General Decay for a Viscoelastic Equation with Acoustic Boundary Conditions and a Logarithmic Nonlinearity

Abstract

In this work, we investigate the stability of solutions in a situation where the logarithmic source term competes with the viscoelastic dissipation under acoustic boundary conditions. We assume minimal conditions on the relaxation function g, namely, $g^{\prime }\left(t\right)\le -\xi \left(t\right)H\left(g\left(t\right)\right)$, where H is a strictly increasing and strictly convex function near the origin, and $\xi \left(t\right)$ is a non-increasing function. Under these general assumptions, we establish a general decay estimate for the solution. This result extends and improves some previous results.

1. Introduction

In this paper, we consider the following viscoelastic wave equation with acoustic boundary conditions and a logarithmic nonlinearity:

$$\begin{array}{cccc}\hfill & u_{tt}-\Delta u+{\int }_0^{t}g(t-s)\Delta u\left(s\right)ds={k\left|u\right|}^{\gamma -2}uln\left|u\right|,\hfill & \hfill & \mathrm{in}\Omega \times (0,\infty ),\hfill \\ \hfill & u=0,\hfill & \hfill & \mathrm{on}{\Gamma }_0\times (0,\infty ),\hfill \\ \hfill & {\displaystyle \frac{\partial u}{\partial \nu }}-{\int }_0^{t}g(t-s){\displaystyle \frac{\partial u}{\partial \nu }}\left(s\right)ds=y_t,\hfill & \hfill & \mathrm{on}{\Gamma }_1\times (0,\infty ),\hfill \\ \hfill & u_t+p\left(x\right)y_t+q\left(x\right)y=0,\hfill & \hfill & \mathrm{on}{\Gamma }_1\times (0,\infty ),\hfill \\ \hfill & u(x,0)=u_0\left(x\right),u_t(x,0)=u_1\left(x\right),\hfill & \hfill & x\in \Omega ,\hfill \\ \hfill & y(x,0)=y_0,\hfill & \hfill & x\in {\Gamma }_1,\hfill \end{array}$$

(1)

where $\Omega \subset {\mathbb{R}}^n$ ($n\ge 1)$ is a bounded domain with boundary $\Gamma ={\Gamma }_0\cup {\Gamma }_1$ of class $C^2$. The sets ${\Gamma }_0$ and ${\Gamma }_1$ are closed, disjoint, and have positive measures. The vector $\nu$ denotes the unit outward normal vector to $\Gamma$. The integral term represents a finite memory effect responsible for viscoelastic damping, where g is a positive and non-increasing function known as the relaxation function. The right-hand side of (1) corresponds to a source term. We assume $\gamma >2$, which will be discussed in more detail later, and k is an arbitrary positive constant. Problem (1) is closely related to the wave equation with a memory kernel, a logarithmic nonlinearity, and acoustic boundary conditions. Our main interest is to investigate its energy decay behavior.

Over the past few decades, viscoelastic wave equations under various boundary conditions have been extensively studied. In particular, many researchers have made significant progress in developing results for equations with Dirichlet boundary conditions and source terms. Berrimi and Messaoudi [1] established the local existence of solutions to the viscoelastic wave equation with the following nonlinear source term

$$u_{tt}-\Delta u+{\int }_0^{t}g(t-s)\Delta u\left(s\right)ds={\left|u\right|}^{\gamma }u,$$

(2)

where $\gamma >0$, and proved that the solution is bounded and global for certain initial data and suitable conditions on

$$g^{\prime }\left(t\right)\le -\xi g^p\left(t\right),1\le p<{\displaystyle \frac{3}{2}}$$

(3)

and $\xi$ is a positive constant. Subsequently, Messaoudi [2] proved a general decay rate of solutions to Equation (2) under the assumption that the relaxation function g satisfies

$$g^{\prime }\left(t\right)\le -\xi \left(t\right)g\left(t\right),\forall t\ge 0$$

(4)

and

$$\left|{\displaystyle \frac{{\xi }^{\prime }\left(t\right)}{\xi \left(t\right)}}\right|\le k_0,\xi \left(t\right)>0,{\xi }^{\prime }\left(t\right)\le 0,\forall t>0,{\int }_0^{\infty }\xi \left(t\right)dt=+\infty .$$

(5)

Later, Baaziz et al. [3] extended the earlier result for Equation (2) with a nonlinear feedback localized on a part of the boundary, where the relaxation function g satisfies

$$g^{\prime }\left(t\right)\le -\xi \left(t\right)g^p\left(t\right),1\le p<{\displaystyle \frac{3}{2}}.$$

(6)

Moreover, Messaoudi and Tatar [4] considered the following quasilinear viscoelastic wave equation under condition (3):

$$|u_t{|}^{\rho }{u}_{tt}-\Delta u-\Delta u_{tt}+{\int }_0^{t}g(t-s)\Delta u\left(s\right)ds=a_0{\left|u\right|}^{\gamma }u,$$

(7)

where $\rho ,a_0,\gamma >0.$ When the relaxation function g satisfied (4), Liu [5] improved the results obtained by Messaoudi and Tatar [4], in which only the exponential and polynomial decay rates were considered. In addition, Mustafa [6] investigated the quasilinear viscoelastic wave Equation (7) for relaxation function g of a more general type than those in (4). He assumed the condition on g proposed in his earlier work [7]:

$$g^{\prime }\left(t\right)\le -\xi \left(t\right)H\left(g\left(t\right)\right),$$

(8)

where H is a strictly increasing and strictly convex $C^2$ function near the origin with $H\left(0\right)=H^{\prime }\left(0\right)=0$, and $\xi \left(t\right)$ is a positive and non-increasing function.

On the other hand, in recent decades, the study of viscoelastic wave equations with acoustic boundary conditions has advanced significantly. Beale and Rosencrans [8] pioneered the mathematical modeling of acoustic boundary conditions, making a significant contribution to the extension of existing research on boundary conditions for the wave equation. Acoustic boundary conditions refer not simply to prescribing fixed or free boundaries at the interface, but rather to conditions that reflect acoustic properties such as sound transmission, absorption, and reflection. For problem (1) with $k=0$, Park and Park [9] derived general energy decay rates under acoustic boundary conditions, assuming that g satisfies (4), $\xi \left(t\right)>0$, and, in addition, ${\int }_0^{\infty }g\left(s\right)ds<{\displaystyle \frac{1}{2}}$. Liu [10] subsequently generalized the results of [9] to an arbitrary rate of decay, not necessarily with an exponential or polynomial one, and without the assumption that ${\int }_0^{\infty }g\left(s\right)ds<{\displaystyle \frac{1}{2}}$. More recently, Yoon et al. [11] established more general decay results under the minimal conditions (8). For $k\ne 0$, Tahamtani et al. [12] considered the following problem with acoustic boundary conditions:

$$u_{tt}-\Delta u+\alpha \left(t\right){\int }_0^{t}g(t-s)\Delta u\left(s\right)ds+u_t+u=kuln\left|u\right|$$

(9)

and proved the global existence of solutions together with a general decay result, assuming that the relaxation function g satisfies condition (4).

Equations with a logarithmic nonlinear source term have been applied in theoretical mathematical research and physics such as quantum mechanics, optics, and nuclear physics, as well as various other fields. Consequently, logarithmic source terms have recently attracted considerable attention from many researchers. When $g=0$, Di et al. [13] studied a strongly damped wave equation with a logarithmic nonlinearity by employing the modified potential well method:

$$u_{tt}-\Delta u-\Delta u_t={\left|u\right|}^{\gamma -2}uln\left|u\right|,$$

(10)

where $\gamma >2$. Yüksekkaya and Pişkin [14] proved the global existence of solutions for the strongly damped wave Equation (10) with a strong delay term, using the well-depth method. Wu et al. [15] considered the global existence and blow-up of weak solutions for the damped p-Laplacian-type wave equation, which extends the problem studied in [16] ($p=2,\gamma =2$):

$$u_{tt}-{div\left(\right|\nabla u|}^{p-2}\nabla u)-\Delta u_t+u_t={\left|u\right|}^{\gamma -2}uln\left|u\right|,$$

(11)

where $\gamma \ge p>2$. Boumaza et al. [17] considered the p-Laplacian hyperbolic-type equation with weak and strong damping terms and a logarithmic nonlinearity:

$$u_{tt}-{div\left(\right|\nabla u|}^{p-2}\nabla u)-\omega \Delta u_t+u_t={\left|u\right|}^{p-2}uln\left|u\right|,$$

with $\omega \ge 0$, which corresponds to the case $\gamma =p$ in (11). They proved global existence, infinite-time blow-up, and asymptotic behavior of solutions when $E\left(0\right)<d$ and $E\left(0\right)=d$. Furthermore, they studied the infinite-time blow-up of solutions in the case $E\left(0\right)>0$ with $\omega =0$.

