General Stability for the Viscoelastic Wave Equation with Nonlinear Time-Varying Delay, Nonlinear Damping and Acoustic Boundary Conditions

Abstract

This paper is focused on energy decay rates for the viscoelastic wave equation that includes nonlinear time-varying delay, nonlinear damping at the boundary, and acoustic boundary conditions. We derive general decay rate results without requiring the condition $a_2>0$ and without imposing any restrictive growth assumption on the damping term $f_1,$ using the multiplier method and some properties of the convex functions. Here we investigate the relaxation function $\psi$, namely ${\psi }^{\prime }\left(t\right)\le -\mu \left(t\right)G\left(\psi \left(t\right)\right)$, where G is a convex and increasing function near the origin, and $\mu$ is a positive nonincreasing function. Moreover, the energy decay rates depend on the functions $\mu$ and G, as well as the function F defined by $f_0$, which characterizes the growth behavior of $f_1$ at the origin.

1. Introduction

In this paper, we study the energy decay rates for the viscoelastic wave equation with nonlinear time-varying delay, nonlinear damping at the boundary, and acoustic boundary conditions

$$\begin{array}{ccc}& & u_{tt}(x,t)-\Delta u(x,t)+{\int }_0^t\psi (t-s)\Delta u(x,s)ds=0,\mathrm{in}\Omega \times (0,\infty ),\hfill \end{array}$$

(1)

$$\begin{array}{ccc}& & u(x,t)=0,\mathrm{on}{\Gamma }_0\times (0,\infty ),\hfill \end{array}$$

(2)

$$\begin{array}{ccc}& & \frac{\partial u}{\partial \nu }(x,t)-{\int }_0^t\psi (t-s)\frac{\partial u}{\partial \nu }(x,s)ds+a_1{f}_1\left(u_t(x,t)\right)+a_2{f}_2\left(u_t(x,t-\varrho \left(t\right))\right)\hfill \\ & & =w_t(x,t),\mathrm{on}{\Gamma }_1\times (0,\infty ),\hfill \end{array}$$

(3)

$$\begin{array}{ccc}& & u_t(x,t)+h\left(x\right)w_t(x,t)+m\left(x\right)w(x,t)=0,\mathrm{on}{\Gamma }_1\times (0,\infty ),\hfill \end{array}$$

(4)

$$\begin{array}{ccc}& & u(x,0)=u_0\left(x\right),u_t(x,0)=u_1\left(x\right),\mathrm{in}\Omega ,\hfill \end{array}$$

(5)

$$\begin{array}{ccc}& & u_t(x,t)=j_0(x,t),\mathrm{in}{\Gamma }_1\times (-\varrho \left(0\right),0),\hfill \end{array}$$

(6)

where $\Omega$ is a bounded domain in ${\mathbb{R}}^n(n\ge 1)$ with smooth boundary $\Gamma$ of class $C^2$; $\Gamma ={\Gamma }_0\cup {\Gamma }_1$, where ${\Gamma }_0$ and ${\Gamma }_1$ are closed and disjoint; $w(x,t)$ is the normal displacement into the domain of a point $x\in {\Gamma }_1$ at time t; and $h,m:{\Gamma }_1\to \mathbb{R}$ are essential bounded functions that represent resistivity and spring constant per unit area, respectively. $f_1,f_2:\mathbb{R}\to \mathbb{R}$ are given functions, and $f_1$ represents the nonlinear frictional damping. $a_1,a_2$ are real numbers with $a_1>0,a_2\ne 0.$ The integral term is the memory responsible for the viscoelastic damping. The functions $\psi$ and $\varrho \left(t\right)$ represent the kernel of the memory term and the time-varying delay, respectively. $\nu$ is the outward unit normal vector to $\Gamma .$ The initial data $(u_0,u_1,j_0)$ belong to a suitable space. Boundary conditions (3) and (4) are called acoustic boundary conditions.

In the past decades, the non-delayed wave equation with a viscoelastic term has garnered significant attention in the field of partial differential equations. Research on the energy decay rate of the solution to the viscoelastic wave equation is vital in various fields, contributing to technological advancements, safety assurance, environmental protection, energy efficiency, and academic exploration. The stability of solutions for such equations has recently been studied by many authors (see [1,2,3] and references therein). When $a_1=a_2=0,$ models (1)–(5) are pertinent to noise control and suppression in practical applications. The noise propagates through some acoustic medium, like air, in a room that is defined by a bounded domain $\Omega$ and whose floor, walls, and ceiling are determined by the boundary conditions [4,5]. Under the conditions that ${\int }_0^{\infty }\psi \left(s\right)ds<\frac{1}{2}$ and ${\psi }^{\prime }\left(t\right)\le -\mu \left(t\right)\psi \left(t\right),$ for $t\ge 0,$ Park and Park [6] considered the general decay for problems (1)–(5). Liu [7] improved the research of [6] by achieving arbitrary rates of decay, which may not necessarily be an exponential or a polynomial one. Recently, Yoon et al. [8] generalized the work of [6,7] without the assumption condition ${\int }_0^{\infty }\psi \left(s\right)ds<\frac{1}{2}.$ The assumption on relaxation function $\psi$ has been weakened compared to the conditions assumed in previous literature [6,7].

Numerous phenomena are influenced by both the current state and the previous occurrences of the system. There has been a notable increase in the research on the equation with delay effects, which frequently arise in various physical, biological, chemical, medical, and economic problems [9,10,11]. However, the delay effects can generally be considered a cause of instability. In order to stabilize a system containing delay terms, additional control terms will be necessary. Kirane and Said-Houari [12] showed the global existence and asymptotic stability for the following wave equation with memory and constant delay,

$$\begin{array}{c}\hfill u_{tt}(x,t)-\Delta u(x,t)+{\int }_0^t\psi (t-s)\Delta u(x,s)ds+a_1{u}_t(x,t)+a_2{u}_t(x,t-\varrho )=0,\end{array}$$

where $a_1,a_2$, and $\varrho$ are positive constants. They used the damping term $a_1{u}_t(x,t)$ to control the delay term in obtaining the decay estimate of the energy. They proved that its energy was exponentially decaying when $a_2\le a_1.$ Dai and Yang [13] investigated the exponential decay of an unsolved problem proposed by Kirane and Said-Houari [12], namely, the problem with $a_1=0.$ In the case of constant weight and constant delay, the delay term typically considers the past history of strain, only up to some finite time $\varrho \left(t\right)\equiv \varrho .$ Nicaise and Pignotti [14] investigated the following wave equation with internal time-varying delay instead of constant delay,

$$\begin{array}{c}\hfill u_{tt}(x,t)-\Delta u(x,t)+a_1{u}_t(x,t)+a_2{u}_t(x,t-\varrho \left(t\right))=0,\end{array}$$

where $\varrho \left(t\right)>0,$ $a_1$, and $a_2$ are real numbers with $a_1>0.$ They proved the exponential stability result for the wave equation under the condition $|a_2|<\sqrt{1-{\zeta }_0}{a}_1,$ where the constant ${\zeta }_0$ satisfies ${\varrho }^{\prime }\left(t\right)\le {\zeta }_0<1,\forall t>0.$ Liu [15] studied the following wave equation involving memory and time-varying delay:

$$\begin{array}{c}\hfill u_{tt}(x,t)-\Delta u(x,t)+\alpha \left(t\right){\int }_0^t\psi (t-s)\Delta u(x,s)ds+a_1{u}_t(x,t)+a_2{u}_t(x,t-\varrho \left(t\right))=0.\end{array}$$

Systems with time-varying delays have been extensively considered by many authors (see [16,17,18,19,20,21,22] and references therein). Recently, Zennir [23] considered the stability for solutions of plate equations with a time-varying delay and weak viscoelasticity in ${\mathbb{R}}^n.$ Moreover, Benaissa et al. [24] proved the global existence and stability for solutions of the following wave equation with a time-varying delay in the weakly nonlinear feedback,

$$\begin{array}{c}\hfill u_{tt}(x,t)-\Delta u(x,t)+a_1\sigma \left(t\right)f_1\left(u_t(x,t)\right)+a_2\sigma \left(t\right)f_2\left(u_t(x,t-\varrho \left(t\right))\right)=0,\end{array}$$

where $\varrho \left(t\right)>0,$ $a_1$, and $a_2$ are positive real numbers, and $f_1,f_2$ satisfy some conditions. This result extended the previous work [10,14]. Park [25] investigated the decay result of the energy for a von Karman equation with time-varying delay by dropping the restriction $a_2>0$ under the same conditions as $\varrho ,$ $f_1$, and $f_2$ in [24]. For the viscoelastic problem with time-varying delay in the nonlinear internal or boundary feedback, we also refer to [26,27]. As far as we know, there are few results for the viscoelastic wave equation with a nonlinear time-varying delay. Recently, Djeradi et al. [28] and Mukiawa et al. [29] showed the stability of the thermoelastic laminated beam and thermoelastic Timoshenko beam with nonlinear time-varying delay, respectively. The papers introduced so far have studied the energy decay rate of the solution for the equation with nonlinear time-varying delay in the Dirichlet boundary condition.

Motivated by these results, we study the general decay rates of the solution for problems (1)–(6) with a nonlinear time-varying delay term, nonlinear damping at the boundary, and acoustic boundary conditions. Research on the energy decay rate of solutions for the viscoelastic wave equation with nonlinear time-delay terms plays a critical role in various application areas, including stability assessment, understanding complex behaviors, advancing neuroscience, disaster preparedness, and improving energy efficiency. We consider the general assumption on the relaxation function $\psi ,$

$$\begin{array}{c}\hfill {\psi }^{\prime }\left(t\right)\le -\mu \left(t\right)G\left(\psi \left(t\right)\right),\end{array}$$

(7)

where $\mu :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ is a positive nonincreasing function, and G is linear or is a strictly increasing and strictly convex function. We derive the general decay rate results without requiring the condition $a_2>0$ and without imposing any restrictive growth assumption on the damping term $f_1.$ The energy decay rates depend on the functions $\mu$ and $G,$ as well as the function F defined by $f_0$, which represents the growth $f_1$ at the origin. Our result improves upon previous work [6,7,8].

This paper is composed of the following. In Section 2, we prepare some notations and materials needed for our work. In Section 3, we introduce some technical lemmas to prove our stability result. In Section 4, we state and prove the general energy decay.

2. Preliminaries

In this section, we present some materials required for our results. Throughout this paper, we use the notation

$$\begin{array}{ccc}& & V=\{u\in H^1(\Omega ):u=0\mathrm{on}{\Gamma }_0\}.\hfill \end{array}$$

For simplicity, we denote ${\parallel ·\parallel }_{L^2(\Omega )}$ and ${\parallel ·\parallel }_{L^2\left({\Gamma }_1\right)}$ by $\parallel ·\parallel$ and ${\parallel ·\parallel }_{{\Gamma }_1}$, respectively.

The Poincaré inequality holds in V; that is, there exist the positive constants ${\lambda }_0$ and ${\lambda }_1$ such that

$${\parallel u\parallel }^2\le {\lambda }_0{\parallel \nabla u\parallel }^2{\mathrm{and}\parallel u\parallel }_{{\Gamma }_1}^2\le {\lambda }_1{\parallel \nabla u\parallel }^2\mathrm{for}\mathrm{all}u\in V.$$

(8)

As in [1,3,8,26,30], we consider the following assumptions for $\psi ,f_1,f_2,\varrho ,h$, and $m.$

(H1) $\psi :[0,\infty )\to {\mathbb{R}}^{+}$ is a differentiable function satisfying

$$1-{\int }_0^{\infty }\psi \left(s\right)ds=l>0,$$

(9)

and there exists a $C^1$ function $G:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ that is linear or is a strictly convex and strictly increasing $C^2$ function on $(0,r_0],r_0\le \psi \left(0\right)$ such that

$${\psi }^{\prime }\left(t\right)\le -\mu \left(t\right)G\left(\psi \left(t\right)\right),\forall t\ge 0,$$

(10)

where $G\left(0\right)=G^{\prime }\left(0\right)=0$, and $\mu$ is a positive nonincreasing differentiable function. The function G was first introduced in [31]. These are weaker conditions on G than those introduced in [31].

(H2) $f_1:\mathbb{R}\to \mathbb{R}$ is a nondecreasing $C^0$ function such that there exists a strictly increasing function $f_0\in C^1\left({\mathbb{R}}^{+}\right),$ with $f_0\left(0\right)=0,$ and positive constants $c_0,c_1$, and $\epsilon$ such that

$$\begin{array}{ccc}& & f_0\left(\right|s\left|\right)\le |f_1\left(s\right)|\le f_0^{-1}\left(\right|s\left|\right)\mathrm{for}\mathrm{all}\left|s\right|\le \epsilon ,\hfill \end{array}$$

(11)

$$\begin{array}{ccc}& & c_0\left|s\right|\le |f_1\left(s\right)|\le c_1\left|s\right|\mathrm{for}\mathrm{all}\left|s\right|\ge \epsilon .\hfill \end{array}$$

(12)

Moreover, we assume that the function F, defined by $F\left(s\right)=\sqrt{s}{f}_0\left(\sqrt{s}\right)$, is a strictly convex $C^2$ function on $(0,r_1]$, for some $r_1>0$, when $f_0$ is nonlinear.

