigl\vert y (x, 1, t) \bigr\vert ^{ m (x)} \,dx \\& \quad \leq \frac{ \vert \beta \vert }{ m_{1} } \int _{\Omega } \vert u_{t} \vert ^{ m (x)} \,dx + \frac{ \vert \beta \vert ( m_{2} -1 ) }{ m_{2} } \int _{\Omega } \bigl\vert y (x, 1, t) \bigr\vert ^{ m (x)} \,dx . \end{aligned}$$

(3.11)

Combining (3.9), (3.10), and (3.11), we obtain

$$\begin{aligned} E'(t) \leq & - \biggl( \alpha - \frac{ \xi }{2} - \frac{ \vert \beta \vert }{ m_{1} } \biggr) \int _{\Omega } \vert u_{t} \vert ^{ m (x)} \,dx - \biggl( \frac{ \xi }{2} - \frac{ \vert \beta \vert ( m_{2} -1 )}{ m_{2} } \biggr) \int _{\Omega } \bigl\vert y (x, 1, t) \bigr\vert ^{ m (x)} \,dx \\ & {}+ \frac{1}{2} \bigl( k ' \Box \Delta u \bigr) - \frac{ k (t) }{2} \Vert \Delta u \Vert _{2}^{2} . \end{aligned}$$

From this, (3.3), and $(A_{2})$, we finish the proof. □

Lemma 3.3

Let $(A_{1})$, $(A_{2})$, $(A_{3})$ hold. If

$$\begin{aligned} I( 0 ) >0 \quad \textit{and} \quad B_{ p_{1} }^{ p_{1} } \biggl( \frac{ 2 p_{1} E(0) }{ k_{l} ( p_{1} -2 ) } \biggr)^{ \frac{ p_{1} -2 }{2} } + B_{ p_{2}}^{ p_{2} } \biggl( \frac{ 2 p_{1} E(0) }{ k_{l} ( p_{1} -2 ) } \biggr)^{ \frac{ p_{2} -2 }{2} } < \frac{ k_{l} }{ \gamma } , \end{aligned}$$

(3.12)

then

$$ I(t) >0 \quad \textit{for all } t \geq 0. $$

(3.13)

Proof

Suppose that there exists $0 < t_{0} \leq T$ with $I ( t_{0} ) \leq 0$. Let $T_{m}$ be the first time satisfying

$$ I (T_{m}) =0 . $$

(3.14)

Then $I (t) \geq 0$ for $t \in [0, T_{m} ]$. Thus, from (3.6), (3.7), and (3.8), we observe

$$\begin{aligned} k_{l} \Vert \Delta u \Vert _{2}^{2} \leq &\biggl( 1 - \int ^{t}_{0} k(s) \,ds \biggr) \Vert \Delta u \Vert _{2}^{2} \\ \leq& \frac{ 2 p_{1} }{ p_{1} -2} J(t) \leq \frac{ 2 p_{1} }{ p_{1} -2} E(t) \leq \frac{ 2 p_{1} }{ p_{1} -2} E(0) \end{aligned}$$

(3.15)

for $t \in [0, T_{m} ]$. Using (3.1), (3.15), and (3.12), we see

$$\begin{aligned} \gamma \int _{\Omega } \vert u \vert ^{ p (x)} \,dx \leq & \gamma \bigl( B_{ p_{1} }^{ p_{1} } \Vert \Delta u \Vert _{2}^{ p_{1} -2 } + B_{ p_{2}}^{ p_{2} } \Vert \Delta u \Vert _{2}^{ p_{2} -2} \bigr) \Vert \Delta u \Vert _{2}^{2} \\ \leq & \gamma \biggl\{ B_{ p_{1} }^{ p_{1} } \biggl( \frac{ 2 p_{1} E(0) }{ k_{l} ( p_{1} -2 ) } \biggr)^{ \frac{ p_{1} -2 }{2} } + B_{ p_{2}}^{ p_{2} } \biggl( \frac{ 2 p_{1} E(0) }{ k_{l} ( p_{1} -2 ) } \biggr)^{ \frac{ p_{2} -2 }{2} } \biggr\} \Vert \Delta u \Vert _{2}^{2} \\ < & k_{l} \Vert \Delta u \Vert _{2}^{2} \leq \biggl( 1 - \int ^{t}_{0} k (s) \,ds \biggr) \Vert \Delta u \Vert _{2}^{2} \end{aligned}$$

(3.16)

for $t \in [0, T_{m} ]$, which implies

$$\begin{aligned} I (t) >0 \quad \text{for } t \in [ 0, T_{m} ] . \end{aligned}$$

This contradicts (3.14). □

Remark 3.1

Let the conditions of Lemma 3.3 hold. From the result of Lemma 3.3 and the same argument of (3.15), we also find

$$\begin{aligned} \bigl\Vert \Delta u (t) \bigr\Vert _{2}^{2} < \frac{ 2 p_{1} E(t) }{ k_{l} ( p_{1} -2 ) } \leq \frac{ 2 p_{1} E(0) }{ k_{l} ( p_{1} -2 ) } . \end{aligned}$$

(3.17)

From Lemma 3.2 and Lemma 3.3, we have the global existence result.

Theorem 3.1

(Global existence)

Assume that $(A_{1})$, $(A_{2})$, $(A_{3})$, and (3.12) hold. Then the solution $( u, y )$ to problem (2.7)–(2.12) is global.

Proof

It suffices to show that $\|u_{t}\|_{2}^{2} + \| \Delta u \|_{2}^{2} + \|\Delta \chi ( u ) \|_{2}^{2} + \int _{\Omega } \int ^{1}_{0} | y (x, \rho , t ) |^{ m (x) } \,d\rho \,dx$ is bounded independent of t. From (3.7), (3.6), (3.13), (3.8), we have

$$\begin{aligned}& \min \biggl\{ \frac{1}{ 4 } , \frac{ \xi \tau }{2} \biggr\} \biggl( \Vert u_{t} \Vert _{2}^{2} + \bigl\Vert \Delta \chi ( u ) \bigr\Vert _{2}^{2} + \int _{\Omega } \int ^{1}_{0} \bigl\vert y (x, \rho , t ) \bigr\vert ^{ m (x) } \,d\rho \,dx \biggr) \\ & \quad \leq \frac{1}{ 2 } \Vert u_{t} \Vert _{2}^{2} + \frac{1}{4} \bigl\Vert \Delta \chi ( u ) \bigr\Vert _{2}^{2} + \frac{ \xi \tau }{ 2 } \int _{\Omega } \int ^{1}_{0} \bigl\vert y (x, \rho , t ) \bigr\vert ^{ m (x) } \,d\rho \,dx \\ & \quad < E(t) \leq E(0) . \end{aligned}$$

From this and (3.17), we have

$$\begin{aligned} \Vert u_{t} \Vert _{2}^{2} + \Vert \Delta u \Vert _{2}^{2} + \bigl\Vert \Delta \chi ( u ) \bigr\Vert _{2}^{2} + \int ^{1}_{0} \int _{\Omega } \bigl\vert y (x, \rho , t ) \bigr\vert ^{ m (x) } \,dx \,d\rho \leq \biggl( \frac{ 1 }{c_{0}} + \frac{ 2 p_{1} }{ k_{l} ( p_{1} -2 ) } \biggr) E(0), \end{aligned}$$

where $c_{0} = \min \{ \frac{1}{ 4 } , \frac{ \xi \tau }{2} \}$. □

General decay results

In this section, we derive general decay results for both cases when $m_{1} \geq 2$ and when $1 < m_{1} < 2$ by following the ideas in [20] and [23] with some necessary modification.

First, we let

$$ k_{ \eta }(t) = \eta k(t) - k'(t) \quad \text{and} \quad {\mathcal{C}}_{\eta } = \int ^{\infty }_{0} \frac{ k^{2}(s) }{ k_{\eta }(s) } \,ds \quad \text{for } 0 < \eta < 1 , $$

then, by the arguments of [14, 23], we have the following lemma.

Lemma 4.1

Let $(A_{3})$ be satisfied. Then, for $v \in L^{2}_{\mathrm{loc}} ( [0, \infty ), L^{2}(\Omega ) )$, it holds

$$\begin{aligned} \int _{\Omega } \biggl( \int ^{t}_{0} k ( t-s ) \bigl( v (t) - v (s) \bigr) \,ds \biggr)^{2} \,dx \leq {\mathcal{C}}_{\eta } ( k_{\eta } \Box v ) (t). \end{aligned}$$

(4.1)

Now, we define

$$ L(t) = N E(t) + N_{1} \Phi (t) + N_{2} \Psi (t) + \Upsilon (t) , $$

where $N > 0$, $N_{i} >0$, $i = 1, 2$,

$$\begin{aligned}& \Phi (t) = \int _{\Omega } u u_{t} \,dx , \\& \Psi (t) = - \int ^{t}_{0} k (t-s) \int _{\Omega } \bigl( u(t) -u(s) \bigr) u_{t} \,dx \,ds , \end{aligned}$$

and

$$ \Upsilon (t) = \tau \int _{\Omega } \int _{0}^{1} e^{ - \rho \tau } \bigl\vert y (x, \rho , t ) \bigr\vert ^{ m (x)} \,d\rho \,dx. $$

Lemma 4.2

Assume that $(A_{1})$, $(A_{2})$, $(A_{3})$, and (3.12) hold. Then $L(t)$ is equivalent to $E(t)$.

Proof

From (3.6), (3.13), and (3.7), we have

$$\begin{aligned}& \bigl\vert L(t) - N E(t) \bigr\vert \\& \quad \leq \frac{ N_{1} + N_{2} }{2} \Vert u_{t} \Vert _{2}^{2} + \frac{ N_{1} B_{2}^{2} }{2} \Vert \Delta u \Vert _{2}^{2} + \frac{ N_{2} B_{2}^{2} ( 1 - k_{l}) }{2} ( k \Box \Delta u ) \\& \qquad {}+ \tau \int _{\Omega } \int _{0}^{1} \bigl\vert y (x, \rho , t ) \bigr\vert ^{ m (x)} \,d\rho \,dx \\& \quad \leq \frac{ N_{1} + N_{2} }{2} \Vert u_{t} \Vert _{2}^{2} + \frac{ N_{2} B_{2}^{2} ( 1 - k_{l}) }{2} ( k \Box \Delta u ) \\& \qquad {}+ \frac{ N_{1} B_{2}^{2} p_{1} }{ k_{l} ( p_{1} -2 ) } \biggl\{ J(t) - \frac{1}{ p_{1} } I(t) - \frac{ p_{1} -2 }{ 2 p_{1} } ( k \Box \Delta u) \biggr\} + \tau \int _{\Omega } \int _{0}^{1} \bigl\vert y (x, \rho , t ) \bigr\vert ^{ m (x)} \,d\rho \,dx \\& \quad \leq \frac{ N_{1} + N_{2} }{2} \Vert u_{t} \Vert _{2}^{2} + \frac{ N_{2} B_{2}^{2} ( 1 - k_{l} ) }{2} ( k \Box \Delta u ) + \frac{ N_{1} B_{2}^{2} p_{1} }{ k_{l} ( p_{1} -2 ) } J(t) \\& \qquad {}+ \tau \int _{\Omega } \int _{0}^{1} \bigl\vert y (x, \rho , t ) \bigr\vert ^{ m (x)} \,d\rho \,dx \\& \quad \leq \frac{ N_{1} + N_{2} }{2} \Vert u_{t} \Vert _{2}^{2} + \biggl( \frac{ 2 N_{2} B_{2}^{2} ( 1 - k_{l} ) p_{1} }{ 2 ( p_{1} -2 ) } + \frac{ N_{1} B_{2}^{2} p_{1} }{k_{l} ( p_{1} -2 ) } \biggr) J(t) \\& \qquad {}+ \tau \int _{\Omega } \int _{0}^{1} \bigl\vert y (x, \rho , t ) \bigr\vert ^{ m (x)} \,d\rho \,dx \\& \quad \leq \max \biggl\{ N_{1} + N_{2} , \frac{ 2 N_{2} B_{2}^{2} ( 1 - k_{l} ) p_{1} }{ 2 ( p_{1} -2 ) } + \frac{ N_{1} B_{2}^{2} p_{1} }{ k_{l} ( p_{1} -2 ) } , \frac{ 2 }{ \xi } \biggr\} E(t) . \end{aligned}$$

Taking $N > \max \{ N_{1} + N_{2} , \frac{ 2 N_{2} B_{2}^{2} ( 1 - k_{l} ) p_{1} }{ 2 ( p_{1} -2 ) } + \frac{ N_{1} B_{2}^{2} p_{1} }{ k_{l} ( p_{1} -2 ) } , \frac{ 2 }{ \xi } \}$, we finish the proof. □