For $\gamma =2$, Al-Gharabli et al. [18] considered the viscoelastic plate equation

$$u_{tt}+{\Delta }^{2}u+u-{\int }_0^{t}g(t-s){\Delta }^{2}u\left(s\right)ds=kuln\left|u\right|,$$

(12)

with a relaxation function g satisfying (6). In addition, Al-Gharabli et al. [19] investigated a viscoelastic plate equation with a velocity-dependent material density and a logarithmic nonlinearity:

$$|u_t{|}^{\rho }{u}_{tt}+{\Delta }^{2}u+{\Delta }^2{u}_{tt}-{\int }_0^{t}g(t-s){\Delta }^{2}u\left(s\right)ds=kuln\left|u\right|,$$

where the relaxation function g satisfies condition (6). Al-Gharabli [20] also studied the same problem (12) with a relaxation function g satisfying (8). He observed that condition (6) with $1\le p<2$ is only a special case of (8) and established more general and explicit decay results. Under condition (8), the stability of solutions for the Balakrishnan–Taylor viscoelastic equation with nonlinear frictional damping and a logarithmic source term was studied in [21,22] and the references therein.

For $\gamma >2$, Ha and Park [23] investigated the viscoelastic wave equation

$$u_{tt}-\Delta u+{\int }_0^{t}g(t-s)\Delta u\left(s\right)ds={\left|u\right|}^{\gamma -2}uln\left|u\right|,$$

under condition (8) and established a general decay estimate for its solutions. Recently, Guo and Zhang [24] studied the general decay of a variable coefficient viscoelastic wave equation with a logarithmic source term and acoustic boundary conditions, for which the kernel g satisfies

$$g^{\prime }\left(t\right)+\xi \left(t\right)g\left(t\right)\ge 0,$$

and

$$e^{\alpha t}[g^{\prime }\left(t\right)+\xi \left(t\right)g\left(t\right)]\in L^1\left({\mathbb{R}}^{+}\right),\mathrm{for}\alpha >0.$$

Motivated by these previous studies, for a viscoelastic wave equation with acoustic boundary conditions and logarithmic nonlinearities under the minimal condition (8), we derive general decay rates of the solutions of (1).

This paper is organized as follows: Section 2 introduces some assumptions and preliminary material essential to our main result. In Section 3, we establish the global existence of solutions. Section 4 presents several technical lemmas required for the proof of the main theorem. Finally, in Section 5, we prove the general decay of solutions, and in Section 6, we present the conclusions of the study.

2. Preliminaries

In this section, we present some material to be used throughout this paper. We use standard functional spaces: ${\parallel ·\parallel }_2$ and ${\parallel ·\parallel }_{{\Gamma }_1}$ denote the $L^2(\Omega )$-norm and $L^2\left({\Gamma }_1\right)$-norm, respectively: $(u,v)={\int }_{\Omega }uvdx$, ${(u,v)}_{{\Gamma }_1}={\int }_{{\Gamma }_1}uvd\Gamma$. We consider the closed subspace of the Sobolev space $H^1(\Omega )$ to be defined by

$$V=\{u\in H^1(\Omega ),u=0\mathrm{on}{\Gamma }_0\},$$

endowed with the norm ${\parallel \nabla u\parallel }_2$. Let $B_r$ be the optimal constant, and let $\tilde{\lambda }$ be the smallest positive constant such that

$${\parallel u\parallel }_r\le B_r{\parallel \nabla u\parallel }_2{\mathrm{and}\parallel u\parallel }_{{\Gamma }_1}^2\le \tilde{\lambda }{\parallel \nabla u\parallel }_2^2,\forall u\in V,$$

(13)

where $2\le r<\infty$ if $n=1,2$, and $2\le r\le {\displaystyle \frac{2n}{n-2}}$ if $n\ge 3$.

We now state the assumptions for problem (1).

(A1)

The exponent $\gamma$ satisfies

$$2<\gamma <\infty \mathrm{if}n=1,2;2<\gamma <{\displaystyle \frac{2(n-1)}{n-2}}\mathrm{if}n\ge 3.$$

(A2)

The function $g:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ is a non-increasing differentiable function satisfying

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

(A3)

There exists a positive function $H\in C^1\left({\mathbb{R}}^{+}\right),$ which is either linear or a strictly increasing and strictly convex $C^2$ function on $(0,r_1]$, where $r_1\le g\left(0\right)$, and it satisfies $H\left(0\right)=H^{\prime }\left(0\right)=0$, such that

$$g^{\prime }\left(t\right)\le -\xi \left(t\right)H\left(g\left(t\right)\right),\forall t\ge 0,$$

where $\xi \left(t\right)$ is a non-increasing differentiable function.

(A4)

Assume that p and q are essentially bounded functions such that $p\left(x\right),q\left(x\right)>0$ for all $x\in {\Gamma }_1$. That is, there exist positive constants $p_i$ and $q_i(i=0,1)$ such that

$$p_0\le p\left(x\right)\le p_1,q_0\le q\left(x\right)\le q_1,\forall x\in {\Gamma }_1.$$

The widely recognized Jensen’s inequality will be crucial in deriving our main result.

Remark 1.

If F is convex function on $[b_0,b_1],f:\Omega \to [b_0,b_1]$ and h are integrable functions on $\Omega . h\left(x\right)\ge 0$, and ${\int }_{\Omega }h\left(x\right)dx=\kappa >0$, then Jensen’s inequality states that

$$F\left({\displaystyle \frac{1}{\kappa }}{\int }_{\Omega }f\left(x\right)h\left(x\right)dx\right)\le {\displaystyle \frac{1}{\kappa }}{\int }_{\Omega }F\left(f\left(x\right)\right)h\left(x\right)dx.$$

Remark 2

([7]). If H is a strictly increasing and strictly convex $C^2$ function on $(0,r_1]$, with $H\left(0\right)=H^{\prime }\left(0\right)=0$, then it has an extension $\overline{H}$ that is strictly increasing and a strictly convex $C^2$ function on $(0,\infty )$. For instance, if $H\left(r_1\right)=a$, $H^{\prime }\left(r_1\right)=b$, and $H^{″}\left(r_1\right)=c$, we can define $\overline{H}$, for $t>r_1$, by

$$\overline{H}\left(t\right)={\displaystyle \frac{c}{2}}{t}^2+(b-c{r}_1)t+(a+{\displaystyle \frac{c}{2}}{r}_1^2-b{r}_1).$$

Lemma 1

([25]). For each $\mu >0$, the following inequalities hold:

$$|ln\tau |\le {\displaystyle \frac{{\tau }^{-\mu }}{e\mu }}\mathrm{for}0<\tau <1,0\le ln\tau \le {\displaystyle \frac{{\tau }^{\mu }}{e\mu }}\mathrm{for}\tau \ge 1.$$

(14)

