Polynomial Decay Rate for Kirchhoff Type in Viscoelasticity with Logarithmic Nonlinearity and Not Necessarily Decreasing Kernel

Abstract

This paper describes a polynomial decay rate of the solution of the Kirchhoff type in viscoelasticity with logarithmic nonlinearity, where an asymptotically-stable result of the global solution is obtained taking into account that the kernel is not necessarily decreasing.

1. Introduction

Consider the quasilinear viscoelastic problem:

$$\begin{array}{c}{\left|u_t\right|}^{\rho }{u}_{tt}+M\left(t\right){\mathsf{\Delta }}^{2}u\left(t\right)+{\mathsf{\Delta }}^2{u}_{tt}\left(t\right)\hfill \\ {\displaystyle -{\int }_0^{t}h\left(t-s\right){\mathsf{\Delta }}^{2}u\left(s\right)ds=u{\left|u\right|}^{\gamma -2}ln{\left|u\right|}^k,\mathrm{in}\mathsf{\Omega }\times (0,\infty ),}\hfill \end{array}$$

(1)

$$u\left(x,t\right)={\displaystyle \frac{\partial u}{\partial v}}\left(x,t\right)=0\mathrm{in}\partial \mathsf{\Omega }\times (0,\infty ),$$

(2)

$$u\left(x,0\right)=u^0,u_t\left(x,0\right)=u^1\mathrm{for}x\in \mathsf{\Omega }$$

(3)

with:

$$M\left(t\right):=\left({\xi }_0+{\xi }_1{∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^{2\beta }\right),\beta \ge 1,$$

(4)

where $\mathsf{\Omega }\subset {\mathbb{R}}^N$ is a bounded domain with a sufficiently smooth boundary $\partial \mathsf{\Omega },\gamma \ge 2$, $k,$ ${\xi }_0$ and ${\xi }_1$ are positive constants, $\rho \ge 0$ for $N=1,2$ and $0\le \rho \le \frac{4}{N-2}$ for $N\ge 3$, and h is a relaxation function, which will be specified later. From the physical point of view, the problem (1)–(3) is related to the spillover problem with memory and the panel flutter equation. Viscoelasticity has an important role in the study of biological phenomena, and it can have a strong effect on some factors that have a correlation with biological phenomenain general. A viscoelastic material will return back to its original size after an influential force has been detached, even though it will remain for a specific time in its original shape; see [1,2,3,4,5,6,7,8,9,10,11].

Recently, viscoelasticity problems have been handled carefully in several papers, and other results relating the global existence and decay of the global solution have been found (see [1,2,3,4,5,6,7,10,11,12,13,14,15,16,17,18,19,20,21,22,23]), where the function of the relaxation was supposed to be either exponential decay or polynomial decay. In [7], the mathematicians searched for a viscoelastic equation for more general decaying kernels and obtained some general decay results, from which the common exponential or polynomial rates were more specific samples of them. Later, many papers, using almost identical techniques, achieved analogous general decay outcomes. See [11,24]. Our decay rate obtained in the third section is less general than that obtained in [7,8], where a common decay rate outcome was established in order to let the functions of the relaxation satisfy:

$$g^{\prime }(t)\le -H(g(t)),t\ge 0,H(0)=0$$

(5)

a positive function $H\in C^1({\mathbb{R}}^{+})$, where H is either linear or a strictly-increasing and strictly-convex $C^2$ function on $(0,r],1>r$. In [7], the exponential decay for the viscoelastic wave equation of the solution with an unnecessarily decreasing kernel was handled. More specifically, the mathematicians had an exponential decay outcome as time tended to infinity $h^{\prime }(t)+\gamma h(t)\ge 0$ for all $t\ge 0$ provided that $\left(h^{\prime }(t)+\gamma h(t)\right)e^{\alpha t}\in L^1\left(0,\infty \right)$ for some $\alpha >0$. Their proof depended on another technique, the so-called “Lyapunov functional”. In this present work, we follow up with the same steps of the previous result in [7,10] for a new class of Kirchhoff hyperbolic equations on bounded domain polynomial decay of the Kirchhoff type in viscoelasticity combined with the right-hand side defined as a logarithmic nonlinearity, and the kernel is not necessarily decreasing taking into account that the similar conditions in the last ones in ([7,10]) are considered. The logarithmic nonlinearity is of much use in physics, since it naturally appears in inflation cosmology and supersymmetric filed theories, quantum mechanics, and nuclear physics (see [5,14]). This kind of problem can be applied in many different areas of physics such as nuclear physics, optics, and geophysics (see [3,15]).

The outline of the paper is as follows. In the second section, we give some basic concepts related to our problem given by (1)–(3). In Section 3, we prove our principle result.

2. Preliminaries and Assumptions

We use the standard Lebesgue space $L^2\left(\mathsf{\Omega }\right)$ and Sobolev space $H_0^1\left(\mathsf{\Omega }\right)$. For a Hilbert space X, we denote ${\left(.,.\right)}_X$ and ${∥.∥}_X$, the inner product and norm of X, respectively. For simplicity, we denote ${\left(.,.\right)}_{L^2\left(\mathsf{\Omega }\right)}$ and ${∥.∥}_{L^2\left(\mathsf{\Omega }\right)}$ by $\left(.,.\right)$ and $∥.∥$, respectively.

As in previous work in ([7,10]), we impose the following conditions on the relaxation function h. Namely, we suppose that the kernel $h\left(t\right)$ is a $C^1({\mathbb{R}}_{+},{\mathbb{R}}_{+})$ function satisfying

$\left(\mathbf{A}\mathbf{1}\right)$${\displaystyle {\int }_0^{\infty }h\left(s\right)ds<{\xi }_0.}$

$\left(\mathbf{A}\mathbf{2}\right)$${\displaystyle {\int }_0^{\infty }{h}^{1-\tau }(s)ds<\infty ,}$ for $0<\tau <2-r<1$ where $1<r<{\displaystyle \frac{3}{2}}.$

$\left(\mathbf{A}\mathbf{3}\right)$ There exists a positive differentiable function $\xi \left(t\right)$ such that:

$$h^{\prime }\left(t\right)+\xi \left(t\right)h^r\left(t\right)\ge 0$$

and:

$$e^{\alpha t}\left[h^{\prime }\left(t\right)+\xi \left(t\right)h^r\left(t\right)\right]\in L^1\left({\mathbb{R}}_{+}\right),$$

where $\left(1\le r<{\displaystyle \frac{3}{2}}\right)$ and for $\alpha >0,$ and $\xi \left(t\right)$ satisfies some positive constant $L,$

$$\left|\frac{{\xi }^{\prime }\left(t\right)}{\xi \left(t\right)}\right|\le L,{\xi }^{\prime }\left(t\right)\le 0,{\int }_0^{\infty }\xi (s)ds=\infty ,t\ge 0.$$

(6)

Furthermore, where $1<r<{\displaystyle \frac{3}{2}}$ for any $t\ge 0,$ there exists a positive constant $C_r$ depending only on $r,$ such that:

$$\frac{t}{{\left({\displaystyle 1+{\int }_0^t\xi (s)ds}\right)}^{{\displaystyle \frac{1}{2(r-1)}}}}\le C_r,\forall t\ge 0.$$

(7)

Notation: We denote by $l,\overline{l},l_{\alpha },{\overline{l}}_{\alpha }$ and h the following expressions:

$$\left\{\begin{array}{c}\xi \left(t\right)l\left(t\right):=h^{\prime }\left(t\right)+\xi \left(t\right)h^r\left(t\right),\hfill \\ {\displaystyle \overline{l}:={\int }_0^{\infty }l(s)ds,l_{\alpha }:=e^{\alpha t}l\left(t\right),}\hfill \\ {\displaystyle {\overline{l}}_{\alpha }:={\int }_0^{\infty }{l}_{\alpha }(s)ds,\overline{h}:={\int }_0^{\infty }h\left(s\right)ds.}\hfill \end{array}\right.$$

(8)

and we recall the binary notation:

$$\left(h\square w\right)\left(t\right):={\int }_0^{t}h\left(t-s\right){∥w\left(x,s\right)-w\left(x,t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^{2}ds.$$

Lemma 1.

The classical energy associated with (1)–(3) is defined by:

$$\begin{array}{ccc}E\left(t\right)\hfill & :\hfill & =\frac{1}{\rho +2}{∥u_t\left(t\right)∥}_{\rho +2}^{\rho +2}+\frac{1}{2}\left({\xi }_0+\frac{{\xi }_1}{\left(\beta +1\right)}{∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^{2\beta }\right){∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2\hfill \\ & & +\frac{1}{2}{∥\mathsf{\Delta }{u}_t\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2-\frac{1}{\gamma }\left({\int }_{\mathsf{\Omega }}{\left|u\left(t\right)\right|}^{\gamma }ln{\left|u\right|}^{k}dx\right)+\frac{k}{{\gamma }^2}{∥u\left(t\right)∥}_{\gamma }^{\gamma }.\hfill \end{array}$$

(9)

and its derivative is:

$$\frac{d}{dt}\left\{E\left(t\right)\right\}={\int }_0^{t}h(t-s){\left(\mathsf{\Delta }u(s),\mathsf{\Delta }{u}_t\left(t\right)\right)}_{L^2\left(\mathsf{\Omega }\right)}ds.$$

(10)

Proof. 

Multiplying (1) by $u_t$ and by integration over $\left(\mathsf{\Omega }\right),$ we have:

$$\begin{array}{cc}& {\left({\left|u_t\right|}^{\rho }{u}_{tt}\left(t\right),u_t\left(t\right)\right)}_{L^2\left(\mathsf{\Omega }\right)}+{\left(M\left(t\right){\mathsf{\Delta }}^{2}u\left(t\right),u_t\left(t\right)\right)}_{L^2\left(\mathsf{\Omega }\right)}\hfill \\ & +{\left({\mathsf{\Delta }}^2{u}_{tt}\left(t\right),u_t\left(t\right)\right)}_{L^2\left(\mathsf{\Omega }\right)}-{\left({\int }_0^{t}h(t-s){\mathsf{\Delta }}^{2}u(s)ds,u_t\left(t\right)\right)}_{L^2\left(\mathsf{\Omega }\right)}\hfill \\ \hfill =& {\left(u{\left|u\right|}^{\gamma -2}ln{\left|u\right|}^k,u_t\left(t\right)\right)}_{L^2\left(\mathsf{\Omega }\right)}.\hfill \end{array}$$

(11)

By using:

$$u_{tt}\left(t\right)u_t\left(t\right)=\frac{1}{2}\frac{d}{dt}\left\{u_t^2\left(t\right)\right\},$$

we get by direct calculation:

$$\begin{array}{ccc}\hfill {\left({\left|u_t\right|}^{\rho }{u}_{tt}\left(t\right),u_t\left(t\right)\right)}_{L^2\left(\mathsf{\Omega }\right)}& =& {\int }_{\mathsf{\Omega }}{\left|u_t\right|}^{\rho }\frac{1}{2}\frac{d}{dt}\left\{u_t^2\left(t\right)\right\}dx\hfill \\ & =& \frac{1}{2}{\int }_{\mathsf{\Omega }}\frac{d}{dt}\left\{{\left|u_t\right|}^{\rho }{u}_t^2\left(t\right)\right\}dx-\frac{1}{2}{\int }_{\mathsf{\Omega }}\frac{d}{dt}\left\{{\left|u_t\right|}^{\rho }\right\}{u}_t^2\left(t\right)dx\hfill \\ & =& \frac{1}{2}\frac{d}{dt}\left\{{\int }_{\mathsf{\Omega }}{\left|u_t\right|}^{\rho +2}dx\right\}-\frac{\rho }{2}{\int }_{\mathsf{\Omega }}{\left|u_t\right|}^{\rho -2}{u}_{tt}\left(t\right)u_t\left(t\right)u_t^2\left(t\right)dx\hfill \\ & =& \frac{1}{2}\frac{d}{dt}\left\{{∥u_t\left(t\right)∥}_{\rho +2}^{\rho +2}\right\}-\frac{\rho }{2}{\left({\left|u_t\right|}^{\rho }{u}_{tt}\left(t\right),u_t\left(t\right)\right)}_{L^2\left(\mathsf{\Omega }\right)},\hfill \end{array}$$

then:

$${\left({\left|u_t\right|}^{\rho }{u}_{tt}\left(t\right),u_t\left(t\right)\right)}_{L^2\left(\mathsf{\Omega }\right)}=\frac{1}{\rho +2}\frac{d}{dt}\left\{{∥u_t\left(t\right)∥}_{\rho +2}^{\rho +2}\right\}.$$

(12)

By using integration by parts, we have:

$$\begin{array}{cc}& {\left(M\left(t\right){\mathsf{\Delta }}^{2}u\left(t\right),u_t\left(t\right)\right)}_{L^2\left(\mathsf{\Omega }\right)}\hfill \\ \hfill =& \left({\xi }_0+{\xi }_1{∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^{2\beta }\right){\int }_{\mathsf{\Omega }}{\mathsf{\Delta }}^{2}u\left(t\right)u_t\left(t\right)dx\hfill \\ \hfill =& \left({\xi }_0+{\xi }_1{∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^{2\beta }\right){\int }_{\mathsf{\Omega }}\mathsf{\Delta }u\left(t\right)\mathsf{\Delta }{u}_t\left(t\right)dx\hfill \\ \hfill =& \left({\xi }_0+{\xi }_1{∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^{2\beta }\right)\frac{1}{2}\frac{d}{dt}\left\{{\int }_{\mathsf{\Omega }}{\left|\mathsf{\Delta }u\left(t\right)\right|}^{2}dx\right\}\hfill \\ \hfill =& \frac{{\xi }_0}{2}\frac{d}{dt}\left\{{∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2\right\}+\frac{{\xi }_1}{2}{∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^{2\beta }\frac{d}{dt}\left\{{∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2\right\}\hfill \\ \hfill =& \frac{{\xi }_0}{2}\frac{d}{dt}\left\{{∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2\right\}+\frac{{\xi }_1}{2\left(\beta +1\right)}\frac{d}{dt}\left\{{∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^{2\left(\beta +1\right)}\right\}\hfill \\ \hfill =& \frac{d}{dt}\left\{\frac{1}{2}\left({\xi }_0+\frac{{\xi }_1}{\left(\beta +1\right)}{∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^{2\beta }\right){∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2\right\}.\hfill \end{array}$$

(13)

Integrating by parts, we have:

$$\begin{array}{ccc}\hfill {\left({\mathsf{\Delta }}^2{u}_{tt}\left(t\right),u_t\left(t\right)\right)}_{L^2\left(\mathsf{\Omega }\right)}& =\hfill & {\left(\mathsf{\Delta }{u}_{tt}\left(t\right),\mathsf{\Delta }{u}_t\left(t\right)\right)}_{L^2\left(\mathsf{\Omega }\right)}\hfill \\ & =\hfill & \frac{1}{2}\frac{d}{dt}\left\{{∥\mathsf{\Delta }{u}_t\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2\right\}.\hfill \end{array}$$

(14)

Also integrating by parts, we get:

$$\begin{array}{cc}& -{\left({\int }_0^{t}h(t-s){\mathsf{\Delta }}^{2}u(s)ds,u_t\left(t\right)\right)}_{L^2\left(\mathsf{\Omega }\right)}\hfill \\ =\hfill & -{\int }_0^{t}h(t-s){\left(\mathsf{\Delta }u(s),\mathsf{\Delta }{u}_t\left(t\right)\right)}_{L^2\left(\mathsf{\Omega }\right)}ds.\hfill \end{array}$$

(15)

By using:

$${\left|u\right|}^{\gamma -2}u\left(t\right)u_t\left(t\right)=\frac{1}{\gamma }\frac{d}{dt}\left\{{\left|u\left(t\right)\right|}^{\gamma }\right\},$$

we get:

$$\begin{array}{cc}& {\left(u{\left|u\right|}^{\gamma -2}ln{\left|u\right|}^k,u_t\left(t\right)\right)}_{L^2\left(\mathsf{\Omega }\right)}\hfill \\ =\hfill & {\int }_{\mathsf{\Omega }}\frac{1}{\gamma }\frac{d}{dt}\left\{{\left|u\left(t\right)\right|}^{\gamma }\right\}ln{\left|u\right|}^{k}dx\hfill \\ =\hfill & \frac{1}{\gamma }{\int }_{\mathsf{\Omega }}\frac{d}{dt}\left\{{\left|u\left(t\right)\right|}^{\gamma }ln{\left|u\right|}^k\right\}dx-\frac{1}{\gamma }{\int }_{\mathsf{\Omega }}{\left|u\left(t\right)\right|}^{\gamma }\frac{d}{dt}\left\{ln{\left|u\right|}^k\right\}dx\hfill \\ =\hfill & \frac{1}{\gamma }\frac{d}{dt}\left\{{\int }_{\mathsf{\Omega }}{\left|u\left(t\right)\right|}^{\gamma }ln{\left|u\right|}^{k}dx\right\}-\frac{k}{\gamma }{\int }_{\mathsf{\Omega }}{\left|u\left(t\right)\right|}^{\gamma -2}u\left(t\right)u_t\left(t\right)dx\hfill \\ =\hfill & \frac{1}{\gamma }\frac{d}{dt}\left\{{\int }_{\mathsf{\Omega }}{\left|u\left(t\right)\right|}^{\gamma }ln{\left|u\right|}^{k}dx\right\}-\frac{k}{{\gamma }^2}{\int }_{\mathsf{\Omega }}\frac{d}{dt}\left\{{\left|u\left(t\right)\right|}^{\gamma }\right\}dx\hfill \\ =\hfill & \frac{d}{dt}\left\{\frac{1}{\gamma }\left({\int }_{\mathsf{\Omega }}{\left|u\left(t\right)\right|}^{\gamma }ln{\left|u\right|}^{k}dx\right)-\frac{k}{{\gamma }^2}{∥u\left(t\right)∥}_{\gamma }^{\gamma }\right\}.\hfill \end{array}$$