(H3) $f_2:\mathbb{R}\to \mathbb{R}$ is an odd nondecreasing $C^1$ function such that there exist positive constants $c_2,c_3$, and $c_4$ that satisfy

$$|f_2^{\prime }\left(s\right)|\le c_2,c_{3}s{f}_2\left(s\right)\le F_2\left(s\right)\le c_{4}s{f}_1\left(s\right),\mathrm{for}s\in \mathbb{R},$$

(13)

where $F_2\left(s\right)={\int }_0^s{f}_2\left(t\right)dt$.

(H4) $\varrho \in W^{2,\infty }\left([0,T]\right)$ is a function such that

$$0<{\varrho }_1\le \varrho \left(t\right)\le {\varrho }_2\mathrm{and}{\varrho }^{\prime }\left(t\right)\le {\varrho }_3<1\mathrm{for}\mathrm{all}t>0,$$

(14)

where $T,$ ${\varrho }_1$, and ${\varrho }_2$ are positive constants. Moreover, the weight of dissipation and the delay satisfy

$$0<|a_2|<\frac{c_3(1-{\varrho }_3)}{c_4(1-c_3{\varrho }_3)}{a}_1.$$

(15)

(H5) We assume that $h,m\in C\left({\Gamma }_1\right)$, $h\left(x\right)>0$, and $m\left(x\right)>0$ for all $x\in {\Gamma }_1$. Then, there exist positive constants $h_i$ and $m_i(i=1,2)$ such that

$$h_1\le h\left(x\right)\le h_2,m_1\le m\left(x\right)\le m_2\mathrm{for}\mathrm{all}x\in {\Gamma }_1.$$

(16)

Remark 1. 

1. The assumption (H2) implies that $s{f}_1\left(s\right)>0,$ for all $s\ne 0.$

2. The assumption (11) of function $f_1$ has been weakened compared to the condition assumed in [24,25].

3. Since $f_2$ is an odd nondecreasing function, $F_2$ is an even and convex function. Furthermore, it is satisfied that $F_2\left(s\right)={\int }_0^s{f}_2\left(t\right)dt\le s{f}_2\left(s\right).$ From (13), we find that $c_3\le 1.$

Remark 2 

([3]). 1. By (H1), we obtain ${\displaystyle \underset{t\to +\infty }{lim}\psi \left(t\right)=0}$. Then, there exists $t_0\ge 0$ large enough that

$$\psi \left(t_0\right)=r_0\Rightarrow \psi \left(t\right)\le r_0,\forall t\ge t_0.$$

(17)

Given ψ and μ are positive nonincreasing continuous functions, G is a positive continuous function, and for (10), we have, for some positive constant $c_5$,

$${\psi }^{\prime }\left(t\right)\le -\mu \left(t\right)G\left(\psi \left(t\right)\right)\le -c_5\psi \left(t\right),\forall t\in [0,t_0].$$

(18)

2. If G is a strictly convex and strictly increasing $C^2$ function on $(0,r_0],$ with $G\left(0\right)=G^{\prime }\left(0\right)=0$, then it has an extension $\overline{G}$, which is a strictly convex and strictly increasing $C^2$ function on $(0,\infty ).$ The same remark can be established for $\overline{F}.$

We recall the well-known Jensen inequality, which plays a pivotal role in proving our main result. If $\varphi$ is a convex function on $[a,b]$, $p:\Omega \to [a,b]$ and k represents integrable functions on $\Omega$ such that $k\left(x\right)\ge 0$ and ${\int }_{\Omega }k\left(x\right)dx=k_0>0$, then Jensen’s inequality holds:

$$\varphi \left[\frac{1}{k_0}{\int }_{\Omega }p\left(x\right)k\left(x\right)dx\right]\le \frac{1}{k_0}{\int }_{\Omega }\varphi \left[p\left(x\right)\right]k\left(x\right)dx.$$

(19)

Let $H^{*}$ be the conjugate of the convex function H defined by ${\displaystyle H^{*}\left(s\right)=\underset{r\ge 0}{sup}(sr-H\left(r\right)),}$ then

$$sr\le H^{*}\left(s\right)+H\left(r\right),\forall s,r\ge 0.$$

(20)

Moreover, due to the argument provided in [32], it holds that

$$H^{*}\left(s\right)=s{\left(H^{\prime }\right)}^{-1}\left(s\right)-H\left({\left(H^{\prime }\right)}^{-1}\left(s\right)\right),\forall s\ge 0.$$

(21)

As in [10,14], we introduce the following new function:

$$v(x,\kappa ,t)=u_t(x,t-\kappa \varrho \left(t\right)),\mathrm{for}(x,\kappa ,t)\in {\Gamma }_1\times (0,1)\times (0,\infty ).$$

Then, problems (1)–(6) can be expressed as follows:

$$\begin{array}{ccc}& & u_{tt}(x,t)-\Delta u(x,t)+{\int }_0^t\psi (t-s)\Delta u(x,s)ds=0,\mathrm{in}\Omega \times (0,\infty ),\hfill \end{array}$$

(22)

$$\begin{array}{ccc}& & \varrho \left(t\right)v_t(x,\kappa ,t)+(1-\kappa {\varrho }^{\prime }\left(t\right))v_{\kappa }(x,\kappa ,t)=0,\mathrm{in}{\Gamma }_1\times (0,1)\times (0,\infty ),\hfill \end{array}$$

(23)

$$\begin{array}{ccc}& & u(x,t)=0,\mathrm{in}{\Gamma }_0\times (0,\infty ),\hfill \end{array}$$

(24)

$$\begin{array}{ccc}& & \frac{\partial u}{\partial \nu }(x,t)-{\int }_0^t\psi (t-s)\frac{\partial u}{\partial \nu }(x,s)ds+a_1{f}_1\left(u_t(x,t)\right)+a_2{f}_2\left(v(x,1,t)\right)=w_t(x,t),\mathrm{on}{\Gamma }_1\times (0,\infty ),\hfill \end{array}$$

(25)

$$\begin{array}{ccc}& & u_t(x,t)+h\left(x\right)w_t(x,t)+m\left(x\right)w(x,t)=0,\mathrm{on}{\Gamma }_1\times (0,\infty ),\hfill \end{array}$$

(26)

$$\begin{array}{ccc}& & u(x,0)=u_0\left(x\right),u_t(x,0)=u_1\left(x\right),\mathrm{in}\Omega ,\hfill \end{array}$$

(27)

$$\begin{array}{ccc}& & v(x,\kappa ,0)=j_0(x,-\kappa \varrho \left(0\right)),\mathrm{in}{\Gamma }_1\times (0,1).\hfill \end{array}$$

(28)

We state the global existence result that can be established by the arguments of [24,33].

Theorem 1. 

Let initial data $(u_0,u_1)\in (V\cap H^2(\Omega ))\times V$ and $j_0\in L^2({\Gamma }_1\times (0,1))$. Suppose that (H1)–(H5) hold. Then, for any $T>0$, there exists a unique pair of functions $(u,w,v)$ that are the solution to problems (22)–(28) in the class

$$\begin{array}{ccc}& & u\in L^{\infty }(0,T;V\cap H^2(\Omega )),u_t\in L^{\infty }(0,T;V),u_{tt}\in L^{\infty }(0,T;L^2(\Omega )),\hfill \\ & & v\in L^{\infty }(0,T;L^2({\Gamma }_1\times (0,1))),w,w_t\in L^2(0,\infty ;L^2\left({\Gamma }_1\right)).\hfill \end{array}$$

As in [6,25], we introduce the energy for problems (22)–(28),

$$\begin{array}{ccc}\hfill & & E\left(t\right)=\frac{1}{2}\parallel u_t{\left(t\right)\parallel }^2+\frac{1}{2}\left(1-{\int }_0^t\psi \left(s\right)ds\right){\parallel \nabla u\left(t\right)\parallel }^2+\frac{1}{2}(\psi \circ \nabla u)\left(t\right)\hfill \\ & & +\frac{1}{2}{\int }_{{\Gamma }_1}m\left(x\right)w^2\left(t\right)d\Gamma +\frac{\zeta \varrho \left(t\right)}{2}{\int }_{{\Gamma }_1}{\int }_0^1{F}_2\left(v(x,\kappa ,t)\right)d\kappa d\Gamma ,\hfill \end{array}$$

(29)

where

$$(\psi \circ \nabla u)\left(t\right)={\int }_0^t\psi (t-s){\parallel \nabla u\left(t\right)-\nabla u\left(s\right)\parallel }^{2}ds$$

and

$$\begin{array}{c}\hfill \frac{2|a_2|(1-c_3)}{c_3(1-{\varrho }_3)}<\zeta <\frac{2(a_1-|a_2\left|c_4\right)}{c_4}.\end{array}$$

(30)

Thanks to (15), this makes sense.

To show the main results of this paper, we need the following lemma.

Lemma 1. 

Assume that (H3)–(H5) hold. Then, there exist positive constants ${\gamma }_0$ and ${\gamma }_1$ satisfying

$$\begin{array}{ccc}\hfill & & E^{\prime }\left(t\right)\le \frac{1}{2}({\psi }^{\prime }\circ \nabla u)\left(t\right)-\frac{1}{2}\psi \left(t\right)\parallel \nabla u\left(t\right){\parallel }^2-h_1\left|\right|w_t\left(t\right){\left|\right|}_{{\Gamma }_1}^2\hfill \\ & & -{\gamma }_0{\int }_{{\Gamma }_1}{f}_1\left(u_t\left(t\right)\right)u_t\left(t\right)d\Gamma -{\gamma }_1{\int }_{{\Gamma }_1}{f}_2\left(v(x,1,t)\right)v(x,1,t)d\Gamma .\hfill \end{array}$$

(31)

Proof. 

Multiplying by $u_t\left(t\right)$ in (22), using Green’s formula, (25), and (26), we have

$$\begin{array}{ccc}\hfill & & \frac{1}{2}\frac{d}{dt}\left[\parallel u_t{\left(t\right)\parallel }^2+\left(1-{\int }_0^t\psi \left(s\right)ds\right){\parallel \nabla u\left(t\right)\parallel }^2+(\psi \circ \nabla u)\left(t\right)+{\int }_{{\Gamma }_1}m\left(x\right)w^2\left(t\right)d\Gamma \right]\hfill \\ & & =\frac{1}{2}({\psi }^{\prime }\circ \nabla u)\left(t\right)-\frac{1}{2}\psi \left(t\right){\parallel \nabla u\left(t\right)\parallel }^2-{\int }_{{\Gamma }_1}h\left(x\right)w_t^2\left(t\right)d\Gamma \hfill \\ & & -a_1{\int }_{{\Gamma }_1}{f}_1\left(u_t\left(t\right)\right)u_t\left(t\right)d\Gamma -a_2{\int }_{{\Gamma }_1}{f}_2\left(v(x,1,t)\right)u_t\left(t\right)d\Gamma ,\hfill \end{array}$$

(32)

where we used the relation

$$\begin{array}{ccc}& & -{\int }_{\Omega }\nabla u_t\left(t\right){\int }_0^t\psi (t-s)\nabla u\left(s\right)dsdx\hfill \\ & & =\frac{d}{dt}\left[\frac{1}{2}(\psi \circ \nabla u)\left(t\right)-\frac{1}{2}{\int }_0^t\psi \left(s\right)ds{\parallel \nabla u\left(t\right)\parallel }^2\right]-\frac{1}{2}({\psi }^{\prime }\circ \nabla u)\left(t\right)+\frac{1}{2}\psi \left(t\right){\parallel \nabla u\left(t\right)\parallel }^2.\hfill \end{array}$$

From (29) and (32), we have

$$\begin{array}{ccc}\hfill & & E^{\prime }\left(t\right)=\frac{1}{2}({\psi }^{\prime }\circ \nabla u)\left(t\right)-\frac{1}{2}\psi \left(t\right){\parallel \nabla u\left(t\right)\parallel }^2-{\int }_{{\Gamma }_1}h\left(x\right)w_t^2\left(t\right)d\Gamma \hfill \\ & & -a_1{\int }_{{\Gamma }_1}{f}_1\left(u_t\left(t\right)\right)u_t\left(t\right)d\Gamma -a_2{\int }_{{\Gamma }_1}{f}_2\left(v(x,1,t)\right)u_t\left(t\right)d\Gamma \hfill \\ & & +\frac{\zeta {\varrho }^{\prime }\left(t\right)}{2}{\int }_{{\Gamma }_1}{\int }_0^1{F}_2\left(v(x,\kappa ,t)\right)d\kappa d\Gamma +\frac{\zeta \varrho \left(t\right)}{2}{\int }_{{\Gamma }_1}{\int }_0^1{f}_2\left(v(x,\kappa ,t)\right)v_t(x,\kappa ,t)d\kappa d\Gamma ,\hfill \end{array}$$