The energy functional associated with (1) is defined as

$$\begin{array}{cc}\hfill E\left(t\right):=& {\displaystyle \frac{1}{2}}\parallel u_t{\parallel }_2^2+{\displaystyle \frac{1}{2}}\left(1-{\int }_0^{t}g\left(s\right)ds\right){\parallel \nabla u\parallel }_2^2+{\displaystyle \frac{1}{2}}(g\circ \nabla u)\left(t\right)+{\displaystyle \frac{1}{2}}{\int }_{{\Gamma }_1}q\left(x\right)y^{2}d\Gamma \hfill \\ \hfill & -{\displaystyle \frac{k}{\gamma }}{\int }_{\Omega }{\left|u\right|}^{\gamma }ln\left|u\right|dx+{\displaystyle \frac{k}{{\gamma }^2}}{\parallel u\parallel }_{\gamma }^{\gamma },\hfill \end{array}$$

(15)

where

$$(g\circ v)\left(t\right)={\int }_{\Omega }{\int }_0^{t}g(t-s){|v\left(t\right)-v\left(s\right)|}^{2}dsdx.$$

Then, we can easily find

$$E^{\prime }\left(t\right)={\displaystyle \frac{1}{2}}(g^{\prime }\circ \nabla u)\left(t\right)-{\displaystyle \frac{1}{2}}g\left(t\right){\parallel \nabla u\parallel }_2^2-{\int }_{{\Gamma }_1}p\left(x\right)y_t^{2}d\Gamma \le 0.$$

(16)

For completeness, the local existence result for problem (1) can be established by combining the arguments in [12,18,19,23,25,26]. As the proof is essentially similar to that of Theorem 3.1 in [23], we omit it here.

Theorem 1.

Suppose that assumptions $\left(A1\right),\left(A2\right)$ and $\left(A4\right)$ hold. For initial data $(u_0,u_1)\in (V\cap H^2(\Omega ))\times V$ and $y_0\in L^2\left({\Gamma }_1\right)$, there exists a unique pair of functions $(u,y)$ that solves problem (1) in the class

$$\begin{array}{cc}\hfill & u\in L^{\infty }((0,T);V\cap H^2(\Omega )),u_t\in L^{\infty }((0,T);V),\hfill \\ \hfill & u_{tt}\in L^{\infty }((0,T);L^2(\Omega )),y,y_t\in L^2((0,T);L^2\left({\Gamma }_1\right)).\hfill \end{array}$$

3. Global Existence

In order to deal with the logarithmic source term, we define

$$\begin{array}{cc}\hfill J\left(u\right(t\left)\right):=J\left(t\right)=& {\displaystyle \frac{1}{2}}\left(1-{\int }_0^{t}g\left(s\right)ds\right){\parallel \nabla u\parallel }_2^2-{\displaystyle \frac{1}{\gamma }}{\int }_{\Omega }{\left|u\right|}^{\gamma }{ln\left|u\right|}^{k}dx+{\displaystyle \frac{k}{{\gamma }^2}}{\parallel u\parallel }_{\gamma }^{\gamma }\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}{\int }_{{\Gamma }_1}q\left(x\right)y^{2}d\Gamma +{\displaystyle \frac{1}{2}}(g\circ \nabla u)\left(t\right),\hfill \end{array}$$

$$\begin{array}{c}\hfill I\left(u\left(t\right)\right):=I\left(t\right)=\left(1-{\int }_0^{t}g\left(s\right)ds\right){\parallel \nabla u\parallel }_2^2-{\int }_{\Omega }{\left|u\right|}^{\gamma }ln{\left|u\right|}^{k}dx.\end{array}$$

Therefore, it is straightforward to see that

$$E\left(t\right)={\displaystyle \frac{1}{2}}{\parallel u_t\parallel }_2^2+J\left(t\right),$$

$$J\left(t\right)={\displaystyle \frac{1}{\gamma }}I\left(t\right)+{\displaystyle \frac{\gamma -2}{2\gamma }}\left(1-{\int }_0^{t}g\left(s\right)ds\right){\parallel \nabla u\parallel }_2^2+{\displaystyle \frac{k}{{\gamma }^2}}{\parallel u\parallel }_{\gamma }^{\gamma }+{\displaystyle \frac{1}{2}}{\int }_{{\Gamma }_1}q\left(x\right)y^{2}d\Gamma +{\displaystyle \frac{1}{2}}(g\circ \nabla u)\left(t\right).$$

Lemma 2.

Assuming that the initial data $(u_0,u_1)\in H_0^1(\Omega )\times L^2(\Omega )$ satisfy, for some $\mu >0$,

$$I\left(0\right)>0,{\displaystyle \frac{2k{B}_{\gamma +\mu }^{\gamma +\mu }}{le\mu }}{\left({\displaystyle \frac{2\gamma E\left(0\right)}{l(\gamma -2)}}\right)}^{{\displaystyle \frac{\gamma +\mu -2}{2}}}<1,$$

(17)

then $I\left(t\right)>0$ for all $t\in [0,T)$.

Proof. 

Since $I\left(0\right)>0$, we infer from continuity that there exists $T^{*}\le T$ such that $I\left(t\right)\ge 0$ for all $t\in [0,T^{*})$. This implies that

$$J\left(t\right)\ge {\displaystyle \frac{\gamma -2}{2\gamma }}\left(1-{\int }_0^{t}g\left(s\right)ds\right){\parallel \nabla u\parallel }_2^2\ge {\displaystyle \frac{l(\gamma -2)}{2\gamma }}{\parallel \nabla u\parallel }_2^2.$$

Hence,

$${\parallel \nabla u\parallel }_2^2\le {\displaystyle \frac{2\gamma }{l(\gamma -2)}}J\left(t\right)\le {\displaystyle \frac{2\gamma }{l(\gamma -2)}}E\left(t\right)\le {\displaystyle \frac{2\gamma }{l(\gamma -2)}}E\left(0\right).$$

(18)

Moreover, owing to assumption $\left(A1\right)$, we can choose $\mu >0$ such that

$$2<\gamma +\mu <\infty \mathrm{if}n=1,2;2<\gamma +\mu <{\displaystyle \frac{2n}{n-2}}\mathrm{if}n\ge 3,$$

where ${\displaystyle \frac{2n}{n-2}}$ is the critical Sobolev exponent for the embedding $H_0^1(\Omega )\hookrightarrow L^p(\Omega )$ in dimension n. Since $\gamma +\mu <{\displaystyle \frac{2n}{n-2}}$ for $n\ge 3$, the Sobolev embedding

$$H_0^1(\Omega )\hookrightarrow L^{\gamma +\mu }(\Omega )$$

is continuous, and hence, the constant $B_{\gamma +\mu }$ exists for such $\mu$. From (14) and (18), we obtain

$$\begin{array}{cc}\hfill {\int }_{\Omega }{\left|u\right|}^{\gamma }ln{\left|u\right|}^{k}dx& \le {\displaystyle \frac{k}{e\mu }}{\int }_{\left|u\right|\ge 1}{\left|u\right|}^{\gamma +\mu }dx\hfill \\ \hfill & \le {\displaystyle \frac{k{B}_{\gamma +\mu }^{\gamma +\mu }}{e\mu }}{(\parallel \nabla u\parallel }_2^2{)}^{{\displaystyle \frac{\gamma +\mu -2}{2}}}{\parallel \nabla u\parallel }_2^2\hfill \\ \hfill & \le {\displaystyle \frac{k{B}_{\gamma +\mu }^{\gamma +\mu }}{e\mu }}{\left({\displaystyle \frac{2\gamma E\left(0\right)}{l(\gamma -2)}}\right)}^{{\displaystyle \frac{\gamma +\mu -2}{2}}}{\parallel \nabla u\parallel }_2^2.\hfill \end{array}$$

(19)

Using (17), we deduce that

$$I\left(t\right)\ge \left(l-{\displaystyle \frac{k{B}_{\gamma +\mu }^{\gamma +\mu }}{e\mu }}{\left({\displaystyle \frac{2\gamma E\left(0\right)}{l(\gamma -2)}}\right)}^{{\displaystyle \frac{\gamma +\mu -2}{2}}}\right){\parallel \nabla u\parallel }_2^2>0,\forall t\in [0,T^{*}).$$

Thus, $T^{*}$ can be extended to $T^{*}=T,$ and the proof of Lemma 2 is complete. □

4. Technical Lemmas

In this section, we establish several lemmas needed to prove our main result.