Lemma 4.3

The function ϒ satisfies

$$\begin{aligned} \Upsilon ' (t) \leq & \int _{\Omega } \vert u_{t} \vert ^{ m (x) } \,dx - \tau e^{ - \tau } \int _{\Omega } \int ^{1}_{0} \bigl\vert y (x, \rho , t ) \bigr\vert ^{ m (x) } \,d\rho \,dx . \end{aligned}$$

(4.2)

Proof

Using (2.9) and $y (x, 0, t ) = u_{t}(x,t)$, we get

$$\begin{aligned} \Upsilon ' (t) = & \tau \int _{\Omega } \int ^{1}_{0} e^{ - \rho \tau } m (x) \bigl\vert y (x, \rho , t ) \bigr\vert ^{ m (x) -2} y (x, \rho , t ) y_{t} (x, \rho , t ) \,d\rho \,dx \\ = & - \int _{\Omega } \int ^{1}_{0} e^{ - \rho \tau } \frac{ \partial }{\partial \rho } \bigl\vert y (x, \rho , t ) \bigr\vert ^{ m (x) } \,d\rho \,dx \\ = & - \int _{\Omega } e^{ - \tau } \bigl\vert y (x, 1, t ) \bigr\vert ^{ m (x) } \,dx + \int _{\Omega } \bigl\vert y (x, 0 , t ) \bigr\vert ^{ m (x) } \,dx \\ &{}- \tau \int _{\Omega } \int ^{1}_{0} e^{ - \rho \tau } \bigl\vert y (x, \rho , t ) \bigr\vert ^{ m (x) } \,d\rho \,dx \\ \leq & \int _{\Omega } \vert u_{t} \vert ^{ m (x) } \,dx - \tau e^{ - \tau } \int _{\Omega } \int ^{1}_{0} \bigl\vert y (x, \rho , t ) \bigr\vert ^{ m (x) } \,d\rho \,dx . \end{aligned}$$

 □

Lemma 4.4

Let ${ g(t) = \int ^{\infty }_{t} k(s) \,ds }$. The following function

$$ \Lambda (t) = \int ^{t}_{0} g(t-s) \bigl\Vert \Delta u (s) \bigr\Vert _{2}^{2} \,ds $$

satisfies

$$\begin{aligned} \Lambda '(t) \leq 2 (1- k_{l} ) \Vert \Delta u \Vert _{2}^{2} - \frac{1}{2}(k \Box \Delta u) . \end{aligned}$$

(4.3)

Proof

Noting $g'(t) = - k (t)$ and using Young’s inequality, we see

$$\begin{aligned} \Lambda '(t) = & g(0) \Vert \Delta u \Vert _{2}^{2} - \int ^{t}_{0} k(t-s) \bigl\Vert \Delta u (s) \bigr\Vert _{2}^{2} \,ds \\ = & \biggl( \int ^{\infty }_{0} k(s) \,ds \biggr) \Vert \Delta u \Vert _{2}^{2} - ( k \Box \Delta u) - \int ^{t}_{0} k(t-s) \bigl\Vert \Delta u (t) \bigr\Vert _{2}^{2} \,ds \\ & {}- 2 \int _{\Omega } \int ^{t}_{0} k(t-s) \Delta u(t) \bigl( \Delta u(s) - \Delta u(t) \bigr) \,ds \,dx \\ \leq & \biggl( \int ^{\infty }_{t} k(s) \,ds \biggr) \Vert \Delta u \Vert _{2}^{2} - \frac{1}{2} ( k \Box \Delta u) + 2 \biggl( \int ^{t}_{0} k (s) \,ds \biggr) \Vert \Delta u \Vert _{2}^{2} \\ \leq & 2 ( 1 - k_{l} ) \Vert \Delta u \Vert _{2}^{2} - \frac{1}{2} ( k \Box \Delta u) . \end{aligned}$$

 □

From here, c and $C_{i}$ denote generic constants, $c_{\delta } > 0$ denotes a generic constant depending on $\delta >0$, and $c_{\delta } (x) = \frac{ m (x) -1}{ \delta ^{ \frac{1}{ m (x) -1 } } ( m (x))^{ \frac{ m (x)}{ m (x) -1 } } }$. We note that $c_{\delta } (x)$ is bounded on Ω for fixed $\delta >0$, that is, $| c_{\delta } (x) | \leq c_{\delta }$ for all $x \in \Omega$.

General decay for the case ${m_{1} \geq 2 }$

In this subsection, we derive a general decay result for the case $m_{1} \geq 2$.

Lemma 4.5

Assume that $(A_{1})$, $(A_{2})$, $(A_{3})$, and (3.12) hold. If $m_{1} \geq 2$, then Φ satisfies

$$\begin{aligned} \Phi '(t) \leq & \Vert u_{t} \Vert _{2}^{2} - \frac{ k_{l} }{4} \Vert \Delta u \Vert _{2}^{2} - \bigl\Vert \Delta \chi (u) \bigr\Vert _{2}^{2} + \gamma \int _{\Omega } \vert u \vert ^{ p (x)} \,dx \\ &{} + \frac{ {\mathcal{C}}_{\eta } }{ 2 k_{l} } ( k_{\eta } \Box \Delta u ) + \alpha C_{1} \int _{\Omega } \vert u_{t} \vert ^{ m (x) } \,dx + \vert \beta \vert C_{1} \int _{\Omega } \bigl\vert y (x, 1, t ) \bigr\vert ^{ m (x)} \,dx . \end{aligned}$$

(4.4)

Proof

Using (2.7)–(2.12), we get

$$\begin{aligned} \Phi '(t) = & \Vert u_{t} \Vert _{2}^{2} - \biggl( 1- \int ^{t}_{0} k(s) \,ds \biggr) \Vert \Delta u \Vert _{2}^{2} - \bigl\Vert \Delta \chi (u) \bigr\Vert _{2}^{2} + \gamma \int _{\Omega } \vert u \vert ^{ p (x)} \,dx \\ & {}+ \int ^{t}_{0} k(t-s) \int _{\Omega } \bigl( \Delta u(s) - \Delta u(t) \bigr) \Delta u (t) \,dx \,ds \\ &{} - \alpha \int _{\Omega } u \vert u_{t} \vert ^{ m (x) -2 } u_{t} \,dx -\beta \int _{\Omega } u \bigl\vert y (x, 1, t) \bigr\vert ^{ m (x) -2 } y (x, 1, t) \,dx . \end{aligned}$$

(4.5)

Using Young’s inequality and (4.1), we have

$$\begin{aligned}& \int ^{t}_{0} k(t-s) \int _{\Omega } \bigl( \Delta u(s) - \Delta u(t) \bigr) \Delta u (t) \,dx \,ds \\& \quad \leq \frac{ k_{l} }{2} \Vert \Delta u \Vert _{2}^{2} + \frac{ 1 }{ 2 k_{l} } \int _{\Omega } \biggl( \int ^{t}_{0} k ( t-s ) \bigl( \Delta u(s) - \Delta u(t) \bigr) \,ds \biggr)^{2} \,dx \\& \quad \leq \frac{ k_{l} }{2} \Vert \Delta u \Vert _{2}^{2} + \frac{ {\mathcal{C}}_{\eta } }{ 2 k_{l} } ( k_{\eta } \Box \Delta u ) . \end{aligned}$$

(4.6)

From (4.5) and (4.6), one sees

$$\begin{aligned} \Phi '(t) \leq & \Vert u_{t} \Vert _{2}^{2} - \frac{ k_{l} }{2} \Vert \Delta u \Vert _{2}^{2} - \bigl\Vert \Delta \chi (u) \bigr\Vert _{2}^{2} + \gamma \int _{\Omega } \vert u \vert ^{ p (x)} \,dx + \frac{ {\mathcal{C}}_{\eta } }{ 2 k_{l}} ( k_{\eta } \Box \Delta u ) \\ & {}- \underbrace{ \alpha \int _{\Omega } u \vert u_{t} \vert ^{ m (x) -2 } u_{t} \,dx }_{I_{1}} - \underbrace{ \beta \int _{\Omega } u \bigl\vert y (x, 1, t) \bigr\vert ^{ m (x) -2 } y (x, 1, t) \,dx }_{I_{2}} . \end{aligned}$$

(4.7)

Using Young’s inequality with $\frac{1}{ m (x) } + \frac{ m (x) -1 }{ m (x) } =1$, (3.1), and (3.17), we find

$$\begin{aligned} - I_{1} \leq & \alpha \delta _{1} \int _{\Omega } \vert u \vert ^{ m (x) } \,dx + \alpha \int _{\Omega } c_{\delta _{1}} (x) \vert u_{t} \vert ^{ m (x) } \,dx \\ \leq & \alpha \delta _{1} C_{E(0)} \Vert \Delta u \Vert _{2}^{2} + \alpha \int _{\Omega } c_{\delta _{1} }(x) \vert u_{t} \vert ^{ m (x) } \,dx \end{aligned}$$

(4.8)

for $\delta _{1} >0$, where $C_{E(0)} = B_{ m_{1} }^{ m_{1} } ( \frac{ 2 p_{1} E(0) }{ k_{l} ( p_{1} -2 ) } )^{ \frac{ m_{1} -2 }{2} } + B_{ m_{2}}^{ m_{2} } ( \frac{ 2 p_{1} E(0) }{ k_{l} ( p_{1} -2 ) } )^{ \frac{ m_{2} -2 }{2} }$.

Similarly, using (2.5), we have

$$\begin{aligned} - I_{2} \leq & \vert \beta \vert \delta _{1} C_{E(0)} \Vert \Delta u \Vert _{2}^{2} + \vert \beta \vert \int _{\Omega } c_{\delta _{1} }(x) \bigl\vert y (x,1, t) \bigr\vert ^{ m (x) } \,dx \\ \leq & \alpha \delta _{1} C_{E(0)} \Vert \Delta u \Vert _{2}^{2} + \vert \beta \vert \int _{\Omega } c_{\delta _{1} }(x) \bigl\vert y (x,1, t) \bigr\vert ^{ m (x) } \,dx . \end{aligned}$$

(4.9)

Combining (4.7), (4.8), (4.9) and taking $\delta _{1} = \frac{ k_{l} }{ 8 \alpha C_{E(0)}}$, we obtain (4.4). □

Lemma 4.6

Assume that $(A_{1})$, $(A_{2})$, $(A_{3})$, and (3.12) hold. If $m_{1} \geq 2$, then Ψ satisfies

$$\begin{aligned} \Psi '(t) \leq & - \biggl( \int ^{t}_{0} k(s) \,ds - \delta \biggr) \Vert u_{t} \Vert _{2}^{2} + \delta C_{3} \Vert \Delta u \Vert _{2}^{2} + \biggl( \frac{ C_{4} ( 1+ {\mathcal{C}}_{\eta }) }{ \delta } + {\mathcal{C}}_{ \eta } \biggr) (k_{\eta } \Box \Delta u) \\ &{} + \delta C_{5} ( k \Box \Delta u ) + \alpha \int _{\Omega } c_{ \delta }(x) \vert u_{t} \vert ^{ m (x) } \,dx + \vert \beta \vert \int _{\Omega } c_{ \delta }(x) \bigl\vert y (x,1, t) \bigr\vert ^{ m (x) } \,dx \end{aligned}$$

(4.10)

for any $\delta >0$.