(33)

where $F_2\left(t\right)={\int }_0^t{f}_2\left(s\right)ds.$ In (23), we multiply by $f_2\left(v(x,\kappa ,t)\right)$ and integrate over ${\Gamma }_1\times (0,1)$ to obtain

$$\begin{array}{ccc}& & \frac{\zeta \varrho \left(t\right)}{2}{\int }_{{\Gamma }_1}{\int }_0^1{f}_2\left(v(x,\kappa ,t)\right)v_t(x,\kappa ,t)d\kappa d\Gamma \hfill \\ & & =-\frac{\zeta }{2}{\int }_{{\Gamma }_1}\left[(1-{\varrho }^{\prime }\left(t\right))F_2\left(v(x,1,t)\right)-F_2\left(v(x,0,t)\right)+{\int }_0^1{\varrho }^{\prime }\left(t\right)F_2\left(v(x,\kappa ,t)\right)d\kappa \right]d\Gamma .\hfill \end{array}$$

Applying this to (33) and noting that $v(x,0,t)=u_t(x,t)$, it follows that

$$\begin{array}{ccc}\hfill & & E^{\prime }\left(t\right)=\frac{1}{2}({\psi }^{\prime }\circ \nabla u)\left(t\right)-\frac{1}{2}\psi \left(t\right){\parallel \nabla u\left(t\right)\parallel }^2-{\int }_{{\Gamma }_1}h\left(x\right)w_t^2\left(t\right)d\Gamma -a_1{\int }_{{\Gamma }_1}{f}_1\left(u_t\left(t\right)\right)u_t\left(t\right)d\Gamma \hfill \\ & & -a_2{\int }_{{\Gamma }_1}{f}_2\left(v(x,1,t)\right)u_t\left(t\right)d\Gamma -\frac{\zeta }{2}{\int }_{{\Gamma }_1}\left[(1-{\varrho }^{\prime }\left(t\right))F_2\left(v(x,1,t)\right)-F_2\left(u_t(x,t)\right)\right]d\Gamma .\hfill \end{array}$$

(34)

From (13) and (14), we obtain

$$\begin{array}{ccc}\hfill & & -\frac{\zeta }{2}{\int }_{{\Gamma }_1}\left[(1-{\varrho }^{\prime }\left(t\right))F_2\left(v(x,1,t)\right)-F_2\left(u_t(x,t)\right)\right]d\Gamma \hfill \\ & & \le -\frac{\zeta c_3}{2}(1-{\varrho }_3){\int }_{{\Gamma }_1}{f}_2\left(v(x,1,t)\right)v(x,1,t)d\Gamma +\frac{\zeta c_4}{2}{\int }_{{\Gamma }_1}{f}_1\left(u_t\left(t\right)\right)u_t\left(t\right)d\Gamma .\hfill \end{array}$$

(35)

Substituting (35) into (34), we obtain

$$\begin{array}{ccc}\hfill & & E^{\prime }\left(t\right)\le \frac{1}{2}({\psi }^{\prime }\circ \nabla u)\left(t\right)-\frac{1}{2}\psi \left(t\right){\parallel \nabla u\left(t\right)\parallel }^2-{\int }_{{\Gamma }_1}h\left(x\right)w_t^2\left(t\right)d\Gamma \hfill \\ & & -\left(a_1-\frac{\zeta c_4}{2}\right){\int }_{{\Gamma }_1}{f}_1\left(u_t\left(t\right)\right)u_t\left(t\right)d\Gamma -\frac{\zeta c_3}{2}(1-{\varrho }_3){\int }_{{\Gamma }_1}{f}_2\left(v(x,1,t)\right)v(x,1,t)d\Gamma \hfill \\ & & -a_2{\int }_{{\Gamma }_1}{f}_2\left(v(x,1,t)\right)u_t\left(t\right)d\Gamma .\hfill \end{array}$$

(36)

Now, we estimate the last term in the right-hand side of (36). The definition of $F_2$ and (21) give

$$\begin{array}{c}\hfill F_2^{*}\left(s\right)=s{f}_2^{-1}\left(s\right)-F_2\left(f_2^{-1}\left(s\right)\right),\mathrm{for}s\ge 0.\end{array}$$

(37)

When $f_2\left(v(x,1,t)\right)<0$ and $u_t\left(t\right)\ge 0$, using (20) and (37) with $s=-f_2\left(v(x,1,t)\right)$ and $r=u_t\left(t\right),$ we obtain (see details in [25])

$$\begin{array}{ccc}\hfill & & a_2{\int }_{{\Gamma }_1}(-f_2\left(v(x,1,t)\right))u_t\left(t\right)d\Gamma \hfill \\ & & \le |a_2|{\int }_{{\Gamma }_1}\left(-f_2\left(v(x,1,t)\right)(-v(x,1,t))-F_2(-v(x,1,t))+F_2\left(u_t\left(t\right)\right)\right)d\Gamma \hfill \\ & & =|a_2|{\int }_{{\Gamma }_1}\left(f_2\left(v(x,1,t)\right)v(x,1,t)-F_2\left(v(x,1,t)\right)+F_2\left(u_t\left(t\right)\right)\right)d\Gamma ,\hfill \end{array}$$

(38)

where we used the fact that $f_2$ is odd and $F_2$ is even. When $f_2\left(v(x,1,t)\right)\ge 0$ and $u_t\left(t\right)<0,$ with $s=f_2\left(v(x,1,t)\right)$ and $r=-u_t\left(t\right),$ we obtain

$$\begin{array}{ccc}\hfill & & a_2{\int }_{{\Gamma }_1}{f}_2\left(v(x,1,t)\right)(-u_t\left(t\right))d\Gamma \hfill \\ & & \le |a_2|{\int }_{{\Gamma }_1}\left(f_2\left(v(x,1,t)\right)\left(v(x,1,t)\right)-F_2\left(v(x,1,t)\right)+F_2(-u_t\left(t\right))\right)d\Gamma \hfill \\ & & =|a_2|{\int }_{{\Gamma }_1}\left(f_2\left(v(x,1,t)\right)v(x,1,t)-F_2\left(v(x,1,t)\right)+F_2\left(u_t\left(t\right)\right)\right)d\Gamma .\hfill \end{array}$$

(39)

From (38) and (39), for the case $f_2\left(v(x,1,t)\right)u_t\left(t\right)\le 0,$ we have

$$\begin{array}{c}\hfill -a_2{\int }_{{\Gamma }_1}{f}_2\left(v(x,1,t)\right)u_t\left(t\right)d\Gamma \le \left|a_2\right|{\int }_{{\Gamma }_1}\left(f_2\left(v(x,1,t)\right)v(x,1,t)-F_2\left(v(x,1,t)\right)+F_2\left(u_t\left(t\right)\right)\right)d\Gamma .\end{array}$$

(40)

Similarly, (40) holds when $f_2\left(v(x,1,t)\right)u_t\left(t\right)\ge 0.$ Hence, using (13) and (40), we see that

$$\begin{array}{ccc}\hfill & & -a_2{\int }_{{\Gamma }_1}{f}_2\left(v(x,1,t)\right)u_t\left(t\right)d\Gamma \hfill \\ & & \le |a_2|\left((1-c_3){\int }_{{\Gamma }_1}{f}_2\left(v(x,1,t)\right)v(x,1,t)d\Gamma +c_4{\int }_{{\Gamma }_1}{f}_1\left(u_t\left(t\right)\right)u_t\left(t\right)d\Gamma \right).\hfill \end{array}$$

(41)

By using (16), (36), and (41), and by selecting $\zeta$ satisfying (30), we obtain the desired inequality (31) where ${\gamma }_0=a_1-\frac{\zeta c_4}{2}-\left|a_2\right|c_4>0$ and ${\gamma }_1=\frac{\zeta c_3}{2}(1-{\varrho }_3)-\left|a_2\right|(1-c_3)>0.$ □

3. Technical Lemmas

In this section, we prove the following lemmas to obtain the general decay rates of the solution to problems (22)–(28).

Lemma 2. 

Under the assumption (H1), the functional ${\Phi }_1$ defined by

$${\Phi }_1\left(t\right)={\int }_{\Omega }u\left(t\right)u_t\left(t\right)dx+{\int }_{{\Gamma }_1}u\left(t\right)w\left(t\right)d\Gamma +\frac{1}{2}{\int }_{{\Gamma }_1}h\left(x\right)w^2\left(t\right)d\Gamma$$

satisfies

$$\begin{array}{ccc}\hfill & & {\Phi }_1^{\prime }\left(t\right)\le \parallel u_t{\left(t\right)\parallel }^2-\frac{l}{2}{\parallel \nabla u\left(t\right)\parallel }^2+\frac{2C\left(\xi \right)}{l}(i\circ \nabla u)\left(t\right)+\frac{8{\lambda }_1}{l}{\parallel w_t\left(t\right)\parallel }_{{\Gamma }_1}^2\hfill \\ & & +\frac{a_1{a}_3}{l}{\int }_{{\Gamma }_1}{f}_1^2\left(u_t\left(t\right)\right)d\Gamma +\frac{|a_2|a_3}{l}{\int }_{{\Gamma }_1}{f}_2^2\left(v(x,1,t)\right)d\Gamma -{\int }_{{\Gamma }_1}m\left(x\right)w^2\left(t\right)d\Gamma ,\hfill \end{array}$$

(42)

for any $0<\xi <1$, where

$$\begin{array}{c}\hfill i\left(t\right)=\xi \psi \left(t\right)-{\psi }^{\prime }\left(t\right)\mathrm{and}C\left(\xi \right)={\int }_0^{\infty }\frac{{\psi }^2\left(s\right)}{i\left(s\right)}ds.\end{array}$$

(43)

Proof. 

Using Equations (22) and (24)–(26), and utilizing (9) and Young’s inequality, we obtain

$$\begin{array}{ccc}& & {\Phi }_1^{\prime }\left(t\right)=\parallel u_t{\left(t\right)\parallel }^2-\left(1-{\int }_0^t\psi \left(s\right)ds\right){\parallel \nabla u\left(t\right)\parallel }^2+{\int }_0^t\psi (t-s)(\nabla u\left(s\right)-\nabla u\left(t\right),\nabla u\left(t\right))ds\hfill \\ & & -a_1{\int }_{{\Gamma }_1}{f}_1\left(u_t\left(t\right)\right)u\left(t\right)d\Gamma -a_2{\int }_{{\Gamma }_1}{f}_2\left(v(x,1,t)\right)u\left(t\right)d\Gamma +2{\int }_{{\Gamma }_1}u\left(t\right)w_t\left(t\right)d\Gamma -{\int }_{{\Gamma }_1}m\left(x\right)w^2\left(t\right)d\Gamma \hfill \\ & & \le \parallel u_t{\left(t\right)\parallel }^2-\frac{7l}{8}{\parallel \nabla u\left(t\right)\parallel }^2+\frac{2}{l}{\int }_{\Omega }{\left({\int }_0^t\psi (t-s)|\nabla u\left(s\right)-\nabla u\left(t\right)|ds\right)}^{2}dx\hfill \\ & & -a_1{\int }_{{\Gamma }_1}{f}_1\left(u_t\left(t\right)\right)u\left(t\right)d\Gamma -a_2{\int }_{{\Gamma }_1}{f}_2\left(v(x,1,t)\right)u\left(t\right)d\Gamma +2{\int }_{{\Gamma }_1}u\left(t\right)w_t\left(t\right)d\Gamma -{\int }_{{\Gamma }_1}m\left(x\right)w^2\left(t\right)d\Gamma .\hfill \end{array}$$

Using the Cauchy–Schwarz inequality and (43), we have (see [3,34])

$$\begin{array}{c}\hfill {\int }_{\Omega }{\left({\int }_0^t\psi (t-s)|\nabla u\left(s\right)-\nabla u\left(t\right)|ds\right)}^{2}dx\le \left({\int }_0^t\frac{{\psi }^2\left(s\right)}{i\left(s\right)}ds\right)(i\circ \nabla u)\left(t\right)\le C\left(\xi \right)(i\circ \nabla u)\left(t\right).\end{array}$$

(44)