(16)

By replacing (12)–(16) in (11), we get:

$$\begin{array}{cc}\hfill & \frac{1}{\rho +2}\frac{d}{dt}\left\{{∥u_t\left(t\right)∥}_{\rho +2}^{\rho +2}\right\}\hfill \\ \hfill & +\frac{d}{dt}\left\{\frac{1}{2}\left({\xi }_0+\frac{{\xi }_1}{\left(\beta +1\right)}{∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^{2\beta }\right){∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2\right\}\hfill \\ \hfill & +\frac{1}{2}\frac{d}{dt}\left\{{∥\mathsf{\Delta }{u}_t\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2\right\}\hfill \\ \hfill & -{\int }_0^{t}h(t-s){\left(\mathsf{\Delta }u(s),\mathsf{\Delta }{u}_t\left(t\right)\right)}_{L^2\left(\mathsf{\Omega }\right)}ds\hfill \\ =\hfill & \frac{d}{dt}\left\{\frac{1}{\gamma }\left({\int }_{\mathsf{\Omega }}{\left|u\left(t\right)\right|}^{\gamma }ln{\left|u\right|}^{k}dx\right)-\frac{k}{{\gamma }^2}{∥u\left(t\right)∥}_{\gamma }^{\gamma }\right\},\hfill \end{array}$$

(17)

then (17) is equivalent to:

$$\begin{array}{cc}\hfill & \frac{d}{dt}\left\{\frac{1}{\rho +2}{∥u_t\left(t\right)∥}_{\rho +2}^{\rho +2}+\frac{1}{2}\left({\xi }_0+\frac{{\xi }_1}{\left(\beta +1\right)}{∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^{2\beta }\right){∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2\right.\hfill \\ \hfill & \left.+\frac{1}{2}{∥\mathsf{\Delta }{u}_t\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2-\frac{1}{\gamma }\left({\int }_{\mathsf{\Omega }}{\left|u\left(t\right)\right|}^{\gamma }ln{\left|u\right|}^{k}dx\right)+\frac{k}{{\gamma }^2}{∥u\left(t\right)∥}_{\gamma }^{\gamma }\right\}\hfill \\ =\hfill & {\int }_0^{t}h(t-s){\left(\mathsf{\Delta }u(s),\mathsf{\Delta }{u}_t\left(t\right)\right)}_{L^2\left(\mathsf{\Omega }\right)}ds,\hfill \end{array}$$

(18)

and by using (9) in (18), we get (10).

The proof is complete. □

Lemma 2.

The modified energy for (1)–(3) is defined by:

$$\begin{array}{ccc}e\left(t\right)\hfill & :\hfill & =\frac{1}{\rho +2}{∥u_t\left(t\right)∥}_{\rho +2}^{\rho +2}+\frac{1}{2}\left({\xi }_0-{\int }_0^{t}h\left(s\right)ds\right){∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2\hfill \\ & & +\frac{{\xi }_1}{2\left(\beta +1\right)}{∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^{2\left(\beta +1\right)}+\frac{1}{2}{∥\mathsf{\Delta }{u}_t\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2\hfill \\ & & -\frac{1}{\gamma }\left({\int }_{\mathsf{\Omega }}{\left|u\left(t\right)\right|}^{\gamma }ln{\left|u\right|}^{k}dx\right)+\frac{k}{{\gamma }^2}{∥u\left(t\right)∥}_{\gamma }^{\gamma }+\frac{1}{2}\left(h\square \mathsf{\Delta }u\right)\left(t\right).\hfill \end{array}$$

(19)

and its derivative satisfies the following:

$$\begin{array}{ccc}\frac{d}{dt}\left\{e\left(t\right)\right\}\hfill & =\hfill & \frac{1}{2}\left(h^{\prime }\square \mathsf{\Delta }u\right)\left(t\right)-\frac{1}{2}h\left(t\right){∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2\hfill \\ & \le \hfill & 0.\hfill \end{array}$$

(20)

Proof. 

Now, for the estimation terms, we have:

$$\begin{array}{cc}& -{\int }_0^{t}h\left(t-s\right){\left(\mathsf{\Delta }u\left(s\right),\mathsf{\Delta }{u}_t\left(t\right)\right)}_{L^2\left(\mathsf{\Omega }\right)}ds\hfill \\ =& -{\int }_0^{t}h\left(t-s\right)\left[{\int }_{\mathsf{\Omega }}\mathsf{\Delta }u\left(x,s\right)\mathsf{\Delta }{u}_t\left(x,t\right)dx\right]ds,\hfill \end{array}$$

and using:

$$-\mathsf{\Delta }u\left(x,s\right)\mathsf{\Delta }{u}_t\left(x,t\right)=\frac{1}{2}\frac{d}{dt}\left\{{\left|\mathsf{\Delta }u\left(x,s\right)-\mathsf{\Delta }u\left(x,t\right)\right|}^2\right\}-\frac{1}{2}\frac{d}{dt}\left\{{\left|\mathsf{\Delta }u\left(x,t\right)\right|}^2\right\},$$

then:

$$\begin{array}{cc}\hfill & -{\int }_0^{t}h\left(t-s\right){\left(\mathsf{\Delta }u\left(s\right),\mathsf{\Delta }{u}_t\left(t\right)\right)}_{L^2\left(\mathsf{\Omega }\right)}ds\hfill \\ =\hfill & {\int }_0^{t}h\left(t-s\right){\int }_{\mathsf{\Omega }}\left(\frac{1}{2}\frac{d}{dt}\left\{{\left|\mathsf{\Delta }u\left(x,s\right)-\mathsf{\Delta }u\left(x,t\right)\right|}^2\right\}\right)dxds\hfill \\ \hfill & -{\int }_0^{t}h\left(t-s\right){\int }_{\mathsf{\Omega }}\left(\frac{1}{2}\frac{d}{dt}\left\{{\left|\mathsf{\Delta }u\left(x,t\right)\right|}^2\right\}\right)dxds\hfill \\ =\hfill & \frac{1}{2}{\int }_0^{t}h\left(t-s\right)\left(\frac{d}{dt}\left\{{\int }_{\mathsf{\Omega }}{\left|\mathsf{\Delta }u\left(x,s\right)-\mathsf{\Delta }u\left(x,t\right)\right|}^{2}dx\right\}\right)ds\hfill \\ \hfill & -\frac{1}{2}{\int }_0^{t}h\left(t-s\right)\left(\frac{d}{dt}\left\{{∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2\right\}\right)ds.\hfill \end{array}$$

(21)

Using the direct account, we find:

$$\begin{array}{cc}\hfill & \frac{1}{2}{\int }_0^{t}h\left(t-s\right)\frac{d}{dt}\left\{{\int }_{\mathsf{\Omega }}{\left|\mathsf{\Delta }u\left(x,s\right)-\mathsf{\Delta }u\left(x,t\right)\right|}^{2}dx\right\}ds\hfill \\ =\hfill & \frac{1}{2}{\int }_0^t\frac{d}{dt}\left\{h\left(t-s\right)\left({\int }_{\mathsf{\Omega }}{\left|\mathsf{\Delta }u\left(x,s\right)-\mathsf{\Delta }u\left(x,t\right)\right|}^{2}dx\right)\right\}ds\hfill \\ \hfill & -\frac{1}{2}{\int }_0^t{h}^{\prime }\left(t-s\right)\left({\int }_{\mathsf{\Omega }}{\left|\mathsf{\Delta }u\left(x,s\right)-\mathsf{\Delta }u\left(x,t\right)\right|}^{2}dx\right)ds\hfill \\ =\hfill & \frac{1}{2}\frac{d}{dt}\left\{{\int }_0^{t}h\left(t-s\right){\int }_{\mathsf{\Omega }}{\left|\mathsf{\Delta }u\left(x,s\right)-\mathsf{\Delta }u\left(x,t\right)\right|}^{2}dxds\right\}\hfill \\ \hfill & -\frac{1}{2}{\int }_0^t{h}^{\prime }\left(t-s\right)\left({\int }_{\mathsf{\Omega }}{\left|\mathsf{\Delta }u\left(x,s\right)-\mathsf{\Delta }u\left(x,t\right)\right|}^{2}dx\right)ds\hfill \\ =\hfill & \frac{1}{2}\frac{d}{dt}\left\{\left(h\square \mathsf{\Delta }u\right)\left(t\right)\right\}-\frac{1}{2}\left(h^{\prime }\square \mathsf{\Delta }u\right)\left(t\right),\hfill \end{array}$$

(22)

and:

$$\begin{array}{cc}\hfill & -\frac{1}{2}{\int }_0^{t}h\left(t-s\right)\left(\frac{d}{dt}\left\{{∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2\right\}\right)ds\hfill \\ =\hfill & -\frac{1}{2}\left({\int }_0^{t}h\left(t-s\right)ds\right)\left(\frac{d}{dt}\left\{{∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2\right\}\right)\hfill \\ =\hfill & -\frac{1}{2}\left({\int }_0^{t}h\left(s\right)ds\right)\frac{d}{dt}\left\{{∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2\right\}\hfill \\ =\hfill & -\frac{1}{2}\frac{d}{dt}\left\{\left({\int }_0^{t}h\left(s\right)ds\right){∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2\right\}+\frac{1}{2}h\left(t\right){∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2.\hfill \end{array}$$

(23)

By replacing (22) and (23) in (21), we get:

$$\begin{array}{cc}\hfill & -{\int }_0^{t}h\left(t-s\right){\left(\mathsf{\Delta }u\left(s\right),\mathsf{\Delta }{u}_t\left(t\right)\right)}_{L^2\left(\mathsf{\Omega }\right)}ds\hfill \\ =\hfill & \frac{1}{2}\frac{d}{dt}\left\{\left(h\square \mathsf{\Delta }u\right)\left(t\right)\right\}-\frac{1}{2}\left(h^{\prime }\square \mathsf{\Delta }u\right)\left(t\right)\hfill \\ \hfill & -\frac{1}{2}\frac{d}{dt}\left\{\left({\int }_0^{t}h\left(s\right)ds\right){∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2\right\}+\frac{1}{2}h\left(t\right){∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2\hfill \\ =\hfill & \frac{d}{dt}\left\{\frac{1}{2}\left(h\square \mathsf{\Delta }u\right)\left(t\right)-\frac{1}{2}\left({\int }_0^{t}h\left(s\right)ds\right){∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2\right\}\hfill \\ \hfill & -\frac{1}{2}\left(h^{\prime }\square \mathsf{\Delta }u\right)\left(t\right)+\frac{1}{2}h\left(t\right){∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2.\hfill \end{array}$$

(24)

By replacing (24) into (18), we get:

$$\begin{array}{cc}\hfill & \frac{d}{dt}\left\{\frac{1}{\rho +2}{∥u_t\left(t\right)∥}_{\rho +2}^{\rho +2}+\frac{1}{2}\left({\xi }_0+\frac{{\xi }_1}{\left(\beta +1\right)}{∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^{2\beta }\right){∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2\right.\hfill \\ \hfill & \left.+\frac{1}{2}{∥\mathsf{\Delta }{u}_t\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2-\frac{1}{\gamma }\left({\int }_{\mathsf{\Omega }}{\left|u\left(t\right)\right|}^{\gamma }ln{\left|u\right|}^{k}dx\right)+\frac{k}{{\gamma }^2}{∥u\left(t\right)∥}_{\gamma }^{\gamma }\right\}\hfill \\ \hfill & +\frac{d}{dt}\left\{\frac{1}{2}\left(h\square \mathsf{\Delta }u\right)\left(t\right)-\frac{1}{2}\left({\int }_0^{t}h\left(s\right)ds\right){∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2\right\}\hfill \\ \hfill & -\frac{1}{2}\left(h^{\prime }\square \mathsf{\Delta }u\right)\left(t\right)+\frac{1}{2}h\left(t\right){∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2\hfill \\ =\hfill & 0,\hfill \end{array}$$

(25)

then (25) is equivalent to:

$$\begin{array}{cc}\hfill & \frac{d}{dt}\left\{\frac{1}{\rho +2}{∥u_t\left(t\right)∥}_{\rho +2}^{\rho +2}+\frac{1}{2}\left({\xi }_0-{\int }_0^{t}h\left(s\right)ds\right){∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2\right.\hfill \\ \hfill & +\frac{{\xi }_1}{2\left(\beta +1\right)}{∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^{2\left(\beta +1\right)}+\frac{1}{2}{∥\mathsf{\Delta }{u}_t\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2\hfill \\ \hfill & \left.-\frac{1}{\gamma }\left({\int }_{\mathsf{\Omega }}{\left|u\left(t\right)\right|}^{\gamma }ln{\left|u\right|}^{k}dx\right)+\frac{k}{{\gamma }^2}{∥u\left(t\right)∥}_{\gamma }^{\gamma }+\frac{1}{2}\left(h\square \mathsf{\Delta }u\right)\left(t\right)\right\}\hfill \\ =\hfill & \frac{1}{2}\left(h^{\prime }\square \mathsf{\Delta }u\right)\left(t\right)-\frac{1}{2}h\left(t\right){∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2,\hfill \end{array}$$

(26)

by using (19) in (26), we get (20).

Then, the proof is complete. □

3. General Decay and Polynomial Decay

In the previous work, it was supposed that $h^{{}^{\prime }}\left(t\right)\le 0.$ Therefore, from (20), we see that $e^{{}^{\prime }}\left(t\right)\le 0.$ This implies that $e\left(t\right)\le e\left(0\right),$ for all $t\ge 0.$ In our case, we are not assuming that $h^{\prime }\left(t\right)\le 0.$ In fact, we are allowing the function $h\left(t\right)$ to oscillate.

To prove our result, we need to introduce the following auxiliary functional:

$$\mathrm{\Phi }\left(t\right):=\frac{1}{\rho +1}{\int }_{\mathsf{\Omega }}{\left|u_t\right|}^{\rho }{u}_t\left(t\right)u\left(t\right)dx,$$

(27)

and:

$$\begin{array}{ccc}\mathrm{\Psi }\left(t\right)\hfill & :\hfill & ={\int }_{\mathsf{\Omega }}{\int }_0^t{L}_{\alpha }\left(t-s\right){\left|\mathsf{\Delta }u\left(t\right)-\mathsf{\Delta }u(s)\right|}^{2}dsdx\hfill \\ & :\hfill & =(L_{\alpha }\square \mathsf{\Delta }u)\left(t\right),\hfill \end{array}$$

(28)

where:

$$\begin{array}{ccc}{L}_{\alpha }\left(t\right)\hfill & :\hfill & =e^{-\alpha t}{\int }_t^{+\infty }{L}_{\alpha }(s)ds\hfill \\ & :\hfill & =e^{-\alpha t}{\int }_t^{+\infty }l(s)e^{\alpha s}ds\hfill \end{array}$$

(29)

and:

$${\displaystyle {\int }_0^{\infty }{L}_{\alpha }^{1-\tau }(s)ds<\infty ,\mathrm{for}0<\tau <2-r<1\mathrm{where}1<r<{\displaystyle \frac{3}{2}}.}$$

(30)

$l\left(t\right)$ is defined in (8). Further, we consider the functional:

$$\begin{array}{ccc}V\left(t\right)\hfill & :\hfill & =e\left(t\right)+\epsilon \xi \left(t\right)\mathrm{\Phi }\left(t\right)+\epsilon \xi \left(t\right){\left(\mathsf{\Delta }{u}_t\left(t\right),\mathsf{\Delta }u\left(t\right)\right)}_{L^2(\mathsf{\Omega })}\hfill \\ & & +\eta \xi \left(t\right)\mathrm{\Psi }\left(t\right)-\eta \xi \left(t\right)\left({\int }_0^t{L}_{\alpha }\left(s\right)ds\right){∥\mathsf{\Delta }u\left(t\right)∥}_{L^2(\mathsf{\Omega })}^2\hfill \\ & & +2\eta \xi \left(t\right)\left\{{\int }_{\mathsf{\Omega }}\mathsf{\Delta }u\left(t\right){\int }_0^t{L}_{\alpha }\left(t-s\right)\mathsf{\Delta }u\left(s\right)dsdx\right\}\hfill \\ & & -2k_1\xi \left(0\right)\left({\int }_0^t{∥u_s\left(s\right)∥}_{\rho +2}^{\rho +2}ds\right)-2k_2\xi \left(0\right)\left({\int }_0^t{∥\mathsf{\Delta }u\left(s\right)∥}_{L^2(\mathsf{\Omega })}^{2}ds\right)\hfill \\ & & -2k_3\xi \left(0\right)\left({\int }_0^t{∥\mathsf{\Delta }{u}_s\left(s\right)∥}_{L^2(\mathsf{\Omega })}^{2}ds\right)-k_4\xi \left(0\right)\left({\int }_0^t{∥u\left(s\right)∥}_{\gamma }^{\gamma }ds\right)\hfill \\ & & -2k_5\xi \left(0\right)\left({\int }_0^t\left(L_{\alpha }^r\square \mathsf{\Delta }u\right)\left(s\right)ds\right),\hfill \end{array}$$

(31)

where $\left(1\le r<{\displaystyle \frac{3}{2}}\right)$ and for some positive constants $\epsilon ,\eta ,k_1,k_2,k_3,k_4$, and $k_5$ to be determined later.

Proposition 1.