Lemma 3.

Under assumptions $\left(A1\right)$ and (13), the functional $\Psi \left(t\right)$ defined by

$$\Psi \left(t\right):={\int }_{\Omega }u{u}_{t}dx+{\int }_{{\Gamma }_1}uyd\Gamma +{\displaystyle \frac{1}{2}}{\int }_{{\Gamma }_1}p\left(x\right)y^{2}d\Gamma ,$$

(20)

satisfies the following estimate along the solution of (1), for all $t\ge 0$,

$${\Psi }^{\prime }\left(t\right)\le \parallel u_t{\parallel }_2^2-{\displaystyle \frac{l}{2}}{\parallel \nabla u\parallel }_2^2+{\displaystyle \frac{(2-l)}{2l}}(g\circ \nabla u)\left(t\right)+{\displaystyle \frac{4\tilde{\lambda }}{l}}\parallel y_t{\parallel }_{{\Gamma }_1}^2-{\int }_{{\Gamma }_1}q\left(x\right)y^{2}d\Gamma +{\int }_{\Omega }{\left|u\right|}^{\gamma }ln{\left|u\right|}^{k}dx.$$

(21)

Proof. 

Differentiating (20) with respect to t and applying (1) yields

$$\begin{array}{cc}\hfill {\Psi }^{\prime }\left(t\right)=& \parallel u_t{\parallel }_2^2-{\parallel \nabla u\parallel }_2^2+{\int }_{\Omega }\nabla u\left(t\right){\int }_0^{t}g(t-s)\nabla u\left(s\right)dsdx+{\int }_{\Omega }{\left|u\right|}^{\gamma }ln{\left|u\right|}^{k}dx\hfill \\ \hfill & -{\int }_{{\Gamma }_1}q\left(x\right)y^{2}d\Gamma +2{\int }_{{\Gamma }_1}u{y}_{t}d\Gamma .\hfill \end{array}$$

By using $\left(A2\right)$ and (13), together with Hölder’s and Young’s inequalities for ${\delta }_1$, $\eta >0$, we obtain

$$2{\int }_{{\Gamma }_1}u{y}_{t}d\Gamma \le {\delta }_1\tilde{\lambda }{\parallel \nabla u\parallel }_2^2+{\displaystyle \frac{1}{{\delta }_1}}{\parallel y_t\parallel }_{{\Gamma }_1}^2$$

and

$$\begin{array}{cc}\hfill {\int }_{\Omega }\nabla u\left(t\right)& {\int }_0^{t}g(t-s)\nabla u\left(s\right)dsdx\hfill \\ \hfill & \le {\displaystyle \frac{1}{2}}\left({\int }_0^{t}g\left(s\right)ds\right){\parallel \nabla u\parallel }_2^2+{\displaystyle \frac{1}{2}}{\int }_{\Omega }{\int }_0^{t}g(t-s){\left|\nabla u\left(s\right)\right|}^{2}dsdx\hfill \\ \hfill & \le {\displaystyle \frac{1-l}{2}}{\parallel \nabla u\parallel }_2^2+{\displaystyle \frac{1}{2}}(1+\eta ){\int }_{\Omega }{\int }_0^{t}g(t-s){\left|\nabla u\left(t\right)\right|}^{2}dsdx\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}\left(1+{\displaystyle \frac{1}{\eta }}\right){\int }_{\Omega }{\int }_0^{t}g(t-s){|\nabla u\left(t\right)-\nabla u\left(s\right)|}^{2}dsdx\hfill \\ \hfill & \le {\displaystyle \frac{1-l}{2}}(2+\eta ){\parallel \nabla u\parallel }_2^2+{\displaystyle \frac{1}{2}}\left(1+{\displaystyle \frac{1}{\eta }}\right)(g\circ \nabla u)\left(t\right).\hfill \end{array}$$

Inserting the above inequalities into the expression for ${\Psi }^{\prime }\left(t\right),$ we get

$$\begin{array}{cc}\hfill {\Psi }^{\prime }\left(t\right)\le & \parallel u_t{\parallel }_2^2-\left(1-{\delta }_1\tilde{\lambda }-{\displaystyle \frac{1-l}{2}}(2+\eta )\right){\parallel \nabla u\parallel }_2^2+{\displaystyle \frac{1}{{\delta }_1}}{\parallel y_t\parallel }_{{\Gamma }_1}^2-{\int }_{{\Gamma }_1}q\left(x\right)y^{2}d\Gamma \hfill \\ \hfill & +{\int }_{\Omega }{\left|u\right|}^{\gamma }ln{\left|u\right|}^{k}dx+{\displaystyle \frac{1}{2}}\left(1+{\displaystyle \frac{1}{\eta }}\right)(g\circ \nabla u)\left(t\right).\hfill \end{array}$$

Finally, choosing ${\delta }_1={\displaystyle \frac{l}{4\tilde{\lambda }}}$ and $\eta ={\displaystyle \frac{l}{2(1-l)}}$ completes the proof of Lemma 3. □

Lemma 4.

Under the assumption of Lemma 2, the functional $\chi \left(t\right)$ defined by

$$\chi \left(t\right):=-{\int }_{\Omega }{u}_t{\int }_0^{t}g(t-s)((u\left(t\right)-u\left(s\right))dsdx,$$

(22)

satisfies the following estimate, along the solution of (1), for all $t\ge 0$,

$$\begin{array}{cc}\hfill {\chi }^{\prime }\left(t\right)\le & -\left({\int }_0^{t}g\left(s\right)ds-\delta \right)\parallel u_t{\parallel }_2^2+\delta \parallel y_t{\parallel }_{{\Gamma }_1}^2+\delta \left(k{C}_{E\left(0\right)}+1+2{(1-l)}^2\right){\parallel \nabla u\parallel }_2^2\hfill \\ \hfill & -{\displaystyle \frac{g\left(0\right)}{4\delta }}{B}_2^2(g^{\prime }\circ \nabla u)\left(t\right)+\left(2\delta +{\displaystyle \frac{1}{2\delta }}+{\displaystyle \frac{\tilde{\lambda }}{4\delta }}+{\displaystyle \frac{k{B}_2^2}{4\delta }}\right)\left({\int }_0^{t}g\left(s\right)ds\right)(g\circ \nabla u)\left(t\right),\hfill \end{array}$$

(23)

where, for small constants ${\mu }_1>0$ and ${\mu }_2>0$ satisfying

$$2<2(\gamma -1-{\mu }_1)<{\displaystyle \frac{2n}{n-2}},2<2(\gamma -1+{\mu }_2)<{\displaystyle \frac{2n}{n-2}}(n\ge 3),$$

we define

$$C_{E\left(0\right)}={\left({\displaystyle \frac{B_{2(\gamma -1-{\mu }_1)}^{\gamma -1-{\mu }_1}}{e{\mu }_1}}\right)}^2{\left({\displaystyle \frac{2\gamma E\left(0\right)}{l(\gamma -2)}}\right)}^{\gamma -2-{\mu }_1}+{\left({\displaystyle \frac{B_{2(\gamma -1+{\mu }_2)}^{\gamma -1+{\mu }_2}}{e{\mu }_2}}\right)}^2{\left({\displaystyle \frac{2\gamma E\left(0\right)}{l(\gamma -2)}}\right)}^{\gamma -2+{\mu }_2}.$$

Proof. 