Applying Young’s inequality and (8), we obtain, for $\eta >0$,

$$\begin{array}{ccc}& & |-a_1{\int }_{{\Gamma }_1}{f}_1\left(u_t\left(t\right)\right)u\left(t\right)d\Gamma |\le \eta a_1{\lambda }_1{\parallel \nabla u\left(t\right)\parallel }^2+\frac{a_1}{4\eta }{\int }_{{\Gamma }_1}{f}_1^2\left(u_t\left(t\right)\right)d\Gamma ,\hfill \end{array}$$

(45)

$$\begin{array}{ccc}& & |-a_2{\int }_{{\Gamma }_1}{f}_2\left(v(x,1,t)\right)u\left(t\right)d\Gamma |\le \eta |a_2|{\lambda }_1{\parallel \nabla u\left(t\right)\parallel }^2+\frac{|a_2|}{4\eta }{\int }_{{\Gamma }_1}{f}_2^2\left(v(x,1,t)\right)d\Gamma ,\hfill \end{array}$$

(46)

and

$$2{\int }_{{\Gamma }_1}u\left(t\right)w_t\left(t\right)d\Gamma \le \frac{l}{8}{\parallel \nabla u\left(t\right)\parallel }^2+\frac{8{\lambda }_1}{l}{\parallel w_t\left(t\right)\parallel }_{{\Gamma }_1}^2.$$

(47)

Combining estimates (44)–(47), we see that

$$\begin{array}{ccc}& & {\Phi }_1^{\prime }\left(t\right)\le \parallel u_t{\left(t\right)\parallel }^2-(\frac{3l}{4}-\eta a_1{\lambda }_1-\eta |a_2|{\lambda }_1{)\parallel \nabla u\left(t\right)\parallel }^2+\frac{2C\left(\xi \right)}{l}(i\circ \nabla u)\left(t\right)+\frac{8{\lambda }_1}{l}{\parallel w_t\left(t\right)\parallel }_{{\Gamma }_1}^2\hfill \\ & & +\frac{a_1}{4\eta }{\int }_{{\Gamma }_1}{f}_1^2\left(u_t\left(t\right)\right)d\Gamma +\frac{|a_2|}{4\eta }{\int }_{{\Gamma }_1}{f}_2^2\left(v(x,1,t)\right)d\Gamma -{\int }_{{\Gamma }_1}m\left(x\right)w^2\left(t\right)d\Gamma .\hfill \end{array}$$

Setting $a_3=(a_1+|a_2\left|\right){\lambda }_1$ and choosing $\eta =\frac{l}{4a_3}$ leads to (42). □

Lemma 3. 

Under the assumption (H1), the functional ${\Phi }_2$ defined by

$${\Phi }_2\left(t\right)=-{\int }_{\Omega }{u}_t\left(t\right){\int }_0^t\psi (t-s)(u\left(t\right)-u\left(s\right))dsdx$$

satisfies

$$\begin{array}{ccc}\hfill & & {\Phi }_2^{\prime }\left(t\right)\le -\left({\int }_0^t\psi \left(s\right)ds-\delta \right)\parallel u_t{\left(t\right)\parallel }^2+\delta {\parallel \nabla u\left(t\right)\parallel }^2+\frac{C_1(1+C\left(\xi \right))}{\delta }(i\circ \nabla u)\left(t\right)\hfill \\ & & +\delta {\lambda }_1\parallel w_t{\left(t\right)\parallel }_{{\Gamma }_1}^2+\delta a_1{\lambda }_1{\int }_{{\Gamma }_1}{f}_1^2\left(u_t\left(t\right)\right)d\Gamma +\delta \left|a_2\right|{\lambda }_1{\int }_{{\Gamma }_1}{f}_2^2\left(v(x,1,t)\right)d\Gamma ,\hfill \end{array}$$

(48)

for any $0<\delta <1$.

Proof. 

Using Equations (22), (24), and (25), we obtain

$$\begin{array}{ccc}& & {\Phi }_2^{\prime }\left(t\right)=\left(1-{\int }_0^t\psi \left(s\right)ds\right){\int }_{\Omega }\nabla u·{\int }_0^t\psi (t-s)(\nabla u\left(t\right)-\nabla u\left(s\right))dsdx\hfill \\ & & +{\int }_{\Omega }{\left({\int }_0^t\psi (t-s)(\nabla u\left(t\right)-\nabla u\left(s\right))ds\right)}^{2}dx-{\int }_{{\Gamma }_1}{w}_t\left(t\right){\int }_0^t\psi (t-s)(u\left(t\right)-u\left(s\right))dsd\Gamma \hfill \\ & & +a_1{\int }_{{\Gamma }_1}{f}_1\left(u_t\left(t\right)\right){\int }_0^t\psi (t-s)(u\left(t\right)-u\left(s\right))dsd\Gamma +a_2{\int }_{{\Gamma }_1}{f}_2\left(v(x,1,t)\right){\int }_0^t\psi (t-s)(u\left(t\right)-u\left(s\right))dsd\Gamma \hfill \\ & & -{\int }_{\Omega }{u}_t\left(t\right){\int }_0^t{\psi }^{\prime }(t-s)(u\left(t\right)-u\left(s\right))dsdx-\left({\int }_0^t\psi \left(s\right)ds\right){\parallel u_t\left(t\right)\parallel }^2\hfill \\ & & ={\vartheta }_1+{\vartheta }_2+\cdots +{\vartheta }_6-\left({\int }_0^t\psi \left(s\right)ds\right){\parallel u_t\left(t\right)\parallel }^2.\hfill \end{array}$$

By Young’s inequality, (8), and (44), we obtain, for $\delta >0$,

$$\begin{array}{ccc}& & {\vartheta }_1\le \delta {\parallel \nabla u\left(t\right)\parallel }^2+\frac{C\left(\xi \right)}{4\delta }(i\circ \nabla u)\left(t\right),\hfill \\ & & {\vartheta }_2\le C\left(\xi \right)(i\circ \nabla u)\left(t\right),\hfill \\ & & |{\vartheta }_3|\le \delta {\lambda }_1{\parallel w_t\left(t\right)\parallel }_{{\Gamma }_1}^2+\frac{C\left(\xi \right)}{4\delta }(i\circ \nabla u)\left(t\right),\hfill \\ & & |{\vartheta }_4|\le \delta a_1{\lambda }_1{\int }_{{\Gamma }_1}{f}_1^2\left(u_t\left(t\right)\right)d\Gamma +\frac{a_{1}C\left(\xi \right)}{4\delta }(i\circ \nabla u)\left(t\right),\hfill \\ & & |{\vartheta }_5|\le \delta |a_2|{\lambda }_1{\int }_{{\Gamma }_1}{f}_2^2\left(v(x,1,t)\right)d\Gamma +\frac{|a_2|C\left(\xi \right)}{4\delta }(i\circ \nabla u)\left(t\right).\hfill \end{array}$$

Using Young’s inequality, (8), (9), (43), and (44), we see that

$$\begin{array}{ccc}& & {\vartheta }_6={\int }_{\Omega }{u}_t\left(t\right){\int }_0^{t}i(t-s)(u\left(t\right)-u\left(s\right))dsdx-\xi {\int }_{\Omega }{u}_t\left(t\right){\int }_0^t\psi (t-s)(u\left(t\right)-u\left(s\right))dsdx\hfill \\ & & \le \delta \parallel u_t{\left(t\right)\parallel }^2+\frac{1}{2\delta }{\int }_{\Omega }{\left({\int }_0^{t}i(t-s)|u\left(s\right)-u\left(t\right)|ds\right)}^{2}dx+\frac{{\xi }^2}{2\delta }{\int }_{\Omega }{\left({\int }_0^t\psi (t-s)|u\left(t\right)-u\left(s\right)|ds\right)}^{2}dx\hfill \\ & & \le \delta \parallel u_t{\left(t\right)\parallel }^2+\frac{{\lambda }_0(\psi \left(0\right)+\xi )}{2\delta }(i\circ \nabla u)\left(t\right)+\frac{{\lambda }_0{\xi }^{2}C\left(\xi \right)}{2\delta }(i\circ \nabla u)\left(t\right).\hfill \end{array}$$

Combining all above estimates and taking $C_1=max\{\frac{{\lambda }_0(\psi \left(0\right)+\xi )}{2},\delta +\frac{1+{\lambda }_0{\xi }^2}{2}+\frac{a_1+\left|a_2\right|}{4}\},$ the desired inequality (48) is established. □

Lemma 4. 

Under the assumptions (H3) and (H4), the functional ${\Phi }_3$ defined by

$${\Phi }_3\left(t\right)=\varrho \left(t\right){\int }_{{\Gamma }_1}{\int }_0^1{e}^{-\kappa \varrho \left(t\right)}{F}_2\left(v(x,\kappa ,t)\right)d\kappa d\Gamma$$

satisfies

$$\begin{array}{ccc}\hfill & & {\Phi }_3^{\prime }\left(t\right)\le -e^{-{\varrho }_2}\varrho \left(t\right){\int }_{{\Gamma }_1}{\int }_0^1{F}_2\left(v(x,\kappa ,t)\right)d\kappa d\Gamma -c_3(1-{\varrho }_3)e^{-{\varrho }_2}{\int }_{{\Gamma }_1}{f}_2\left(v(x,1,t)\right)v(x,1,t)d\Gamma \hfill \\ & & +c_4{\int }_{{\Gamma }_1}{f}_1\left(u_t\left(t\right)\right)u_t\left(t\right)d\Gamma .\hfill \end{array}$$

(49)

Proof. 

Using Equation (23), integration by parts, (13), and (14), we obtain (see [26])

$$\begin{array}{ccc}& & {\Phi }_3^{\prime }\left(t\right)={\varrho }^{\prime }\left(t\right){\int }_{{\Gamma }_1}{\int }_0^1{e}^{-\kappa \varrho \left(t\right)}{F}_2\left(v(x,\kappa ,t)\right)d\kappa d\Gamma -\varrho \left(t\right){\int }_{{\Gamma }_1}{\int }_0^1\kappa {\varrho }^{\prime }\left(t\right)e^{-\kappa \varrho \left(t\right)}{F}_2\left(v(x,\kappa ,t)\right)d\kappa d\Gamma \hfill \\ & & -{\int }_{{\Gamma }_1}{\int }_0^1{e}^{-\kappa \varrho \left(t\right)}(1-\kappa {\varrho }^{\prime }\left(t\right))\frac{d}{d\kappa }{F}_2\left(v(x,\kappa ,t)\right)d\kappa d\Gamma \hfill \\ & & =-{\Phi }_3\left(t\right)-e^{-\varrho \left(t\right)}{\int }_{{\Gamma }_1}(1-{\varrho }^{\prime }\left(t\right))F_2\left(v(x,1,t)\right)d\Gamma +{\int }_{{\Gamma }_1}{F}_2\left(u_t(x,t)\right)d\Gamma \hfill \\ & & \le -e^{-{\varrho }_2}\varrho \left(t\right){\int }_{{\Gamma }_1}{\int }_0^1{F}_2\left(v(x,\kappa ,t)\right)d\kappa d\Gamma -c_3(1-{\varrho }_3)e^{-{\varrho }_2}{\int }_{{\Gamma }_1}{f}_2\left(v(x,1,t)\right)v(x,1,t)d\Gamma \hfill \\ & & +c_4{\int }_{{\Gamma }_1}{f}_1\left(u_t\left(t\right)\right)u_t\left(t\right)d\Gamma .\hfill \end{array}$$

□

Lemma 5 

([3]). Under the assumption (H1), the functional ${\Phi }_4$ defined by

$${\Phi }_4\left(t\right)={\int }_{\Omega }{\int }_0^t{G}_2(t-s){|\nabla u\left(s\right)|}^{2}dsdx$$

satisfies

$${\Phi }_4^{\prime }\left(t\right)\le 3(1-l){\parallel \nabla u\left(t\right)\parallel }^2-\frac{1}{2}(\psi \circ \nabla u)\left(t\right),$$

(50)

where $G_2\left(t\right)={\int }_t^{\infty }\psi \left(s\right)ds.$

Next, let us define the perturbed modified energy by

$$\begin{array}{c}\hfill L\left(t\right)=NE\left(t\right)+N_1{\Phi }_1\left(t\right)+N_2{\Phi }_2\left(t\right)+{\Phi }_3\left(t\right)+b_{1}E\left(t\right),\end{array}$$

(51)

where $N,N_1,N_2$, and $b_1$ are some positive constants.

As in [6,26], for a large enough $N>0$, there exist positive constants ${\beta }_1$ and ${\beta }_2$ such that

$$\begin{array}{c}\hfill {\beta }_{1}E\left(t\right)\le L\left(t\right)\le {\beta }_{2}E\left(t\right).\end{array}$$

Lemma 6. 