Let:

$$\begin{array}{ccc}H\left(t\right)\hfill & :\hfill & =V\left(t\right)+2k_1\xi \left(0\right)\left({\int }_0^t{∥u_s\left(s\right)∥}_{\rho +2}^{\rho +2}ds\right)+2k_2\xi \left(0\right)\left({\int }_0^t{∥\mathsf{\Delta }u\left(s\right)∥}_{L^2(\mathsf{\Omega })}^{2}ds\right)\hfill \\ & & +2k_3\xi \left(0\right)\left({\int }_0^t{∥\mathsf{\Delta }{u}_s\left(s\right)∥}_{L^2(\mathsf{\Omega })}^{2}ds\right)+k_4\xi \left(0\right)\left({\int }_0^t{∥u\left(s\right)∥}_{\gamma }^{\gamma }ds\right)\hfill \\ & & +2k_5\xi \left(0\right)\left({\int }_0^t\left(L_{\alpha }^r\square \mathsf{\Delta }u\right)\left(s\right)ds\right)\hfill \\ & :\hfill & =e\left(t\right)+\epsilon \xi \left(t\right)\mathrm{\Phi }\left(t\right)+\epsilon \xi \left(t\right){\left(\mathsf{\Delta }{u}_t\left(t\right),\mathsf{\Delta }u\left(t\right)\right)}_{L^2(\mathsf{\Omega })}\hfill \\ & & +\eta \xi \left(t\right)\mathrm{\Psi }\left(t\right)-\eta \xi \left(t\right)\left({\int }_0^t{L}_{\alpha }\left(s\right)ds\right){∥\mathsf{\Delta }u\left(t\right)∥}_{L^2(\mathsf{\Omega })}^2\hfill \\ & & +2\eta \xi \left(t\right)\left\{{\int }_{\mathsf{\Omega }}\mathsf{\Delta }u\left(t\right){\int }_0^t{L}_{\alpha }\left(t-s\right)\mathsf{\Delta }u\left(s\right)dsdx\right\},\hfill \end{array}$$

(32)

then there exist positive constants $C_1$ and $C_2$ such that:

$$C_1\left\{E\left(t\right)+\left(h\square \mathsf{\Delta }u\right)\left(t\right)\right\}\le H\left(t\right)\le C_2\left\{E\left(t\right)+\left(h\square \mathsf{\Delta }u\right)\left(t\right)+\mathrm{\Psi }\left(t\right)\right\}.$$

(33)

Proof. 

For the function $\mathrm{\Phi }\left(t\right)$ definite in (27), by using Holder inequality $\left(\mathrm{for}p={\displaystyle \frac{\rho +2}{\rho +1}}\mathrm{and}q=\rho +2\right)$, Young’s inequality $\left(\mathrm{for}\epsilon ={\epsilon }_1\right),{∥u_t\left(t\right)∥}_{\rho +2}^{\rho }\le {\left[\left(\rho +2\right)e\left(0\right)\right]}^{\frac{\rho }{\left(\rho +2\right)}},$ embedding $H^1\hookrightarrow L^{2(\rho +1)}$, and the Poincaré inequality, we get:

$$\begin{array}{ccc}\mathrm{\Phi }\left(t\right)\hfill & \le \hfill & \frac{1}{\rho +1}{∥u_t\left(t\right)∥}_{\rho +2}^{\rho +1}{∥u\left(t\right)∥}_{\rho +2}\hfill \\ & \le \hfill & \frac{{\epsilon }_1}{2{\left(\rho +1\right)}^2}{∥u_t\left(t\right)∥}_{\rho +2}^{2\left(\rho +1\right)}+\frac{1}{2{\epsilon }_1}{∥u\left(t\right)∥}_{\rho +2}^2\hfill \\ & =\hfill & \frac{{\epsilon }_1}{2{\left(\rho +1\right)}^2}{∥u_t\left(t\right)∥}_{\rho +2}^{\rho }{∥u_t\left(t\right)∥}_{\rho +2}^{\rho +2}+\frac{1}{2{\epsilon }_1}{∥u\left(t\right)∥}_{\rho +2}^2\hfill \\ & \le \hfill & \frac{{\epsilon }_1{\left[\left(\rho +2\right)e\left(0\right)\right]}^{\frac{\rho }{\left(\rho +2\right)}}}{2{\left(\rho +1\right)}^2}{∥u_t\left(t\right)∥}_{\rho +2}^{\rho +2}+\frac{C_0{C}_p}{2{\epsilon }_1}{∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2,\hfill \end{array}$$

(34)

where $C_0$ comes from the embedding $H^1\hookrightarrow L^{2(\rho +1)}$ and the $C_p$ constant Poincaré inequality.

By multiplying (34) by $\epsilon \xi \left(t\right)$ and using $0<\xi \left(t\right)\le \xi \left(0\right),$ we get:

$$\begin{array}{cc}\hfill & \epsilon \xi \left(t\right)\mathrm{\Phi }\left(t\right)\hfill \\ \le \hfill & \frac{\epsilon \xi \left(0\right){\epsilon }_1{\left[\left(\rho +2\right)e\left(0\right)\right]}^{\frac{\rho }{\left(\rho +2\right)}}}{2{\left(\rho +1\right)}^2}{∥u_t\left(t\right)∥}_{\rho +2}^{\rho +2}+\frac{\epsilon \xi \left(0\right)C_0{C}_p}{2{\epsilon }_1}{∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2.\hfill \end{array}$$

(35)

By using Young’s inequality $\left(\mathrm{for}\epsilon ={\epsilon }_2\right)$ and using $0<\xi \left(t\right)\le \xi \left(0\right),$ we get:

$$\begin{array}{cc}\hfill & \epsilon \xi \left(t\right){\left(\mathsf{\Delta }{u}_t\left(t\right),\mathsf{\Delta }u\left(t\right)\right)}_{L^2(\mathsf{\Omega })}\hfill \\ \le \hfill & \frac{\epsilon \xi \left(0\right){\epsilon }_2}{2}{∥\mathsf{\Delta }{u}_t\left(t\right)∥}_{L^2(\mathsf{\Omega })}^2+\frac{\epsilon \xi \left(0\right)}{2{\epsilon }_2}{∥\mathsf{\Delta }u\left(t\right)∥}_{L^2(\mathsf{\Omega })}^2.\hfill \end{array}$$

(36)

Note that from (8) and (29), we get:

$$\begin{array}{ccc}{\int }_0^t{L}_{\alpha }(s)ds\hfill & \le \hfill & \frac{1}{\alpha }{\int }_0^{\infty }l(s)e^{\alpha s}ds\hfill \\ & :\hfill & =\frac{{\overline{l}}_{\alpha }}{\alpha },\hfill \end{array}$$

(37)

then, by multiplying (37) by $\xi \left(t\right)$ and using $0<\xi \left(t\right)\le \xi \left(0\right),$ we get:

$$-\eta \xi \left(t\right)\left({\int }_0^t{L}_{\alpha }(s)ds\right){∥\mathsf{\Delta }u\left(t\right)∥}_{L^2(\mathsf{\Omega })}^2\le \frac{\eta \xi \left(0\right){\overline{l}}_{\alpha }}{\alpha }{∥\mathsf{\Delta }u\left(t\right)∥}_{L^2(\mathsf{\Omega })}^2.$$

(38)

By using Young’s inequality $\left(\mathrm{for}\epsilon ={\displaystyle \frac{{\epsilon }_3}{2}}\right)$ and (28), we get:

$$\begin{array}{cc}\hfill & {\int }_{\mathsf{\Omega }}\mathsf{\Delta }u\left(t\right){\int }_0^t{L}_{\alpha }\left(t-s\right)\mathsf{\Delta }u(s)dsdx\hfill \\ =\hfill & {\int }_0^t{L}_{\alpha }\left(t-s\right)\left[{\int }_{\mathsf{\Omega }}\mathsf{\Delta }u\left(s\right)\mathsf{\Delta }u\left(t\right)dx\right]ds\hfill \\ =\hfill & {\int }_0^t{L}_{\alpha }\left(t-s\right)\left[{\int }_{\mathsf{\Omega }}\left[\mathsf{\Delta }u\left(s\right)-\mathsf{\Delta }u(t)+\mathsf{\Delta }u(t)\right]\mathsf{\Delta }u(t)dx\right]ds\hfill \\ =\hfill & {\int }_0^t{L}_{\alpha }\left(t-s\right)\left[{\int }_{\mathsf{\Omega }}\left[\mathsf{\Delta }u\left(s\right)-\mathsf{\Delta }u(t)\right]\mathsf{\Delta }u(t)dx\right]ds\hfill \\ \hfill & +{\int }_0^t{L}_{\alpha }\left(t-s\right)\left[{\int }_{\mathsf{\Omega }}{\left|\mathsf{\Delta }u\left(t\right)\right|}^{2}dx\right]ds\hfill \\ \le \hfill & \frac{1}{4{\epsilon }_3}{\int }_0^t{L}_{\alpha }\left(t-s\right)\left[{\int }_{\mathsf{\Omega }}{\left|\mathsf{\Delta }u\left(s\right)-\mathsf{\Delta }u(t)\right|}^{2}dx\right]ds\hfill \\ \hfill & +{\epsilon }_3{\int }_0^t{L}_{\alpha }\left(t-s\right)\left[{\int }_{\mathsf{\Omega }}{\left|\mathsf{\Delta }u(t)\right|}^{2}dx\right]ds\hfill \\ \hfill & +\left({\int }_0^t{L}_{\alpha }(s)ds\right){∥\mathsf{\Delta }u\left(t\right)∥}_{L^2(\mathsf{\Omega })}^2\hfill \\ =\hfill & \frac{1}{4{\epsilon }_3}\mathrm{\Psi }\left(t\right)+\left(1+{\epsilon }_3\right)\left({\int }_0^t{L}_{\alpha }(s)ds\right){∥\mathsf{\Delta }u\left(t\right)∥}_{L^2(\mathsf{\Omega })}^2.\hfill \end{array}$$

(39)

By multiplying (39) by $\xi \left(t\right)$ and using $0<\xi \left(t\right)\le \xi \left(0\right)$ and (37), we get:

$$\begin{array}{cc}\hfill & 2\eta \xi \left(t\right)\left\{{\int }_{\mathsf{\Omega }}\mathsf{\Delta }u\left(t\right){\int }_0^t{L}_{\alpha }\left(t-s\right)\mathsf{\Delta }u(s)dsdx\right\}\hfill \\ \le \hfill & \frac{\eta }{2{\epsilon }_3}\xi \left(t\right)\mathrm{\Psi }\left(t\right)+\frac{2\eta \left(1+{\epsilon }_3\right)\xi \left(0\right){\overline{l}}_{\alpha }}{\alpha }{∥\mathsf{\Delta }u\left(t\right)∥}_{L^2(\mathsf{\Omega })}^2.\hfill \end{array}$$

(40)

By using (19), (35), (36), (38) and (40) in (32), we get:

$$\begin{array}{ccc}H\left(t\right)\hfill & \ge \hfill & \left\{\frac{1}{\rho +2}-\frac{\epsilon \xi \left(0\right){\epsilon }_1{\left[\left(\rho +2\right)e\left(0\right)\right]}^{\frac{\rho }{\left(\rho +2\right)}}}{2{\left(\rho +1\right)}^2}\right\}{∥u_t\left(t\right)∥}_{\rho +2}^{\rho +2}\hfill \\ & & +\left\{\frac{\left({\xi }_0-\overline{h}\right)}{2}-\frac{\epsilon \xi \left(0\right)C_0{C}_p}{2{\epsilon }_1}-\frac{\epsilon \xi \left(0\right)}{2{\epsilon }_2}\right.\hfill \\ & & -\left.\frac{\eta \xi \left(0\right){\overline{l}}_{\alpha }}{\alpha }-\frac{2\eta \left(1+{\epsilon }_3\right)\xi \left(0\right){\overline{l}}_{\alpha }}{\alpha }\right\}{∥\mathsf{\Delta }u\left(t\right)∥}_{L^2(\mathsf{\Omega })}^2\hfill \\ & & +\frac{{\xi }_1}{2\left(\beta +1\right)}{∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^{2\left(\beta +1\right)}+\left\{\frac{1}{2}-\frac{\epsilon \xi \left(0\right){\epsilon }_2}{2}\right\}{∥\mathsf{\Delta }{u}_t\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2\hfill \\ & & -\frac{1}{\gamma }\left({\int }_{\mathsf{\Omega }}{\left|u\left(t\right)\right|}^{\gamma }ln{\left|u\right|}^{k}dx\right)+\frac{k}{{\gamma }^2}{∥u\left(t\right)∥}_{\gamma }^{\gamma }+\frac{1}{2}\left(h\square \mathsf{\Delta }u\right)\left(t\right)\hfill \\ & & +\eta \left(1-\frac{1}{2{\epsilon }_3}\right)\xi \left(t\right)\mathrm{\Psi }\left(t\right).\hfill \end{array}$$

(41)

Clearly, choosing:

$$\left\{\begin{array}{c}{\epsilon }_1:={\displaystyle \frac{{\left(\rho +1\right)}^2}{\left(\rho +2\right)\epsilon \xi \left(0\right){\left[\left(\rho +2\right)e\left(0\right)\right]}^{\frac{\rho }{\left(\rho +2\right)}}}},\hfill \\ {\epsilon }_2:={\displaystyle \frac{1}{2\epsilon \xi \left(0\right)}},\hfill \\ {\epsilon }_3:=1,\hfill \\ \epsilon <{\displaystyle \frac{\left(\rho +1\right)\sqrt{\left({\xi }_0-\overline{h}\right)}}{\xi \left(0\right)\sqrt{3C_0{C}_p\left(\rho +2\right){\left[\left(\rho +2\right)e\left(0\right)\right]}^{\frac{\rho }{\left(\rho +2\right)}}+6{\left(\rho +1\right)}^2}}},\hfill \\ \eta <{\displaystyle \frac{\alpha \left({\xi }_0-\overline{h}\right)}{30\xi \left(0\right){\overline{l}}_{\alpha }}}.\hfill \end{array}\right.$$

(42)

By using (43) in (42), we get:

$$\begin{array}{ccc}\hfill H\left(t\right)& \ge & \frac{1}{2\left(\rho +2\right)}{∥u_t\left(t\right)∥}_{\rho +2}^{\rho +2}+\frac{\left({\xi }_0-\overline{h}\right)}{6}{∥\mathsf{\Delta }u\left(t\right)∥}_{L^2(\mathsf{\Omega })}^2\hfill \\ & & +\frac{{\xi }_1}{2\left(\beta +1\right)}{∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^{2\left(\beta +1\right)}+\frac{1}{4}{∥\mathsf{\Delta }{u}_t\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2\hfill \\ & & -\frac{1}{\gamma }\left({\int }_{\mathsf{\Omega }}{\left|u\left(t\right)\right|}^{\gamma }ln{\left|u\right|}^{k}dx\right)+\frac{k}{{\gamma }^2}{∥u\left(t\right)∥}_{\gamma }^{\gamma }+\frac{1}{2}\left(h\square \mathsf{\Delta }u\right)\left(t\right).\hfill \end{array}$$

Then, $\exists C_1>0,$ where:

$$H\left(t\right)\ge C_1\left\{E\left(t\right)+\left(h\square \mathsf{\Delta }u\right)\left(t\right)\right\}.$$

(43)

On the other hand, by replacing (19), (35), (36), (38) and (40) in (32) and taking ${\epsilon }_1={\epsilon }_2={\epsilon }_3=1,$ we get:

$$\begin{array}{ccc}\hfill H\left(t\right)& \le & \left\{\frac{1}{\rho +2}+\frac{\epsilon \xi \left(0\right){\left[\left(\rho +2\right)e\left(0\right)\right]}^{\frac{\rho }{\left(\rho +2\right)}}}{2{\left(\rho +1\right)}^2}\right\}{∥u_t\left(t\right)∥}_{\rho +2}^{\rho +2}\hfill \\ & & +\left\{\frac{\left({\xi }_0+\epsilon \xi \left(0\right)C_0{C}_p+\epsilon \xi \left(0\right)\right)}{2}+\frac{4\eta \xi \left(0\right){\overline{l}}_{\alpha }}{\alpha }\right\}{∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2\hfill \\ & & +\frac{{\xi }_1}{2\left(\beta +1\right)}{∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^{2\left(\beta +1\right)}\hfill \\ & & +\frac{\left\{1+\epsilon \xi \left(0\right)\right\}}{2}{∥\mathsf{\Delta }{u}_t\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2-\frac{1}{\gamma }\left({\int }_{\mathsf{\Omega }}{\left|u\left(t\right)\right|}^{\gamma }ln{\left|u\right|}^{k}dx\right)\hfill \\ & & +\frac{k}{{\gamma }^2}{∥u\left(t\right)∥}_{\gamma }^{\gamma }+\frac{1}{2}\left(h\square \mathsf{\Delta }u\right)\left(t\right)+\frac{3\eta }{2}\xi \left(0\right)\mathrm{\Psi }\left(t\right).\hfill \end{array}$$

Then, $\exists C_2>0,$ where:

$$H\left(t\right)\le C_2\left\{E\left(t\right)+\left(h\square \mathsf{\Delta }u\right)\left(t\right)+\mathrm{\Psi }\left(t\right)\right\},$$

(44)

by using (43) and (44), we get (33).

The proof is complete. □

Lemma 3.