Differentiating (22) with respect to t and applying (1) yields

$$\begin{array}{cc}\hfill {\chi }^{\prime }\left(t\right)& =-\left({\int }_0^{t}g\left(s\right)ds\right){\parallel u_t\parallel }_2^2+{\int }_{\Omega }\nabla u{\int }_0^{t}g(t-s)(\nabla u\left(t\right)-\nabla u\left(s\right))dsdx\hfill \\ \hfill & -{\int }_{\Omega }\left({\int }_0^{t}g(t-s)\nabla u\left(s\right)ds\right)\left({\int }_0^{t}g(t-s)(\nabla u\left(t\right)-\nabla u\left(s\right))ds\right)dx\hfill \\ \hfill & -{\int }_{{\Gamma }_1}{y}_t{\int }_0^{t}g(t-s)(u\left(t\right)-u\left(s\right))dsd\Gamma -{\int }_{\Omega }{u}_t{\int }_0^t{g}^{\prime }(t-s)(u\left(t\right)-u\left(s\right))dsdx\hfill \\ \hfill & -k{\int }_{\Omega }{\left|u\right|}^{\gamma -2}uln\left|u\right|{\int }_0^{t}g(t-s)(u\left(t\right)-u\left(s\right))dsdx\hfill \\ \hfill & :=-\left({\int }_0^{t}g\left(s\right)ds\right){\parallel u_t\parallel }_2^2+\sum _{i=1}^5{I}_i.\hfill \end{array}$$

(24)

By applying $\left(A2\right)$, (13), and Young’s inequality, and by performing integration by parts for the integral on the right-hand side of (24), for $\delta >0$, we have the following:

$$\begin{array}{ccc}& & I_1\le \delta {\parallel \nabla u\parallel }_2^2+{\displaystyle \frac{1}{4\delta }}\left({\int }_0^{t}g\left(s\right)ds\right)(g\circ \nabla u)\left(t\right),\hfill \\ & & |I_2|\le \delta {\int }_{\Omega }{\left({\int }_0^{t}g(t-s)\nabla u\left(s\right)ds\right)}^{2}dx+{\displaystyle \frac{1}{4\delta }}{\int }_{\Omega }{\left({\int }_0^{t}g(t-s)(\nabla u\left(t\right)-\nabla u\left(s\right))ds\right)}^{2}dx\hfill \\ & & \le \left(2\delta +{\displaystyle \frac{1}{4\delta }}\right)\left({\int }_0^{t}g\left(s\right)ds\right)(g\circ \nabla u)\left(t\right)+2\delta {(1-l)}^2{\parallel \nabla u\parallel }_2^2,\hfill \\ & & |I_3|\le \delta \parallel y_t{\parallel }_{{\Gamma }_1}^2+{\displaystyle \frac{\tilde{\lambda }}{4\delta }}\left({\int }_0^{t}g\left(s\right)ds\right)(g\circ \nabla u)\left(t\right),\hfill \\ & & |I_4|\le \delta \parallel u_t{\parallel }_2^2-{\displaystyle \frac{g\left(0\right)}{4\delta }}{B}_2^2(g^{\prime }\circ \nabla u)\left(t\right),\hfill \\ & & |I_5|\le k\delta {\int }_{\Omega }{\left({\left|u\right|}^{\gamma -1}ln\left|u\right|\right)}^{2}dx+{\displaystyle \frac{k{B}_2^2}{4\delta }}\left({\int }_0^{t}g\left(s\right)ds\right)(g\circ \nabla u)\left(t\right).\hfill \end{array}$$

Let ${\Omega }_1=\{x\in \Omega :|u(x,t)|<1\}$ and ${\Omega }_2=\{x\in \Omega :|u(x,t)|\ge 1\}$. Since $2<2(\gamma -1)<{\displaystyle \frac{2n}{n-2}}(n\ge 3)$, we can choose small constants ${\mu }_1>0$ and ${\mu }_2>0$ such that

$$2<2(\gamma -1-{\mu }_1)<{\displaystyle \frac{2n}{n-2}}\mathrm{and}2<2(\gamma -1+{\mu }_2)<{\displaystyle \frac{2n}{n-2}}.$$

Then, by Lemma 1 and (18), we obtain

$$\begin{array}{cc}\hfill & {\int }_{\Omega }{\left({\left|u\right|}^{\gamma -1}ln\left|u\right|\right)}^{2}dx\hfill \\ \hfill & \le {\left({\displaystyle \frac{1}{e{\mu }_1}}\right)}^2{\int }_{{\Omega }_1}{\left|u\right|}^{2(\gamma -1-{\mu }_1)}dx+{\left({\displaystyle \frac{1}{e{\mu }_2}}\right)}^2{\int }_{{\Omega }_2}{\left|u\right|}^{2(\gamma -1+{\mu }_2)}dx\hfill \\ \hfill & \le {\left({\displaystyle \frac{B_{2(\gamma -1-{\mu }_1)}^{\gamma -1-{\mu }_1}}{e{\mu }_1}}\right)}^2{\left({\displaystyle \frac{2\gamma E\left(0\right)}{l(\gamma -2)}}\right)}^{\gamma -2-{\mu }_1}{\parallel \nabla u\parallel }_2^2+{\left({\displaystyle \frac{B_{2(\gamma -1+{\mu }_2)}^{\gamma -1+{\mu }_2}}{e{\mu }_2}}\right)}^2{\left({\displaystyle \frac{2\gamma E\left(0\right)}{l(\gamma -2)}}\right)}^{\gamma -2+{\mu }_2}{\parallel \nabla u\parallel }_2^2\hfill \\ \hfill & \le C_{E\left(0\right)}{\parallel \nabla u\parallel }_2^2.\hfill \end{array}$$

Combining these inequalities with (24) establishes Lemma 4. □

Now, we define the perturbed modified energy functional

$$L\left(t\right):=ME\left(t\right)+M_1\Psi \left(t\right)+M_2\chi \left(t\right),$$

where M, $M_1$, and $M_2$ are positive constants.

Lemma 5.

Under the assumption of Lemma 2, for a sufficiently large M, the functionals $E\left(t\right)$ and $L\left(t\right)$ are equivalent.

Proof. 

By applying $\left(A2\right),\left(A4\right)$, (13), (15), (20), (22), Lemma 2, and Young’s inequality, we obtain

$$\begin{array}{cc}\hfill \left|L\right(t)-ME(t\left)\right|& \le M_1|\Psi \left(t\right)|+M_2\left|\chi \left(t\right)\right|\hfill \\ \hfill & \le {\displaystyle \frac{M_1}{2}}\parallel u_t{\parallel }_2^2+{\displaystyle \frac{M_1{B}_2^2}{2}}{\parallel \nabla u\parallel }_2^2+{\displaystyle \frac{M_1\tilde{\lambda }}{2}}{\parallel \nabla u\parallel }_2^2+{\displaystyle \frac{M_1}{2}}{\parallel y\parallel }_{{\Gamma }_1}^2\hfill \\ \hfill & +{\displaystyle \frac{M_1{p}_1}{2}}{\parallel y\parallel }_{{\Gamma }_1}^2+{\displaystyle \frac{M_2}{2}}{\parallel u_t\parallel }_2^2+{\displaystyle \frac{M_2{B}_2^2}{2}}\left({\int }_0^{t}g\left(s\right)ds\right)(g\circ \nabla u)\left(t\right)\hfill \\ \hfill & \le {\displaystyle \frac{M_1+M_2}{2}}\parallel u_t{\parallel }_2^2+{\displaystyle \frac{M_1(B_2^2+\tilde{\lambda })}{2}}{\parallel \nabla u\parallel }_2^2+{\displaystyle \frac{M_2{B}_2^2}{2}}\left({\int }_0^{t}g\left(s\right)ds\right)(g\circ \nabla u)\left(t\right)\hfill \\ \hfill & +{\displaystyle \frac{M_1(p_1+1)}{2}}{\parallel y\parallel }_{{\Gamma }_1}^2\hfill \\ \hfill & \le C_{1}E\left(t\right),\hfill \end{array}$$

where $C_1$ is a positive constant depending on $M_1$, $M_2$, $p_1,B_2^2$, and $\tilde{\lambda }$. By choosing a sufficiently large M, we complete the proof of Lemma 5. □

Lemma 6.

Under the assumption of Lemma 2, there exist positive constants m and $C_2$ such that

$$L^{\prime }\left(t\right)\le -mE\left(t\right)+C_2(g\circ \nabla u)\left(t\right)\mathrm{for}t\ge t_0>0.$$

(25)

Proof. 