Assume that (H1) and (H3)–(H5) hold. Then, there exist positive constants ${\beta }_3,{\beta }_4$, and ${\beta }_5$ such that

$$\begin{array}{c}\hfill L^{\prime }\left(t\right)\le -{\beta }_{3}E\left(t\right)+{\beta }_4{\int }_{t_0}^t\psi \left(s\right){\int }_{\Omega }{|\nabla u\left(t\right)-\nabla u(t-s)|}^{2}dxds+{\beta }_5{\int }_{{\Gamma }_1}{f}_1^2\left(u_t\left(t\right)\right)d\Gamma ,\forall t\ge t_0,\end{array}$$

(52)

where $t_0$ was introduced in (17).

Proof. 

Let ${\psi }_0={\int }_0^{t_0}\psi \left(s\right)ds$. Using the fact that $i\left(t\right)=\xi \psi \left(t\right)-{\psi }^{\prime }\left(t\right)$ and combining (31), (42), (48), (49), and (51), we obtain, for all $t\ge t_0$,

$$\begin{array}{ccc}\hfill & & L^{\prime }\left(t\right)\le \frac{\xi N}{2}(\psi \circ \nabla u)\left(t\right)-\left(\frac{l{N}_1}{2}-\delta N_2\right){\parallel \nabla u\left(t\right)\parallel }^2-\left({\psi }_0{N}_2-\delta N_2-N_1\right){\parallel u_t\left(t\right)\parallel }^2\hfill \\ & & -\left(\frac{N}{2}-\frac{2C\left(\xi \right)N_1}{l}-\frac{C_1(1+C\left(\xi \right))N_2}{\delta }\right)(i\circ \nabla u)\left(t\right)-N_1{\int }_{{\Gamma }_1}m\left(x\right)w^2\left(t\right)d\Gamma +b_1{E}^{\prime }\left(t\right)\hfill \\ & & -\left(h_{1}N-\frac{8{\lambda }_1{N}_1}{l}-\delta {\lambda }_1{N}_2\right){\parallel w_t\left(t\right)\parallel }_{{\Gamma }_1}^2-e^{-{\varrho }_2}\varrho \left(t\right){\int }_{{\Gamma }_1}{\int }_0^1{F}_2\left(v(x,\kappa ,t)\right)d\kappa d\Gamma \hfill \\ & & -\left({\gamma }_{0}N-c_4\right){\int }_{{\Gamma }_1}{f}_1\left(u_t\left(t\right)\right)u_t\left(t\right)d\Gamma -\left({\gamma }_{1}N+c_3(1-{\varrho }_3)e^{-{\varrho }_2}\right){\int }_{{\Gamma }_1}{f}_2\left(v(x,1,t)\right)v(x,1,t)d\Gamma \hfill \\ & & +\left(\frac{a_1{a}_3{N}_1}{l}+\delta a_1{\lambda }_1{N}_2\right){\int }_{{\Gamma }_1}{f}_1^2\left(u_t\left(t\right)\right)d\Gamma +\left(\frac{|a_2|a_3{N}_1}{l}+\delta \left|a_2\right|{\lambda }_1{N}_2\right){\int }_{{\Gamma }_1}{f}_2^2\left(v(x,1,t)\right)d\Gamma .\hfill \end{array}$$

(53)

From (13), we find that

$$\begin{array}{c}\hfill {\int }_{{\Gamma }_1}{f}_2^2\left(v(x,1,t)\right)d\Gamma \le c_2{\int }_{{\Gamma }_1}{f}_2\left(v(x,1,t)\right)v(x,1,t)d\Gamma .\end{array}$$

(54)

Applying (54) to (53) and taking $\delta =\frac{l}{4N_2}$, we obtain, for all $t\ge t_0$,

$$\begin{array}{ccc}& & L^{\prime }\left(t\right)\le \frac{\xi N}{2}(\psi \circ \nabla u)\left(t\right)-\left(\frac{l{N}_1}{2}-\frac{l}{4}\right){\parallel \nabla u\left(t\right)\parallel }^2-\left({\psi }_0{N}_2-N_1-\frac{l}{4}\right){\parallel u_t\left(t\right)\parallel }^2\hfill \\ & & -\left(\frac{N}{2}-\frac{4C_1{N}_2^2}{l}-C\left(\xi \right)\left[\frac{2N_1}{l}+\frac{4C_1{N}_2^2}{l}\right]\right)(i\circ \nabla u)\left(t\right)-N_1{\int }_{{\Gamma }_1}m\left(x\right)w^2\left(t\right)d\Gamma \hfill \\ & & -\left(h_{1}N-\frac{8{\lambda }_1{N}_1}{l}-\frac{l{\lambda }_1}{4}\right){\parallel w_t\left(t\right)\parallel }_{{\Gamma }_1}^2-e^{-{\varrho }_2}\varrho \left(t\right){\int }_{{\Gamma }_1}{\int }_0^1{F}_2\left(v(x,\kappa ,t)\right)d\kappa d\Gamma \hfill \\ & & -\left({\gamma }_{0}N-c_4\right){\int }_{{\Gamma }_1}{f}_1\left(u_t\left(t\right)\right)u_t\left(t\right)d\Gamma +\left(\frac{a_1{a}_3{N}_1}{l}+\frac{a_{1}l{\lambda }_1}{4}\right){\int }_{{\Gamma }_1}{f}_1^2\left(u_t\left(t\right)\right)d\Gamma +b_1{E}^{\prime }\left(t\right)\hfill \\ & & -\left({\gamma }_{1}N+c_3(1-{\varrho }_3)e^{-{\varrho }_2}-\frac{|a_2|a_3{c}_2{N}_1}{l}-\frac{|a_2|c_{2}l{\lambda }_1}{4}\right){\int }_{{\Gamma }_1}{f}_2\left(v(x,1,t)\right)v(x,1,t)d\Gamma .\hfill \end{array}$$

We choose $N_1$ large enough so that

$$\frac{l{N}_1}{2}-\frac{l}{4}>4(1-l),$$

then $N_2$ large enough so that

$${\psi }_0{N}_2-N_1-\frac{l}{4}>1.$$

Using the fact that $\frac{\xi {\psi }^2\left(s\right)}{i\left(s\right)}<\psi \left(s\right)$ and the Lebesgue dominated convergence theorem, we deduce that

$$\xi C\left(\xi \right)={\int }_0^{\infty }\frac{\xi {\psi }^2\left(s\right)}{i\left(s\right)}ds\to 0\mathrm{as}\xi \to 0.$$

Hence, there is $0<{\xi }_0<1$ such that if $\xi <{\xi }_0$, then

$$\xi C\left(\xi \right)\left[\frac{2N_1}{l}+\frac{4C_1{N}_2^2}{l}\right]<\frac{1}{8}.$$

Finally, selecting $\xi =\frac{1}{2N}$ and choosing N large enough so that

$$N>max\left\{\frac{16C_1{N}_2^2}{l},\frac{1}{h_1}\left(\frac{8{\lambda }_1{N}_1}{l}+\frac{l{\lambda }_1}{4}\right),\frac{c_4}{{\gamma }_0},\frac{1}{{\gamma }_1}\left(\frac{|a_2|a_3{c}_2{N}_1}{l}+\frac{|a_2|c_{2}l{\lambda }_1}{4}-c_3(1-{\varrho }_3)e^{-{\varrho }_2}\right)\right\},$$

we obtain

$$\begin{array}{ccc}\hfill & & L^{\prime }\left(t\right)\le -\parallel u_t{\left(t\right)\parallel }^2-4(1-l){\parallel \nabla u\left(t\right)\parallel }^2+\frac{1}{4}(\psi \circ \nabla u)\left(t\right)-N_1{\int }_{{\Gamma }_1}m\left(x\right)w^2\left(t\right)d\Gamma \hfill \\ & & -e^{-{\varrho }_2}\varrho \left(t\right){\int }_{{\Gamma }_1}{\int }_0^1{F}_2\left(v(x,\kappa ,t)\right)d\kappa d\Gamma +{\beta }_5{\int }_{{\Gamma }_1}{f}_1^2\left(u_t\left(t\right)\right)d\Gamma +b_1{E}^{\prime }\left(t\right),\forall t\ge t_0,\hfill \end{array}$$

(55)

where ${\beta }_5=\frac{a_1{a}_3{N}_1}{l}+\frac{a_{1}l{\lambda }_1}{4}.$ Using (18) and (31), we find that, for any $t\ge t_0$,

$$\begin{array}{c}\hfill {\int }_0^{t_0}\psi \left(s\right){\int }_{\Omega }{|\nabla u\left(t\right)-\nabla u(t-s)|}^{2}dxds\le -\frac{1}{c_5}{\int }_0^{t_0}{\psi }^{\prime }\left(s\right){\int }_{\Omega }{|\nabla u\left(t\right)-\nabla u(t-s)|}^{2}dxds\le -\frac{2}{c_5}{E}^{\prime }\left(t\right).\end{array}$$

(56)

Combining (29), (55), and (56) and making a suitable choice for $b_1,$ we obtain the estimate (52). □

To evaluate the two terms on the right side of (52), we establish the following lemmas.

Lemma 7 

([1]). Assume that (H2) holds and $max\{r_1,f_0\left(r_1\right)\}<\epsilon ,$ where ε was introduced in (11). Then, there exist positive constants $C_2,C_3$, and $C_4$ such that

$$\begin{array}{cc}& {\displaystyle {\int }_{{\Gamma }_1}{f}_1^2\left(u_t\left(t\right)\right)d\Gamma \le C_2{\int }_{{\Gamma }_1}{f}_1\left(u_t\left(t\right)\right)u_t\left(t\right)d\Gamma ,\mathit{if}{f}_0\mathit{is}\mathit{linear},}\end{array}$$

(57)

$$\begin{array}{cc}& {\displaystyle {\int }_{{\Gamma }_1}{f}_1^2\left(u_t\left(t\right)\right)d\Gamma \le C_3{F}^{-1}\left(\chi \left(t\right)\right)-C_3{E}^{\prime }\left(t\right),\mathit{if}{f}_0\mathit{is}\mathit{nonlinear},}\end{array}$$

(58)

where

$$\chi \left(t\right)=\frac{1}{|{\Gamma }_{11}|}{\int }_{{\Gamma }_{11}}{f}_1\left(u_t\left(t\right)\right)u_t\left(t\right)d\Gamma \le -C_4{E}^{\prime }\left(t\right),$$

(59)

${\Gamma }_{11}=\{x\in {\Gamma }_1:|u_t\left(t\right)|\le {\epsilon }_1\}$ and $0<{\epsilon }_1=min\{r_1,f_0\left(r_1\right)\}.$

Lemma 8. 

Assume that (H1) and (H3)–(H5) hold and that $f_0$ is linear. Then, the energy functional satisfies

$$\begin{array}{c}\hfill {\int }_0^{\infty }E\left(s\right)ds<\infty .\end{array}$$

(60)

Proof. 

We introduce the functional

$$\mathcal{L}\left(t\right)=L\left(t\right)+{\Phi }_4\left(t\right),$$

which is nonnegative. From (50) and (55), we see that, for all $t\ge t_0$,

$$\begin{array}{ccc}& & {\mathcal{L}}^{\prime }\left(t\right)\le -\parallel u_t{\left(t\right)\parallel }^2-(1-l){\parallel \nabla u\left(t\right)\parallel }^2-\frac{1}{4}(\psi \circ \nabla u)\left(t\right)-N_1{\int }_{{\Gamma }_1}m\left(x\right)w^2\left(t\right)d\Gamma \hfill \\ & & -e^{-{\varrho }_2}\varrho \left(t\right){\int }_{{\Gamma }_1}{\int }_0^1{F}_2\left(v(x,\kappa ,t)\right)d\kappa d\Gamma +{\beta }_5{\int }_{{\Gamma }_1}{f}_1^2\left(u_t\left(t\right)\right)d\Gamma +b_1{E}^{\prime }\left(t\right).\hfill \end{array}$$

Applying (29), (31), and (57), we have

$${\mathcal{L}}^{\prime }\left(t\right)\le -d_{1}E\left(t\right)+\left(b_1-\frac{{\beta }_5{C}_2}{{\gamma }_0}\right)E^{\prime }\left(t\right),$$

where $d_1$ is some positive constant. Selecting a suitable choice for $b_1,$ we obtain

$${\mathcal{L}}^{\prime }\left(t\right)\le -d_{1}E\left(t\right).$$

This implies that

$$d_1{\int }_{t_0}^{t}E\left(s\right)ds\le \mathcal{L}\left(t_0\right)-\mathcal{L}\left(t\right)\le \mathcal{L}\left(t_0\right)<\infty .$$

□

Next, we define ${\rm Y}\left(t\right)$ by

$$\begin{array}{c}\hfill {\rm Y}\left(t\right):=-{\int }_{t_0}^t{\psi }^{\prime }\left(s\right){\int }_{\Omega }{|\nabla u\left(t\right)-\nabla u(t-s)|}^{2}dxds\le -2E^{\prime }\left(t\right).\end{array}$$

(61)

Lemma 9. 