For $r>1$ and $0<\theta <1$, we have:

$$\begin{array}{cc}& {\int }_0^{t}h\left(t-s\right){∥w(s)∥}_{L^2\left(\mathsf{\Omega }\right)}^{2}ds\hfill \\ \le & {\left({\int }_0^t{h}^{1-\theta }\left(t-s\right){∥w(s)∥}_{L^2\left(\mathsf{\Omega }\right)}^{2}ds\right)}^{\frac{1}{r}}\times {\left({\int }_0^t{h}^{\frac{(r-1+\theta )}{r-1}}\left(t-s\right){∥w(s)∥}_{L^2\left(\mathsf{\Omega }\right)}^{2}ds\right)}^{\frac{r-1}{r}},\hfill \end{array}$$

for any $w\in L^2\left(\mathsf{\Omega }\right).$

Proof. 

It suffices to note that:

$$\begin{array}{cc}& {\int }_0^{t}h\left(t-s\right){∥w(s)∥}_{L^2\left(\mathsf{\Omega }\right)}^{2}ds\hfill \\ =& {\int }_0^t{h}^{\frac{{}^{(1-\theta )}}{r}}\left(t-s\right){∥w(s)∥}_{L^2\left(\mathsf{\Omega }\right)}^{\frac{2}{r}}{h}^{\frac{(r-1+\theta )}{r}}\left(t-s\right){∥w(s)∥}_{L^2\left(\mathsf{\Omega }\right)}^{\frac{2(r-1)}{r}}ds,\hfill \end{array}$$

and applying Holder’s inequality for

$$p=r,q={\displaystyle \frac{r}{r-1}},r>1.$$

The proof of the lemma is completed. □

Lemma 4.

Let $v\in L^{\infty }\left(\left(0,T\right);L^2\left(\mathsf{\Omega }\right)\right)$ be such that $v_x\in L^{\infty }\left(\left(0,t\right);L^2\left(\mathsf{\Omega }\right)\right)$ and h be a continuous function on $\left[0,T\right]$ and suppose that $0<\theta <1$ and $\rho >1.$ Then, there exists a constant $C>0$ such that:

$$\begin{array}{cc}& {\int }_0^{t}h\left(t-s\right){∥v_x\left(.,t\right)-v_x\left(.,s\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^{2}ds\hfill \\ \le & C{\left(\underset{0<s<T}{sup}{∥v\left(.,s\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2{\int }_0^t{h}^{1-\theta }(s)ds\right)}^{\frac{\rho -1}{\rho -1+\theta }}\hfill \\ & \times {\left({\int }_0^t{h}^{\rho }\left(t-s\right){∥v_x\left(.,t\right)-v_x\left(.,s\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^{2}ds\right)}^{\frac{\theta }{\rho -1+\theta }}.\hfill \end{array}$$

Proof. 

By using Lemma 3 with $r:={\displaystyle \frac{\left(\rho -1+\theta \right)}{\left(\rho -1\right)}},$ we obtain:

$$\begin{array}{cc}\hfill & {\int }_0^{t}h\left(t-s\right){∥v_x\left(.,t\right)-v_x\left(.,s\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^{2}ds\hfill \\ \le \hfill & {\left({\int }_0^t{h}^{1-\theta }\left(t-s\right){∥v_x\left(.,t\right)-v_x\left(.,s\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^{2}ds\right)}^{\frac{\rho -1}{\rho -1+\theta }}\hfill \\ \hfill & \times {\left({\int }_0^t{h}^{\rho }\left(t-s\right){∥v_x\left(.,t\right)-v_x\left(.,s\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2\right)}^{\frac{\theta }{\rho -1+\theta }}.\hfill \end{array}$$

(45)

It is easy to see that:

$$\begin{array}{cc}\hfill & {\int }_0^t{h}^{1-\theta }\left(t-s\right){∥v_x\left(.,t\right)-v_x\left(.,s\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^{2}ds\hfill \\ \le \hfill & C\underset{0<s<T}{sup}{∥v_x\left(.,s\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2{\int }_0^t{h}^{1-\theta }(s)ds.\hfill \end{array}$$

(46)

By combining (45) and (46), the proof of the lemma is completed. □

Lemma 5.

Let $v\in L^{\infty }\left(\left(0,T\right);L^2\left(\mathsf{\Omega }\right)\right)$ be such that $v_x\in L^{\infty }\left(\left(0,T\right);L^2\left(\mathsf{\Omega }\right)\right)$ and h be a continuous function on $\left[0,T\right]$ and suppose that $\rho >1$. Then, there exists a constant $C>0$ such that:

$$\begin{array}{cc}\hfill & {\int }_0^{t}h\left(t-s\right){∥v_x\left(.,t\right)-v_x\left(.,s\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^{2}ds\hfill \\ \le \hfill & c{\left(t{∥v_x\left(.,t\right)∥}_H^2+{\int }_0^t{∥v_x\left(.,s\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^{2}ds\right)}^{\frac{\rho -1}{\rho }}\hfill \\ \hfill & \times {\left({\int }_0^t{h}^{\rho }\left(t-s\right){∥v_x\left(.,t\right)-v_x\left(.,s\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^{2}ds\right)}^{\frac{1}{\rho }}.\hfill \end{array}$$

(47)

Proof. 

We use (45) for $\theta =1$ to arrive at:

$$\begin{array}{cc}& {\int }_0^{t}h\left(t-s\right){∥v_x\left(.,t\right)-v_x\left(.,s\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^{2}ds\hfill \\ \le & {\left({\int }_0^t{∥v_x\left(.,t\right)-v_x\left(.,s\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^{2}ds\right)}^{\frac{\rho -1}{\rho }}\hfill \\ & \times {\left({\int }_0^t{h}^{\rho }\left(t-s\right){∥v_x\left(.,t\right)-v_x\left(.,s\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^{2}ds\right)}^{\frac{1}{\rho }}.\hfill \end{array}$$

It suffices to note that:

$$\begin{array}{cc}\hfill & {\int }_0^t{∥v_x\left(.,t\right)-v_x\left(.,s\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^{2}ds\hfill \\ \le \hfill & 2t{∥v_x\left(.,t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2+2{\int }_0^t{∥v_x\left(.,s\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^{2}ds,\hfill \end{array}$$

(48)

to obtain (47).

This completes the proof. □

Let:

$$\epsilon :=min\left\{{\displaystyle \frac{\left(\rho +1\right)\sqrt{\left({\xi }_0-\overline{h}\right)}}{\xi \left(0\right)\sqrt{3C_0{C}_p\left(\rho +2\right){\left[\left(\rho +2\right)e\left(0\right)\right]}^{\frac{\rho }{\left(\rho +2\right)}}+6{\left(\rho +1\right)}^2}}},{\displaystyle \frac{\left({\xi }_0-\overline{h}\right)}{4\left({\displaystyle {\int }_0^t{h}^{2-r}(s)ds}\right)}}\right\}$$

and:

$${\overline{l}}_{\alpha }<{\displaystyle \frac{\alpha \left({\xi }_0-\overline{h}\right)}{12\alpha +6L}}\epsilon .$$

Now, we are in a position to state and prove our first result.

Theorem 1.

Assume that the hypotheses (A1)–(A3) hold, the initial data $(u_0,u_1)$ satisfy $E\left(0\right)>0$, and ${\overline{l}}_{\alpha }$ is as above.

Then, the classical energy $E\left(t\right)$ of (1)–(3) decays to zero exponentially and polynomially. That is, there exist positive constants $C_1,C_2$, and $C_3$ such that:

$$E(t)\le \left\{\begin{array}{c}{\displaystyle \frac{V\left(0\right)}{C_1}}{e}^{-{\displaystyle \frac{C_3}{C_2}}\left({\displaystyle {\int }_0^t\xi \left(s\right)ds}\right)},r=1,t\ge 0,\hfill \\ C{\left({\displaystyle 1+{\int }_0^t\xi (s)ds}\right)}^{{\displaystyle \frac{-1}{r-1}}},1<r<{\displaystyle \frac{3}{2}},t\ge 0.\hfill \end{array}\right.$$

(49)

Proof. 

Now, a differentiation of $V\left(t\right)$ definite in (31) with respect to time gives:

$$\begin{array}{cc}\hfill & \frac{d}{dt}\left\{V\left(t\right)+k_1\xi \left(0\right)\left({\int }_0^t{∥u_s\left(s\right)∥}_{\rho +2}^{\rho +2}ds\right)+k_2\xi \left(0\right)\left({\int }_0^t{∥\mathsf{\Delta }u\left(s\right)∥}_{L^2(\mathsf{\Omega })}^{2}ds\right)\right.\hfill \\ \hfill & +k_3\xi \left(0\right)\left({\int }_0^t{∥\mathsf{\Delta }{u}_s\left(s\right)∥}_{L^2(\mathsf{\Omega })}^{2}ds\right)+k_4\xi \left(0\right)\left({\int }_0^t{∥u\left(s\right)∥}_{\gamma }^{\gamma }ds\right)\hfill \\ \hfill & +\left.k_5\xi \left(0\right)\left({\int }_0^t\left(L_{\alpha }^r\square \mathsf{\Delta }u\right)\left(s\right)ds\right)\right\}\hfill \\ =\hfill & \frac{d}{dt}\left\{e\left(t\right)\right\}+\epsilon \frac{d}{dt}\left\{\xi \left(t\right)\mathrm{\Phi }\left(t\right)+\xi \left(t\right){\left(\mathsf{\Delta }{u}_t\left(t\right),\mathsf{\Delta }u\left(t\right)\right)}_{L^2(\mathsf{\Omega })}\right\}\hfill \\ \hfill & +\eta \frac{d}{dt}\left\{\xi \left(t\right)\psi \left(t\right)-\xi \left(t\right)\left({\int }_0^t{L}_{\alpha }\left(s\right)ds\right){∥\mathsf{\Delta }u\left(t\right)∥}_{L^2(\mathsf{\Omega })}^2\right.\hfill \\ \hfill & +\left.2\xi \left(t\right){\int }_{\mathsf{\Omega }}\mathsf{\Delta }u\left(t\right){\int }_0^t{L}_{\alpha }\left(t-s\right)\mathsf{\Delta }u\left(s\right)dsdx\right\}\hfill \\ \hfill & -k_1\xi \left(0\right){∥u_t\left(t\right)∥}_{\rho +2}^{\rho +2}-k_2\xi \left(0\right){∥\mathsf{\Delta }u\left(t\right)∥}_{L^2(\mathsf{\Omega })}^2-k_3\xi \left(0\right){∥\mathsf{\Delta }{u}_t\left(t\right)∥}_{L^2(\mathsf{\Omega })}^2\hfill \\ \hfill & -k_4\xi \left(0\right){∥u\left(t\right)∥}_{\gamma }^{\gamma }-k_5\xi \left(0\right)\left(L_{\alpha }^r\square \mathsf{\Delta }u\right)\left(t\right).\hfill \end{array}$$

(50)

By a differentiation of (27), we have:

$$\begin{array}{cc}\hfill & \frac{d}{dt}\left\{\xi \left(t\right)\mathrm{\Phi }\left(t\right)\right\}\hfill \\ =\hfill & \frac{1}{\rho +1}{\xi }^{\prime }\left(t\right)\left({\int }_{\mathsf{\Omega }}{\left|u_t\right|}^{\rho }{u}_t\left(t\right)u\left(t\right)dx\right)+\frac{1}{\rho +1}\xi \left(t\right)\left({\int }_{\mathsf{\Omega }}{\left|u_t\right|}^{\rho }{u}_t\left(t\right)u_t\left(t\right)dx\right)\hfill \\ \hfill & +\frac{1}{\rho +1}\xi \left(t\right)\left({\int }_{\mathsf{\Omega }}\frac{d}{dt}\left\{{\left|u_t\right|}^{\rho }\right\}{u}_t\left(t\right)u\left(t\right)dx\right)\hfill \\ \hfill & +\frac{1}{\rho +1}\xi \left(t\right)\left({\int }_{\mathsf{\Omega }}{\left|u_t\right|}^{\rho }{u}_{tt}\left(t\right)u\left(t\right)dx\right)\hfill \\ =\hfill & \frac{1}{\rho +1}{\xi }^{\prime }\left(t\right)\left({\int }_{\mathsf{\Omega }}{\left|u_t\right|}^{\rho }{u}_t\left(t\right)u\left(t\right)dx\right)+\frac{1}{\rho +1}\xi \left(t\right){∥u_t\left(t\right)∥}_{\rho +2}^{\rho +2}\hfill \\ \hfill & +\frac{\rho }{\rho +1}\xi \left(t\right)\left({\int }_{\mathsf{\Omega }}{\left|u_t\right|}^{\rho -2}{u}_{tt}\left(t\right)u_t\left(t\right)u_t\left(t\right)u\left(t\right)dx\right)\hfill \\ \hfill & +\frac{1}{\rho +1}\xi \left(t\right)\left({\int }_{\mathsf{\Omega }}{\left|u_t\right|}^{\rho }{u}_{tt}\left(t\right)u\left(t\right)dx\right)\hfill \\ =\hfill & \frac{1}{\rho +1}{\xi }^{\prime }\left(t\right)\left({\int }_{\mathsf{\Omega }}{\left|u_t\right|}^{\rho }{u}_t\left(t\right)u\left(t\right)dx\right)+\frac{1}{\rho +1}\xi \left(t\right){∥u_t\left(t\right)∥}_{\rho +2}^{\rho +2}\hfill \\ \hfill & +\xi \left(t\right)\left({\int }_{\mathsf{\Omega }}{\left|u_t\right|}^{\rho }{u}_{tt}\left(t\right)u\left(t\right)dx\right).\hfill \end{array}$$

(51)

By using $u$ as a solution in (1)–(3), we get:

$$\begin{array}{cc}\hfill & \xi \left(t\right)\left({\int }_{\mathsf{\Omega }}{\left|u_t\right|}^{\rho }{u}_{tt}\left(t\right)u\left(t\right)dx\right)\hfill \\ =\hfill & -\xi \left(t\right){\int }_{\mathsf{\Omega }}\left({\xi }_0+{\xi }_1{∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^{2\beta }\right){\mathsf{\Delta }}^{2}u\left(t\right)u\left(t\right)dx\hfill \\ \hfill & -\xi \left(t\right)\left({\int }_{\mathsf{\Omega }}{\mathsf{\Delta }}^2{u}_{tt}\left(t\right)u\left(t\right)dx\right)\hfill \\ \hfill & +\xi \left(t\right){\int }_{\mathsf{\Omega }}u\left(t\right)\left[{\displaystyle {\int }_0^{t}h\left(t-s\right){\mathsf{\Delta }}^{2}u\left(s\right)ds}\right]dx\hfill \\ \hfill & +\xi \left(t\right)\left({\int }_{\mathsf{\Omega }}u{\left|u\right|}^{\gamma -2}ln{\left|u\right|}^{k}u\left(t\right)dx\right).\hfill \end{array}$$

(52)

Integrating by parts, we get:

$$\begin{array}{cc}\hfill & -\xi \left(t\right){\int }_{\mathsf{\Omega }}\left({\xi }_0+{\xi }_1{∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^{2\beta }\right){\mathsf{\Delta }}^{2}u\left(t\right)u\left(t\right)dx\hfill \\ =\hfill & -\xi \left(t\right)\left({\xi }_0+{\xi }_1{∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^{2\beta }\right){\int }_{\mathsf{\Omega }}{\mathsf{\Delta }}^{2}u\left(t\right)u\left(t\right)dx\hfill \\ =\hfill & -\xi \left(t\right)\left({\xi }_0+{\xi }_1{∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^{2\beta }\right){∥\mathsf{\Delta }u\left(t\right)∥}_{L^2(\mathsf{\Omega })}^2\hfill \\ =\hfill & -{\xi }_0\xi \left(t\right){∥\mathsf{\Delta }u\left(t\right)∥}_{L^2(\mathsf{\Omega })}^2-{\xi }_1\xi \left(t\right){∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^{2\left(\beta +1\right)}.\hfill \end{array}$$

(53)

Also integrating by parts, we get:

$$\begin{array}{cc}\hfill & -\xi \left(t\right)\left({\int }_{\mathsf{\Omega }}{\mathsf{\Delta }}^2{u}_{tt}\left(t\right)u\left(t\right)dx\right)\hfill \\ =\hfill & -\xi \left(t\right)\left({\int }_{\mathsf{\Omega }}\mathsf{\Delta }{u}_{tt}\left(t\right)\mathsf{\Delta }u\left(t\right)dx\right)\hfill \\ =\hfill & -\xi \left(t\right)\frac{d}{dt}\left\{{\left(\mathsf{\Delta }{u}_t\left(t\right),\mathsf{\Delta }u\left(t\right)\right)}_{L^2(\mathsf{\Omega })}\right\}+\xi \left(t\right){∥\mathsf{\Delta }{u}_t\left(t\right)∥}_{L^2(\mathsf{\Omega })}^2\hfill \\ =\hfill & -\frac{d}{dt}\left\{\xi \left(t\right){\left(\mathsf{\Delta }{u}_t\left(t\right),\mathsf{\Delta }u\left(t\right)\right)}_{L^2(\mathsf{\Omega })}\right\}+{\xi }^{\prime }\left(t\right){\left(\mathsf{\Delta }{u}_t\left(t\right),\mathsf{\Delta }u\left(t\right)\right)}_{L^2(\mathsf{\Omega })}\hfill \\ \hfill & +\xi \left(t\right){∥\mathsf{\Delta }{u}_t\left(t\right)∥}_{L^2(\mathsf{\Omega })}^2.\hfill \end{array}$$

(54)

By using integration by parts, we get:

$$\begin{array}{cc}\hfill & \xi \left(t\right){\int }_{\mathsf{\Omega }}u\left(t\right)\left[{\displaystyle {\int }_0^{t}h\left(t-s\right){\mathsf{\Delta }}^{2}u\left(s\right)ds}\right]dx\hfill \\ =\hfill & \xi \left(t\right)\left\{{\int }_{\mathsf{\Omega }}\mathsf{\Delta }u\left(t\right){\int }_0^{t}h\left(t-s\right)\mathsf{\Delta }u\left(s\right)dsdx\right\}.\hfill \end{array}$$