Proof

Using (2.7)–(2.12), we have

$$\begin{aligned} \Psi '(t) = & - \biggl( \int ^{t}_{0} k(s) \,ds \biggr) \Vert u_{t} \Vert _{2}^{2} - \int _{\Omega } u_{t} \int ^{t}_{0} k'(t-s) \bigl( u(t) - u(s) \bigr) \,ds \,dx \\ & {}+ \biggl( 1 - \int ^{t}_{0} k(s) \,ds \biggr) \int _{\Omega } \Delta u \int ^{t}_{0} k(t-s) \bigl( \Delta u(t) - \Delta u(s) \bigr) \,ds \,dx \\ & {}+ \int _{\Omega } \biggl( \int ^{t}_{0} k(t-s) \bigl( \Delta u(t) - \Delta u(s) \bigr) \,ds \biggr)^{2} \,dx \\ & {}- \int _{\Omega } \bigl[ u, \chi (u)\bigr] \int ^{t}_{0} k(t-s) \bigl( u(t) - u(s) \bigr) \,ds \,dx \\ & {}- \gamma \int _{\Omega } \vert u \vert ^{ p (x) -2} u \int ^{t}_{0} k(t-s) \bigl( u(t) - u(s) \bigr) \,ds \,dx \\ &{} + \alpha \int _{\Omega } \vert u_{t} \vert ^{ m (x) -2} u_{t} \int ^{t}_{0} k(t-s) \bigl( u(t) - u(s) \bigr) \,ds \,dx \\ &{} + \beta \int _{\Omega } \bigl\vert y (x,1,t) \bigr\vert ^{ m (x) -2} y (x,1,t) \int ^{t}_{0} k(t-s) \bigl( u(t) - u(s) \bigr) \,ds \,dx \\ : = & - \biggl( \int ^{t}_{0} k(s) \,ds \biggr) \Vert u_{t} \Vert _{2}^{2} + \sum _{i =1}^{5} J_{i} \\ & {}+ \underbrace{ \alpha \int _{\Omega } \vert u_{t} \vert ^{ m (x) -2} u_{t} \int ^{t}_{0} k(t-s) \bigl( u(t) - u(s) \bigr) \,ds \,dx }_{J_{6}} \\ &{} + \underbrace{ \beta \int _{\Omega } \bigl\vert y (x,1,t) \bigr\vert ^{ m (x) -2} y (x,1,t) \int ^{t}_{0} k(t-s) \bigl( u(t) - u(s) \bigr) \,ds \,dx }_{J_{7}} . \end{aligned}$$

(4.11)

Using Young’s inequality, $k ' = \eta k - k_{\eta }$, (4.1) we get

$$\begin{aligned} \vert J_{1} \vert \leq & \delta \Vert u_{t} \Vert _{2}^{2} + \frac{1}{4 \delta } \biggl\Vert \int ^{t}_{0} k'(t-s) \bigl( u(t) - u(s) \bigr) \,ds \biggr\Vert _{2}^{2} \\ \leq &\delta \Vert u_{t} \Vert _{2}^{2} + \frac{1}{2 \delta } \biggl( \biggl\Vert \int ^{t}_{0} k_{\eta }(t-s) \bigl( u(t) - u(s) \bigr) \,ds \biggr\Vert _{2}^{2} + \eta ^{2} \biggl\Vert \int ^{t}_{0} k(t-s) \bigl( u(t) - u(s) \bigr) \,ds \biggr\Vert _{2}^{2} \biggr) \\ \leq& \delta \Vert u_{t} \Vert _{2}^{2} + \frac{1}{2 \delta } \biggl\{ \biggl( \int ^{t}_{0} k_{\eta }(s) \,ds \biggr) ( k_{\eta } \Box u) + \eta ^{2} { \mathcal{C}}_{\eta } (k_{\eta } \Box u) \biggr\} \\ \leq &\delta \Vert u_{t} \Vert _{2}^{2} + \frac{ B_{2}^{2} ( \eta (1- k_{l} ) + k(0) ) }{2 \delta } (k_{\eta } \Box \Delta u) + \frac{ B_{2}^{2} {\mathcal{C}}_{\eta } }{ 2 \delta } (k_{ \eta } \Box \Delta u ) , \end{aligned}$$

(4.12)

$$\begin{aligned} \vert J_{2} \vert \leq \delta \Vert \Delta u \Vert _{2}^{2} + \frac{ {\mathcal{C}}_{\eta } }{ 4 \delta } (k_{\eta } \Box \Delta u ) , \end{aligned}$$

(4.13)

and

$$\begin{aligned} \vert J_{3} \vert \leq {\mathcal{C}}_{\eta } (k_{\eta } \Box \Delta u ) . \end{aligned}$$

(4.14)

Using (3.17), we infer

$$\begin{aligned} \vert J_{4} \vert \leq & c \Vert u \Vert _{H^{2}(\Omega )} \bigl\Vert \chi ( u ) \bigr\Vert _{W^{2, \infty } (\Omega )} \biggl\Vert \int ^{t}_{0} k(t-s) \bigl( u(t) - u(s) \bigr) \,ds \biggr\Vert _{2} \\ \leq & c B_{2} \Vert \Delta u \Vert _{2}^{3} \bigl( {\mathcal{C}}_{\eta } ( k_{ \eta } \Box \Delta u ) \bigr)^{ \frac{1}{2} } \\ \leq & c B_{2} \biggl( \frac{ 2 p_{1} E(0) }{ k_{l} ( p_{1} -2 ) } \biggr) \Vert \Delta u \Vert _{2} \bigl( {\mathcal{C}}_{\eta } ( k_{\eta } \Box \Delta u ) \bigr)^{ \frac{1}{2} } \\ \leq & \delta \Vert \Delta u \Vert _{2}^{2} + \frac{ c }{ 4 \delta } { \mathcal{C}}_{\eta } ( k_{\eta } \Box \Delta u ) , \end{aligned}$$

(4.15)

here we used the Karman bracket property (see p. 270 in [10])

$$ \big\| [ v_{1} , v_{2} ] \big\| \leq c \| v_{1} \|_{H^{2}(\Omega)} \| v_{2} \|_{W^{2, \infty} (\Omega)} \quad \text{for } v_{1} \in H^{2}(\Omega), v_{2} \in W^{2, \infty } (\Omega) $$

and

$$ \big\| \chi (u) \big\| _{W^{2, \infty} (\Omega)} \leq c \| u \|_{H^{2}(\Omega)}^{2} . $$

Using (3.1), (3.17), (4.1), we deduce

$$\begin{aligned} \vert J_{5} \vert \leq & \gamma \delta \int _{\Omega } \vert u \vert ^{2 ( p (x) -1) } \,dx + \frac{\gamma }{ 4 \delta } \int _{\Omega } \biggl( \int ^{t}_{0} k(t-s) \bigl( u(t) - u(s) \bigr) \,ds \biggr)^{2} \,dx \\ \leq & \gamma \delta {\overline{C}}_{E(0)} \Vert \Delta u \Vert _{2}^{2} + \frac{\gamma B_{2}^{2} {\mathcal{C}}_{\eta } }{ 4 \delta } ( k_{\eta } \Box \Delta u ) ), \end{aligned}$$

(4.16)

where ${\overline{C}}_{E(0)} = B_{ 2( p_{1} -1) }^{ 2 ( p_{1} -1) } ( \frac{ 2 p_{1} E(0) }{ k_{l} ( p_{1} -2 ) } )^{ \frac{ 2 p_{1} - 4 }{2} } + B_{ 2( p_{2} -1 )}^{ 2 ( p_{2} -1) } ( \frac{ 2 p_{1} E(0) }{ k_{l} ( p_{1} -2 ) } )^{ \frac{ 2 p_{2} - 4 }{2} }$.

Using the similar calculation of (4.8) and Hölder’s inequality, we get

$$\begin{aligned} \vert J_{6} \vert \leq & \alpha \delta \int _{\Omega } \biggl\vert \int ^{t}_{0} k(t-s) \bigl( u(t) - u(s) \bigr) \,ds \biggr\vert ^{ m (x)} \,dx + \alpha \int _{\Omega } c_{ \delta }(x) \vert u_{t} \vert ^{ m (x) } \,dx \\ \leq & \alpha \delta \int _{\Omega } \biggl( \int ^{t}_{0} k (s) \,ds \biggr)^{ m (x) -1} \biggl( \int ^{t}_{0} k(t-s) \bigl\vert u(t) - u(s) \bigr\vert ^{ m (x)} \,ds \biggr) \,dx \\ &{} + \alpha \int _{\Omega } c_{\delta }(x) \vert u_{t} \vert ^{ m (x) } \,dx \\ \leq & \alpha \delta ( 1 - k_{l} )^{ m_{1} -1} \int _{\Omega } \biggl( \int ^{t}_{0} k(t-s) \bigl\vert u(t) - u(s) \bigr\vert ^{ m (x)} \,ds \biggr) \,dx + \alpha \int _{\Omega } c_{\delta }(x) \vert u_{t} \vert ^{ m (x) } \,dx \\ \leq & \alpha \delta ( 1 - k_{l} )^{ m_{1} -1} \hat{C}_{E(0)} \int ^{t}_{0} k(t-s) \bigl\Vert \Delta u(t) - \Delta u(s) \bigr\Vert _{2}^{2} \,ds + \alpha \int _{ \Omega } c_{\delta }(x) \vert u_{t} \vert ^{ m (x) } \,dx \\ = & \alpha \delta ( 1 - k_{l} )^{ m_{1} -1} \hat{C}_{E(0)} ( k \Box \Delta u) + \alpha \int _{\Omega } c_{\delta }(x) \vert u_{t} \vert ^{ m (x) } \,dx , \end{aligned}$$

(4.17)

where $\hat{C}_{E(0)} = B_{ m_{1} }^{ m_{1} } ( \frac{ 8 p_{1} E(0) }{ k_{l} ( p_{1} -2 ) } )^{ \frac{ m_{1} -2 }{2} } + B_{ m_{2}}^{ m_{2} } ( \frac{ 8 p_{1} E(0) }{ k_{l} ( p_{1} -2 ) } )^{ \frac{ m_{2} -2 }{2} }$.

Similarly, we also have

$$\begin{aligned} \vert J_{7} \vert \leq \vert \beta \vert \delta ( 1 - k_{l} )^{ m_{1} -1} \hat{C}_{E(0)} ( k \Box \Delta u) + \vert \beta \vert \int _{\Omega } c_{\delta }(x) \bigl\vert y ( x, 1, t) \bigr\vert ^{ m (x) } \,dx . \end{aligned}$$

(4.18)

Applying the estimates of $J_{i}$ to (4.11), we obtain (4.10). □

Lemma 4.7

Assume that $(A_{1})$, $(A_{2})$, $(A_{3})$, and (3.12) hold. Moreover, we assume that

$$\begin{aligned} B_{ p_{1} }^{ p_{1} } \biggl( \frac{ 2 p_{1} E(0) }{ k_{l} ( p_{1} -2 ) } \biggr)^{ \frac{ p_{1} -2 }{2} } + B_{ p_{2}}^{ p_{2} } \biggl( \frac{ 2 p_{1} E(0) }{ k_{l} ( p_{1} -2 ) } \biggr)^{ \frac{ p_{2} -2 }{2} } < \frac{ k_{l} }{4 \gamma }. \end{aligned}$$

(4.19)

If $m_{1} \geq 2$, then there exists $\lambda >0$ such that

$$\begin{aligned} L'(t) \leq - \lambda E(t) - 3 (1- k_{l} ) \Vert \Delta u \Vert _{2}^{2} + \frac{1}{2} ( k \Box \Delta u )\quad \textit{for } t \geq k^{-1}( \varepsilon ). \end{aligned}$$

(4.20)

Proof

From $(A_{3})$, there exists $t_{ \varepsilon } > 0$ with $k( t_{ \varepsilon } ) = \varepsilon$, that is, $t_{ \varepsilon } = k^{-1}( \varepsilon )$. We put $\int _{0}^{ t_{ \varepsilon } } k(s) \,ds = k_{ \varepsilon }$. Using (3.8), (4.4), (4.10), (4.2), and $k' = \eta k -k_{\eta }$, we have