Assume that (H1) and (H2) hold and that G is nonlinear. Then, the solution to (22)–(28) satisfies the estimates

$$\begin{array}{cc}& {\displaystyle {\int }_{t_0}^t\psi \left(s\right){\int }_{\Omega }{|\nabla u\left(t\right)-\nabla u(t-s)|}^{2}dxds\le \frac{1}{\theta }{\overline{G}}^{-1}\left(\frac{\theta {\rm Y}\left(t\right)}{\mu \left(t\right)}\right),\forall t\ge t_0,\mathit{if}{f}_0\mathit{is}\mathit{linear},}\end{array}$$

(62)

$$\begin{array}{cc}& {\displaystyle {\int }_{t_0}^t\psi \left(s\right){\int }_{\Omega }{|\nabla u\left(t\right)-\nabla u(t-s)|}^{2}dxds\le \frac{t-t_0}{\theta }{\overline{G}}^{-1}\left(\frac{\theta {\rm Y}\left(t\right)}{(t-t_0)\mu \left(t\right)}\right),\forall t>t_0,\mathit{if}{f}_0\mathit{is}\mathit{nonlinear},}\end{array}$$

(63)

where $\theta \in (0,1)$, and $\overline{G}$ is an extension of G such that $\overline{G}$ is a strictly convex and strictly increasing $C^2$ function on $(0,\infty ).$

Proof. 

First, we prove the estimate (62) when $f_0$ is linear. For $0<\theta <1,$ we define $I\left(t\right)$ by

$$I\left(t\right):=\theta {\int }_{t_0}^t{\int }_{\Omega }{|\nabla u\left(t\right)-\nabla u(t-s)|}^{2}dxds.$$

By (60), $\theta$ is taken so small that, for all $t\ge t_0$,

$$I\left(t\right)<1.$$

(64)

Since G is strictly convex on $(0,r_0]$, then

$$\begin{array}{c}\hfill G\left(q\zeta \right)\le qG\left(\zeta \right),\end{array}$$

(65)

where $0\le q\le 1$ and $\zeta \in (0,r_0]$. Using the fact that $\mu$ is a positive nonincreasing function and applying (10), (64), (65), and Jensen’s inequality (19), we find that (see details in [1,3])

$$\begin{array}{ccc}& & {\rm Y}\left(t\right)\ge \frac{\mu \left(t\right)}{\theta I\left(t\right)}{\int }_{t_0}^{t}I\left(t\right)G\left(\psi \left(s\right)\right){\int }_{\Omega }\theta {|\nabla u\left(t\right)-\nabla u(t-s)|}^{2}dxds\hfill \\ & & \ge \frac{\mu \left(t\right)}{\theta I\left(t\right)}{\int }_{t_0}^{t}G\left(I\left(t\right)\psi \left(s\right)\right){\int }_{\Omega }\theta {|\nabla u\left(t\right)-\nabla u(t-s)|}^{2}dxds\hfill \\ & & \ge \frac{\mu \left(t\right)}{\theta }\overline{G}\left(\theta {\int }_{t_0}^t\psi \left(s\right){\int }_{\Omega }{|\nabla u\left(t\right)-\nabla u(t-s)|}^{2}dxds\right).\hfill \end{array}$$

(66)

Since $\overline{G}$ is strictly increasing, we obtain

$${\int }_{t_0}^t\psi \left(s\right){\int }_{\Omega }{|\nabla u\left(t\right)-\nabla u(t-s)|}^{2}dxds\le \frac{1}{\theta }{\overline{G}}^{-1}\left(\frac{\theta {\rm Y}\left(t\right)}{\mu \left(t\right)}\right).$$

Now, we show the estimate (63) when $f_0$ is nonlinear. Since we cannot guarantee (60), we define the following function:

$$I_1\left(t\right):=\frac{\theta }{t-t_0}{\int }_{t_0}^t{\int }_{\Omega }{|\nabla u\left(t\right)-\nabla u(t-s)|}^{2}dxds,\forall t>t_0.$$

Using the fact that $E^{\prime }\left(t\right)\le 0$ and (29), we have

$$\begin{array}{c}\hfill I_1\left(t\right)\le \frac{2\theta }{t-t_0}{\int }_{t_0}^t{\left(\right||\nabla u\left(t\right)||}^2+{\left|\right|\nabla u(t-s)\left|\right|}^2)ds\le \frac{8\theta E\left(0\right)}{l}.\end{array}$$

Choose $\theta$ small enough so that, for all $t>t_0,$

$$\begin{array}{c}\hfill I_1\left(t\right)<1.\end{array}$$

(67)

Similar to (67), using (10), (65), (67), and Jensen’s inequality (19), we obtain

$$\begin{array}{ccc}& & {\rm Y}\left(t\right)=\frac{t-t_0}{\theta I_1\left(t\right)}{\int }_{t_0}^t{I}_1\left(t\right)(-{\psi }^{\prime }\left(s\right)){\int }_{\Omega }\frac{\theta }{t-t_0}{|\nabla u\left(t\right)-\nabla u(t-s)|}^{2}dxds\hfill \\ & & \ge \frac{(t-t_0)\mu \left(t\right)}{\theta I_1\left(t\right)}{\int }_{t_0}^{t}G\left(I_1\left(t\right)\psi \left(s\right)\right){\int }_{\Omega }\frac{\theta }{t-t_0}{|\nabla u\left(t\right)-\nabla u(t-s)|}^{2}dxds\hfill \\ & & \ge \frac{(t-t_0)\mu \left(t\right)}{\theta }\overline{G}\left(\frac{\theta }{t-t_0}{\int }_{t_0}^t\psi \left(s\right){\int }_{\Omega }{|\nabla u\left(t\right)-\nabla u(t-s)|}^{2}dxds\right).\hfill \end{array}$$

This implies that

$$\begin{array}{c}\hfill {\int }_{t_0}^t\psi \left(s\right){\int }_{\Omega }{|\nabla u\left(t\right)-\nabla u(t-s)|}^{2}dxds\le \frac{t-t_0}{\theta }{\overline{G}}^{-1}\left(\frac{\theta {\rm Y}\left(t\right)}{(t-t_0)\mu \left(t\right)}\right).\end{array}$$

□

4. General Decay of the Energy

In this section, we state and prove the main result of our work.

Theorem 2. 

Assume that (H1)–(H5) hold and that $f_0$ is linear. Then, there exist positive constants $k_1,k_2,k_3$, and $k_4$ such that the energy functional satisfies, for all $t\ge t_0,$

$$\begin{array}{cc}& {\displaystyle E\left(t\right)\le k_2{e}^{-k_1{\int }_{t_0}^t\mu \left(s\right)ds},\mathit{if}G\mathit{is}\mathit{linear},}\end{array}$$

(68)

$$\begin{array}{cc}& {\displaystyle E\left(t\right)\le k_4{G}_1^{-1}\left(k_3{\int }_{t_0}^t\mu \left(s\right)ds\right),\mathit{if}G\mathit{is}\mathit{nonlinear},}\end{array}$$

(69)

where $G_1\left(t\right)={\int }_t^{r_0}\frac{1}{s{G}^{\prime }\left(s\right)}ds$ is strictly decreasing and convex on $(0,r_0].$

Proof. 

Now, we consider the following two cases.

Case 1: $G\left(t\right)$ is linear. Multiplying (52) by the positive nonincreasing function $\mu \left(t\right)$ and using (10), (31), and (57), we obtain

$$\begin{array}{ccc}& & \mu \left(t\right)L^{\prime }\left(t\right)\le -{\beta }_3\mu \left(t\right)E\left(t\right)+{\beta }_4{\int }_{t_0}^t\mu \left(s\right)\psi \left(s\right){\int }_{\Omega }{|\nabla u\left(t\right)-\nabla u(t-s)|}^{2}dxds+{\beta }_5\mu \left(t\right){\int }_{{\Gamma }_1}{f}_1^2\left(u_t\left(t\right)\right)d\Gamma \hfill \\ & & \le -{\beta }_3\mu \left(t\right)E\left(t\right)-{\beta }_4{\int }_{t_0}^t{\psi }^{\prime }\left(s\right){\int }_{\Omega }{|\nabla u\left(t\right)-\nabla u(t-s)|}^{2}dxds+{\beta }_5{C}_2\mu \left(0\right){\int }_{{\Gamma }_1}{f}_1\left(u_t\left(t\right)\right)u_t\left(t\right)d\Gamma \hfill \\ & & \le -{\beta }_3\mu \left(t\right)E\left(t\right)-C_5{E}^{\prime }\left(t\right),\hfill \end{array}$$

where $C_5=2{\beta }_4+\frac{{\beta }_5{C}_2\mu \left(0\right)}{{\gamma }_0}$ is a positive constant. Since $\mu \left(t\right)$ is nonincreasing, we have

$${(\mu L+C_{5}E)}^{\prime }\left(t\right)\le -{\beta }_3\mu \left(t\right)E\left(t\right),\forall t\ge t_0.$$

Since $\mu \left(t\right)L\left(t\right)+C_{5}E\left(t\right)\sim E\left(t\right)$, for some positive constants $k_1$ and $k_2,$ we obtain

$$E\left(t\right)\le k_2{e}^{-k_1{\int }_{t_0}^t\mu \left(s\right)ds}.$$

Case 2: $G\left(t\right)$ is nonlinear. This case is obtained through the ideas presented in [3] as follows.

Using (31), (52), (57), and (62), we obtain

$$L^{\prime }\left(t\right)\le -{\beta }_{3}E\left(t\right)+\frac{{\beta }_4}{\theta }{\overline{G}}^{-1}\left(\frac{\theta {\rm Y}\left(t\right)}{\mu \left(t\right)}\right)-\frac{{\beta }_5{C}_2}{{\gamma }_0}{E}^{\prime }\left(t\right),\forall t\ge t_0.$$

(70)

Let $L_1\left(t\right)=L\left(t\right)+\frac{{\beta }_5{C}_2}{{\gamma }_0}E\left(t\right)\sim E\left(t\right),$ and then (70) becomes

$$L_1^{\prime }\left(t\right)\le -{\beta }_{3}E\left(t\right)+\frac{{\beta }_4}{\theta }{\overline{G}}^{-1}\left(\frac{\theta {\rm Y}\left(t\right)}{\mu \left(t\right)}\right),\forall t\ge t_0.$$

(71)

For $0<{\epsilon }_0<r_0$, using (71) and the fact that $E^{\prime }\le 0,{\overline{G}}^{\prime }>0$ and ${\overline{G}}^{\prime \prime }>0$, we find that the functional $L_2$, defined by

$$L_2\left(t\right):={\overline{G}}^{\prime }\left({\epsilon }_0\frac{E\left(t\right)}{E\left(0\right)}\right)L_1\left(t\right)\sim E\left(t\right),$$

satisfies

$$L_2^{\prime }\left(t\right)\le -{\beta }_{3}E\left(t\right){\overline{G}}^{\prime }\left({\epsilon }_0\frac{E\left(t\right)}{E\left(0\right)}\right)+\frac{{\beta }_4}{\theta }{\overline{G}}^{\prime }\left({\epsilon }_0\frac{E\left(t\right)}{E\left(0\right)}\right){\overline{G}}^{-1}\left(\frac{\theta {\rm Y}\left(t\right)}{\mu \left(t\right)}\right),\forall t\ge t_0.$$

(72)

With $s={\overline{G}}^{\prime }\left({\epsilon }_0\frac{E\left(t\right)}{E\left(0\right)}\right)$ and $r={\overline{G}}^{-1}\left(\frac{\theta {\rm Y}\left(t\right)}{\mu \left(t\right)}\right)$, using (20), (21), and (72), we obtain

$$L_2^{\prime }\left(t\right)\le -{\beta }_{3}E\left(t\right)G^{\prime }\left({\epsilon }_0\frac{E\left(t\right)}{E\left(0\right)}\right)+\frac{{\epsilon }_0{\beta }_4}{\theta }\frac{E\left(t\right)}{E\left(0\right)}{G}^{\prime }\left({\epsilon }_0\frac{E\left(t\right)}{E\left(0\right)}\right)+\frac{{\beta }_4{\rm Y}\left(t\right)}{\mu \left(t\right)},$$

where we have used that ${\epsilon }_0\frac{E\left(t\right)}{E\left(0\right)}<r_0$ and ${\overline{G}}^{\prime }=G^{\prime }$ on $(0,r_0].$ Multiplying this by $\mu \left(t\right)$ and using (61), we obtain

$$\mu \left(t\right)L_2^{\prime }\left(t\right)\le -\left({\beta }_{3}E\left(0\right)-\frac{{\epsilon }_0{\beta }_4}{\theta }\right)\frac{\mu \left(t\right)E\left(t\right)}{E\left(0\right)}{G}^{\prime }\left({\epsilon }_0\frac{E\left(t\right)}{E\left(0\right)}\right)-2{\beta }_4{E}^{\prime }\left(t\right).$$