(55)

By direct calculation, we get:

$$\xi \left(t\right)\left({\int }_{\mathsf{\Omega }}u{\left|u\right|}^{\gamma -2}ln{\left|u\right|}^{k}u\left(t\right)dx\right)=\xi \left(t\right)\left({\int }_{\mathsf{\Omega }}{\left|u\left(t\right)\right|}^{\gamma }ln{\left|u\right|}^{k}dx\right).$$

(56)

By replacing (53)–(56) in (52), we get:

$$\begin{array}{cc}\hfill & \xi \left(t\right)\left({\int }_{\mathsf{\Omega }}{\left|u_t\right|}^{\rho }{u}_{tt}\left(t\right)u\left(t\right)dx\right)\hfill \\ =\hfill & -{\xi }_0\xi \left(t\right){∥\mathsf{\Delta }u\left(t\right)∥}_{L^2(\mathsf{\Omega })}^2-{\xi }_1\xi \left(t\right){∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^{2\left(\beta +1\right)}\hfill \\ \hfill & -\frac{d}{dt}\left\{\xi \left(t\right){\left(\mathsf{\Delta }{u}_t\left(t\right),\mathsf{\Delta }u\left(t\right)\right)}_{L^2(\mathsf{\Omega })}\right\}\hfill \\ \hfill & +{\xi }^{\prime }\left(t\right){\left(\mathsf{\Delta }{u}_t\left(t\right),\mathsf{\Delta }u\left(t\right)\right)}_{L^2(\mathsf{\Omega })}+\xi \left(t\right){∥\mathsf{\Delta }{u}_t\left(t\right)∥}_{L^2(\mathsf{\Omega })}^2\hfill \\ \hfill & +\xi \left(t\right)\left\{{\int }_{\mathsf{\Omega }}\mathsf{\Delta }u\left(t\right){\int }_0^{t}h\left(t-s\right)\mathsf{\Delta }u\left(s\right)dsdx\right\}\hfill \\ \hfill & +\xi \left(t\right)\left({\int }_{\mathsf{\Omega }}{\left|u\left(t\right)\right|}^{\gamma }ln{\left|u\right|}^{k}dx\right).\hfill \end{array}$$

(57)

By replacing (57) in (51), we get:

$$\begin{array}{cc}\hfill & \frac{d}{dt}\left\{\xi \left(t\right)\mathrm{\Phi }\left(t\right)\right\}\hfill \\ =\hfill & \frac{1}{\rho +1}{\xi }^{\prime }\left(t\right)\left({\int }_{\mathsf{\Omega }}{\left|u_t\right|}^{\rho }{u}_t\left(t\right)u\left(t\right)dx\right)+\frac{1}{\rho +1}\xi \left(t\right){∥u_t\left(t\right)∥}_{\rho +2}^{\rho +2}\hfill \\ \hfill & -{\xi }_0\xi \left(t\right){∥\mathsf{\Delta }u\left(t\right)∥}_{L^2(\mathsf{\Omega })}^2-{\xi }_1\xi \left(t\right){∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^{2\left(\beta +1\right)}\hfill \\ \hfill & -\frac{d}{dt}\left\{\xi \left(t\right){\left(\mathsf{\Delta }{u}_t\left(t\right),\mathsf{\Delta }u\left(t\right)\right)}_{L^2(\mathsf{\Omega })}\right\}\hfill \\ \hfill & +{\xi }^{\prime }\left(t\right){\left(\mathsf{\Delta }{u}_t\left(t\right),\mathsf{\Delta }u\left(t\right)\right)}_{L^2(\mathsf{\Omega })}+\xi \left(t\right){∥\mathsf{\Delta }{u}_t\left(t\right)∥}_{L^2(\mathsf{\Omega })}^2\hfill \\ \hfill & +\xi \left(t\right)\left\{{\int }_{\mathsf{\Omega }}\mathsf{\Delta }u\left(t\right){\int }_0^{t}h\left(t-s\right)\mathsf{\Delta }u\left(s\right)dsdx\right\}+\xi \left(t\right)\left({\int }_{\mathsf{\Omega }}{\left|u\left(t\right)\right|}^{\gamma }ln{\left|u\right|}^{k}dx\right),\hfill \end{array}$$

(58)

then (58) is equivalent to:

$$\begin{array}{cc}\hfill & \frac{d}{dt}\left\{\xi \left(t\right)\mathrm{\Phi }\left(t\right)+\xi \left(t\right){\left(\mathsf{\Delta }{u}_t\left(t\right),\mathsf{\Delta }u\left(t\right)\right)}_{L^2(\mathsf{\Omega })}\right\}\hfill \\ =\hfill & \frac{1}{\rho +1}{\xi }^{\prime }\left(t\right)\left({\int }_{\mathsf{\Omega }}{\left|u_t\right|}^{\rho }{u}_t\left(t\right)u\left(t\right)dx\right)+\frac{1}{\rho +1}\xi \left(t\right){∥u_t\left(t\right)∥}_{\rho +2}^{\rho +2}\hfill \\ \hfill & -{\xi }_0\xi \left(t\right){∥\mathsf{\Delta }u\left(t\right)∥}_{L^2(\mathsf{\Omega })}^2-{\xi }_1\xi \left(t\right){∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^{2\left(\beta +1\right)}\hfill \\ \hfill & +{\xi }^{\prime }\left(t\right){\left(\mathsf{\Delta }{u}_t\left(t\right),\mathsf{\Delta }u\left(t\right)\right)}_{L^2(\mathsf{\Omega })}+\xi \left(t\right){∥\mathsf{\Delta }{u}_t\left(t\right)∥}_{L^2(\mathsf{\Omega })}^2\hfill \\ \hfill & +\xi \left(t\right)\left\{{\int }_{\mathsf{\Omega }}\mathsf{\Delta }u\left(t\right){\int }_0^{t}h\left(t-s\right)\mathsf{\Delta }u\left(s\right)dsdx\right\}+\xi \left(t\right)\left({\int }_{\mathsf{\Omega }}{\left|u\right|}^{\gamma }ln{\left|u\right|}^{k}dx\right)\hfill \end{array}$$

(59)

and:

$$\frac{d}{dt}\left\{\xi \left(t\right)\mathrm{\Psi }\left(t\right)\right\}={\xi }^{\prime }\left(t\right)\mathrm{\Psi }\left(t\right)+\xi \left(t\right)\frac{d}{dt}\left\{\mathrm{\Psi }\left(t\right)\right\}.$$

(60)

On the other hand, we have:

$$\begin{array}{ccc}\frac{d}{dt}\left\{\mathrm{\Psi }\left(t\right)\right\}\hfill & =\hfill & {\int }_{\mathsf{\Omega }}{\int }_0^t\frac{d}{dt}\left\{L_{\alpha }\left(t-s\right)\right\}{\left|\mathsf{\Delta }u\left(t\right)-\mathsf{\Delta }u(s)\right|}^{2}dsdx\hfill \\ & & +{\int }_{\mathsf{\Omega }}{\int }_0^t{L}_{\alpha }\left(t-s\right)\frac{d}{dt}\left\{{\left|\mathsf{\Delta }u\left(t\right)-\mathsf{\Delta }u(s)\right|}^2\right\}dsdx.\hfill \end{array}$$

(61)

By using:

$$\begin{array}{ccc}\hfill \frac{d}{dt}\left\{L_{\alpha }\left(t-s\right)\right\}& =& \frac{d}{dt}\left\{{\displaystyle e^{-\alpha \left(t-s\right)}{\int }_{\left(t-s\right)}^{+\infty }l\left(y\right)e^{\alpha y}dy}\right\}\hfill \\ & =& -\alpha e^{-\alpha \left(t-s\right)}{\int }_{\left(t-s\right)}^{+\infty }l\left(y\right)e^{\alpha y}dy-e^{-\alpha \left(t-s\right)}l\left(t-s\right)e^{\alpha \left(t-s\right)}\hfill \\ & =& -\alpha L_{\alpha }\left(t-s\right)-l\left(t-s\right),\hfill \end{array}$$

we get:

$$\begin{array}{cc}\hfill & {\int }_{\mathsf{\Omega }}{\int }_0^t\frac{d}{dt}\left\{L_{\alpha }\left(t-s\right)\right\}{\left|\mathsf{\Delta }u\left(t\right)-\mathsf{\Delta }u(s)\right|}^{2}dsdx\hfill \\ =\hfill & {\int }_{\mathsf{\Omega }}{\int }_0^t\left\{-\alpha L_{\alpha }\left(t-s\right)-l\left(t-s\right)\right\}{\left|\mathsf{\Delta }u\left(t\right)-\mathsf{\Delta }u(s)\right|}^{2}dsdx\hfill \\ =\hfill & -\alpha {\int }_{\mathsf{\Omega }}{\int }_0^t{L}_{\alpha }\left(t-s\right){\left|\mathsf{\Delta }u\left(t\right)-\mathsf{\Delta }u(s)\right|}^{2}dsdx\hfill \\ \hfill & -{\int }_{\mathsf{\Omega }}{\int }_0^{t}l\left(t-s\right){\left|\mathsf{\Delta }u\left(t\right)-\mathsf{\Delta }u(s)\right|}^{2}dsdx\hfill \\ =\hfill & -\alpha \mathrm{\Psi }\left(t\right)-(l\square \mathsf{\Delta }u)\left(t\right).\hfill \end{array}$$

(62)

By direct calculation, we get:

$$\begin{array}{cc}\hfill & {\int }_{\mathsf{\Omega }}{\int }_0^t{L}_{\alpha }\left(t-s\right)\frac{d}{dt}\left\{{\left|\mathsf{\Delta }u\left(t\right)-\mathsf{\Delta }u(s)\right|}^2\right\}dsdx\hfill \\ =\hfill & 2{\int }_{\mathsf{\Omega }}{\int }_0^t{L}_{\alpha }\left(t-s\right)\mathsf{\Delta }{u}_t\left(t\right)\left(\mathsf{\Delta }u\left(t\right)-\mathsf{\Delta }u(s)\right)dsdx\hfill \\ =\hfill & 2\left({\int }_0^t{L}_{\alpha }(s)ds\right){\int }_{\mathsf{\Omega }}\mathsf{\Delta }u\left(t\right).\mathsf{\Delta }{u}_t\left(t\right)dx\hfill \\ \hfill & -2\left\{{\int }_{\mathsf{\Omega }}\mathsf{\Delta }{u}_t\left(t\right){\int }_0^t{L}_{\alpha }\left(t-s\right)\mathsf{\Delta }u(s)dsdx\right\}.\hfill \end{array}$$

(63)

Replacing (62) and (63) in (61), we get:

$$\begin{array}{ccc}\frac{d}{dt}\left\{\mathrm{\Psi }\left(t\right)\right\}\hfill & =\hfill & -\alpha \mathrm{\Psi }\left(t\right)-(l\square \mathsf{\Delta }u)\left(t\right)+2\left({\int }_0^t{L}_{\alpha }(s)ds\right){\int }_{\mathsf{\Omega }}\mathsf{\Delta }u\left(t\right).\mathsf{\Delta }{u}_t\left(t\right)dx\hfill \\ & & -2\left\{{\int }_{\mathsf{\Omega }}\mathsf{\Delta }{u}_t\left(t\right){\int }_0^t{L}_{\alpha }\left(t-s\right)\mathsf{\Delta }u(s)dsdx\right\}.\hfill \end{array}$$

(64)

By replacing (64) in (60), we get:

$$\begin{array}{cc}\hfill & \frac{d}{dt}\left\{\xi \left(t\right)\mathrm{\Psi }\left(t\right)\right\}\hfill \\ =\hfill & {\xi }^{\prime }\left(t\right)\mathrm{\Psi }\left(t\right)-\alpha \xi \left(t\right)\mathrm{\Psi }\left(t\right)-\xi \left(t\right)(l\square \mathsf{\Delta }u)\left(t\right)\hfill \\ \hfill & +2\left({\int }_0^t{L}_{\alpha }(s)ds\right)\xi \left(t\right){\int }_{\mathsf{\Omega }}\mathsf{\Delta }u\left(t\right).\mathsf{\Delta }{u}_t\left(t\right)dx\hfill \\ \hfill & -2\xi \left(t\right)\left\{{\int }_{\mathsf{\Omega }}\mathsf{\Delta }{u}_t\left(t\right){\int }_0^t{L}_{\alpha }\left(t-s\right)\mathsf{\Delta }u(s)dsdx\right\}.\hfill \end{array}$$

(65)

By using direct calculation, we get:

$$\begin{array}{cc}\hfill & 2\left({\int }_0^t{L}_{\alpha }(s)ds\right)\xi \left(t\right){\int }_{\mathsf{\Omega }}\mathsf{\Delta }u\left(t\right).\mathsf{\Delta }{u}_t\left(t\right)dx\hfill \\ =\hfill & \left({\int }_0^t{L}_{\alpha }(s)ds\right)\xi \left(t\right)\frac{d}{dt}\left\{{∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2\right\}\hfill \\ =\hfill & \frac{d}{dt}\left\{\left({\int }_0^t{L}_{\alpha }(s)ds\right)\xi \left(t\right){∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2\right\}\hfill \\ \hfill & -\frac{d}{dt}\left\{\left({\int }_0^t{L}_{\alpha }(s)ds\right)\xi \left(t\right)\right\}{∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2\hfill \\ =\hfill & \frac{d}{dt}\left\{\left({\int }_0^t{L}_{\alpha }(s)ds\right)\xi \left(t\right){∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2\right\}\hfill \\ \hfill & -\left\{L_{\alpha }\left(t\right)\xi \left(t\right)+\left({\int }_0^t{L}_{\alpha }(s)ds\right){\xi }^{\prime }\left(t\right)\right\}{∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2.\hfill \end{array}$$

(66)

By using direct calculation and ${\displaystyle \frac{d}{dt}}\left\{L_{\alpha }\left(t\right)\right\}=-\alpha L_{\alpha }\left(t\right)-l\left(t\right)$, we get:

$$\begin{array}{cc}\hfill & -2\xi \left(t\right)\left\{{\int }_{\mathsf{\Omega }}\mathsf{\Delta }{u}_t\left(t\right){\int }_0^t{L}_{\alpha }\left(t-s\right)\mathsf{\Delta }u(s)dsdx\right\}\hfill \\ =\hfill & -2\xi \left(t\right)\left\{{\int }_{\mathsf{\Omega }}\frac{d}{dt}\left\{\mathsf{\Delta }u\left(t\right)\right\}{\int }_0^t{L}_{\alpha }\left(t-s\right)\mathsf{\Delta }u(s)dsdx\right\}\hfill \\ =\hfill & -2\frac{d}{dt}\left\{\xi \left(t\right)\left({\int }_{\mathsf{\Omega }}\mathsf{\Delta }u\left(t\right){\int }_0^t{L}_{\alpha }\left(t-s\right)\mathsf{\Delta }u(s)dsdx\right)\right\}\hfill \\ \hfill & +2{\xi }^{\prime }\left(t\right)\left\{{\int }_{\mathsf{\Omega }}\mathsf{\Delta }u\left(t\right){\int }_0^t{L}_{\alpha }\left(t-s\right)\mathsf{\Delta }u(s)dsdx\right\}\hfill \\ \hfill & -2\alpha \xi \left(t\right)\left\{{\int }_{\mathsf{\Omega }}\mathsf{\Delta }u\left(t\right){\int }_0^t{L}_{\alpha }\left(t-s\right)\mathsf{\Delta }u(s)dsdx\right\}\hfill \\ \hfill & -2\xi \left(t\right)\left\{{\int }_{\mathsf{\Omega }}\mathsf{\Delta }u\left(t\right){\int }_0^{t}l\left(t-s\right)\mathsf{\Delta }u(s)dsdx\right\}\hfill \\ \hfill & +2L_{\alpha }\left(0\right)\xi \left(t\right){∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2.\hfill \end{array}$$

(67)

Using (66) and (67) in (65), we see that:

$$\begin{array}{cc}\hfill & \frac{d}{dt}\left\{\xi \left(t\right)\mathrm{\Psi }\left(t\right)-\left({\int }_0^t{L}_{\alpha }(s)ds\right)\xi \left(t\right){∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2\right.\hfill \\ \hfill & \left.+2\xi \left(t\right)\left({\int }_{\mathsf{\Omega }}\mathsf{\Delta }u\left(t\right){\int }_0^t{L}_{\alpha }\left(t-s\right)\mathsf{\Delta }u(s)dsdx\right)\right\}\hfill \\ =\hfill & \left\{{\xi }^{\prime }\left(t\right)-\alpha \xi \left(t\right)\right\}\mathrm{\Psi }\left(t\right)-\xi \left(t\right)(l\square \mathsf{\Delta }u)\left(t\right)\hfill \\ \hfill & -\left\{L_{\alpha }\left(t\right)\xi \left(t\right)+\left({\int }_0^t{L}_{\alpha }(s)ds\right){\xi }^{\prime }\left(t\right)-2L_{\alpha }\left(0\right)\xi \left(t\right)\right\}{∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2\hfill \\ \hfill & +2\left({\xi }^{\prime }\left(t\right)-\alpha \xi \left(t\right)\right)\left\{{\int }_{\mathsf{\Omega }}\mathsf{\Delta }u\left(t\right){\int }_0^t{L}_{\alpha }\left(t-s\right)\mathsf{\Delta }u(s)dsdx\right\}\hfill \\ \hfill & -2\xi \left(t\right)\left\{{\int }_{\mathsf{\Omega }}\mathsf{\Delta }u\left(t\right){\int }_0^{t}l\left(t-s\right)\mathsf{\Delta }u(s)dsdx\right\}.\hfill \end{array}$$