$$\begin{aligned} L'(t) \leq & - \biggl\{ N_{2} \biggl( \int _{0}^{t} k(s) \,ds - \delta \biggr) - N_{1} \biggr\} \Vert u_{t} \Vert _{2}^{2} - \biggl\{ \frac{ N_{1} k_{l} }{4} - N_{2} \delta C_{3} \biggr\} \Vert \Delta u \Vert _{2}^{2} \\ &{} - N_{1} \bigl\Vert \Delta \chi (u) \bigr\Vert _{2}^{2} + N_{1} \gamma \int _{\Omega } \vert u \vert ^{ p (x)} \,dx + \biggl( \frac{ N \eta }{2} + N_{2} \delta C_{5} \biggr) ( k \Box \Delta u ) \\ & {}- \biggl\{ \frac{ N }{4} - \frac{ N_{2} C_{4}}{ \delta } + \frac{ N }{4} - {\mathcal{C}}_{\eta } \biggl( \frac{ N_{1} }{ 2 k_{l} } + \frac{ N_{2} C_{4} }{ \delta } + N_{2} \biggr) \biggr\} ( k_{\eta } \Box \Delta u ) \\ & {}- \int _{\Omega } \bigl( N C_{0} - N_{1} \alpha C_{1} - N_{2} \alpha c_{\delta }(x) - 1 \bigr) \vert u_{t} \vert ^{ m (x)} \,dx \\ &{} - \int _{\Omega } \bigl( N C_{0} - N_{1} \vert \beta \vert C_{1} - N_{2} \vert \beta \vert c_{\delta }(x) \bigr) \bigl\vert y ( x, 1, t) \bigr\vert ^{ m (x)} \,dx \\ &{} - \tau e^{ - \tau } \int _{\Omega } \int ^{1}_{0} \bigl\vert y (x, \rho , t ) \bigr\vert ^{ m (x) } \,d\rho \,dx \\ \leq & - \lambda E(t) - \biggl\{ N_{2} ( k_{ \varepsilon } - \delta ) - N_{1} - \frac{ \lambda }{2} \biggr\} \Vert u_{t} \Vert _{2}^{2} \\ &{} - \biggl\{ \frac{ N_{1} k_{l} }{4} - N_{2} \delta C_{3} - \frac{ \lambda }{2} \biggl( 1- \int ^{t}_{0} k(s) \,ds \biggr) \biggr\} \Vert \Delta u \Vert _{2}^{2} - \biggl( N_{1} - \frac{ \lambda }{4} \biggr) \bigl\Vert \Delta \chi (u) \bigr\Vert _{2}^{2} \\ &{} + \gamma \biggl( N_{1} - \frac{ \lambda }{ p_{2} } \biggr) \int _{ \Omega } \vert u \vert ^{ p (x)} \,dx + \biggl( \frac{ N \eta }{2} + N_{2} \delta C_{5} + \frac{ \lambda }{2} \biggr) ( k \Box \Delta u ) \\ &{} - \biggl\{ \frac{ N }{4} - \frac{ N_{2} C_{4}}{ \delta } + \frac{ N }{4} - {\mathcal{C}}_{\eta } \biggl( \frac{ N_{1} }{ 2 k_{l} } + \frac{ N_{2} C_{4} }{ \delta } + N_{2} \biggr) \biggr\} ( k_{\eta } \Box \Delta u ) \\ &{} - \int _{\Omega } \bigl( N C_{0} - N_{1} \alpha C_{1} - N_{2} \alpha c_{\delta }(x) - 1 \bigr) \vert u_{t} \vert ^{ m (x)} \,dx \\ &{} - \int _{\Omega } \bigl( N C_{0} - N_{1} \vert \beta \vert C_{1} - N_{2} \vert \beta \vert c_{\delta }(x) \bigr) \bigl\vert y ( x, 1, t) \bigr\vert ^{ m (x)} \,dx \\ &{} - \biggl( \tau e^{ - \tau } - \frac{ \lambda \xi \tau }{2} \biggr) \int _{\Omega } \int ^{1}_{0} \bigl\vert y (x, \rho , t ) \bigr\vert ^{ m (x) } \,d\rho \,dx \end{aligned}$$

(4.21)

for $\lambda >0$ and $t \geq t_{ \varepsilon }$. Using estimate (3.16) and taking $\delta = \frac{ k_{l} }{ 4 N_{2} C_{5}}$, we get

$$\begin{aligned} L'(t) \leq & - \lambda E(t) - \biggl\{ N_{2} k_{ \varepsilon } - \frac{ k_{l} }{ 4 C_{5}} - N_{1} - \frac{ \lambda }{2} \biggr\} \Vert u_{t} \Vert _{2}^{2} \\ & {}- \biggl[ N_{1} \biggl\{ \frac{ k_{l} }{4} - \gamma \biggl( B_{ p_{1} }^{ p_{1} } \biggl( \frac{ 2 p_{1} E(0) }{ k_{l} ( p_{1} -2 ) } \biggr)^{ \frac{ p_{1} -2 }{2} } + B_{ p_{2}}^{ p_{2} } \biggl( \frac{ 2 p_{1} E(0) }{ k_{l} ( p_{1} -2 ) } \biggr)^{ \frac{ p_{2} -2 }{2} } \biggr) \biggr\} \\ &{} - \frac{ k_{l} C_{3} }{ 4 C_{5}} - \frac{ \lambda }{2} \biggr] \Vert \Delta u \Vert _{2}^{2} \\ &{} - \biggl( N_{1} - \frac{ \lambda }{4} \biggr) \bigl\Vert \Delta \chi (u) \bigr\Vert _{2}^{2} + \biggl( \frac{ N \eta }{2} + \frac{ k_{l} }{ 4 } + \frac{ \lambda }{2} \biggr) ( k \Box \Delta u ) \\ &{} - \biggl\{ \frac{ N }{4} - \frac{ 4 N_{2}^{2} C_{4} C_{5}}{ k_{l} } + \frac{ N }{4} - {\mathcal{C}}_{\eta } \biggl( \frac{ N_{1} }{ 2 k_{l} } + \frac{ 4 N_{2}^{2} C_{4} C_{5} }{ k_{l} } + N_{2} \biggr) \biggr\} ( k_{ \eta } \Box \Delta u ) \\ &{} - ( N C_{0} - N_{1} \alpha C_{1} - N_{2} \alpha c_{\delta } - 1 ) \int _{\Omega } \vert u_{t} \vert ^{ m (x)} \,dx \\ &{} - \bigl( N C_{0} - N_{1} \vert \beta \vert C_{1} - N_{2} \vert \beta \vert c_{ \delta } \bigr) \int _{\Omega } \bigl\vert y ( x, 1, t) \bigr\vert ^{ m (x)} \,dx \\ &{} - \biggl( \tau e^{ - \tau } - \frac{ \lambda \xi \tau }{2} \biggr) \int _{\Omega } \int ^{1}_{0} \bigl\vert y (x, \rho , t ) \bigr\vert ^{ m (x) } \,d\rho \,dx \quad \text{for } t \geq t_{ \varepsilon } . \end{aligned}$$

(4.22)

From (4.19), we know

$$ \frac{ k_{l} }{4} - \gamma \biggl( B_{ p_{1} }^{ p_{1} } \biggl( \frac{ 2 p_{1} E(0) }{ k_{l} ( p_{1} -2 ) } \biggr)^{ \frac{ p_{1} -2 }{2} } + B_{ p_{2}}^{ p_{2} } \biggl( \frac{ 2 p_{1} E(0) }{ k_{l} ( p_{1} -2 ) } \biggr)^{ \frac{ p_{2} -2 }{2} } \biggr) > 0 .$$

Firstly, we take $N_{1} > \frac{\lambda }{4}$ large enough to get

$$\begin{aligned}& N_{1} \biggl\{ \frac{ k_{l} }{4} - \gamma \biggl( B_{ p_{1} }^{ p_{1} } \biggl( \frac{ 2 p_{1} E(0) }{ k_{l} ( p_{1} -2 ) } \biggr)^{ \frac{ p_{1} -2 }{2} } + B_{ p_{2}}^{ p_{2} } \biggl( \frac{ 2 p_{1} E(0) }{ k_{l} ( p_{1} -2 ) } \biggr)^{ \frac{ p_{2} -2 }{2} } \biggr) \biggr\} - \frac{ k_{l} C_{3} }{ 4 C_{5}} \\& \quad > 4 ( 1 - k_{l} ) , \end{aligned}$$

(4.23)

and then choose $N_{2} >0$ satisfying

$$\begin{aligned} N_{2} k_{ \varepsilon } - \frac{ k_{l} }{ 4 C_{5}} - N_{1} > 1 . \end{aligned}$$

(4.24)

Noting $\frac{ \eta k^{2} (s) }{ k_{\eta } (s)} < k(s)$ and making use of the Lebesgue dominated convergence theorem, we have

$$ \lim_{\eta \to 0^{+}} \eta {\mathcal{C}}_{\eta } = \lim _{\eta \to 0^{+}} \int ^{\infty }_{0} \frac{\eta k^{2}(s)}{k_{\eta }(s)} \,ds =0. $$

Thus, there exists $0 < \eta _{0} < 1$ satisfying

$$ \eta {\mathcal{C}}_{\eta } < \frac{1}{ 16 ( \frac{ N_{1} }{ 2 k_{l} } + \frac{ 4 N_{2}^{2} C_{4} C_{5} }{ k_{l} } + N_{2} ) } \quad \text{for } \eta < \eta _{0}. $$

(4.25)

Secondly, we take $N > 0$ large enough again to get

$$\begin{aligned}& \frac{1}{ 4 N } < \eta _{0},\qquad \frac{ N }{4} - \frac{ 4 N_{2}^{2} C_{4} C_{5}}{ k_{l} } >0 , \end{aligned}$$

(4.26)

$$\begin{aligned}& N C_{0} - N_{1} \alpha C_{1} - N_{2} \alpha c_{\delta } - 1 >0, \end{aligned}$$

(4.27)

and

$$\begin{aligned} N C_{0} - N_{1} \vert \beta \vert C_{1} - N_{2} \vert \beta \vert c_{\delta } >0. \end{aligned}$$

(4.28)

Thirdly, selecting $\eta = \frac{1}{ 4 N } < \eta _{0}$, we get

$$ \frac{ N \eta }{2} + \frac{ k_{l} }{ 4 } = \frac{1}{8} + \frac{ k_{l} }{ 4 } < \frac{3}{8} $$

(4.29)

and

$$ \frac{N}{4} - {\mathcal{C}}_{\eta } \biggl( \frac{ N_{1} }{ 2 k_{l} } + \frac{ 4 N_{2}^{2} C_{4} C_{5} }{ k_{l} } + N_{2} \biggr) > \frac{N}{4} - \frac{1}{ 16 \eta } =0 , $$

(4.30)

here we used (4.25). From (4.22), (4.23), (4.24), (4.26), (4.27), (4.28), (4.29), and (4.30), we get

$$\begin{aligned} L'(t) \leq & - \lambda E(t) - \biggl\{ 1 - \frac{ \lambda }{2} \biggr\} \Vert u_{t} \Vert _{2}^{2} - \biggl\{ 4 ( 1 - k_{l} ) - \frac{ \lambda }{2} \biggr\} \Vert \Delta u \Vert _{2}^{2} \\ & {}+ \biggl( \frac{ 3 }{ 8 } + \frac{ \lambda }{2} \biggr) ( k \Box \Delta u ) - \biggl( \tau e^{ - \tau } - \frac{ \lambda \xi \tau }{2} \biggr) \int _{\Omega } \int ^{1}_{0} \bigl\vert y (x, \rho , t ) \bigr\vert ^{ m (x) } \,d\rho \,dx \end{aligned}$$

for $t \geq t_{ \varepsilon }$. Finally, selecting $\lambda >0$ satisfying

$$ \lambda \leq \min \biggl\{ 2 ( 1 - k_{l} ) , \frac{1}{4}, \frac{ 2 \tau e^{- \tau }}{ \xi \tau } \biggr\} , $$

we obtain (4.20). □

Lemma 4.8

Assume that $(A_{1})$, $(A_{2})$, $(A_{3})$, and (4.19) hold and let $m_{1} \geq 2$. Then

$$\begin{aligned} 0 < \int ^{\infty }_{0} E(s) \,ds < \infty . \end{aligned}$$

(4.31)

Proof

From (4.20) and (4.3), we see

$$\begin{aligned} \bigl( L(t) + \Lambda (t) \bigr) ' \leq - \lambda E(t) \quad \text{for } t \geq t_{ \varepsilon }, \end{aligned}$$

(4.32)

and

$$\begin{aligned} 0< \int ^{t}_{t_{\varepsilon }} E(s) \,ds \leq - \frac{1}{ \lambda } \int ^{t}_{t_{\varepsilon }}\bigl( L'(s) + \Lambda '(s) \bigr) \,ds \leq \frac{ L(t_{\varepsilon }) + \Lambda (t_{\varepsilon }) }{ \lambda } < \infty ,\quad \forall t \geq t_{\varepsilon } , \end{aligned}$$

which gives

$$\begin{aligned} 0 < \int ^{\infty }_{0} E(s) \,ds = \int _{0}^{t_{\varepsilon }} E(s) \,ds + \int ^{\infty }_{t_{\varepsilon }} E(s) \,ds < \infty . \end{aligned}$$

 □

Theorem 4.1

Assume that $(A_{1})$, $(A_{2})$, $(A_{3})$, and (4.19) hold and let $m_{1} \geq 2$. Then there exist $c_{i} , \omega _{i} >0$, $i =1,2$, such that, for $t \geq k^{-1}(\varepsilon )$,

$$\begin{aligned} E(t) \leq c_{1} \exp \biggl( - \omega _{1} \int ^{t}_{k^{-1}( \varepsilon )} \zeta (s) \,ds \biggr) \quad \textit{in the case }K \textit{ is linear } \end{aligned}$$

(4.33)

and

$$ E(t) \leq c_{2} \tilde{K}^{-1} \biggl( \omega _{2} \int ^{t}_{k^{-1}( \varepsilon )} \zeta (s) \,ds \biggr)\quad \textit{in the case } K \textit{ is nonlinear }, $$

where

$$ \tilde{K}(s) = \int ^{\varepsilon }_{s} \frac{1}{ \tau K'(\tau )} \,d \tau . $$

(4.34)