By defining $L_3\left(t\right)=\mu \left(t\right)L_2\left(t\right)+2{\beta }_{4}E\left(t\right)$, we see that, for some positive constants ${\gamma }_2$ and ${\gamma }_3$,

$${\gamma }_2{L}_3\left(t\right)\le E\left(t\right)\le {\gamma }_3{L}_3\left(t\right).$$

(73)

With a suitable choice of ${\epsilon }_0$, we obtain, for some positive constant $d_2,$

$$L_3^{\prime }\left(t\right)\le -d_2\mu \left(t\right)\frac{E\left(t\right)}{E\left(0\right)}{G}^{\prime }\left({\epsilon }_0\frac{E\left(t\right)}{E\left(0\right)}\right)=-d_2\mu \left(t\right)G_2\left(\frac{E\left(t\right)}{E\left(0\right)}\right),\forall t\ge t_0,$$

(74)

where $G_2\left(t\right)=t{G}^{\prime }\left({\epsilon }_{0}t\right)$. Applying the strict convexity of G on $(0,r_0]$ and $G_2^{\prime }\left(t\right)=G^{\prime }\left({\epsilon }_{0}t\right)+{\epsilon }_{0}t{G}^{\prime \prime }\left({\epsilon }_{0}t\right)$, we see that $G_2\left(t\right),G_2^{\prime }\left(t\right)>0$ on $(0,1]$. Finally, defining

$$Q\left(t\right)=\frac{{\gamma }_2{L}_3\left(t\right)}{E\left(0\right)}$$

and using (73), we have

$$Q\left(t\right)\le \frac{E\left(t\right)}{E\left(0\right)}\le 1\mathrm{and}Q\left(t\right)\sim E\left(t\right).$$

(75)

From (74), (75), and the fact that $G_2^{\prime }\left(t\right)>0$ on $(0,1],$ we arrive at

$$Q^{\prime }\left(t\right)\le -k_3\mu \left(t\right)G_2\left(Q\left(t\right)\right),\forall t\ge t_0,$$

where $k_3=\frac{d_2{\gamma }_2}{E\left(0\right)}$ is a positive constant. Integrating this over $(t_0,t)$ and using variable transformation, we find that (see details in [3])

$${\int }_t^{t_0}\frac{{\epsilon }_0{Q}^{\prime }\left(s\right)}{{\epsilon }_{0}Q\left(s\right)G^{\prime }\left({\epsilon }_{0}Q\left(s\right)\right)}ds\ge k_3{\int }_{t_0}^t\mu \left(s\right)ds⟹{\int }_{{\epsilon }_{0}Q\left(t\right)}^{{\epsilon }_{0}Q\left(t_0\right)}\frac{1}{s{G}^{\prime }\left(s\right)}ds\ge k_3{\int }_{t_0}^t\mu \left(s\right)ds.$$

Since ${\epsilon }_0<r_0$ and $Q\left(t\right)\le 1,$ for all $t\ge t_0$, we have

$$\begin{array}{c}\hfill G_1\left({\epsilon }_{0}Q\left(t\right)\right)={\int }_{{\epsilon }_{0}Q\left(t\right)}^{r_0}\frac{1}{s{G}^{\prime }\left(s\right)}ds\ge k_3{\int }_{t_0}^t\mu \left(s\right)ds⟹Q\left(t\right)\le \frac{1}{{\epsilon }_0}{G}_1^{-1}\left(k_3{\int }_{t_0}^t\mu \left(s\right)ds\right),\end{array}$$

(76)

where $G_1\left(t\right)={\int }_t^{r_0}\frac{1}{s{G}^{\prime }\left(s\right)}ds$. Here, we have used the fact that $G_1$ is a strictly decreasing function on $(0,r_0].$ Therefore, using (75) and (76), the estimate (69) is established. □

Theorem 3. 

Assume that (H1)–(H5) hold and that $f_0$ is nonlinear. Then, there exist positive constants ${\alpha }_1,{\alpha }_2,{\alpha }_3$, and ${\alpha }_4$ such that the energy functional satisfies

$$E\left(t\right)\le {\alpha }_2{F}_1^{-1}\left({\alpha }_1{\int }_{t_0}^t\mu \left(s\right)ds\right),\forall t\ge t_0,\mathrm{if}G\mathrm{is}\mathrm{linear},$$

(77)

where $F_1\left(t\right)={\int }_t^{r_1}\frac{1}{s{F}^{\prime }\left(s\right)}ds$ and

$$E\left(t\right)\le {\alpha }_4(t-t_0)K_1^{-1}\left(\frac{{\alpha }_3}{(t-t_0){\int }_{t_1}^t\mu \left(s\right)ds}\right),\forall t\ge t_1,\mathrm{if}G\mathrm{is}\mathrm{nonlinear},$$

(78)

where $K_1\left(t\right)=t{K}^{\prime }\left({\epsilon }_{2}t\right),0<{\epsilon }_2<r_2=min\{r_0,r_1\}\mathrm{and}K={\left({\overline{G}}^{-1}+{\overline{F}}^{-1}\right)}^{-1}.$

Proof. Case 1: $G\left(t\right)$ is linear. Multiplying (52) by the positive nonincreasing function $\mu \left(t\right)$ and using (10), (31), and (58), we obtain

$$\begin{array}{ccc}\hfill & & \mu \left(t\right)L^{\prime }\left(t\right)\le -{\beta }_3\mu \left(t\right)E\left(t\right)+{\beta }_5{C}_3\mu \left(t\right)F^{-1}\left(\chi \left(t\right)\right)-C_6{E}^{\prime }\left(t\right),\hfill \end{array}$$

(79)

where $C_6=2{\beta }_4+{\beta }_5{C}_3\mu \left(0\right)$ is a positive constant. Since $\mu \left(t\right)$ is nonincreasing, (79) becomes

$$F_3^{\prime }\left(t\right)\le -{\beta }_3\mu \left(t\right)E\left(t\right)+{\beta }_5{C}_3\mu \left(t\right)F^{-1}\left(\chi \left(t\right)\right),\forall t\ge t_0,$$

(80)

where $F_3\left(t\right)=\mu \left(t\right)L\left(t\right)+C_{6}E\left(t\right)\sim E\left(t\right).$ For $0<{\epsilon }_1<r_1$, using (80) and the fact that $E^{\prime }\le 0,F^{\prime }>0$ and $F^{\prime \prime }>0$ on $(0,r_1]$, the functional $F_4$, defined by

$$F_4\left(t\right):=F^{\prime }\left({\epsilon }_1\frac{E\left(t\right)}{E\left(0\right)}\right)F_3\left(t\right)\sim E\left(t\right),$$

satisfies

$$\begin{array}{c}\hfill F_4^{\prime }\left(t\right)\le -{\beta }_3\mu \left(t\right)E\left(t\right)F^{\prime }\left({\epsilon }_1\frac{E\left(t\right)}{E\left(0\right)}\right)+{\beta }_5{C}_3\mu \left(t\right)F^{\prime }\left({\epsilon }_1\frac{E\left(t\right)}{E\left(0\right)}\right)F^{-1}\left(\chi \left(t\right)\right).\end{array}$$

Given (20) and (21) with $s=F^{\prime }\left({\epsilon }_1\frac{E\left(t\right)}{E\left(0\right)}\right)$ and $r=F^{-1}\left(\chi \left(t\right)\right)$, using (59), we obtain that

$$\begin{array}{ccc}& & F_4^{\prime }\left(t\right)\le -{\beta }_3\mu \left(t\right)E\left(t\right)F^{\prime }\left({\epsilon }_1\frac{E\left(t\right)}{E\left(0\right)}\right)+{\epsilon }_1{\beta }_5{C}_3\frac{\mu \left(t\right)E\left(t\right)}{E\left(0\right)}{F}^{\prime }\left({\epsilon }_1\frac{E\left(t\right)}{E\left(0\right)}\right)+{\beta }_5{C}_3\mu \left(0\right)\chi \left(t\right)\hfill \\ & & \le -\left({\beta }_{3}E\left(0\right)-{\epsilon }_1{\beta }_5{C}_3\right)\frac{\mu \left(t\right)E\left(t\right)}{E\left(0\right)}{F}^{\prime }\left({\epsilon }_1\frac{E\left(t\right)}{E\left(0\right)}\right)-{\beta }_5{C}_3{C}_4\mu \left(0\right)E^{\prime }\left(t\right),\forall t\ge t_0.\hfill \end{array}$$

Let $F_5\left(t\right)=F_4\left(t\right)+{\beta }_5{C}_3{C}_4\mu \left(0\right)E\left(t\right)$; then it satisfies, for positive constants ${\gamma }_4$ and ${\gamma }_5$,

$${\gamma }_4{F}_5\left(t\right)\le E\left(t\right)\le {\gamma }_5{F}_5\left(t\right).$$

(81)

Consequently, with a suitable choice of ${\epsilon }_1,$ we have, for some positive constant $d_3,$

$$F_5^{\prime }\left(t\right)\le -d_3\mu \left(t\right)\frac{E\left(t\right)}{E\left(0\right)}{F}^{\prime }\left({\epsilon }_1\frac{E\left(t\right)}{E\left(0\right)}\right)=-d_3\mu \left(t\right)F_0\left(\frac{E\left(t\right)}{E\left(0\right)}\right),\forall t\ge t_0,$$

(82)

where $F_0\left(t\right)=t{F}^{\prime }\left({\epsilon }_{1}t\right)$. From the strict convexity of F on $(0,r_1],$ we obtain $F_0\left(t\right),F_0^{\prime }\left(t\right)>0$ on $(0,1]$. Let

$$J\left(t\right)=\frac{{\gamma }_4{F}_5\left(t\right)}{E\left(0\right)},$$

and from (81) and (82), we obtain

$$J\left(t\right)\le \frac{E\left(t\right)}{E\left(0\right)}\le 1\mathrm{and}{J}^{\prime }\left(t\right)\le -{\alpha }_1\mu \left(t\right)F_0\left(J\left(t\right)\right),\forall t\ge t_0,$$

where ${\alpha }_1=\frac{d_3{\gamma }_4}{E\left(0\right)}$ is a positive constant. Then, similar to (76), the integration over $(t_0,t)$ and variable transformation yield

$$\begin{array}{c}\hfill J\left(t\right)\le \frac{1}{{\epsilon }_1}{F}_1^{-1}\left({\alpha }_1{\int }_{t_0}^t\mu \left(s\right)ds\right),\end{array}$$

(83)

where $F_1\left(t\right)={\int }_t^{r_1}\frac{1}{s{F}^{\prime }\left(s\right)}ds,$ which is a strictly decreasing function on $(0,r_1].$ Combining (81) and (83), the estimate (77) is proved.