(68)

Taking into account (20), (59), and (68) in (50), we obtain:

$$\begin{array}{cc}\hfill & \frac{d}{dt}\left\{V\left(t\right)+k_1\xi \left(0\right)\left({\int }_0^t{∥u_s\left(s\right)∥}_{\rho +2}^{\rho +2}ds\right)+k_2\xi \left(0\right)\left({\int }_0^t{∥\mathsf{\Delta }u\left(s\right)∥}_{L^2(\mathsf{\Omega })}^{2}ds\right)\right.\hfill \\ \hfill & +k_3\xi \left(0\right)\left({\int }_0^t{∥\mathsf{\Delta }{u}_s\left(s\right)∥}_{L^2(\mathsf{\Omega })}^{2}ds\right)+k_4\xi \left(0\right)\left({\int }_0^t{∥u\left(s\right)∥}_{\gamma }^{\gamma }ds\right)\hfill \\ \hfill & \left.+k_5\xi \left(0\right)\left({\int }_0^t\left(L_{\alpha }^r\square \mathsf{\Delta }u\right)\left(s\right)ds\right)\right\}\hfill \\ =\hfill & \frac{1}{2}\left(h^{\prime }\square \mathsf{\Delta }u\right)\left(t\right)-\frac{1}{2}h\left(t\right){∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2+\frac{\epsilon }{\rho +1}{\xi }^{\prime }\left(t\right)\left({\int }_{\mathsf{\Omega }}{\left|u_t\right|}^{\rho }{u}_t\left(t\right)u\left(t\right)dx\right)\hfill \\ \hfill & +\frac{\epsilon }{\rho +1}\xi \left(t\right){∥u_t\left(t\right)∥}_{\rho +2}^{\rho +2}-\epsilon {\xi }_0\xi \left(t\right){∥\mathsf{\Delta }u\left(t\right)∥}_{L^2(\mathsf{\Omega })}^2-\epsilon {\xi }_1\xi \left(t\right){∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^{2\left(\beta +1\right)}\hfill \\ \hfill & +\epsilon {\xi }^{\prime }\left(t\right){\left(\mathsf{\Delta }{u}_t\left(t\right),\mathsf{\Delta }u\left(t\right)\right)}_{L^2(\mathsf{\Omega })}+\epsilon \xi \left(t\right){∥\mathsf{\Delta }{u}_t\left(t\right)∥}_{L^2(\mathsf{\Omega })}^2\hfill \\ \hfill & +\epsilon \xi \left(t\right)\left\{{\int }_{\mathsf{\Omega }}\mathsf{\Delta }u\left(t\right){\int }_0^{t}h\left(t-s\right)\mathsf{\Delta }u\left(s\right)dsdx\right\}+\epsilon \xi \left(t\right)\left({\int }_{\mathsf{\Omega }}{\left|u\left(t\right)\right|}^{\gamma }ln{\left|u\right|}^{k}dx\right)\hfill \\ \hfill & +\eta \left\{{\xi }^{\prime }\left(t\right)-\alpha \xi \left(t\right)\right\}\mathrm{\Psi }\left(t\right)-\eta \xi \left(t\right)(l\square \mathsf{\Delta }u)\left(t\right)\hfill \\ \hfill & -\eta \left\{L_{\alpha }\left(t\right)\xi \left(t\right)+\left({\int }_0^t{L}_{\alpha }(s)ds\right){\xi }^{\prime }\left(t\right)-2L_{\alpha }\left(0\right)\xi \left(t\right)\right\}{∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2\hfill \\ \hfill & +2\eta \left({\xi }^{\prime }\left(t\right)-\alpha \xi \left(t\right)\right)\left\{{\int }_{\mathsf{\Omega }}\mathsf{\Delta }u\left(t\right){\int }_0^t{L}_{\alpha }\left(t-s\right)\mathsf{\Delta }u(s)dsdx\right\}\hfill \\ \hfill & -2\eta \xi \left(t\right)\left\{{\int }_{\mathsf{\Omega }}\mathsf{\Delta }u\left(t\right){\int }_0^{t}l\left(t-s\right)\mathsf{\Delta }u(s)dsdx\right\}\hfill \\ \hfill & -k_1\xi \left(0\right){∥u_t\left(t\right)∥}_{\rho +2}^{\rho +2}-k_2\xi \left(0\right){∥\mathsf{\Delta }u\left(t\right)∥}_{L^2(\mathsf{\Omega })}^2-k_3\xi \left(0\right){∥\mathsf{\Delta }{u}_t\left(t\right)∥}_{L^2(\mathsf{\Omega })}^2\hfill \\ \hfill & -k_4\xi \left(0\right){∥u\left(t\right)∥}_{\gamma }^{\gamma }-k_5\xi \left(0\right)\left(L_{\alpha }^r\square \mathsf{\Delta }u\right)\left(t\right).\hfill \end{array}$$

(69)

Next, we use the estimate (69).

By multiplying (34) by $\epsilon {\xi }^{\prime }\left(t\right)$ and $\left|{\displaystyle \frac{{\xi }^{\prime }\left(t\right)}{\xi \left(t\right)}}\right|\le L,$ we get:

$$\begin{array}{cc}\hfill & \frac{\epsilon }{\rho +1}{\xi }^{\prime }\left(t\right)\left({\int }_{\mathsf{\Omega }}{\left|u_t\right|}^{\rho }{u}_t\left(t\right)u\left(t\right)dx\right)\hfill \\ \le \hfill & \frac{\epsilon L{\delta }_1{\left[\left(\rho +2\right)e\left(0\right)\right]}^{\frac{\rho }{\left(\rho +2\right)}}}{2{\left(\rho +1\right)}^2}\xi \left(t\right){∥u_t\left(t\right)∥}_{\rho +2}^{\rho +2}+\frac{\epsilon L{C}_0{C}_p}{2{\delta }_1}\xi \left(t\right){∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2.\hfill \end{array}$$

(70)

By using Young’s inequality $\left(\mathrm{for}\epsilon ={\delta }_2\right)$ and $\left|{\displaystyle \frac{{\xi }^{\prime }\left(t\right)}{\xi \left(t\right)}}\right|\le L$, we get:

$$\begin{array}{cc}\hfill & \epsilon {\xi }^{\prime }\left(t\right){\left(\mathsf{\Delta }{u}_t\left(t\right),\mathsf{\Delta }u\left(t\right)\right)}_{L^2(\mathsf{\Omega })}\hfill \\ \le \hfill & \frac{\epsilon L{\delta }_2}{2}\xi \left(t\right){∥\mathsf{\Delta }{u}_t\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2+\frac{\epsilon L}{2{\delta }_2}\xi \left(t\right){∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2.\hfill \end{array}$$

(71)

By using Young’s inequality $\left(\mathrm{for}\epsilon ={\eta }_1\right)$ and $\left({\displaystyle {\int }_0^{t}h\left(s\right)ds}\right)\le \overline{h}$, we get:

$$\begin{array}{cc}& \xi (t){\int }_{\mathsf{\Omega }}\mathsf{\Delta }u(t)\left({\int }_0^{t}h\left(t-s\right)\mathsf{\Delta }u(s)ds\right)dx\hfill \\ \le & \frac{1}{4}\xi (t){∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2\hfill \\ & +\xi (t){\int }_{\mathsf{\Omega }}{\left({\int }_0^{t}h\left(t-s\right)(\left|\mathsf{\Delta }u(s)-\mathsf{\Delta }u(t)\right|+\left|\mathsf{\Delta }u(t)\right|)ds\right)}^{2}dx\hfill \\ \le & \frac{1}{4}\xi (t){∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2+(1+{\eta }_1){\overline{h}}^2\xi (t){∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2\hfill \\ & +\left(1+\frac{1}{{\eta }_1}\right)\xi (t)\left({\int }_0^t{h}^{2-r}(s)ds\right)\left(h^r\square \mathsf{\Delta }u\right)\left(t\right)\hfill \\ =& \left(\frac{1}{4}+(1+{\eta }_1){\overline{h}}^2\right)\xi (t){∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2\hfill \\ & +\left(1+\frac{1}{{\eta }_1}\right)\xi (t)\left({\int }_0^t{h}^{2-r}(s)ds\right)\left(h^r\square \mathsf{\Delta }u\right)\left(t\right).\hfill \end{array}$$

Choosing ${\eta }_1:={\displaystyle \frac{\left({\xi }_0-\overline{h}\right)}{\overline{h}}},$ hence $\left(1+{\eta }_1\right){\overline{h}}^2=\overline{h}$ and $\left(1+{\displaystyle \frac{1}{{\eta }_1}}\right)={\displaystyle \frac{1}{\left({\xi }_0-\overline{h}\right)}},$ we get:

$$\begin{array}{cc}\hfill & \epsilon \xi (t){\int }_{\mathsf{\Omega }}\mathsf{\Delta }u(t)\left({\int }_0^{t}h\left(t-s\right)\mathsf{\Delta }u(s)ds\right)dx\hfill \\ \le \hfill & \epsilon \left(\frac{1}{4}+\overline{h}\right)\xi (t){∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2\hfill \\ \hfill & +{\displaystyle \frac{\epsilon }{\left({\xi }_0-\overline{h}\right)}}\xi (t)\left({\int }_0^t{h}^{2-r}(s)ds\right)\left(h^r\square \mathsf{\Delta }u\right)\left(t\right).\hfill \end{array}$$

(72)

By using Young’s inequality $\left(\mathrm{for}\epsilon =1\right)$ and $\left({\displaystyle {\int }_0^t{L}_{\alpha }(s)ds}\right)\le {\displaystyle \frac{{\overline{l}}_{\alpha }}{\alpha }}$, we get:

$$\begin{array}{cc}\hfill & {\int }_{\mathsf{\Omega }}\mathsf{\Delta }u(t)\left({\int }_0^t{L}_{\alpha }\left(t-s\right)\mathsf{\Delta }u(s)ds\right)dx\hfill \\ \le \hfill & \frac{1}{4}{∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2\hfill \\ \hfill & +{\int }_{\mathsf{\Omega }}{\left({\int }_0^t{L}_{\alpha }\left(t-s\right)(\left|\mathsf{\Delta }u(s)-\mathsf{\Delta }u(t)\right|+\left|\mathsf{\Delta }u(t)\right|)ds\right)}^{2}dx\hfill \\ \le \hfill & \frac{1}{4}{∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2+2{\left({\displaystyle \frac{{\overline{l}}_{\alpha }}{\alpha }}\right)}^2{∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2\hfill \\ \hfill & +2\left({\int }_0^t{L}_{\alpha }^{2-r}(s)ds\right)\left(L_{\alpha }^r\square \mathsf{\Delta }u\right)\left(t\right)\hfill \\ =\hfill & \left(\frac{1}{4}+2{\left({\displaystyle \frac{{\overline{l}}_{\alpha }}{\alpha }}\right)}^2\right){∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2+2\left({\int }_0^t{L}_{\alpha }^{2-r}(s)ds\right)\left(L_{\alpha }^r\square \mathsf{\Delta }u\right)\left(t\right),\hfill \end{array}$$

(73)

Multiplying (73) by $2\eta \left({\xi }^{\prime }\left(t\right)-\alpha \xi \left(t\right)\right)$ and using $\left|{\displaystyle \frac{{\xi }^{\prime }\left(t\right)}{\xi \left(t\right)}}\right|\le L$, we get:

$$\begin{array}{cc}\hfill & 2\eta \left({\xi }^{\prime }\left(t\right)-\alpha \xi \left(t\right)\right){\int }_{\mathsf{\Omega }}\mathsf{\Delta }u(t)\left({\int }_0^t{L}_{\alpha }\left(t-s\right)\mathsf{\Delta }u(s)ds\right)dx\hfill \\ \le \hfill & \eta \left(L+\alpha \right)\left(\frac{1}{2}+4{\left({\displaystyle \frac{{\overline{l}}_{\alpha }}{\alpha }}\right)}^2\right)\xi (t){∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2\hfill \\ \hfill & +4\eta \left(L+\alpha \right)\left({\int }_0^t{L}_{\alpha }^{2-r}(s)ds\right)\xi (t)\left(L_{\alpha }^r\square \mathsf{\Delta }u\right)\left(t\right).\hfill \end{array}$$

(74)

Similarly, By using Young’s inequality $\left(\mathrm{for}\epsilon ={\delta }_3\right),$ we get:

$$\begin{array}{cc}\hfill & -2\eta \xi \left(t\right)\left\{{\int }_{\mathsf{\Omega }}\mathsf{\Delta }u\left(t\right){\int }_0^{t}l\left(t-s\right)\mathsf{\Delta }u(s)dsdx\right\}\hfill \\ \le \hfill & \frac{\eta }{2{\delta }_3}\xi \left(t\right)\left(l\square \mathsf{\Delta }u\right)\left(t\right)+2\eta \left(1+{\delta }_3\right)\overline{l}\xi \left(t\right){∥\mathsf{\Delta }u\left(t\right)∥}_{L^2(\mathsf{\Omega })}^2.\hfill \end{array}$$

(75)

Making use of (70)–(72), (74) and (75) into (69), we get:

$$\begin{array}{cc}\hfill & \frac{d}{dt}\left\{V\left(t\right)+k_1\xi \left(0\right)\left({\int }_0^t{∥u_s\left(s\right)∥}_{\rho +2}^{\rho +2}ds\right)+k_2\xi \left(0\right)\left({\int }_0^t{∥\mathsf{\Delta }u\left(s\right)∥}_{L^2(\mathsf{\Omega })}^{2}ds\right)\right.\hfill \\ \hfill & +k_3\xi \left(0\right)\left({\int }_0^t{∥\mathsf{\Delta }{u}_s\left(s\right)∥}_{L^2(\mathsf{\Omega })}^{2}ds\right)+k_4\xi \left(0\right)\left({\int }_0^t{∥u\left(s\right)∥}_{\gamma }^{\gamma }ds\right)\hfill \\ \hfill & \left.+k_5\xi \left(0\right)\left({\int }_0^t\left(L_{\alpha }^r\square \mathsf{\Delta }u\right)\left(s\right)ds\right)\right\}\hfill \\ \le \hfill & -\left\{k_1-\frac{\epsilon L{\delta }_1{\left[\left(\rho +2\right)e\left(0\right)\right]}^{\frac{\rho }{\left(\rho +2\right)}}}{2{\left(\rho +1\right)}^2}-\frac{\epsilon }{\left(\rho +1\right)}\right\}\xi \left(t\right){∥u_t\left(t\right)∥}_{\rho +2}^{\rho +2}\hfill \\ \hfill & -\left\{k_2+\epsilon \left(\left({\xi }_0-\overline{h}\right)-\frac{L{C}_0{C}_p}{2{\delta }_1}-\frac{L}{2{\delta }_2}-\frac{1}{4}\right)\right.-\frac{\eta {\overline{l}}_{\alpha }L}{\alpha }-2\eta {\overline{l}}_{\alpha }\hfill \\ \hfill & -\left.\eta \left(L+\alpha \right)\left(\frac{1}{2}+4{\left({\displaystyle \frac{{\overline{l}}_{\alpha }}{\alpha }}\right)}^2\right)-2\eta \left(1+{\delta }_3\right)\overline{l}\right\}\xi \left(t\right){∥\mathsf{\Delta }u\left(t\right)∥}_{L^2(\mathsf{\Omega })}^2\hfill \\ \hfill & -\epsilon {\xi }_1\xi \left(t\right){∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^{2\left(\beta +1\right)}-\left\{k_3-\frac{\epsilon L{\delta }_2}{2}-\epsilon \right\}\xi \left(t\right){∥\mathsf{\Delta }{u}_t\left(t\right)∥}_{L^2(\mathsf{\Omega })}^2\hfill \\ \hfill & +\epsilon \xi \left(t\right)\left({\int }_{\mathsf{\Omega }}{\left|u\right|}^{\gamma }ln{\left|u\right|}^{k}dx\right)-k_4\xi \left(t\right){∥u\left(t\right)∥}_{\gamma }^{\gamma }\hfill \\ \hfill & -\left\{\frac{1}{2}-{\displaystyle \frac{\epsilon }{\left({\xi }_0-\overline{h}\right)}}\left({\int }_0^t{h}^{2-r}(s)ds\right)\right\}\xi \left(t\right)\left(h^r\square \mathsf{\Delta }u\right)\left(t\right)\hfill \\ \hfill & -\left\{k_5-4\eta \left(L+\alpha \right)\left({\int }_0^t{L}_{\alpha }^{2-r}(s)ds\right)\right\}\xi \left(t\right)\left(L_{\alpha }^r\square \mathsf{\Delta }u\right)\left(t\right)\hfill \\ \hfill & -\left\{\eta \left(1-\frac{1}{2{\delta }_3}\right)-\frac{1}{2}\right\}\xi \left(t\right)\left(l\square \mathsf{\Delta }u\right)\left(t\right).\hfill \end{array}$$

(76)

Finally, we choose:

$$\left\{\begin{array}{c}{\delta }_1:={\displaystyle \frac{3L{C}_0{C}_p}{2\left({\xi }_0-\overline{h}\right)}},\hfill \\ {\delta }_2:={\displaystyle \frac{3L}{2\left({\xi }_0-\overline{h}\right)}},\hfill \\ {\delta }_3:=1\hfill \end{array}\right.$$