Proof

From Lemma 4.5, Lemma 4.6, and Lemma 4.7, the proof is similar to that of [23]. But, for the completeness, we give the proof. Since k and ζ are continuous in t, we have

$$ a_{1} \leq \zeta (t) K\bigl(k(t)\bigr) \leq a_{2} \quad \text{for } t \in [0, t_{ \varepsilon }] $$

for some $a_{1}, a_{2} >0$, and

$$ k'(t) \leq - \zeta (t) K\bigl(k(t)\bigr) \leq - a_{1} \leq -\frac{a_{1}}{k(0)} k(t) \quad \text{for } t \in [0, t_{\varepsilon }]. $$

(4.35)

From (4.20), (4.35), (3.8), we get

$$\begin{aligned} L'(t) \leq & - \lambda E(t) - \frac{ k(0) }{2 a_{1}} \bigl( k' \Box \Delta u \bigr) + \frac{1}{2} \int _{t_{\varepsilon }}^{t} k (s) \bigl\Vert \Delta u (t) - \Delta u (t-s) \bigr\Vert _{2}^{2} \,ds \\ \leq & - \lambda E(t) - \frac{ k(0) }{ a_{1}} E'(t) + \frac{1}{2} \int _{t_{\varepsilon }}^{t} k (s) \bigl\Vert \Delta u (t) - \Delta u (t-s) \bigr\Vert _{2}^{2} \,ds \quad \text{for } t \geq t_{ \varepsilon }. \end{aligned}$$

(4.36)

Let

$$ R (t) = L(t) + \frac{ k(0)}{ a_{1}} E(t) , $$

then $R \sim E$ and

$$\begin{aligned} R'(t) \leq - \lambda E(t) + \frac{1}{2} \int _{t_{\varepsilon }}^{t} k (s) \bigl\Vert \Delta u (t) - \Delta u (t-s) \bigr\Vert _{2}^{2} \,ds \quad \text{for } t \geq t_{ \varepsilon } . \end{aligned}$$

(4.37)

Case 1: K is linear, that is, $K (s) = a s$ for some $a > 0$. Put

$$ {\mathcal{R}}_{1} (t) = \zeta (t) R(t) + \frac{1}{ a } E(t) . $$

From (4.37), (2.6), and (3.8), we have

$$\begin{aligned} {\mathcal{R}}_{1} ' (t) \leq & - \lambda \zeta (t) E(t) + \frac{1}{2} \int ^{t}_{t_{\varepsilon }} \zeta (s) k(s) \bigl\Vert \Delta u(t) - \Delta u(t-s) \bigr\Vert _{2}^{2} \,ds + \frac{1}{ a } E'(t) \\ \leq & - \lambda \zeta (t) E(t) - \frac{1}{2 a } \int ^{t}_{t_{ \varepsilon }} k'(s) \bigl\Vert \Delta u(t) - \Delta u(t-s) \bigr\Vert ^{2} \,ds + \frac{1}{ a } E'(t) \\ \leq & - \lambda \zeta (t) E(t), \quad t \geq t_{\varepsilon } . \end{aligned}$$

(4.38)

This and the relation ${\mathcal{R}}_{1} (t) \sim E(t)$ prove (4.33).

Case 2: K is nonlinear. For $t \geq t_{\varepsilon }$, we put

$$\begin{aligned} \Gamma _{1} (t) = a_{3} \int ^{t}_{t_{\varepsilon }} \bigl\Vert \Delta u(t) - \Delta u(t-s) \bigr\Vert _{2}^{2} \,ds \end{aligned}$$

and

$$\begin{aligned} \Gamma _{2} (t) = - \int ^{t}_{t_{\varepsilon }} k'(s) \bigl\Vert \Delta u(t) - \Delta u(t-s) \bigr\Vert _{2}^{2} \,ds. \end{aligned}$$

From (3.17), (3.8), (4.31), we get

$$\begin{aligned} \int ^{t}_{t_{\varepsilon }} \bigl\Vert \Delta u(t) - \Delta u(t-s) \bigr\Vert _{2}^{2} \,ds \leq & 2 \int ^{t}_{t_{\varepsilon }} \bigl\Vert \Delta u(t) \bigr\Vert _{2}^{2} + \bigl\Vert \Delta u(t-s) \bigr\Vert _{2}^{2} \,ds \\ \leq & \frac{ 4 p_{1} }{ k_{l} ( p_{1} -2 )} \int ^{t}_{t_{ \varepsilon }} \bigl( E(t) + E(t-s) \bigr) \,ds \\ \leq & \frac{ 4 p_{1} }{ k_{l} ( p_{1} -2 )} \biggl( \int ^{t}_{t_{ \varepsilon }} E(s) \,ds + \int _{0}^{t - t_{\varepsilon }} E(s ) \biggr) \,ds ) < \infty . \end{aligned}$$

(4.39)

Thus, there exists $0 < a_{3} < 1$ satisfying

$$ \Gamma _{1} (t) < 1 \quad \text{for } t \geq t_{\varepsilon }. $$

(4.40)

From (3.8), we know

$$ \Gamma _{2} (t) \leq - \bigl(k' \Box \Delta u\bigr) (t) \leq - 2 E'(t). $$

(4.41)

Using $(A_{3})$, (4.40), the relation $\overline{ K } ( \varrho t ) \leq \varrho \overline{ K } ( t )$ for $0 \leq \varrho \leq 1$ and $t \in [0, \infty )$, and Jensen’s inequality, we find

$$\begin{aligned} \Gamma _{2} (t) = & - \frac{1}{ a_{3} \Gamma _{1} (t)} \int ^{t}_{t_{ \varepsilon }} \Gamma _{1} (t) k'(s) a_{3} \bigl\Vert \Delta u(t) - \Delta u(t-s) \bigr\Vert _{2}^{2} \,ds \\ \geq & \frac{1}{ a_{3} \Gamma _{1} (t)} \int ^{t}_{t_{\varepsilon }} \Gamma _{1} (t) \zeta (s) K\bigl(k(s)\bigr) a_{3} \bigl\Vert \Delta u(t) - \Delta u(t-s) \bigr\Vert _{2}^{2} \,ds \\ \geq & \frac{\zeta (t)}{ a_{3} \Gamma _{1} (t)} \int ^{t}_{t_{ \varepsilon }} \overline{ K } \bigl( \Gamma _{1} (t) k(s) \bigr) a_{3} \bigl\Vert \Delta u(t) - \Delta u(t-s) \bigr\Vert _{2}^{2} \,ds \\ \geq & \frac{\zeta (t)}{ a_{3} } \overline{ K } \biggl( a_{3} \int ^{t}_{t_{ \varepsilon }} k(s) \bigl\Vert \Delta u(t) - \Delta u(t-s) \bigr\Vert _{2}^{2} \,ds \biggr), \quad t \geq t_{\varepsilon } . \end{aligned}$$

(4.42)

So, we have

$$\begin{aligned} \int ^{t}_{t_{\varepsilon }} k(s) \bigl\Vert \Delta u(t) - \Delta u(t-s) \bigr\Vert _{2}^{2} \,ds \leq \frac{1}{a_{3}} \overline{K}^{-1} \biggl( \frac{ a_{3} \Gamma _{2} (t)}{ \zeta (t)} \biggr). \end{aligned}$$

Applying this to (4.37), we get

$$\begin{aligned} R '(t) \leq - \lambda E(t) + \frac{1}{2 a_{3} } \overline{K}^{-1} \biggl( \frac{ a_{3} \Gamma _{2}(t)}{ \zeta (t)} \biggr) \quad \text{for } t \geq t_{\varepsilon } . \end{aligned}$$

(4.43)

We know that the convex function K̅ satisfies

$$\begin{aligned} st \leq \overline{ K }^{*}(s) + \overline{ K }(t) \quad \text{for } s, t \geq 0 \end{aligned}$$

(4.44)

and

$$\begin{aligned} \overline{ K }^{*}(s) = s\bigl(\overline{ K }'\bigr)^{-1}(s) - \overline{ K }\bigl(\bigl( \overline{ K }'\bigr)^{-1}(s)\bigr) \quad \text{for } s \geq 0 , \end{aligned}$$

(4.45)

where $\overline{ K }^{*}$ is the conjugate function of K̅.

Let $0 < \mu < \min \{ \varepsilon , 2 a_{3} \lambda E(0) \}$ and ${\mathcal{E}}(t) = \frac{E(t)}{E(0)}$. Using $\overline{K}'(s) >0$, $\overline{K}''(s) >0$, $E'(t) \leq 0$, $\overline{K}(0) = \overline{K}'(0) =0$, (4.43), (4.44), and (4.45), we infer

$$\begin{aligned} \bigl( \overline{K}' \bigl( \mu { \mathcal{E}}(t) \bigr) R (t) \bigr)' \leq & - \lambda \overline{K}' \bigl( \mu {\mathcal{E}}(t) \bigr) E(t) + \frac{1}{2 a_{3} } \overline{K}' \bigl( \mu {\mathcal{E}}(t) \bigr) \overline{K}^{-1} \biggl( \frac{ a_{3} \Gamma _{2}(t)}{ \zeta (t)} \biggr) \\ \leq & - \lambda \overline{K}' \bigl( \mu {\mathcal{E}}(t) \bigr) E(t) + \frac{1}{2 a_{3}} \overline{K}^{*} \bigl( \overline{K}' \bigl( \mu { \mathcal{E}}(t) \bigr) \bigr) + \frac{ \Gamma _{2}(t)}{2 \zeta (t)} \\ \leq & - \lambda \overline{K}' \bigl( \mu {\mathcal{E}}(t) \bigr) E(t) + \frac{ \mu }{ 2 a_{3}} {\mathcal{E}}(t) \overline{K}' \bigl( \mu {\mathcal{E}}(t) \bigr) + \frac{ \Gamma _{2}(t)}{ 2 \zeta (t)} \\ = & - a_{4} {\mathcal{E}}(t) K' \bigl( \mu { \mathcal{E}}(t) \bigr) + \frac{ \Gamma _{2}(t)}{ 2 \zeta (t)} , \end{aligned}$$

(4.46)

where $a_{4} = \lambda E(0) - \frac{ \mu }{ 2 a_{3}} > 0$. Setting

$$\begin{aligned} {\mathcal{R}}_{2} (t)= \zeta (t) \overline{K}' \bigl( \mu {\mathcal{E}}(t) \bigr) R(t) + E(t) , \end{aligned}$$

from (4.46) and (4.41), we get

$$\begin{aligned} {\mathcal{R}}_{2}' (t) \leq - a_{4} \zeta (t) {\mathcal{E}}(t) K' \bigl( \mu { \mathcal{E}}(t) \bigr) + \frac{ \Gamma _{2}(t)}{2} + E'(t) \leq - a_{4} \zeta (t) K_{0} \bigl( {\mathcal{E}}(t) \bigr) \quad \text{for } t\geq t_{ \varepsilon } , \end{aligned}$$

(4.47)

where $K_{0}(s) = s K'( \mu s)$. Since ${\mathcal{R}}_{2} (t) \sim E(t)$, there exist $a_{5}, a_{6} >0$ satisfying

$$ a_{5} {\mathcal{R}}_{2} (t) \leq E(t) \leq a_{6} {\mathcal{R}}_{2} (t). $$

Finally, we let

$$ {\mathcal{L}} (t) = \frac{ a_{5} {\mathcal{R}}_{2} (t)}{E(0)} , $$

(4.48)

then

$$ {\mathcal{L}}(t) \leq {\mathcal{E}}(t) \leq 1 . $$

(4.49)

Since $K_{0}$ is an increasing function on $(0,1]$, from (4.48), (4.47), and (4.49), we deduce

$$ {\mathcal{L}}'(t) \leq - \omega _{2} \zeta (t) K_{0} \bigl( {\mathcal{E}}(t) \bigr) \leq - \omega _{2} \zeta (t) K_{0} \bigl( { \mathcal{L}}(t) \bigr) \quad \text{for } t \geq t_{\varepsilon } , $$

where $\omega _{2} = \frac{ a_{4} a_{5} }{ E(0)}$, and

$$\begin{aligned} \int ^{t}_{t_{\varepsilon }} \omega _{2} \zeta (s) \,ds \leq & - \int ^{t}_{t_{ \varepsilon }} \frac{{\mathcal{L}}'(s)}{ K_{0} ( {\mathcal{L}}(s) ) } \,ds = - \int ^{t}_{t_{\varepsilon }} \frac{{\mathcal{L}}'(s)}{ {\mathcal{L}}(s) K' ( \mu {\mathcal{L}}(s) ) } \,ds = \int ^{\mu {\mathcal{L}}(t_{\varepsilon })}_{\mu {\mathcal{L}}(t)} \frac{ 1 }{ s K' ( s ) } \,ds \\ \leq & \int ^{\varepsilon }_{\mu {\mathcal{L}}(t)} \frac{ 1 }{ s K' ( s ) } \,ds = \tilde{K}\bigl( \mu {\mathcal{L}}(t) \bigr), \end{aligned}$$

here K̃ is the function given in (4.34). Because K̃ is strictly decreasing on $(0, \varepsilon ]$, we obtain

$$\begin{aligned} {\mathcal{L}}(t) \leq \frac{1}{\mu } \tilde{K}^{-1} \biggl( \omega _{2} \int ^{t}_{t_{\varepsilon }} \zeta (s) \,ds \biggr) \quad \text{for } t\geq t_{ \varepsilon } . \end{aligned}$$

 □

General decay for the case ${1 < m_{1} < 2 }$

In this subsection, we derive a general decay result for the case $1 < m_{1} < 2$. We let

$$\begin{aligned} \Omega _{1} = \bigl\{ x \in \Omega : m (x) < 2 \bigr\} , \qquad \Omega _{2} = \bigl\{ x \in \Omega : m (x) \geq 2 \bigr\} \end{aligned}$$

and

$$\begin{aligned} \Omega _{i}^{-} = \bigl\{ x \in \Omega _{i} : \bigl\vert u_{t} (x,t) \bigr\vert < 1 \bigr\} , \qquad \Omega _{i}^{+} = \bigl\{ x \in \Omega _{i} : \bigl\vert u_{t} (x,t) \bigr\vert \geq 1 \bigr\} \end{aligned}$$

for $i =1,2$.