Differentiating L with respect to t and taking (16), (21), and (23) into account, we obtain

$$\begin{array}{cc}\hfill L^{\prime }\left(t\right)\le & -\left[M_2\left({\int }_0^{t}g\left(s\right)ds-\delta \right)-M_1\right]{\parallel u_t\parallel }_2^2+\left[{\displaystyle \frac{M}{2}}-{\displaystyle \frac{M_{2}g\left(0\right)B_2^2}{4\delta }}\right](g^{\prime }\circ \nabla u)\left(t\right)\hfill \\ \hfill & -\left[{\displaystyle \frac{M_{1}l}{2}}-M_2\delta \left(k{C}_{E\left(0\right)}+1+2{(1-l)}^2\right)\right]{\parallel \nabla u\parallel }_2^2\hfill \\ \hfill & +\left[{\displaystyle \frac{M_1(2-l)}{2l}}+M_2\left(2\delta +{\displaystyle \frac{1}{2\delta }}+{\displaystyle \frac{\tilde{\lambda }}{4\delta }}+{\displaystyle \frac{k{B}_2^2}{4\delta }}\right)\left({\int }_0^{t}g\left(s\right)ds\right)\right](g\circ \nabla u)\left(t\right)\hfill \\ \hfill & -M_1{\int }_{{\Gamma }_1}q\left(x\right)y^{2}d\Gamma -\left[M{p}_0-{\displaystyle \frac{4M_1\tilde{\lambda }}{l}}-\delta M_2\right]\parallel y_t{\parallel }_{{\Gamma }_1}^2+M_1{\int }_{\Omega }{\left|u\right|}^{\gamma }ln{\left|u\right|}^{k}dx.\hfill \end{array}$$

Using the definition of $E\left(t\right)$, we obtain, for any $m>0$,

$$\begin{array}{cc}\hfill L^{\prime }\left(t\right)\le & -mE\left(t\right)-\left[M_2\left({\int }_0^{t}g\left(s\right)ds-\delta \right)-M_1-{\displaystyle \frac{m}{2}}\right]{\parallel u_t\parallel }_2^2+\left[{\displaystyle \frac{M}{2}}-{\displaystyle \frac{M_{2}g\left(0\right)B_2^2}{4\delta }}\right](g^{\prime }\circ \nabla u)\left(t\right)\hfill \\ \hfill & -\left[{\displaystyle \frac{M_{1}l}{2}}-M_2\delta \left(k{C}_{E\left(0\right)}+1+2{(1-l)}^2\right)-{\displaystyle \frac{m}{2}}\left(1-{\int }_0^{t}g\left(s\right)ds\right)\right]{\parallel \nabla u\parallel }_2^2\hfill \\ \hfill & +\left[{\displaystyle \frac{M_1(2-l)}{2l}}+M_2\left(2\delta +{\displaystyle \frac{1}{2\delta }}+{\displaystyle \frac{\tilde{\lambda }}{4\delta }}+{\displaystyle \frac{k{B}_2^2}{4\delta }}\right)\left({\int }_0^{t}g\left(s\right)ds\right)+{\displaystyle \frac{m}{2}}\right](g\circ \nabla u)\left(t\right)\hfill \\ \hfill & -\left(M_1-{\displaystyle \frac{m}{2}}\right){\int }_{{\Gamma }_1}q\left(x\right)y^{2}d\Gamma -\left[M{p}_0-{\displaystyle \frac{4M_1\tilde{\lambda }}{l}}-\delta M_2\right]{\parallel y_t\parallel }_{{\Gamma }_1}^2\hfill \\ \hfill & +\left(M_1-{\displaystyle \frac{m}{\gamma }}\right){\int }_{\Omega }{\left|u\right|}^{\gamma }{ln\left|u\right|}^{k}dx+{\displaystyle \frac{km}{{\gamma }^2}}{\parallel u\parallel }_{\gamma }^{\gamma }.\hfill \end{array}$$

From (13) and (18), we find

$${\parallel u\parallel }_{\gamma }^{\gamma }\le B_{\gamma }^{\gamma }{\parallel \nabla u\parallel }_2^{\gamma }\le B_{\gamma }^{\gamma }{\left({\displaystyle \frac{2\gamma E\left(0\right)}{l(\gamma -2)}}\right)}^{{\displaystyle \frac{\gamma -2}{2}}}{\parallel \nabla u\parallel }_2^2.$$

(26)

Letting $g_0:={\int }_0^{t_0}g\left(s\right)ds,$ and selecting $M_1>{\displaystyle \frac{m}{2}}$, we derive the following result from (19) and (26):

$$\begin{array}{cc}\hfill L^{\prime }\left(t\right)\le & -mE\left(t\right)-\left[M_2\left(g_0-\delta \right)-M_1-{\displaystyle \frac{m}{2}}\right]{\parallel u_t\parallel }_2^2+\left[{\displaystyle \frac{M}{2}}-{\displaystyle \frac{M_{2}g\left(0\right)B_2^2}{4\delta }}\right](g^{\prime }\circ \nabla u)\left(t\right)\hfill \\ \hfill & -\left[M_1\left({\displaystyle \frac{l}{2}}-{\displaystyle \frac{k{B}_{\gamma +\mu }^{\gamma +\mu }}{e\mu }}{\left({\displaystyle \frac{2\gamma E\left(0\right)}{l(\gamma -2)}}\right)}^{{\displaystyle \frac{\gamma +\mu -2}{2}}}\right)-M_2\delta \left(k{C}_{E\left(0\right)}+1+2{(1-l)}^2\right)\right.\hfill \\ \hfill & \left.-{\displaystyle \frac{m}{2}}\left(1-{\int }_0^{t}g\left(s\right)ds\right)-{\displaystyle \frac{km{B}_{\gamma }^{\gamma }}{{\gamma }^2}}{\left({\displaystyle \frac{2\gamma E\left(0\right)}{l(\gamma -2)}}\right)}^{{\displaystyle \frac{\gamma -2}{2}}}\right]{\parallel \nabla u\parallel }_2^2\hfill \\ \hfill & +\left[{\displaystyle \frac{M_1(2-l)}{2l}}+M_2\left(2\delta +{\displaystyle \frac{1}{2\delta }}+{\displaystyle \frac{\tilde{\lambda }}{4\delta }}+{\displaystyle \frac{k{B}_2^2}{4\delta }}\right)\left({\int }_0^{t}g\left(s\right)ds\right)+{\displaystyle \frac{m}{2}}\right](g\circ \nabla u)\left(t\right)\hfill \\ \hfill & -\left(M_1-{\displaystyle \frac{m}{2}}\right){\int }_{{\Gamma }_1}q\left(x\right)y^{2}d\Gamma -\left[M{p}_0-{\displaystyle \frac{4M_1\tilde{\lambda }}{l}}-\delta M_2\right]{\parallel y_t\parallel }_{{\Gamma }_1}^2,\forall t\ge t_0.\hfill \end{array}$$