Case 2: $G\left(t\right)$ is nonlinear. This case is obtained by the arguments presented in [1] as follows. Using (52), (58), and (63), we obtain

$$L^{\prime }\left(t\right)\le -{\beta }_{3}E\left(t\right)+\frac{{\beta }_4(t-t_0)}{\theta }{\overline{G}}^{-1}\left(\frac{\theta {\rm Y}\left(t\right)}{(t-t_0)\mu \left(t\right)}\right)+{\beta }_5{C}_3{F}^{-1}\left(\chi \left(t\right)\right)-{\beta }_5{C}_3{E}^{\prime }\left(t\right),\forall t>t_0.$$

(84)

Since ${\displaystyle \underset{t\to \infty }{lim}\frac{1}{t-t_0}=0,}$ there exists $t_1>t_0$ such that

$$\frac{1}{t-t_0}<1,\forall t\ge t_1.$$

(85)

Using the strictly convex and strictly increasing function of $\overline{F}$ and (65) with $q=\frac{1}{t-t_0},$ we see that

$${\overline{F}}^{-1}\left(\chi \left(t\right)\right)\le (t-t_0){\overline{F}}^{-1}\left(\frac{\chi \left(t\right)}{t-t_0}\right),\forall t\ge t_1.$$

(86)

Combining (84) and (86), we arrive at

$$R_1^{\prime }\left(t\right)\le -{\beta }_{3}E\left(t\right)+\frac{{\beta }_4(t-t_0)}{\theta }{\overline{G}}^{-1}\left(\frac{\theta {\rm Y}\left(t\right)}{(t-t_0)\mu \left(t\right)}\right)+{\beta }_5{C}_3(t-t_0){\overline{F}}^{-1}\left(\frac{\chi \left(t\right)}{t-t_0}\right),\forall t\ge t_1,$$

(87)

where $R_1\left(t\right)=L\left(t\right)+{\beta }_5{C}_{3}E\left(t\right)\sim E\left(t\right).$ Let

$$r_2=min\{r_0,r_1\},\phi \left(t\right)=max\left\{\frac{\theta {\rm Y}\left(t\right)}{(t-t_0)\mu \left(t\right)},\frac{\chi \left(t\right)}{t-t_0}\right\}\mathrm{and}K={\left({\overline{G}}^{-1}+{\overline{F}}^{-1}\right)}^{-1},\forall t\ge t_1.$$

(88)

Therefore, (87) reduces to

$$R_1^{\prime }\left(t\right)\le -{\beta }_{3}E\left(t\right)+C_7(t-t_0)K^{-1}\left(\phi \left(t\right)\right),\forall t\ge t_1,$$

(89)

where $C_7=max\{\frac{{\beta }_4}{\theta },{\beta }_5{C}_3\}.$ The strictly increasing and strictly convex properties of $\overline{G}$ and $\overline{F}$ imply that

$$K^{\prime }=\frac{{\overline{G}}^{\prime }{\overline{F}}^{\prime }}{{\overline{G}}^{\prime }+{\overline{F}}^{\prime }}>0\mathrm{and}{K}^{\prime \prime }=\frac{{\overline{G}}^{\prime \prime }{\left({\overline{F}}^{\prime }\right)}^2+{\left({\overline{G}}^{\prime }\right)}^2{\overline{F}}^{\prime \prime }}{{({\overline{G}}^{\prime }+{\overline{F}}^{\prime })}^2}>0,$$

(90)

on $(0,r_2].$

Now, for $0<{\epsilon }_2<r_2,$ using (85), we see that $\frac{{\epsilon }_2}{t-t_0}\frac{E\left(t\right)}{E\left(0\right)}<r_2.$ Defining

$$R_2\left(t\right)=K^{\prime }\left(\frac{{\epsilon }_2}{t-t_0}\frac{E\left(t\right)}{E\left(0\right)}\right)R_1\left(t\right),\forall t\ge t_1,$$

and using (89) and (90), we find that

$$\begin{array}{ccc}\hfill & & R_2^{\prime }\left(t\right)=\left(-\frac{{\epsilon }_2}{{(t-t_0)}^2}\frac{E\left(t\right)}{E\left(0\right)}+\frac{{\epsilon }_2}{t-t_0}\frac{E^{\prime }\left(t\right)}{E\left(0\right)}\right)K^{″}\left(\frac{{\epsilon }_2}{t-t_0}\frac{E\left(t\right)}{E\left(0\right)}\right)R_1\left(t\right)+K^{\prime }\left(\frac{{\epsilon }_2}{t-t_0}\frac{E\left(t\right)}{E\left(0\right)}\right)R_1^{\prime }\left(t\right)\hfill \\ & & \le -{\beta }_{3}E\left(t\right)K^{\prime }\left(\frac{{\epsilon }_2}{t-t_0}\frac{E\left(t\right)}{E\left(0\right)}\right)+C_7(t-t_0)K^{\prime }\left(\frac{{\epsilon }_2}{t-t_0}\frac{E\left(t\right)}{E\left(0\right)}\right)K^{-1}\left(\phi \left(t\right)\right),\forall t\ge t_1.\hfill \end{array}$$

(91)

Using (20) and (21) with $s=K^{\prime }\left(\frac{{\epsilon }_2}{t-t_0}\frac{E\left(t\right)}{E\left(0\right)}\right)$ and $r=K^{-1}\left(\phi \left(t\right)\right)$ and applying (91), we obtain

$$R_2^{\prime }\left(t\right)\le -{\beta }_{3}E\left(t\right)K^{\prime }\left(\frac{{\epsilon }_2}{t-t_0}\frac{E\left(t\right)}{E\left(0\right)}\right)+{\epsilon }_2{C}_7\frac{E\left(t\right)}{E\left(0\right)}{K}^{\prime }\left(\frac{{\epsilon }_2}{t-t_0}\frac{E\left(t\right)}{E\left(0\right)}\right)+C_7(t-t_0)\phi \left(t\right).$$

(92)

From (59), (61), and (88), we obtain

$$(t-t_0)\mu \left(t\right)\phi \left(t\right)\le -C_8{E}^{\prime }\left(t\right),$$

(93)

where $C_8=min\{2\theta ,C_4\mu \left(0\right)\}.$ Multiplying (92) by the positive nonincreasing function $\mu \left(t\right)$ and using (93), we have

$$R_3^{\prime }\left(t\right)\le -\left({\beta }_{3}E\left(0\right)-{\epsilon }_2{C}_7\right)\frac{\mu \left(t\right)E\left(t\right)}{E\left(0\right)}{K}^{\prime }\left(\frac{{\epsilon }_2}{t-t_0}\frac{E\left(t\right)}{E\left(0\right)}\right),\forall t\ge t_1,$$

where $R_3\left(t\right)=\mu \left(t\right)R_2\left(t\right)+C_7{C}_{8}E\left(t\right)\sim E\left(t\right).$ For a suitable choice of ${\epsilon }_2,$ we find that

$$R_3^{\prime }\left(t\right)\le -d_4\frac{\mu \left(t\right)E\left(t\right)}{E\left(0\right)}{K}^{\prime }\left(\frac{{\epsilon }_2}{t-t_0}\frac{E\left(t\right)}{E\left(0\right)}\right),\forall t\ge t_1,$$

(94)

where $d_4$ is a positive constant. An integration of (94) yields

$$\frac{d_4}{E\left(0\right)}{\int }_{t_1}^{t}E\left(s\right)K^{\prime }\left(\frac{{\epsilon }_2}{s-t_0}\frac{E\left(s\right)}{E\left(0\right)}\right)\mu \left(s\right)ds\le {\int }_t^{t_1}{R}_3^{\prime }\left(s\right)ds\le R_3\left(t_1\right).$$

Using (90) and the non-increasing property of $E,$ we see that the map $t\to E\left(t\right)K^{\prime }\left(\frac{{\epsilon }_2}{t-t_0}\frac{E\left(t\right)}{E\left(0\right)}\right)$ is non-increasing and, consequently, we obtain

$$\begin{array}{c}\hfill d_4\frac{E\left(t\right)}{E\left(0\right)}{K}^{\prime }\left(\frac{{\epsilon }_2}{t-t_0}\frac{E\left(t\right)}{E\left(0\right)}\right){\int }_{t_1}^t\mu \left(s\right)ds\le R_3\left(t_1\right),\forall t\ge t_1.\end{array}$$

(95)

Multiplying (95) by $\frac{1}{t-t_0},$ we obtain

$$\begin{array}{c}\hfill d_4{K}_1\left(\frac{1}{t-t_0}\frac{E\left(t\right)}{E\left(0\right)}\right){\int }_{t_1}^t\mu \left(s\right)ds\le \frac{R_3\left(t_1\right)}{t-t_0},\forall t\ge t_1,\end{array}$$

where $K_1\left(s\right)=s{K}^{\prime }\left({\epsilon }_{2}s\right)$, which is strictly increasing. Therefore, we deduce that

$$E\left(t\right)\le {\alpha }_4(t-t_0)K_1^{-1}\left(\frac{{\alpha }_3}{(t-t_0){\int }_{t_1}^t\mu \left(s\right)ds}\right),\forall t\ge t_1,$$

where ${\alpha }_3$ and ${\alpha }_4$ are positive constants. This completes the proof. □

Examples. We provide examples to explain the decay of energy (see [1]).

• Case: $f_0$ and G are linear.

Let $\psi \left(t\right)=a{e}^{-b(1+t)},\mu \left(t\right)=b$, and $G\left(t\right)=t,$ where $b>0$, and $a>0$ is small enough. Assume that $f_0\left(t\right)=ct$ and $F\left(t\right)=\sqrt{t}{f}_0\left(\sqrt{t}\right)=ct.$ Then, we can obtain

$$E\left(t\right)\le k_2{e}^{-k_{1}t},\mathrm{for}\mathrm{all}t\ge t_0.$$

2.

Case: $f_0$ is linear and G is nonlinear.

Let $\psi \left(t\right)=a{e}^{-t^p},\mu \left(t\right)=1$, and $G\left(t\right)=\frac{p^t}{{(ln\left(\frac{a}{t}\right))}^{\frac{1}{p}-1}}$, where $0<p<1$, and $a>0$ is small enough. Assume that $f_0\left(t\right)=ct$ and $F\left(t\right)=\sqrt{t}{f}_0\left(\sqrt{t}\right)=ct.$ Then, G satisfies the condition (H1) on $(0,r_0]$ for any $0<r_0<a.$

$$\begin{array}{c}\hfill G_1\left(t\right)={\int }_t^{r_0}\frac{1}{s{G}^{\prime }\left(s\right)}ds={\int }_t^{r_0}\frac{{[ln\frac{a}{s}]}^{\frac{1}{p}}}{s[1-p+pln\frac{a}{s}]}ds={\int }_{ln\frac{a}{r_0}}^{ln\frac{a}{t}}\frac{u^{\frac{1}{p}}}{1-p+pu}du\le {\left(ln\frac{a}{t}\right)}^{\frac{1}{p}}.\end{array}$$

Then, we can have

$$E\left(t\right)\le k_4{e}^{-k_3{t}^p},\mathrm{for}\mathrm{all}t\ge t_0.$$

3.

Case: $f_0$ is nonlinear and G is linear.

Let $\psi \left(t\right)=a{e}^{-b(1+t)},\mu \left(t\right)=b$, and $G\left(t\right)=t,$ where $b>0$, and $a>0$ is small enough. Assume that $f_0\left(t\right)=c{t}^p,$ where $p>1$ and $F\left(t\right)=\sqrt{t}{f}_0\left(\sqrt{t}\right)=c{t}^{\frac{p+1}{2}}.$ Then,

$$\begin{array}{c}\hfill F_1\left(t\right)={\int }_t^{r_1}\frac{1}{s{F}^{\prime }\left(s\right)}ds={\int }_t^{r_1}\frac{2}{c(p+1)}{s}^{-\frac{p+1}{2}}ds=-{\alpha }_0\left(r_1^{-\frac{p-1}{2}}-t^{-\frac{p-1}{2}}\right)\end{array}$$

and

$$F_1^{-1}\left(t\right)={(r_1^{-\frac{p-1}{2}}+\frac{1}{{\alpha }_0}t)}^{-\frac{2}{p-1}},$$

where ${\alpha }_0=\frac{4}{c(p+1)(p-1)}.$ Therefore, we find that

$$E\left(t\right)\le {({\alpha }_{1}t+{\alpha }_2)}^{-\frac{2}{p-1}},\mathrm{for}\mathrm{all}t\ge t_0.$$

4.

Case: $f_0$ is nonlinear and G is nonlinear.

Let $\psi \left(t\right)=\frac{a}{{(1+t)}^2},\mu \left(t\right)=b$, and $G\left(t\right)=t^{\frac{3}{2}},$ where $b>0$, and $a>0$ is taken so that (9) remains valid. Assume that $f_0\left(t\right)=t^5$ and $F\left(t\right)=t^3.$ Then,

$$K\left(s\right)={(G^{-1}+F^{-1})}^{-1}\left(s\right)={\left(\frac{-1+\sqrt{1+4s}}{2}\right)}^3.$$

Therefore, we see that

$$E\left(t\right)\le \frac{{\alpha }_3}{{(t-t_0)}^{\frac{1}{3}}},\mathrm{for}\mathrm{all}t\ge t_1,$$

where $t_1>t_0.$

5. Conclusions

Numerous phenomena are influenced by both the current state and the previous occurrences of the system. There has been a notable increase in the research on the equation with delay effects, which frequently arise in various physical, biological, chemical, medical, and economic problems. In this paper, we study the energy decay rates for the viscoelastic wave equation with nonlinear time-varying delay, nonlinear damping at the boundary, and acoustic boundary conditions. We consider the relaxation function $\psi$, namely ${\psi }^{\prime }\left(t\right)\le -\mu \left(t\right)G\left(\psi \left(t\right)\right)$, where G is an increasing and convex function near the origin, and $\mu$ is a positive nonincreasing function. We establish general decay rate results without the need for the condition $a_2>0$ and without imposing any limiting growth assumption on the damping term $f_1,$ using the multiplier method and some properties of the convex functions. Moreover, the energy decay rates depend on the functions $\mu$ and G, as well as the function F defined by $f_0$, which characterizes the growth behavior of $f_1$ at the origin.