Lemma 4.9

Assume that $(A_{1})$, $(A_{2})$, $(A_{3})$, and (3.12) hold. If $1 < m_{1} < 2$, then Φ satisfies

$$\begin{aligned} \Phi '(t) \leq & \Vert u_{t} \Vert _{2}^{2} - \frac{ k_{l} }{4} \Vert \Delta u \Vert _{2}^{2} - \bigl\Vert \Delta \chi (u) \bigr\Vert _{2}^{2} + \gamma \int _{\Omega } \vert u \vert ^{ p (x)} \,dx + \frac{ {\mathcal{C}}_{\eta } }{ 2 k_{l} } ( k_{\eta } \Box \Delta u ) \\ & {}+ \alpha C_{6} \biggl\{ \int _{\Omega } \vert u_{t} \vert ^{ m (x) } \,dx + \biggl( \int _{\Omega } \vert u_{t} \vert ^{ m (x) } \,dx \biggr)^{ m_{1} -1 } \biggr\} \\ &{} + \vert \beta \vert C_{6} \biggl\{ \int _{\Omega } \bigl\vert y (x, 1, t ) \bigr\vert ^{ m (x)} \,dx + \biggl( \int _{\Omega } \bigl\vert y (x, 1, t ) \bigr\vert ^{ m (x) } \,dx \biggr)^{ m_{1} -1 } \biggr\} . \end{aligned}$$

(4.50)

Proof

We re-estimate $I_{1}$ and $I_{2}$ in (4.7) for the case $1 < m_{1} <2$. Using Young’s inequality, for $\delta _{2} >0$, we have

$$\begin{aligned} - \alpha \int _{\Omega _{1}} u \vert u_{t} \vert ^{ m (x) -2 } u_{t} \,dx \leq & \alpha \delta _{2} \int _{\Omega _{1}} \vert u \vert ^{2} \,dx + \frac{ \alpha }{ 4 \delta _{2} } \int _{\Omega _{1}} \vert u_{t} \vert ^{2 m (x) -2 } \,dx . \end{aligned}$$

(4.51)

Noting $2 m_{1} -2 < 2 m (x) -2 < m (x) < 2$ for $x \in \Omega _{1}$ and using Hölder’s inequality with $(2 - m_{1}) + ( m_{1} -1) =1$, we get

$$\begin{aligned} \int _{\Omega _{1}} \vert u_{t} \vert ^{2 m (x) -2 } \,dx \leq & \int _{ \Omega _{1}^{-} } \vert u_{t} \vert ^{2 m_{1} -2 } \,dx + \int _{\Omega _{1}^{+} } \vert u_{t} \vert ^{ m (x) } \,dx \\ \leq & \bigl\vert \Omega _{1}^{-} \bigr\vert ^{ 2 - m_{1}} \biggl( \int _{\Omega _{1}^{-} } \vert u_{t} \vert ^{2} \,dx \biggr)^{ m_{1} -1} + \int _{\Omega _{1}^{+} } \vert u_{t} \vert ^{ m (x) } \,dx \\ \leq & \bigl\vert \Omega _{1}^{-} \bigr\vert ^{ 2 - m_{1}} \biggl( \int _{\Omega _{1}^{-} } \vert u_{t} \vert ^{ m (x) } \,dx \biggr)^{ m_{1} -1} + \int _{\Omega _{1}^{+} } \vert u_{t} \vert ^{ m (x) } \,dx \\ \leq & c \biggl\{ \biggl( \int _{\Omega } \vert u_{t} \vert ^{ m (x) } \,dx \biggr)^{ m_{1} -1} + \int _{\Omega } \vert u_{t} \vert ^{ m (x) } \,dx \biggr\} . \end{aligned}$$

(4.52)

Applying (4.52) to (4.51), we see

$$\begin{aligned}& - \alpha \int _{\Omega _{1}} u \vert u_{t} \vert ^{ m (x) -2 } u_{t} \,dx \\& \quad \leq \alpha \delta _{2} B_{2}^{2} \Vert \Delta u \Vert _{2}^{2} + \frac{ \alpha c }{ 4 \delta _{2} } \biggl\{ \biggl( \int _{\Omega } \vert u_{t} \vert ^{ m (x) } \,dx \biggr)^{ m_{1} -1} + \int _{\Omega } \vert u_{t} \vert ^{ m (x) } \,dx \biggr\} . \end{aligned}$$

(4.53)

As the estimates of (4.8), for $\delta _{3} >0$, we find

$$\begin{aligned} - \alpha \int _{\Omega _{2}} u \vert u_{t} \vert ^{ m (x) -2 } u_{t} \,dx \leq & \alpha \delta _{3} \int _{\Omega _{2}} \vert u \vert ^{ m (x) } \,dx + \alpha \int _{\Omega _{2}} c_{\delta _{3}} (x) \vert u_{t} \vert ^{ m (x) } \,dx \\ \leq & \alpha \delta _{3} \tilde{C}_{E(0)} \Vert \Delta u \Vert _{2}^{2} + \alpha \int _{\Omega } c_{\delta _{3} }(x) \vert u_{t} \vert ^{ m (x) } \,dx , \end{aligned}$$

(4.54)

where $\tilde{C}_{E(0)} = B_{ m_{-} }^{ m_{-} } ( \frac{ 2 p_{1} E(0) }{ k_{l} ( p_{1} -2 ) } )^{ \frac{ m_{-} -2 }{2} } + B_{ m_{+}}^{ m_{+} } ( \frac{ 2 p_{1} E(0) }{ k_{l} ( p_{1} -2 ) } )^{ \frac{ m_{+} -2 }{2} }$, here

$$ m_{-} = \operatorname{ess} \inf_{x \in \Omega _{2}} m (x) \geq 2 \quad \text{and} \quad m_{+} = \operatorname{ess} \sup _{x \in \Omega _{2}} m (x) \geq 2 .$$

Combining (4.53) and (4.54) and taking $\delta _{2} = \frac{ k_{l} }{ 16 \alpha B_{2}^{2}}$ and $\delta _{3} = \frac{ k_{l} }{ 16 \alpha \tilde{C}_{E(0)}}$, we have

$$\begin{aligned} - I_{1} \leq & \frac{ k_{l} }{8} \Vert \Delta u \Vert _{2}^{2} + \alpha C_{6} \biggl\{ \biggl( \int _{\Omega } \vert u_{t} \vert ^{ m (x) } \,dx \biggr)^{ m_{1} -1} + \int _{\Omega } \vert u_{t} \vert ^{ m (x) } \,dx \biggr\} . \end{aligned}$$

(4.55)

Similarly, we have

$$\begin{aligned} - I_{2} \leq & \frac{ k_{l} }{8} \Vert \Delta u \Vert _{2}^{2} + \vert \beta \vert C_{6} \biggl\{ \biggl( \int _{\Omega } \bigl\vert y(x,1, t) \bigr\vert ^{ m (x) } \,dx \biggr)^{ m_{1} -1} + \int _{\Omega } \bigl\vert y(x,1, t) \bigr\vert ^{ m (x) } \,dx \biggr\} . \end{aligned}$$

(4.56)

Adapting (4.55) and (4.56) to (4.7), we obtain (4.50). □

Lemma 4.10

Assume that $(A_{1})$, $(A_{2})$, $(A_{3})$, and (3.12) hold. If $1 < m_{1} < 2$, then Ψ satisfies

$$\begin{aligned} \Psi '(t) \leq & - \biggl( \int ^{t}_{0} k(s) \,ds - \delta \biggr) \Vert u_{t} \Vert _{2}^{2} + \delta C_{3} \Vert \Delta u \Vert _{2}^{2} + \biggl( \frac{ C_{4} ( 1+ {\mathcal{C}}_{\eta }) }{ \delta } + {\mathcal{C}}_{ \eta } \biggr) (k_{\eta } \Box \Delta u) \\ & {}+ \delta C_{7} ( k \Box \Delta u ) + \frac{ \alpha C_{8} }{ \delta } \biggl\{ c_{\delta } \int _{\Omega } \vert u_{t} \vert ^{ m (x) } \,dx + \biggl( \int _{\Omega } \vert u_{t} \vert ^{ m (x) } \,dx \biggr)^{ m_{1} -1 } \biggr\} \\ & {}+ \frac{ \vert \beta \vert C_{8} }{ \delta } \biggl\{ c_{\delta } \int _{ \Omega } \bigl\vert y (x, 1, t ) \bigr\vert ^{ m (x)} \,dx + \biggl( \int _{\Omega } \bigl\vert y (x, 1, t ) \bigr\vert ^{ m (x) } \,dx \biggr)^{ m_{1} -1 } \biggr\} \end{aligned}$$

(4.57)

for any $\delta >0$.

Proof

We re-estimate $J_{6}$ and $J_{7}$ in (4.11) for the case $1 < m_{1} <2$. Let $\delta >0$. Using (4.52), we have

$$\begin{aligned}& \alpha \int _{\Omega _{1} } \vert u_{t} \vert ^{ m (x) -2} u_{t} \int ^{t}_{0} k(t-s) \bigl( u(t) - u(s) \bigr) \,ds \,dx \\& \quad \leq \alpha \delta \int _{\Omega _{1}} \biggl\vert \int ^{t}_{0} k(t-s) \bigl( u(t) - u(s) \bigr) \,ds \biggr\vert ^{2} \,dx + \frac{ \alpha }{ 4 \delta } \int _{ \Omega _{1}} \vert u_{t} \vert ^{ 2 m (x) -2 } \,dx \\& \quad \leq \alpha \delta ( 1 - k_{l} ) B_{2}^{2} ( k \Box \Delta u ) + \frac{ \alpha c }{ 4 \delta } \biggl\{ \biggl( \int _{\Omega } \vert u_{t} \vert ^{ m (x) } \,dx \biggr)^{ m_{1} -1} + \int _{\Omega } \vert u_{t} \vert ^{ m (x) } \,dx \biggr\} . \end{aligned}$$

(4.58)

Since $m(x) \geq 2$ on $\Omega _{2}$, we can apply the same argument of (4.17) on $\Omega _{2}$ instead of Ω to obtain

$$\begin{aligned}& \alpha \int _{\Omega _{2} } \vert u_{t} \vert ^{ m (x) -2} u_{t} \int ^{t}_{0} k(t-s) \bigl( u(t) - u(s) \bigr) \,ds \,dx \\& \quad \leq \alpha \delta ( 1 - k_{l} )^{ m_{1} -1} \int _{\Omega _{2}} \biggl( \int ^{t}_{0} k(t-s) \bigl\vert u(t) - u(s) \bigr\vert ^{ m (x)} \,ds \biggr) \,dx + \alpha \int _{\Omega _{2}} c_{\delta }(x) \vert u_{t} \vert ^{ m (x) } \,dx \\& \quad \leq \alpha \delta ( 1 - k_{l} )^{ m_{1} -1} \hat{C}_{E(0)} ( k \Box \Delta u) + \frac{ \alpha }{\delta } \int _{\Omega } \delta c_{ \delta }(x) \vert u_{t} \vert ^{ m (x) } \,dx . \end{aligned}$$

(4.59)