(77)

and:

$$\left\{\begin{array}{c}{k}_1:={\displaystyle \frac{\epsilon }{\rho +1}}\left({\displaystyle \frac{L{\delta }_1{\left[\left(\rho +2\right)e\left(0\right)\right]}^{\frac{\rho }{\left(\rho +2\right)}}}{\left(\rho +1\right)}}+2\right),\hfill \\ k_2:={\displaystyle \frac{\epsilon }{4}}+\left(L+\alpha \right)\left({\displaystyle \frac{1}{2}}+4{\left({\displaystyle \frac{{\overline{l}}_{\alpha }}{\alpha }}\right)}^2\right)+4\overline{l},\hfill \\ k_3:=\epsilon \left(L{\delta }_2+2\right),\hfill \\ k_5:=8\left(L+\alpha \right)\left({\displaystyle {\int }_0^t{L}_{\alpha }^{2-r}(s)ds}\right)\hfill \\ \eta :=1,\hfill \\ \epsilon <{\displaystyle \frac{\left({\xi }_0-\overline{h}\right)}{4\left({\displaystyle {\int }_0^t{h}^{2-r}(s)ds}\right)}}.\hfill \end{array}\right.$$

(78)

Then, if ${\overline{l}}_{\alpha }<{\displaystyle \frac{\alpha \left({\xi }_0-\overline{h}\right)}{12\alpha +6L}}\epsilon ,$ we require from (76) that:

$$\begin{array}{cc}\hfill & \frac{d}{dt}\left\{V\left(t\right)+k_1\xi \left(0\right)\left({\int }_0^t{∥u_s\left(s\right)∥}_{\rho +2}^{\rho +2}ds\right)+k_2\xi \left(0\right)\left({\int }_0^t{∥\mathsf{\Delta }u\left(s\right)∥}_{L^2(\mathsf{\Omega })}^{2}ds\right)\right.\hfill \\ \hfill & +k_3\xi \left(0\right)\left({\int }_0^t{∥\mathsf{\Delta }{u}_s\left(s\right)∥}_{L^2(\mathsf{\Omega })}^{2}ds\right)+k_4\xi \left(0\right)\left({\int }_0^t{∥u\left(s\right)∥}_{\gamma }^{\gamma }ds\right)\hfill \\ \hfill & \left.+k_5\xi \left(0\right)\left({\int }_0^t\left(L_{\alpha }^r\square \mathsf{\Delta }u\right)\left(s\right)ds\right)\right\}\hfill \\ \le \hfill & -\frac{k_1}{2}\xi \left(t\right){∥u_t\left(t\right)∥}_{\rho +2}^{\rho +2}-\frac{\epsilon \left({\xi }_0-\overline{h}\right)}{6}\xi \left(t\right){∥\mathsf{\Delta }u\left(t\right)∥}_{L^2(\mathsf{\Omega })}^2\hfill \\ \hfill & -\epsilon {\xi }_1\xi \left(t\right){∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^{2\left(\beta +1\right)}-\frac{k_3}{2}\xi \left(t\right){∥\mathsf{\Delta }{u}_t\left(t\right)∥}_{L^2(\mathsf{\Omega })}^2\hfill \\ \hfill & +\epsilon \xi \left(t\right)\left({\int }_{\mathsf{\Omega }}{\left|u\right|}^{\gamma }ln{\left|u\right|}^{k}dx\right)-k_4\xi \left(t\right){∥u\left(t\right)∥}_{\gamma }^{\gamma }\hfill \\ \hfill & -\frac{1}{4}\xi \left(t\right)\left(h\square \mathsf{\Delta }u\right)\left(t\right)-\frac{k_5}{2}\xi \left(t\right)\left(L_{\alpha }^r\square \mathsf{\Delta }u\right)\left(t\right).\hfill \end{array}$$

(79)

By using (9) in (79), we get:

$$\begin{array}{cc}\hfill & \frac{d}{dt}\left\{V\left(t\right)+k_1\xi \left(0\right)\left({\int }_0^t{∥u_s\left(s\right)∥}_{\rho +2}^{\rho +2}ds\right)\right.\hfill \\ \hfill & +k_2\xi \left(0\right)\left({\int }_0^t{∥\mathsf{\Delta }u\left(s\right)∥}_{L^2(\mathsf{\Omega })}^{2}ds\right)\hfill \\ \hfill & +k_3\xi \left(0\right)\left({\int }_0^t{∥\mathsf{\Delta }{u}_s\left(s\right)∥}_{L^2(\mathsf{\Omega })}^{2}ds\right)+k_4\xi \left(0\right)\left({\int }_0^t{∥u\left(s\right)∥}_{\gamma }^{\gamma }ds\right)\hfill \\ \hfill & \left.+k_5\xi \left(0\right)\left({\int }_0^t\left(L_{\alpha }^r\square \mathsf{\Delta }u\right)\left(s\right)ds\right)\right\}\hfill \\ \le \hfill & -C_3\xi \left(t\right)\left\{E\left(t\right)+\left(h^r\square \mathsf{\Delta }u\right)\left(t\right)+\left(L_{\alpha }^r\square \mathsf{\Delta }u\right)\left(t\right)\right\},C_3>0.\hfill \end{array}$$

(80)

Case 1.

$r=\mathbf{1}$.

From (80), this is written in the form:

$$\begin{array}{cc}\hfill & \frac{d}{dt}\left\{V\left(t\right)+k_1\xi \left(0\right)\left({\int }_0^t{∥u_s\left(s\right)∥}_{\rho +2}^{\rho +2}ds\right)+k_2\xi \left(0\right)\left({\int }_0^t{∥\mathsf{\Delta }u\left(s\right)∥}_{L^2(\mathsf{\Omega })}^{2}ds\right)\right.\hfill \\ \hfill & +k_3\xi \left(0\right)\left({\int }_0^t{∥\mathsf{\Delta }{u}_s\left(s\right)∥}_{L^2(\mathsf{\Omega })}^{2}ds\right)+k_4\xi \left(0\right)\left({\int }_0^t{∥u\left(s\right)∥}_{\gamma }^{\gamma }ds\right)\hfill \\ \hfill & \left.+k_5\xi \left(0\right)\left({\int }_0^t\left(L_{\alpha }^r\square \mathsf{\Delta }u\right)\left(s\right)ds\right)\right\}\hfill \\ \le \hfill & -C_3\xi \left(t\right)\left\{E\left(t\right)+\left(h\square \mathsf{\Delta }u\right)\left(t\right)+\left(L_{\alpha }\square \mathsf{\Delta }u\right)\left(t\right)\right\}\hfill \end{array}$$

(81)

By virtue of Proposition 1 (the right-hand side inequality) in (81), we find for all $t\ge 0$:

$$\begin{array}{cc}\hfill & \frac{d}{dt}\left\{V\left(t\right)+k_1\xi \left(0\right)\left({\int }_0^t{∥u_s\left(s\right)∥}_{\rho +2}^{\rho +2}ds\right)+k_2\xi \left(0\right)\left({\int }_0^t{∥\mathsf{\Delta }u\left(s\right)∥}_{L^2(\mathsf{\Omega })}^{2}ds\right)\right.\hfill \\ \hfill & +k_3\xi \left(0\right)\left({\int }_0^t{∥\mathsf{\Delta }{u}_s\left(s\right)∥}_{L^2(\mathsf{\Omega })}^{2}ds\right)+k_4\xi \left(0\right)\left({\int }_0^t{∥u\left(s\right)∥}_{\gamma }^{\gamma }ds\right)\hfill \\ \hfill & \left.+k_5\xi \left(0\right)\left({\int }_0^t\left(L_{\alpha }^r\square \mathsf{\Delta }u\right)\left(s\right)ds\right)\right\}\hfill \\ \le \hfill & -\frac{C_3}{C_2}\xi \left(t\right)\left\{V\left(t\right)+k_1\xi \left(0\right)\left({\int }_0^t{∥u_s\left(s\right)∥}_{\rho +2}^{\rho +2}ds\right)+k_2\xi \left(0\right)\left({\int }_0^t{∥\mathsf{\Delta }u\left(s\right)∥}_{L^2(\mathsf{\Omega })}^{2}ds\right)\right.\hfill \\ \hfill & +k_3\xi \left(0\right)\left({\int }_0^t{∥\mathsf{\Delta }{u}_s\left(s\right)∥}_{L^2(\mathsf{\Omega })}^{2}ds\right)+k_4\xi \left(0\right)\left({\int }_0^t{∥u\left(s\right)∥}_{\gamma }^{\gamma }ds\right)\hfill \\ \hfill & \left.+k_5\xi \left(0\right)\left({\int }_0^t\left(L_{\alpha }^r\square \mathsf{\Delta }u\right)\left(s\right)ds\right)\right\}.\hfill \end{array}$$

(82)

By a simple integration in (82) over $\left(0,t\right),$ we find:

$$\begin{array}{cc}& V\left(t\right)+k_1\xi \left(0\right)\left({\int }_0^t{∥u_s\left(s\right)∥}_{\rho +2}^{\rho +2}ds\right)+k_2\xi \left(0\right)\left({\int }_0^t{∥\mathsf{\Delta }u\left(s\right)∥}_{L^2(\mathsf{\Omega })}^{2}ds\right)\hfill \\ & +k_3\xi \left(0\right)\left({\int }_0^t{∥\mathsf{\Delta }{u}_s\left(s\right)∥}_{L^2(\mathsf{\Omega })}^{2}ds\right)+k_4\xi \left(0\right)\left({\int }_0^t{∥u\left(s\right)∥}_{\gamma }^{\gamma }ds\right)\hfill \\ & +k_5\xi \left(0\right)\left({\int }_0^t\left(L_{\alpha }^r\square \mathsf{\Delta }u\right)\left(s\right)ds\right)\hfill \\ \le & V\left(0\right)e^{-{\displaystyle \frac{C_3}{C_2}}\left({\displaystyle {\int }_0^t\xi \left(s\right)ds}\right)},t\ge 0.\hfill \end{array}$$

Notice that by our assumption $E\left(0\right)>0$ in the theorem, we have $V\left(0\right)>0$. Again, by Proposition 1 (the left-hand side inequality), we conclude the assertion of our theorem:

$$E\left(t\right)\le \frac{V\left(0\right)}{C_1}{e}^{-{\displaystyle \frac{C_3}{C_2}}\left({\displaystyle {\int }_0^t\xi \left(s\right)ds}\right)},t\ge 0.$$

Case 2.

$\mathbf{1}<r<{\displaystyle \frac{3}{2}}.$ Therefore, Lemma 4 $\left(\mathrm{for}\theta :=\tau \mathrm{and}\rho :=r\right)$ and $\left(A2\right)$ yield:

$$\begin{array}{ccc}\hfill (h\square \mathsf{\Delta }u)(t)& \le & {\gamma }_1{\left(e\left(0\right){\int }_0^{\infty }{h}^{1-\tau }(s)ds\right)}^{{\displaystyle \frac{r-1}{r-1+\tau }}}{\left((h^r\square \mathsf{\Delta }u)(t)\right)}^{{\displaystyle \frac{\tau }{r-1+\tau }}}\hfill \\ & \le & {\gamma }_2{((h^r\square \mathsf{\Delta }u)(t))}^{{\displaystyle \frac{\tau }{r-1+\tau }}}.\hfill \end{array}$$

Similarly, using Lemma 4 $\left(\mathrm{for}\theta :=\tau \mathrm{and}\rho :=r\right)$ and (30) yields:

$$(L_{\alpha }\square \mathsf{\Delta }u)(t)\le {\gamma }_2^{\prime }{((L_{\alpha }^r\square \mathsf{\Delta }u)(t))}^{{\displaystyle \frac{\tau }{r-1+\tau }}},$$

for some ${\gamma }_1$ and ${\gamma }_2$ positive constants. Therefore, for any $r_1>1,$ we arrive at:

$$\begin{array}{cc}\hfill & {\left\{E\left(t\right)+\left(h\square \mathsf{\Delta }u\right)\left(t\right)+\left(L_{\alpha }\square \mathsf{\Delta }u\right)\left(t\right)\right\}}^{r_1}\hfill \\ \le \hfill & {\gamma }_3{e}^{r_1-1}(0)\left({∥u_t\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2+{∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2+{∥u\left(t\right)∥}_{\gamma +2}^{\gamma +2}\right)\hfill \\ \hfill & +{\gamma }_4{\left(\left(h\square \mathsf{\Delta }u\right)\left(t\right)\right)}^{r_1}+{\gamma }_4{\left(\left(L_{\alpha }\square \mathsf{\Delta }u\right)\left(t\right)\right)}^{r_1}\hfill \\ \le \hfill & {\gamma }_3{e}^{r_1-1}(0)\left({∥u_t\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2+{∥\mathsf{\Delta }u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2+{∥u\left(t\right)∥}_{\gamma +2}^{\gamma +2}\right)\hfill \\ \hfill & +{\gamma }_5{\left(\left(L_{\alpha }^r\square \mathsf{\Delta }u\right)\left(t\right)\right)}^{{\displaystyle \frac{\tau r_1}{r-1+\tau }}}+{\gamma }_5{\left(\left(h^r\square \mathsf{\Delta }u\right)\left(t\right)\right)}^{{\displaystyle \frac{\tau r_1}{r-1+\tau }}}.\hfill \end{array}$$

(83)

By choosing $\tau ={\displaystyle \frac{1}{2}}$ and $r_1=2r-1$ $\left(\mathrm{hence}{\displaystyle \frac{\tau r_1}{r-1+\tau }}=1\right),$ estimate (83) gives, for some ${\gamma }_6>0,$

$$\begin{array}{cc}\hfill & {\left\{E\left(t\right)+\left(h\square \mathsf{\Delta }u\right)\left(t\right)+\left(L_{\alpha }\square \mathsf{\Delta }u\right)\left(t\right)\right\}}^{r_1}\hfill \\ \le \hfill & {\gamma }_6\left\{E\left(t\right)+((h^r\square \mathsf{\Delta }u)(t))+\left(L_{\alpha }^r\square \mathsf{\Delta }u\right)\left(t\right)\right\}.\hfill \end{array}$$

(84)

By combining (84) and (33) into (80), we obtain:

$$\begin{array}{cc}\hfill & \frac{d}{dt}\left\{V\left(t\right)+k_1\xi \left(0\right)\left({\int }_0^t{∥u_s\left(s\right)∥}_{\rho +2}^{\rho +2}ds\right)+k_2\xi \left(0\right)\left({\int }_0^t{∥\mathsf{\Delta }u\left(s\right)∥}_{L^2(\mathsf{\Omega })}^{2}ds\right)\right.\hfill \\ \hfill & +k_3\xi \left(0\right)\left({\int }_0^t{∥\mathsf{\Delta }{u}_s\left(s\right)∥}_{L^2(\mathsf{\Omega })}^{2}ds\right)+k_4\xi \left(0\right)\left({\int }_0^t{∥u\left(s\right)∥}_{\gamma }^{\gamma }ds\right)\hfill \\ \hfill & +\left.k_5\xi \left(0\right)\left({\int }_0^t\left(L_{\alpha }^r\square \mathsf{\Delta }u\right)\left(s\right)ds\right)\right\}\hfill \\ \le \hfill & -{\displaystyle \frac{C_3}{{\gamma }_6}}\xi \left(t\right){\left\{E\left(t\right)+\left(h\square \mathsf{\Delta }u\right)\left(t\right)+\left(L_{\alpha }\square \mathsf{\Delta }u\right)\left(t\right)\right\}}^{r_1}\hfill \\ \le \hfill & -{\displaystyle \frac{C_3}{{\gamma }_6{\left(C_2\right)}^{r_1}}}\xi \left(t\right)\left\{V\left(t\right)+k_1\xi \left(0\right)\left({\int }_0^t{∥u_s\left(s\right)∥}_{\rho +2}^{\rho +2}ds\right)\right.\hfill \\ \hfill & +k_2\xi \left(0\right)\left({\int }_0^t{∥\mathsf{\Delta }u\left(s\right)∥}_{L^2(\mathsf{\Omega })}^{2}ds\right)+k_3\xi \left(0\right)\left({\int }_0^t{∥\mathsf{\Delta }{u}_s\left(s\right)∥}_{L^2(\mathsf{\Omega })}^{2}ds\right)\hfill \\ \hfill & +{\left.k_4\xi \left(0\right)\left({\int }_0^t{∥u\left(s\right)∥}_{\gamma }^{\gamma }ds\right)+k_5\xi \left(0\right)\left({\int }_0^t\left(L_{\alpha }^r\square \mathsf{\Delta }u\right)\left(s\right)ds\right)\right\}}^{r_1}.\hfill \end{array}$$

(85)

A simple integration of (85) over $\left(0,t\right)$ leads to:

$$\begin{array}{cc}\hfill & V\left(t\right)+k_1\xi \left(0\right)\left({\int }_0^t{∥u_s\left(s\right)∥}_{\rho +2}^{\rho +2}ds\right)+k_2\xi \left(0\right)\left({\int }_0^t{∥\mathsf{\Delta }u\left(s\right)∥}_{L^2(\mathsf{\Omega })}^{2}ds\right)\hfill \\ \hfill & +k_3\xi \left(0\right)\left({\int }_0^t{∥\mathsf{\Delta }{u}_s\left(s\right)∥}_{L^2(\mathsf{\Omega })}^{2}ds\right)+k_4\xi \left(0\right)\left({\int }_0^t{∥u\left(s\right)∥}_{\gamma }^{\gamma }ds\right)\hfill \\ \hfill & +k_5\xi \left(0\right)\left({\int }_0^t\left(L_{\alpha }^r\square \mathsf{\Delta }u\right)\left(s\right)ds\right)\hfill \\ \le \hfill & C_1^{*}{\left({\displaystyle 1+{\int }_0^t\xi (s)ds}\right)}^{-{\displaystyle \frac{1}{r_1-1}}},\forall t\ge 0.\hfill \end{array}$$