Taking $\delta ={\displaystyle \frac{l}{4M_2}}$ and

$$C_2:={\displaystyle \frac{M_1(2-l)}{2l}}+M_2\left(2\delta +{\displaystyle \frac{1}{2\delta }}+{\displaystyle \frac{\tilde{\lambda }}{4\delta }}+{\displaystyle \frac{k{B}_2^2}{4\delta }}\right)\left({\int }_0^{t}g\left(s\right)ds\right)+{\displaystyle \frac{m}{2}}>0,$$

we have

$$\begin{array}{cc}\hfill L^{\prime }\left(t\right)\le & -mE\left(t\right)-\left[M_2{g}_0-{\displaystyle \frac{l}{4}}-M_1-{\displaystyle \frac{m}{2}}\right]{\parallel u_t\parallel }_2^2+\left[{\displaystyle \frac{M}{2}}-{\displaystyle \frac{M_2^{2}g\left(0\right)B_2^2}{l}}\right](g^{\prime }\circ \nabla u)\left(t\right)\hfill \\ \hfill & -\left[M_1\left({\displaystyle \frac{l}{2}}-{\displaystyle \frac{k{B}_{\gamma +\mu }^{\gamma +\mu }}{e\mu }}{\left({\displaystyle \frac{2\gamma E\left(0\right)}{l(\gamma -2)}}\right)}^{{\displaystyle \frac{\gamma +\mu -2}{2}}}\right)-{\displaystyle \frac{l}{4}}\left(k{C}_{E\left(0\right)}+1+2{(1-l)}^2\right)\right.\hfill \\ \hfill & \left.-{\displaystyle \frac{m}{2}}\left(1-{\int }_0^{t}g\left(s\right)ds\right)-{\displaystyle \frac{km{B}_{\gamma }^{\gamma }}{{\gamma }^2}}{\left({\displaystyle \frac{2\gamma E\left(0\right)}{l(\gamma -2)}}\right)}^{{\displaystyle \frac{\gamma -2}{2}}}\right]{\parallel \nabla u\parallel }_2^2\hfill \\ \hfill & +C_2(g\circ \nabla u)\left(t\right)-\left(M_1-{\displaystyle \frac{m}{2}}\right){\int }_{{\Gamma }_1}q\left(x\right)y^{2}d\Gamma -\left[M{p}_0-{\displaystyle \frac{4M_1\tilde{\lambda }}{l}}-{\displaystyle \frac{l}{4}}\right]{\parallel y_t\parallel }_{{\Gamma }_1}^2,\forall t\ge t_0.\hfill \end{array}$$

At this point, we carefully choose the constants. First, we fix $M_1>{\displaystyle \frac{m}{2}}$ to be suitably large so that

$$M_1\left({\displaystyle \frac{l}{2}}-{\displaystyle \frac{k{B}_{\gamma +\mu }^{\gamma +\mu }}{e\mu }}{\left({\displaystyle \frac{2\gamma E\left(0\right)}{l(\gamma -2)}}\right)}^{{\displaystyle \frac{\gamma +\mu -2}{2}}}\right)-{\displaystyle \frac{l}{4}}\left(k{C}_{E\left(0\right)}+1+2{(1-l)}^2\right)>0,$$

and then choose an $M_2>{\displaystyle \frac{l}{4g_0}}$ value that is sufficiently large such that

$$M_2{g}_0-{\displaystyle \frac{l}{4}}-M_1>1.$$

Next, we pick a sufficiently large M so that Lemma 5 holds; specifically,

$$\begin{array}{c}\hfill {\displaystyle \frac{M}{2}}-{\displaystyle \frac{M_2^{2}g\left(0\right)B_2^2}{l}}>0,M{p}_0-{\displaystyle \frac{4M_1\tilde{\lambda }}{l}}-{\displaystyle \frac{l}{4}}>0.\end{array}$$

Considering the above inequalities, by selecting a sufficiently small $m>0$, we complete the proof of Lemma 6. □

5. General Decay

In this section, we prove the main result of this work. We define

$$K\left(t\right):=(-g^{\prime }\circ \nabla u)\left(t\right)\le -2E^{\prime }\left(t\right).$$

(27)

Lemma 7.

Under the assumption of Lemma 2 and $\left(A3\right)$, the following estimate holds:

$$(g\circ \nabla u)\left(t\right)\le {\displaystyle \frac{t}{q}}{\left(\overline{H}\right)}^{-1}\left({\displaystyle \frac{qK\left(t\right)}{t\xi \left(t\right)}}\right),$$

(28)

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

Proof. 

To prove (28), we introduce the function $\lambda \left(t\right)$ defined by

$$\lambda \left(t\right):={\displaystyle \frac{q}{t}}{\int }_0^t{\parallel \nabla u\left(t\right)-\nabla u(t-s)\parallel }_2^{2}ds.$$

By using (18), we easily obtain

$$\lambda \left(t\right)\le {\displaystyle \frac{8q\gamma E\left(0\right)}{l(\gamma -2)}},$$

and we choose the constant $q\in (0,1)$ to be sufficiently small so that

$$\lambda \left(t\right)<1,\forall t\ge 0.$$

(29)

Since H is strictly convex on $(0,r_1]$ with $H\left(0\right)=0$, it follows that

$$H\left(\theta x\right)\le \theta H\left(x\right),$$

(30)

for all $0\le \theta \le 1$ and $x\in (0,r_1]$. Using this fact along with hypothesis $\left(A3\right)$, (29), (30), and Jensen’s inequality, we get

$$\begin{array}{cc}\hfill K\left(t\right)& ={\displaystyle \frac{1}{q\lambda \left(t\right)}}{\int }_0^t\lambda \left(t\right)(-g^{\prime }\left(s\right))q{\parallel \nabla u\left(t\right)-\nabla u(t-s)\parallel }_2^{2}ds\hfill \\ \hfill & \ge {\displaystyle \frac{1}{q\lambda \left(t\right)}}{\int }_0^t\lambda \left(t\right)\xi \left(s\right)H\left(g\left(s\right)\right)q{\parallel \nabla u\left(t\right)-\nabla u(t-s)\parallel }_2^{2}ds\hfill \\ \hfill & \ge {\displaystyle \frac{\xi \left(t\right)}{q\lambda \left(t\right)}}{\int }_0^{t}H\left(\lambda \left(t\right)g\left(s\right)\right)q{\parallel \nabla u\left(t\right)-\nabla u(t-s)\parallel }_2^{2}ds\hfill \\ \hfill & \ge {\displaystyle \frac{t\xi \left(t\right)}{q}}\overline{H}\left({\displaystyle \frac{q}{t}}{\int }_0^{t}g\left(s\right){\parallel \nabla u\left(t\right)-\nabla u(t-s)\parallel }_2^{2}ds\right).\hfill \end{array}$$

This completes the proof of (28). □

Theorem 2.

Let $(u_0,u_1)\in (V\cap H^2(\Omega ))\times V$ and $y_0\in L^2\left({\Gamma }_1\right)$ be given. Under the assumption of Lemma 2 and assuming that $\left(A1\right)$ –$\left(A4\right)$ are satisfied, there exist strictly positive constants $c_1,c_2,c_3,$ and $c_4$ such that the solution of (1) satisfies

$$\begin{array}{c}\hfill E\left(t\right)\le c_1{e}^{-c_2{\int }_{t_0}^t\xi \left(s\right)ds},\forall t\ge t_0,\mathit{if}H\mathit{is}\mathit{linear},\end{array}$$

(31)

$$\begin{array}{c}\hfill E\left(t\right)\le c_{3}t{H}_2^{-1}\left({\displaystyle \frac{c_4}{t{\int }_{t_1}^t\xi \left(s\right)ds}}\right),\forall t\ge t_1,\mathit{if}H\mathit{is}\mathit{nonlinear},\end{array}$$

(32)

where $t_1=max\{1,t_0\}$ and $H_2\left(s\right)=s{H}^{\prime }\left(\epsilon s\right)$.

Proof. 

• Case 1: H is linear.

We multiply (25) by $\xi \left(t\right)$ and use $\left(A3\right)$ and (27) to get

$$\begin{array}{cc}\hfill \xi \left(t\right)L^{\prime }\left(t\right)& \le -m\xi \left(t\right)E\left(t\right)+C_2\xi \left(t\right)(g\circ \nabla u)\left(t\right)\le -m\xi \left(t\right)E\left(t\right)-c{E}^{\prime }\left(t\right),\forall t\ge t_0.\hfill \end{array}$$

Therefore, we have

$${(\xi L+cE)}^{\prime }\left(t\right)\le -m\xi \left(t\right)E\left(t\right),\forall t\ge t_0.$$

Hence, using the fact that $\xi L+cE\sim E$ and integrating over $(t_0,t)$, we get (31).

• Case 2: H is nonlinear.