Combining (4.58) and (4.59) and noting that $\delta c_{\delta }(x)$ is bounded on Ω, we have

$$\begin{aligned} \vert J_{6} \vert \leq & \alpha \delta \bigl\{ ( 1 - k_{l} ) B_{2}^{2} + ( 1 - k_{l} )^{ m_{1} -1} \hat{C}_{E(0)} \bigr\} ( k \Box \Delta u ) \\ & {}+ \frac{ \alpha C_{8} }{ \delta } \biggl\{ \biggl( \int _{\Omega } \vert u_{t} \vert ^{ m (x) } \,dx \biggr)^{ m_{1} -1} + c_{\delta } \int _{\Omega } \vert u_{t} \vert ^{ m (x) } \,dx \biggr\} . \end{aligned}$$

(4.60)

Similarly, we find

$$\begin{aligned} \vert J_{7} \vert \leq & \vert \beta \vert \delta \bigl\{ ( 1 - k_{l} ) B_{2}^{2} + ( 1 - k_{l} )^{ m_{1} -1} \hat{C}_{E(0)} \bigr\} ( k \Box \Delta u ) \\ & {}+ \frac{ \vert \beta \vert C_{8} }{ \delta } \biggl\{ \biggl( \int _{\Omega } \bigl\vert y(x,1,t) \bigr\vert ^{ m (x) } \,dx \biggr)^{ m_{1} -1} + c_{\delta } \int _{\Omega } \bigl\vert y(x,1,t) \bigr\vert ^{ m (x) } \,dx \biggr\} . \end{aligned}$$

(4.61)

Substituting (4.12), (4.13), (4.14), (4.15), (4.16), (4.60), and (4.61) into (4.11), we obtain (4.57). □

Lemma 4.11

Assume that $(A_{1})$, $(A_{2})$, $(A_{3})$, and (4.19) hold. If $1 < m_{1} < 2$, there exists $\lambda >0$ such that

$$\begin{aligned} L'(t) \leq - \lambda E(t) - 3 (1- k_{l} ) \Vert \Delta u \Vert _{2}^{2} + \frac{1}{2} ( k \Box \Delta u ) + C_{9} \bigl( - E ' (t)\bigr)^{ m_{1} -1 } \quad \textit{for } t \geq k^{-1}( \varepsilon ). \end{aligned}$$

(4.62)

Proof

From (3.8), (4.2), (4.50), and (4.57), the proof is similar to that of (4.20) by replacing the constants $C_{1}$, $C_{5}$, and $c_{\delta } (x)$ by $C_{6}$, $C_{7}$, and $\frac{ c_{ \delta } C_{8}}{ \delta }$, respectively, adding

$$\begin{aligned}& \biggl( N_{1} \alpha C_{6} + \frac{ N_{2} \alpha C_{8} }{ \delta } \biggr) \biggl( \int _{\Omega } \vert u_{t} \vert ^{ m (x) } \,dx \biggr)^{ m_{1} -1} \\& \quad {}+ \biggl( N_{1} \vert \beta \vert C_{6} + \frac{ N_{2} \vert \beta \vert C_{8} }{ \delta } \biggr) \biggl( \int _{\Omega } \bigl\vert y(x,1,t) \bigr\vert ^{ m (x) } \,dx \biggr)^{ m_{1} -1} , \end{aligned}$$

taking $\delta = \frac{ k_{l} }{ 4 N_{2} C_{7}}$ in (4.21), and using the relation

$$\begin{aligned} \biggl( \int _{\Omega } \vert u_{t} \vert ^{ m (x) } \,dx \biggr)^{ m_{1} -1} + \biggl( \int _{\Omega } \bigl\vert y(x,1,t) \bigr\vert ^{ m (x) } \,dx \biggr)^{ m_{1} -1} \leq 2 \biggl( - \frac{ E '(t)}{ C_{0} } \biggr)^{ m_{1} -1} , \end{aligned}$$

which is seen from (3.8). □

Lemma 4.12

Assume that $(A_{1})$, $(A_{2})$, $(A_{3})$, and (4.19) hold and let $1 < m_{1} < 2$. Then

$$\begin{aligned} \int ^{t_{2}}_{ t_{1} } E (s) \,ds \leq C_{10} ( t_{2} - t_{1} )^{ 2 - m_{1} } \quad \textit{for any } t_{2} \geq t_{1} \geq 0 . \end{aligned}$$

(4.63)

Proof

By virtue of (4.62), estimate (4.32) is replaced by

$$\begin{aligned} \bigl( L(t) + \Lambda (t) \bigr) ' \leq - \lambda E(t) + C_{9} \bigl( - E ' (t) \bigr)^{ m_{1} -1 } \quad \text{for } t \geq t_{\varepsilon } . \end{aligned}$$

(4.64)

Using (3.8), (4.64), and Young’s inequality with $( m_{1} -1) + ( 2 - m_{1} ) =1$, we observe

$$\begin{aligned}& \bigl\{ E^{\frac{2- m_{1} }{ m_{1} -1 }}(t) \bigl( L(t) + \Lambda (t) \bigr) + a_{7} E(t) \bigr\} ' \\& \quad \leq - \lambda E^{\frac{1 }{ m_{1} -1 }}(t) + C_{9} E^{ \frac{2- m_{1} }{ m_{1} -1 }}(t) \bigl( - E ' (t)\bigr)^{ m_{1} -1 } + a_{7} E'(t) \\& \quad \leq - \lambda E^{\frac{1 }{ m_{1} -1 }}(t) + \frac{ (2 - m_{1} ) \lambda }{2} E^{ \frac{1}{ m_{1} -1 } }(t) + ( m_{1} -1 ) C_{9}^{ \frac{1}{ m_{1} -1 } } \biggl( \frac{ \lambda }{2} \biggr)^{ \frac{ m_{1} -2 }{ m_{1} -1 } } \bigl( - E'(t)\bigr) + a_{7} E'(t) \\& \quad \leq - \lambda E^{\frac{1 }{ m_{1} -1 }}(t) + \frac{ \lambda }{2} E^{ \frac{1}{ m_{1} -1 } }(t) + C_{9}^{ \frac{1}{ m_{1} -1 } } \biggl( \frac{ \lambda }{2} \biggr)^{ \frac{ m_{1} -2 }{ m_{1} -1 } } \bigl( - E'(t)\bigr) + a_{7} E'(t) \\& \quad = - \frac{ \lambda }{2} E^{\frac{1 }{ m_{1} -1 }}(t) , \end{aligned}$$

(4.65)

where $a_{7} = C_{9}^{ \frac{1}{ m_{1} -1 } } ( \frac{ \lambda }{2} )^{ \frac{ m_{1} -2 }{ m_{1} -1 } }$, which yields

$$\begin{aligned} \int ^{t}_{ t_{\varepsilon } } E^{ \frac{1 }{ m_{1} -1 } }( s ) \,ds \leq \frac{2}{ \lambda } \bigl\{ E^{\frac{2- m_{1} }{ m_{1} -1 }}( t_{ \varepsilon } ) \bigl( L( t_{\varepsilon } ) + \Lambda ( t_{\varepsilon } ) \bigr) + a_{7} E( t_{\varepsilon } ) \bigr\} : = a_{8} \end{aligned}$$

for all $t \geq t_{\varepsilon }$. So, we get

$$\begin{aligned} 0 < \int ^{\infty }_{0} E^{ \frac{1}{ m_{1} -1 } }(s) \,ds \leq \int ^{ t_{ \varepsilon } }_{0} E^{ \frac{1}{ m_{1} -1 } }(s) \,ds + a_{8} < \infty . \end{aligned}$$

From this and Hölder inequality with $( 2 - m_{1} ) + ( m_{1} -1 ) =1$, we obtain

$$\begin{aligned} \int ^{t_{2}}_{ t_{1} } E ( s ) \,ds \leq ( t_{2} - t_{1} )^{ 2 - m_{1} } \biggl( \int ^{t_{2}}_{ t_{1} } E^{\frac{1 }{ m_{1} -1 }}( s ) \,ds \biggr)^{ m_{1} -1 } \leq C_{10} ( t_{2} - t_{1} )^{ 2 - m_{1} } \end{aligned}$$

for any $t_{2} \geq t_{1} \geq 0$. □

Theorem 4.2

Assume that $(A_{1})$, $(A_{2})$, $(A_{3})$, and (4.19) hold and let $1 < m_{1} < 2$. Then there exist $c_{i} , \omega _{i} >0$, $i =3,4$, and $t_{0} > k^{-1}(\varepsilon )$ satisfying

(i) if K is linear,

$$\begin{aligned} E(t) \leq c_{3} \biggl( \omega _{3} \int ^{t}_{k^{-1}(\varepsilon )} \zeta (s) \,ds \biggr)^{\frac{ 1- m_{1} }{ 2 - m_{1} }}, \quad t > k^{-1}( \varepsilon ) , \end{aligned}$$

(4.66)

(ii) if K is nonlinear,

$$ E(t) \leq c_{4} \bigl( t - k^{-1}( \varepsilon ) \bigr)^{2 - m_{1} } \hat{K}^{-1} \biggl( \omega _{4} \biggl( \bigl( t - k^{-1}(\varepsilon ) \bigr)^{ \frac{2 - m_{1}}{ m_{1} - 1 } } \int ^{t}_{ t_{0} } \zeta (s) \,ds \biggr)^{-1} \biggr), \quad t > t_{0} , $$

where

$$ \hat{K}(s) = s^{ \frac{1}{ m_{1} -1 } } K'( s ) . $$

Proof

Owing to (4.62), estimates (4.36) and (4.37) are replaced by

$$\begin{aligned} L'(t) \leq - \lambda E(t) - \frac{ k(0) }{ a_{1}} E'(t) + \frac{1}{2} \int _{t_{\varepsilon }}^{t} k (s) \bigl\Vert \Delta u (t) - \Delta u (t-s) \bigr\Vert _{2}^{2} \,ds + C_{9} \bigl( - E ' (t)\bigr)^{ m_{1} -1 } \end{aligned}$$

and

$$\begin{aligned} R'(t) \leq - \lambda E(t) + \frac{1}{2} \int _{t_{\varepsilon }}^{t} k (s) \bigl\Vert \Delta u (t) - \Delta u (t-s) \bigr\Vert _{2}^{2} \,ds + C_{9} \bigl( - E ' (t)\bigr)^{ m_{1} -1 } \end{aligned}$$

(4.67)

for $t \geq t_{ \varepsilon }$, respectively.

Case 1: K is linear, that is, $K (s) = a s$ for some $a > 0$. Due to (4.67), estimate (4.38) is replaced by

$$\begin{aligned} {\mathcal{R}}_{1} ' (t) \leq - \lambda \zeta (t) E(t) + C_{9} \zeta (t) \bigl( - E ' (t)\bigr)^{ m_{1} -1 } , \quad t \geq t_{\varepsilon }. \end{aligned}$$

(4.68)

We set

$$\begin{aligned} {\mathcal{R}}_{3}(t) = E^{\frac{2- m_{1} }{ m_{1} -1 }}(t){\mathcal{R}}_{1} (t) + a_{9} E(t) , \end{aligned}$$

where $a_{9} = C_{9}^{ \frac{1}{ m_{1} -1 } } ( \frac{ \lambda }{2} )^{ \frac{ m_{1} -2 }{ m_{1} -1 } } \zeta ( 0 )$, which satisfies ${\mathcal{R}}_{3}(t) \sim E(t)$. Using (4.68) and the same argument of (4.65), we have

$$\begin{aligned} {\mathcal{R}}_{3}'(t) \leq & - \lambda \zeta (t) E^{ \frac{1}{ m_{1} -1 } }(t) + C_{9} \zeta (t) E^{ \frac{2- m_{1} }{ m_{1} -1 }}(t) \bigl( - E ' (t)\bigr)^{ m_{1} -1 } + a_{9} E'(t) \\ \leq & - \frac{ \lambda }{2} \zeta (t) E^{ \frac{1}{ m_{1} -1 } }(t) , \quad t \geq t_{\varepsilon }, \end{aligned}$$

which ensures (4.66).