(86)

Therefore,

$$\begin{array}{cc}& {\displaystyle {\int }_0^{\infty }\left\{V\left(t\right)+k_1\xi \left(0\right)\left({\int }_0^t{∥u_s\left(s\right)∥}_{\rho +2}^{\rho +2}ds\right)+k_2\xi \left(0\right)\left({\int }_0^t{∥\mathsf{\Delta }u\left(s\right)∥}_{L^2(\mathsf{\Omega })}^{2}ds\right)\right.}\hfill \\ & +k_3\xi \left(0\right)\left({\int }_0^t{∥\mathsf{\Delta }{u}_s\left(s\right)∥}_{L^2(\mathsf{\Omega })}^{2}ds\right)+k_4\xi \left(0\right)\left({\int }_0^t{∥u\left(s\right)∥}_{\gamma }^{\gamma }ds\right)\hfill \\ & \left.+k_5\xi \left(0\right)\left({\int }_0^t\left(L_{\alpha }^r\square \mathsf{\Delta }u\right)\left(s\right)ds\right)\right\}dt\hfill \\ \le & {\displaystyle C_1^{*}{\int }_0^{\infty }\frac{1}{{\left({\displaystyle 1+{\int }_0^t\xi (s)ds}\right)}^{{\displaystyle \frac{1}{r_1-1}}}}dt.}\hfill \end{array}$$

Since ${\displaystyle \frac{1}{r_1-1}}>0$ and $\left({\displaystyle 1+{\int }_0^t\xi (s)ds}\right)\to +\infty$ as $t\to +\infty$, we get:

$$\begin{array}{cc}& {\displaystyle {\int }_0^{\infty }\left\{V\left(t\right)+k_1\xi \left(0\right)\left({\int }_0^t{∥u_s\left(s\right)∥}_{\rho +2}^{\rho +2}ds\right)+k_2\xi \left(0\right)\left({\int }_0^t{∥\mathsf{\Delta }u\left(s\right)∥}_{L^2(\mathsf{\Omega })}^{2}ds\right)\right.}\hfill \\ & +k_3\xi \left(0\right)\left({\int }_0^t{∥\mathsf{\Delta }{u}_s\left(s\right)∥}_{L^2(\mathsf{\Omega })}^{2}ds\right)+k_4\xi \left(0\right)\left({\int }_0^t{∥u\left(s\right)∥}_{\gamma }^{\gamma }ds\right)\hfill \\ & \left.+k_5\xi \left(0\right)\left({\int }_0^t\left(L_{\alpha }^r\square \mathsf{\Delta }u\right)\left(s\right)ds\right)\right\}dt\hfill \\ <& \infty .\hfill \end{array}$$

In addition, by using (7) in (86), we have:

$$\begin{array}{cc}\hfill & t\left\{V\left(t\right)+k_1\xi \left(0\right)\left({\int }_0^t{∥u_s\left(s\right)∥}_{\rho +2}^{\rho +2}ds\right)+k_2\xi \left(0\right)\left({\int }_0^t{∥\mathsf{\Delta }u\left(s\right)∥}_{L^2(\mathsf{\Omega })}^{2}ds\right)\right.\hfill \\ \hfill & +k_3\xi \left(0\right)\left({\int }_0^t{∥\mathsf{\Delta }{u}_s\left(s\right)∥}_{L^2(\mathsf{\Omega })}^{2}ds\right)+k_4\xi \left(0\right)\left({\int }_0^t{∥u\left(s\right)∥}_{\gamma }^{\gamma }ds\right)\hfill \\ \hfill & \left.+k_5\xi \left(0\right)\left({\int }_0^t\left(L_{\alpha }^r\square \mathsf{\Delta }u\right)\left(s\right)ds\right)\right\}\hfill \\ \le \hfill & \frac{C_1^{*}t}{{\left({\displaystyle 1+{\int }_0^t\xi (s)ds}\right)}^{{\displaystyle \frac{1}{r_1-1}}}}\hfill \\ \le \hfill & C_r.\hfill \end{array}$$

(87)

Therefore, we obtain:

$$\begin{array}{cc}& \underset{t\ge 0}{sup}t\left\{V\left(t\right)+k_1\xi \left(0\right)\left({\int }_0^t{∥u_s\left(s\right)∥}_{\rho +2}^{\rho +2}ds\right)+k_2\xi \left(0\right)\left({\int }_0^t{∥\mathsf{\Delta }u\left(s\right)∥}_{L^2(\mathsf{\Omega })}^{2}ds\right)\right.\hfill \\ & +k_3\xi \left(0\right)\left({\int }_0^t{∥\mathsf{\Delta }{u}_s\left(s\right)∥}_{L^2(\mathsf{\Omega })}^{2}ds\right)+k_4\xi \left(0\right)\left({\int }_0^t{∥u\left(s\right)∥}_{\gamma }^{\gamma }ds\right)\hfill \\ & \left.+k_5\xi \left(0\right)\left({\int }_0^t\left(L_{\alpha }^r\square \mathsf{\Delta }u\right)\left(s\right)ds\right)\right\}\hfill \\ <& \infty .\hfill \end{array}$$

We use (87) and (89) to get:

$$\begin{array}{cc}& {\int }_0^{\infty }\left\{V\left(t\right)+k_1\xi \left(0\right)\left({\int }_0^t{∥u_s\left(s\right)∥}_{\rho +2}^{\rho +2}ds\right)+k_2\xi \left(0\right)\left({\int }_0^t{∥\mathsf{\Delta }u\left(s\right)∥}_{L^2(\mathsf{\Omega })}^{2}ds\right)\right.\hfill \\ & +k_3\xi \left(0\right)\left({\int }_0^t{∥\mathsf{\Delta }{u}_s\left(s\right)∥}_{L^2(\mathsf{\Omega })}^{2}ds\right)+k_4\xi \left(0\right)\left({\int }_0^t{∥u\left(s\right)∥}_{\gamma }^{\gamma }ds\right)\hfill \\ & \left.+k_5\xi \left(0\right)\left({\int }_0^t\left(L_{\alpha }^r\square \mathsf{\Delta }u\right)\left(s\right)ds\right)\right\}dt\hfill \\ & +\underset{t\ge 0}{sup}t\left\{V\left(t\right)+k_1\xi \left(0\right)\left({\int }_0^t{∥u_s\left(s\right)∥}_{\rho +2}^{\rho +2}ds\right)+k_2\xi \left(0\right)\left({\int }_0^t{∥\mathsf{\Delta }u\left(s\right)∥}_{L^2(\mathsf{\Omega })}^{2}ds\right)\right.\hfill \\ & +k_3\xi \left(0\right)\left({\int }_0^t{∥\mathsf{\Delta }{u}_s\left(s\right)∥}_{L^2(\mathsf{\Omega })}^{2}ds\right)+k_4\xi \left(0\right)\left({\int }_0^t{∥u\left(s\right)∥}_{\gamma }^{\gamma }ds\right)\hfill \\ & \left.+k_5\xi \left(0\right)\left({\int }_0^t\left(L_{\alpha }^r\square \mathsf{\Delta }u\right)\left(s\right)ds\right)\right\}\hfill \\ <& \infty .\hfill \end{array}$$

Then, by using (90) and Lemma 5 $\left(\mathrm{for}\rho :=r\right)$, we have:

$$\begin{array}{cc}& \left(h\square \mathsf{\Delta }u\right)(t)\hfill \\ \le & C_2^{*}{\left({\displaystyle t{∥u_x(x,t)∥}_H^2+{\int }_0^t{∥u_x(x,s)∥}_H^{2}ds}\right)}^{{\displaystyle \frac{r-1}{r}}}\hfill \\ & \times \left({\int }_0^t{h}^r\left(t-s\right){∥\mathsf{\Delta }u(x,t)-\mathsf{\Delta }u(x,s)∥}_H^{2}ds\right)\hfill \\ \le & C_2^{*}\left[t\left\{V\left(t\right)+k_1\xi \left(0\right)\left({\int }_0^t{∥u_s\left(s\right)∥}_{\rho +2}^{\rho +2}ds\right)+k_2\xi \left(0\right)\left({\int }_0^t{∥\mathsf{\Delta }u\left(s\right)∥}_{L^2(\mathsf{\Omega })}^{2}ds\right)\right.\right.\hfill \\ & +k_3\xi \left(0\right)\left({\int }_0^t{∥\mathsf{\Delta }{u}_s\left(s\right)∥}_{L^2(\mathsf{\Omega })}^{2}ds\right)+k_4\xi \left(0\right)\left({\int }_0^t{∥u\left(s\right)∥}_{\gamma }^{\gamma }ds\right)\hfill \\ & \left.+k_5\xi \left(0\right)\left({\int }_0^t\left(L_{\alpha }^r\square \mathsf{\Delta }u\right)\left(s\right)ds\right)\right\}\hfill \\ & +{\int }_0^t\left\{V\left(t\right)+k_1\xi \left(0\right)\left({\int }_0^t{∥u_s\left(s\right)∥}_{\rho +2}^{\rho +2}ds\right)+k_2\xi \left(0\right)\left({\int }_0^t{∥\mathsf{\Delta }u\left(s\right)∥}_{L^2(\mathsf{\Omega })}^{2}ds\right)\right.\hfill \\ & +k_3\xi \left(0\right)\left({\int }_0^t{∥\mathsf{\Delta }{u}_s\left(s\right)∥}_{L^2(\mathsf{\Omega })}^{2}ds\right)+k_4\xi \left(0\right)\left({\int }_0^t{∥u\left(s\right)∥}_{\gamma }^{\gamma }ds\right)\hfill \\ & {\left.\left.+k_5\xi \left(0\right)\left({\int }_0^t\left(L_{\alpha }^r\square \mathsf{\Delta }u\right)\left(s\right)ds\right)\right\}dt\right]}^{{\displaystyle \frac{r-1}{r}}}{\left((h^r\square \mathsf{\Delta }u)\left(t\right)\right)}^{{\displaystyle \frac{1}{r}}}\hfill \\ \le & C_3^{*}{\left((h^r\square \mathsf{\Delta }u)\left(t\right)\right)}^{{\displaystyle \frac{1}{r}}},\hfill \end{array}$$

which implies that:

$$\left(h^r\square \mathsf{\Delta }u\right)\left(t\right)\ge C_4^{*}{\left(\left(h\square \mathsf{\Delta }u\right)\left(t\right)\right)}^r,$$

(88)

Similarly, we get:

$$\left(L_{\alpha }^r\square \mathsf{\Delta }u\right)\left(t\right)\ge C_5^{*}{\left(\left(L_{\alpha }\square \mathsf{\Delta }u\right)\left(t\right)\right)}^r,$$

(89)

for some constant $C_4^{*}>0$ and $C_5^{*}>0.$

Consequently, a combination of (91) and (92) in (80) yields:

$$\begin{array}{cc}\hfill & \frac{d}{dt}\left\{V\left(t\right)+k_1\xi \left(0\right)\left({\int }_0^t{∥u_s\left(s\right)∥}_{\rho +2}^{\rho +2}ds\right)+k_2\xi \left(0\right)\left({\int }_0^t{∥\mathsf{\Delta }u\left(s\right)∥}_{L^2(\mathsf{\Omega })}^{2}ds\right)\right.\hfill \\ \hfill & +k_3\xi \left(0\right)\left({\int }_0^t{∥\mathsf{\Delta }{u}_s\left(s\right)∥}_{L^2(\mathsf{\Omega })}^{2}ds\right)+k_4\xi \left(0\right)\left({\int }_0^t{∥u\left(s\right)∥}_{\gamma }^{\gamma }ds\right)\hfill \\ \hfill & \left.+k_5\xi \left(0\right)\left({\int }_0^t\left(L_{\alpha }^r\square \mathsf{\Delta }u\right)\left(s\right)ds\right)\right\}\hfill \\ \le \hfill & -C_6\xi (t)\left\{E\left(t\right)+{\left(\left(h\square \mathsf{\Delta }u\right)\left(t\right)\right)}^r+{\left(\left(L_{\alpha }\square \mathsf{\Delta }u\right)\left(t\right)\right)}^r\right\},\hfill \end{array}$$

(90)

for some constant $C_6>0.$

On the other hand, we can obtain:

$$\begin{array}{cc}\hfill & {\left\{E\left(t\right)+\left(h\square \mathsf{\Delta }u\right)(t)+\left(\left(L_{\alpha }\square \mathsf{\Delta }u\right)\left(t\right)\right)\right\}}^r\hfill \\ \le \hfill & C_7\left\{E\left(t\right)+{\left(\left(h\square \mathsf{\Delta }u\right)\left(t\right)\right)}^r+{\left(\left(L_{\alpha }\square \mathsf{\Delta }u\right)\left(t\right)\right)}^r\right\},\hfill \end{array}$$

(91)

for all $t\ge 0$ and some constant $C_7>0.$ Combining the last two inequalities and (33), we obtain:

$$\begin{array}{cc}\hfill & \frac{d}{dt}\left\{V\left(t\right)+k_1\xi \left(0\right)\left({\int }_0^t{∥u_s\left(s\right)∥}_{\rho +2}^{\rho +2}ds\right)+k_2\xi \left(0\right)\left({\int }_0^t{∥\mathsf{\Delta }u\left(s\right)∥}_{L^2(\mathsf{\Omega })}^{2}ds\right)\right.\hfill \\ \hfill & +k_3\xi \left(0\right)\left({\int }_0^t{∥\mathsf{\Delta }{u}_s\left(s\right)∥}_{L^2(\mathsf{\Omega })}^{2}ds\right)+k_4\xi \left(0\right)\left({\int }_0^t{∥u\left(s\right)∥}_{\gamma }^{\gamma }ds\right)\hfill \\ \hfill & \left.+k_5\xi \left(0\right)\left({\int }_0^t\left(L_{\alpha }^r\square \mathsf{\Delta }u\right)\left(s\right)ds\right)\right\}\hfill \\ \le \hfill & -C_6\xi (t){\left\{E\left(t\right)+\left(h\square \mathsf{\Delta }u\right)(t)+\left(\left(L_{\alpha }\square \mathsf{\Delta }u\right)\left(t\right)\right)\right\}}^r\hfill \\ \le \hfill & -C_7\xi (t)\left\{V\left(t\right)+k_1\xi \left(0\right)\left({\int }_0^t{∥u_s\left(s\right)∥}_{\rho +2}^{\rho +2}ds\right)+k_2\xi \left(0\right)\left({\int }_0^t{∥\mathsf{\Delta }u\left(s\right)∥}_{L^2(\mathsf{\Omega })}^{2}ds\right)\right.\hfill \\ \hfill & +k_3\xi \left(0\right)\left({\int }_0^t{∥\mathsf{\Delta }{u}_s\left(s\right)∥}_{L^2(\mathsf{\Omega })}^{2}ds\right)+k_4\xi \left(0\right)\left({\int }_0^t{∥u\left(s\right)∥}_{\gamma }^{\gamma }ds\right)\hfill \\ \hfill & {\left.+k_5\xi \left(0\right)\left({\int }_0^t\left(L_{\alpha }^r\square \mathsf{\Delta }u\right)\left(s\right)ds\right)\right\}}^r,\hfill \end{array}$$

(92)

for some constant $C_7>0.$ A simple integration of (92) over $\left(0,t\right)$ gives:

$$\begin{array}{cc}& V\left(t\right)+k_1\xi \left(0\right)\left({\int }_0^t{∥u_s\left(s\right)∥}_{\rho +2}^{\rho +2}ds\right)+k_2\xi \left(0\right)\left({\int }_0^t{∥\mathsf{\Delta }u\left(s\right)∥}_{L^2(\mathsf{\Omega })}^{2}ds\right)\hfill \\ & +k_3\xi \left(0\right)\left({\int }_0^t{∥\mathsf{\Delta }{u}_s\left(s\right)∥}_{L^2(\mathsf{\Omega })}^{2}ds\right)+k_4\xi \left(0\right)\left({\int }_0^t{∥u\left(s\right)∥}_{\gamma }^{\gamma }ds\right)\hfill \\ & +k_5\xi \left(0\right)\left({\int }_0^t\left(L_{\alpha }^r\square \mathsf{\Delta }u\right)\left(s\right)ds\right)\hfill \\ \le & C_8{\left(1+{\int }_0^t\xi (s)ds\right)}^{-{\displaystyle \frac{1}{r-1}}},\forall t\ge 0.\hfill \end{array}$$

Again by Proposition 1 (the left-hand side of the inequality), we conclude with the assertion of our theorem:

$$E\left(t\right)\le C{\left({\displaystyle 1+{\int }_0^t\xi (s)ds}\right)}^{-{\displaystyle \frac{1}{r-1}}},\forall t\ge 0.$$

Therefore, (49) is obtained. This completes the proof. □

4. Conclusions

Our outcomes are natural extensions from the recent preceding ones in ( [7,8,24]), where the authors dealt with a viscoelastic equation and the system in more general decaying kernels and gave some common decay results, from which the usual exponential and polynomial rates are not only general cases. Our decay rate obtained in the third section is less general then that obtained in [7,8] with respect to the right-hand side defined as the logarithmic nonlinearity, which is familiar in physics, since it appears clearly natural in inflation cosmology and supersymmetric filed theories, quantum mechanics, and nuclear physics (see [5,14]). This sort of problem has many applications in several branches of physics such as nuclear physics, optics, and geophysics (see [3,15]). In future work, we will try to extend this study for the evolutionary case of the presented problem, but by using the semigroup theory.