From (28), inequality (25) becomes

$$L^{\prime }\left(t\right)\le -mE\left(t\right)+{\displaystyle \frac{C_{2}t}{q}}{\left(\overline{H}\right)}^{-1}\left({\displaystyle \frac{qK\left(t\right)}{t\xi \left(t\right)}}\right),\forall t\ge t_0.$$

(33)

Then, using the fact that $E^{\prime }\le 0,{\overline{H}}^{\prime }>0,$ and ${\overline{H}}^{″}>0$ on $(0,r_1],$ together with (33), we define the functional $F_1$, which is equivalent to E, for $\epsilon <r_1$, as follows:

$$F_1\left(t\right):={\overline{H}}^{\prime }\left({\displaystyle \frac{\epsilon E\left(t\right)}{E\left(0\right)t}}\right)L\left(t\right)+E\left(t\right),\forall t\ge 1.$$

Therefore, for all $t\ge t_1$, we obtain

$$F_1^{\prime }\left(t\right)\le -mE\left(t\right){\overline{H}}^{\prime }\left({\displaystyle \frac{\epsilon E\left(t\right)}{E\left(0\right)t}}\right)+{\displaystyle \frac{C_{2}t}{q}}{\overline{H}}^{\prime }\left({\displaystyle \frac{\epsilon E\left(t\right)}{E\left(0\right)t}}\right){\left(\overline{H}\right)}^{-1}\left({\displaystyle \frac{qK\left(t\right)}{t\xi \left(t\right)}}\right)+E^{\prime }\left(t\right),$$

(34)

where $t_1=max\{1,t_0\}.$ Let ${\overline{H}}^{*}$ denote the convex conjugate of $\overline{H}$ in the sense of Young [27]; then,

$${\overline{H}}^{*}\left(s\right)=s{\left({\overline{H}}^{\prime }\right)}^{-1}\left(s\right)-\overline{H}\left[{\left({\overline{H}}^{\prime }\right)}^{-1}\left(s\right)\right],\mathrm{if}s\in (0,{\overline{H}}^{\prime }\left(r_1\right)],$$

(35)

and ${\overline{H}}^{*}$ satisfies the following generalized Young’s inequality:

$$AB\le {\overline{H}}^{*}\left(A\right)+\overline{H}\left(B\right),\mathrm{if}A\in (0,{\overline{H}}^{\prime }\left(r_1\right)],B\in (0,r_1].$$

(36)

Let $A={\overline{H}}^{\prime }\left({\displaystyle \frac{\epsilon E\left(t\right)}{E\left(0\right)t}}\right)$ and $B={\overline{H}}^{-1}\left({\displaystyle \frac{qK\left(t\right)}{t\xi \left(t\right)}}\right).$ Then, using (16) and (34)–(36), we get

$$F_1^{\prime }\left(t\right)\le -mE\left(t\right){\overline{H}}^{\prime }\left({\displaystyle \frac{\epsilon E\left(t\right)}{E\left(0\right)t}}\right)+{\displaystyle \frac{\epsilon C_{2}E\left(t\right)}{qE\left(0\right)}}{\overline{H}}^{\prime }\left({\displaystyle \frac{\epsilon E\left(t\right)}{E\left(0\right)t}}\right)+{\displaystyle \frac{C_{2}K\left(t\right)}{\xi \left(t\right)}}+E^{\prime }\left(t\right),\forall t\ge t_1.$$

Then, multiplying by $\xi \left(t\right)$ and using the fact that ${\overline{H}}^{\prime }=H^{\prime }$ on $(0,r_1]$ together with (27), we arrive at

$$\xi \left(t\right)F_1^{\prime }\left(t\right)\le -m\xi \left(t\right)E\left(t\right)H^{\prime }\left({\displaystyle \frac{\epsilon E\left(t\right)}{E\left(0\right)t}}\right)+{\displaystyle \frac{\epsilon C_{2}E\left(t\right)}{E\left(0\right)q}}\xi \left(t\right)H^{\prime }\left({\displaystyle \frac{\epsilon E\left(t\right)}{E\left(0\right)t}}\right)-c{E}^{\prime }\left(t\right),\forall t\ge t_1,$$

where $c>0.$ Using the non-increasing property of $\xi \left(t\right)$, we obtain

$${(\xi F_1+cE)}^{\prime }\left(t\right)\le -m\xi \left(t\right)E\left(t\right)H^{\prime }\left({\displaystyle \frac{\epsilon E\left(t\right)}{E\left(0\right)t}}\right)+{\displaystyle \frac{\epsilon C_{2}E\left(t\right)}{qE\left(0\right)}}\xi \left(t\right)H^{\prime }\left({\displaystyle \frac{\epsilon E\left(t\right)}{E\left(0\right)t}}\right),\forall t\ge t_1.$$

Next, by setting $F_2:=\xi F_1+cE$, which is equivalent to E, we conclude that

$$F_2^{\prime }\left(t\right)\le -\xi \left(t\right)\left({\displaystyle \frac{E\left(t\right)}{E\left(0\right)}}\right)\left(mE\left(0\right)-{\displaystyle \frac{\epsilon C_2}{q}}\right)H^{\prime }\left({\displaystyle \frac{\epsilon E\left(t\right)}{E\left(0\right)t}}\right),\forall t\ge t_1.$$

Now, choosing a sufficiently small $\epsilon$ so that $mE\left(0\right)-{\displaystyle \frac{\epsilon C_2}{q}}>0$, the inequality becomes

$$F_2^{\prime }\left(t\right)\le -k_1\xi \left(t\right)\left({\displaystyle \frac{E\left(t\right)}{E\left(0\right)}}\right)H^{\prime }\left({\displaystyle \frac{\epsilon E\left(t\right)}{E\left(0\right)t}}\right),\forall t\ge t_1,$$

where $k_1=mE\left(0\right)-{\displaystyle \frac{\epsilon C_2}{q}}>0.$ Integrating over $(t_1,t),$ we obtain

$${\int }_{t_1}^t{k}_1\xi \left(s\right)\left({\displaystyle \frac{E\left(s\right)}{E\left(0\right)}}\right)H^{\prime }\left({\displaystyle \frac{\epsilon E\left(s\right)}{E\left(0\right)s}}\right)ds\le -{\int }_{t_1}^t{F}_2^{\prime }\left(s\right)ds\le F_2\left(t_1\right),\forall t\ge t_1.$$

Then, using the fact that $H^{\prime }>0,H^{″}>0,$ and $E^{\prime }\le 0,$ we deduce that the function $t\mapsto E\left(t\right)H^{\prime }\left({\displaystyle \frac{\epsilon E\left(t\right)}{E\left(0\right)t}}\right)$ is non-increasing. Hence, we derive

$$\begin{array}{c}\hfill k_1\left({\displaystyle \frac{E\left(t\right)}{E\left(0\right)}}\right)H^{\prime }\left({\displaystyle \frac{\epsilon E\left(t\right)}{E\left(0\right)t}}\right){\int }_{t_1}^t\xi \left(s\right)ds\le k_2,\forall t\ge t_1,\end{array}$$

for some constant $k_2>0$. For the final step, we define $H_2\left(s\right)=s{H}^{\prime }\left(\epsilon s\right),$ which is strictly increasing. Then, we get

$$k_1{H}_2\left({\displaystyle \frac{E\left(t\right)}{E\left(0\right)t}}\right){\int }_{t_1}^t\xi \left(s\right)ds\le {\displaystyle \frac{k_2}{t}},\forall t\ge t_1.$$

(37)

Finally, inequality (37) leads to the desired estimate (32). □

6. Conclusions

Acoustic boundary conditions are essential for modeling and predicting wave behavior, such as sound transmission, absorption, and reflection. In addition, equations with a logarithmic source term can be applied to various fields, such as entropy models, quantum mechanics, and optics, as they reflect the behavior of a physical system. Therefore, in this work, we deal with a viscoelastic wave equation involving acoustic boundary conditions and a logarithmic nonlinearity under the minimal conditions on the relaxation function g, namely, $g^{\prime }\left(t\right)\le -\xi \left(t\right)H\left(g\left(t\right)\right)$, where H is a strictly increasing and a strictly convex function near the origin, and $\xi \left(t\right)$ is a non-increasing function. With these general assumptions, we establish a general decay estimate of the solution.