Case 2: K is nonlinear. We let

$$\begin{aligned} \Gamma _{3} (t) = \frac{ a_{10} }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \int ^{t}_{t_{ \varepsilon }} \bigl\Vert \Delta u(t) - \Delta u(t-s) \bigr\Vert _{2}^{2} \,ds , \quad t \geq t_{ \varepsilon } . \end{aligned}$$

Using (4.39) and (4.63), we get

$$\begin{aligned}& \frac{ 1 }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \int ^{t}_{t_{ \varepsilon }} \bigl\Vert \Delta u(t) - \Delta u(t-s) \bigr\Vert _{2}^{2} \,ds \\& \quad \leq \frac{ 4 p_{1} }{ k_{l} ( p_{1} -2 ) ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \biggl( \int ^{t}_{t_{\varepsilon }} E(s) \,ds + \int ^{ t - t_{ \varepsilon } }_{0} E( s ) \biggr) \,ds ) \\& \quad \leq \frac{ 8 p_{1} C_{10} }{ k_{l} ( p_{1} -2 ) } < \infty , \quad t \geq t_{\varepsilon } . \end{aligned}$$

Thus, there exists $0 < a_{10} < 1$ satisfying

$$ \Gamma _{3} (t) < 1 \quad \text{for } t \geq t_{\varepsilon }. $$

(4.69)

Using (4.69) and the same argument of (4.42), we can replace estimate (4.42) by

$$\begin{aligned} \Gamma _{2} (t) = & - \frac{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } }{ a_{10} \Gamma _{3} (t)} \int ^{t}_{t_{\varepsilon }} \Gamma _{3} (t) k'(s) \frac{ a_{10} \Vert \Delta u(t) - \Delta u(t-s) \Vert _{2}^{2} }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \,ds \\ \geq & \frac{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } }{ a_{10} \Gamma _{3} (t)} \int ^{t}_{t_{\varepsilon }} \Gamma _{3} (t) \zeta (s) K \bigl( k(s) \bigr) \frac{ a_{10} \Vert \Delta u(t) - \Delta u(t-s) \Vert _{2}^{2} }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \,ds \\ \geq & \frac{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } \zeta (t ) }{ a_{10} \Gamma _{3} (t)} \int ^{t}_{t_{\varepsilon }} \overline{ K } \bigl( \Gamma _{3} (t) k(s) \bigr) \frac{ a_{10} \Vert \Delta u(t) - \Delta u(t-s) \Vert _{2}^{2} }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \,ds \\ \geq & \frac{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } \zeta (t ) }{ a_{10} } \overline{ K } \biggl( \int ^{t}_{t_{\varepsilon }} \frac{ a_{10} k(s) \Vert \Delta u(t) - \Delta u(t-s) \Vert _{2}^{2} }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \,ds \biggr) , \quad t \geq t_{\varepsilon } , \end{aligned}$$

which reads

$$\begin{aligned} \int ^{t}_{t_{\varepsilon }} k(s) \bigl\Vert \Delta u(t) - \Delta u(t-s) \bigr\Vert _{2}^{2} \,ds \leq \frac{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } }{ a_{10} } \overline{K}^{-1} \biggl( \frac{ a_{10} \Gamma _{2} (t)}{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } \zeta (t)} \biggr). \end{aligned}$$

From this and (4.67), we get

$$\begin{aligned} R '(t) \leq - \lambda E(t) + \frac{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } }{ 2 a_{10} } \overline{K}^{-1} \biggl( \frac{ a_{10} \Gamma _{2} (t)}{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } \zeta (t)} \biggr) + C_{9} \bigl( - E ' (t)\bigr)^{ m_{1} -1 }, \quad t \geq t_{\varepsilon } . \end{aligned}$$

(4.70)

Let $0 < \mu < \min \{ \varepsilon , 2 a_{10} \lambda E(0) \}$ and ${\mathcal{E}}(t) = \frac{E(t)}{E(0)}$. Using (4.70) and the same argument of (4.46), we obtain

$$\begin{aligned}& \biggl( \overline{K}' \biggl( \frac{ \mu {\mathcal{E}}(t) }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \biggr) R (t) \biggr)' \\& \quad \leq - \lambda \overline{K}' \biggl( \frac{ \mu {\mathcal{E}}(t) }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \biggr) E(t) + C_{9} \overline{K}' \biggl( \frac{ \mu {\mathcal{E}}(t) }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \biggr) \bigl( - E ' (t)\bigr)^{ m_{1} -1 } \\& \qquad {} + \frac{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } }{ 2 a_{10} } \overline{K}' \biggl( \frac{ \mu {\mathcal{E}}(t) }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \biggr) \overline{K}^{-1} \biggl( \frac{ a_{10} \Gamma _{2} (t) }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } \zeta (t) } \biggr) \\& \quad \leq - \lambda \overline{K}' \biggl( \frac{ \mu {\mathcal{E}}(t) }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \biggr) E(t) + C_{9} \overline{K}' \biggl( \frac{ \mu {\mathcal{E}}(t) }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \biggr) \bigl( - E ' (t)\bigr)^{ m_{1} -1 } \\& \qquad {} + \frac{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } }{ 2 a_{10} } \overline{K}^{*} \biggl( \overline{K}' \biggl( \frac{ \mu {\mathcal{E}}(t) }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \biggr) \biggr) + \frac{ \Gamma _{2} (t) }{ 2 \zeta (t) } \\& \quad \leq - a_{11} {\mathcal{E}}(t) \overline{K}' \biggl( \frac{ \mu {\mathcal{E}}(t) }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \biggr) + \frac{ \Gamma _{2} (t) }{ 2 \zeta (t) } + C_{9} \overline{K}' \biggl( \frac{ \mu {\mathcal{E}}(t) }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \biggr) \bigl( - E ' (t)\bigr)^{ m_{1} -1 } \end{aligned}$$

(4.71)

for $t \geq t_{\varepsilon }$, where $a_{11} = \lambda E(0) - \frac{ \mu }{ 2 a_{10} }$. Letting

$$\begin{aligned} {\mathcal{R}}_{4} (t)= \zeta (t) \overline{K}' \biggl( \frac{ \mu {\mathcal{E}}(t) }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \biggr) R(t) + E(t) , \end{aligned}$$

from (4.71) and (4.41), we get

$$\begin{aligned} {\mathcal{R}}_{4}' (t) \leq &- \frac{ a_{11} }{E(0)} \zeta (t) E(t) { \overline{K}}' \biggl( \frac{ \mu {\mathcal{E}}(t) }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \biggr) \\ &{} + C_{9} \zeta (t) { \overline{K}}' \biggl( \frac{ \mu {\mathcal{E}}(t) }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \biggr) \bigl( - E ' (t)\bigr)^{ m_{1} -1 } \end{aligned}$$

(4.72)

for $t\geq t_{\varepsilon }$. Define

$$\begin{aligned} {\mathcal{L}} (t) = E^{ \frac{2- m_{1} }{ m_{1} -1 } }(t) {\mathcal{R}}_{4} (t) + a_{12} E(t) , \end{aligned}$$

where $a_{12} = C_{9}^{ \frac{1}{ m_{1} -1 } } \zeta (0) ( \frac{ a_{11} }{2 E(0) } )^{ \frac{ m_{1} -2 }{ m_{1} -1 } } { \overline{K}}' ( \mu {\mathcal{E}}(0) )$. Using (4.72) and the same argument of (4.65), we see

$$\begin{aligned} {\mathcal{L}}' (t) \leq & - \frac{ a_{11} }{E(0)} \zeta (t) E^{ \frac{1}{ m_{1} -1 } } (t) {\overline{K}}' \biggl( \frac{ \mu {\mathcal{E}}(t) }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \biggr) \\ & {}+ C_{9} \zeta (t) E^{ \frac{2- m_{1} }{ m_{1} -1 } }(t) \bigl( - E ' (t)\bigr)^{ m_{1} -1 } {\overline{K}}' \biggl( \frac{ \mu {\mathcal{E}}(t) }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \biggr) + a_{12} E '(t) \\ \leq & - \frac{ a_{11} }{2 E(0) } \zeta (t) E^{ \frac{1}{ m_{1} -1 } } (t) { \overline{K}}' \biggl( \frac{ \mu {\mathcal{E}}(t) }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \biggr) \\ & {}+ C_{9}^{ \frac{1}{ m_{1} -1 } } \biggl( \frac{ a_{11} }{2 E(0) } \biggr)^{ \frac{ m_{1} -2 }{ m_{1} -1 } } \zeta (t) \bigl( - E'(t)\bigr) { \overline{K}}' \biggl( \frac{ \mu {\mathcal{E}}(t) }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \biggr) \\ & {} + a_{12} E '(t) , \quad t \geq t_{\varepsilon } . \end{aligned}$$

(4.73)

Thanks to $\lim_{ t \to \infty } \frac{ 1 }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } =0$, there exists $t_{0} > t_{\varepsilon }$ such that

$$\begin{aligned} \frac{ 1 }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } < 1, \quad \forall t > t_{0} , \end{aligned}$$

(4.74)

which ensures ${\mathcal{R}}_{4} (t) \sim E(t) \sim {\mathcal{L}} (t)$ for $t > t_{0}$. Moreover, from (4.73) and (4.74), we deduce

$$\begin{aligned} {\mathcal{L}}' (t) \leq & - \frac{ a_{11} }{2 E(0) } \zeta (t) E^{ \frac{1}{ m_{1} -1 } } (t) K' \biggl( \frac{ \mu {\mathcal{E}}(t) }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \biggr) \\ & {}+ C_{9}^{ \frac{1}{ m_{1} -1 } } \zeta (0) \biggl( \frac{ a_{11} }{2 E(0) } \biggr)^{ \frac{ m_{1} -2 }{ m_{1} -1 } } K ' \bigl( \mu { \mathcal{E}}(0) \bigr) \bigl( - E'(t)\bigr) + a_{12} E'(t) \\ = & - \frac{ a_{11} }{2 E(0) } \zeta (t) E^{ \frac{1}{ m_{1} -1 } } (t) K' \biggl( \frac{ \mu {\mathcal{E}}(t) }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \biggr) , \quad t > t_{0} , \end{aligned}$$

and hence

$$\begin{aligned} \int ^{t}_{t_{0}} \zeta ( s ) E^{ \frac{1}{ m_{1} -1 } } ( s ) K' \biggl( \frac{ \mu {\mathcal{E}}( s ) }{ ( s - t_{\varepsilon } )^{ 2 - m_{1} } } \biggr) \,ds \leq \frac{2 E(0) }{ a_{11} } {\mathcal{L}} ( t_{0} ) , \quad t > t_{0} , \end{aligned}$$

Since

$$\begin{aligned}& \biggl\{ E^{ \frac{1}{ m_{1} -1 } } ( t ) K' \biggl( \frac{ \mu {\mathcal{E}}( t ) }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \biggr) \biggr\} ' \leq 0, \\& E^{ \frac{1}{ m_{1} -1 } } (t) K' \biggl( \frac{ \mu {\mathcal{E}}(t) }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \biggr) \int ^{t}_{t_{0}} \zeta ( s ) \,ds \leq \frac{2 E(0) }{ a_{11} } { \mathcal{L}} ( t_{0} ) , \quad t > t_{0} , \end{aligned}$$

multiplying this by $( \frac{ \mu }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } )^{ \frac{1}{ m_{1} -1 } }$, we have

$$\begin{aligned}& E^{ \frac{1}{ m_{1} -1 } }(0) \biggl( \frac{ \mu {\mathcal{E}}(t )}{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \biggr)^{ \frac{1}{ m_{1} -1 } } K' \biggl( \frac{ \mu {\mathcal{E}}(t) }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \biggr) \int ^{t}_{t_{0}} \zeta ( s ) \,ds \\& \quad \leq \frac{2 E(0) {\mathcal{L}} ( t_{0} ) }{ a_{11} } \biggl( \frac{ \mu }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \biggr)^{ \frac{1}{ m_{1} -1 } } \end{aligned}$$

for $t > t_{0}$. So, letting ${\hat{K}}(s) = s^{ \frac{1}{ m_{1} -1 } } K '(s)$, we have

$$\begin{aligned} E^{ \frac{1}{ m_{1} -1 } }(0) {\hat{K}} \biggl( \frac{ \mu {\mathcal{E}}(t )}{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \biggr) \leq \frac{2 E(0) }{ a_{11} } {\mathcal{L}} ( t_{0} ) \biggl( \frac{ \mu }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \biggr)^{ \frac{1}{ m_{1} -1 } } \biggl( \int ^{t}_{t_{0}} \zeta ( s ) \,ds \biggr)^{-1} \end{aligned}$$

for $t > t_{0}$, which gives

$$\begin{aligned} {\mathcal{E}}(t ) \leq \frac{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } }{ \mu } { \hat{K}}^{-1} \biggl( \omega _{4} \biggl( ( t - t_{\varepsilon } )^{ \frac{2 - m_{1}}{ m_{1} -1 } } \int ^{t}_{t_{0}} \zeta ( s ) \,ds \biggr)^{-1} \biggr) \end{aligned}$$

for $t > t_{0}$ and some $\omega _{4} >0$. □

Conclusion

In this paper, we considered a viscoelastic von Karman equation with damping, source, and time delay terms of variable exponent type. Under assumptions $(A_{1})$, $(A_{2})$, $(A_{3})$, and (3.12), we showed that the local solution of problem (2.7)–(2.11) is global. Moreover, we established very general decay results of the solution for both cases $1 < \operatorname{ess} \inf_{x\in \Omega } m(x) <2$ and $\operatorname{ess} \inf_{x\in \Omega } m(x) \geq 2$ by giving additional condition (4.19) on initial data.