# Asymptotic stability for a class of viscoelastic equations with general
relaxation functions and the time delay

Menglan Liao, Zhong Tan∗ Menglan Liao
School of Mathematical Sciences, Xiamen University, Xiamen, Fujian, 361005,
People’s Republic of China<EMAIL_ADDRESS>Zhong Tan
School of Mathematical Sciences, Xiamen University, Xiamen, Fujian, 361005,
People’s Republic of China<EMAIL_ADDRESS>
###### Abstract.

The goal of the present paper is to study the viscoelastic wave equation with
the time delay

$|u_{t}|^{\rho}u_{tt}-\Delta u-\Delta u_{tt}+\int_{0}^{t}g(t-s)\Delta
u(s)ds+\mu_{1}u_{t}(x,t)+\mu_{2}u_{t}(x,t-\tau)=b|u|^{p-2}u$

under initial boundary value conditions, where $\rho,~{}b,~{}\mu_{1}$ are
positive constants, $\mu_{2}$ is a real number, $\tau>0$ represents the time
delay. By using the multiplier method together with some properties of the
convex functions, the explicit and general stability results of energy are
proved under the general assumption on the relaxation function $g$. This work
generalizes and improves earlier results on the stability of the viscoelastic
equations with the time delay in the literature.

###### Key words and phrases:

Viscoelasticity; delay term; source term; energy decay.

###### 2010 Mathematics Subject Classification:

35B40, 26A51, 93D20.

Supported by the National Natural Science Foundation of China (Grant Nos.
11926316, 11531010, 12071391)

∗ Corresponding author: Zhong Tan

##  1\. Introduction

This paper concerns the following initial boundary value problem with the time
delay

$\begin{cases}|u_{t}|^{\rho}u_{tt}-\Delta u-\Delta
u_{tt}+\int_{0}^{t}g(t-s)\Delta u(s)ds\\\
\quad+\mu_{1}u_{t}(x,t)+\mu_{2}u_{t}(x,t-\tau)=b|u|^{p-2}u&(x,t)\in\Omega\times(0,\infty),\\\
u_{t}(x,t-\tau)=f_{0}(x,t-\tau)&(x,t)\in\Omega\times(0,\tau),\\\
u(x,0)=u_{0}(x),~{}u_{t}(x,0)=u_{1}(x)&x\in\Omega,\\\
u(x,t)=0&(x,t)\in\partial\Omega\times[0,\infty),\end{cases}$ (1.1)

where $\Omega\subset\mathbb{R}^{N}(N\geq 1)$ is a bounded domain with a smooth
boundary $\partial\Omega$, the unknown $u:=u(x,t)$ is a real valued function
defined on $\Omega\times(0,\infty)$, $\rho,~{}b,~{}\mu_{1}$ are positive
constants, $\mu_{2}$ is a real number, $\tau>0$ represents the time delay, $g$
is the kernel of the memory term, and the initial data $(u_{0},u_{1},f_{0})$
are given functions belonging to suitable spaces. In addition, the following
assumptions are given throughout this paper:

$(\mathbf{H_{1}})$ The relaxation function $g:[0,\infty)\to(0,\infty)$ is a
differentiable function satisfying

$1-\int_{0}^{\infty}g(s)ds=l>0,$ (1.2)

and there exists a $C^{1}$ function $G:(0,\infty)\to(0,\infty)$ which is
strictly increasing and strictly convex $C^{2}$ function on $(0,r]$, $r\leq
g(0)$, with $G(0)=G^{\prime}(0)=0$ such that

$g^{\prime}(t)\leq-\zeta(t)G(g(t))\quad\text{for }t\geq 0,$ (1.3)

here $\zeta(t)$ is a positive non-increasing differentiable function.

$(\mathbf{H_{2}})$ $\rho$ and $p$ satisfy

$0<\rho\leq\frac{2}{N-2}\text{ for }N\geq 3\text{ and }\rho>0\text{ for
}N=1,2;$ $2<p\leq\frac{2(N-1)}{N-2}\text{ for }N\geq 3\text{ and }p>2\text{
for }N=1,2.$

It is well known that time delay effects which often appear in many practical
applications may induce some instabilities. Some results on the local
existence and blow-up of solutions to a class of equations with delay have
been obtained, the interested readers can refer to [4, 12, 13, 14, 23] and the
reference therein. Nicaise and Pignotti [20] considered the wave equation with
a delay term in the boundary condition as well as the wave equation with a
delayed velocity term and mixed Dirichlet-Neumann boundary condition in a
bounded and smooth domain, respectively. By introducing suitable energies and
by using some observability inequalities, they proved an exponential stability
of the solution in both cases under suitable assumptions. Kirane and Said-
Houari [11] studied the following viscoelastic wave equation with a delay term
in internal feedback

$u_{tt}-\Delta u+\int_{0}^{t}g(t-s)\Delta
u(s)ds+\mu_{1}u_{t}(x,t)+\mu_{2}u_{t}(x,t-\tau)=0$ (1.4)

here $\mu_{1},~{}\mu_{2}$ are positive constants, under the initial boundary
conditions of problem (1.1). They proved the existence of a unique weak
solution for $\mu_{2}\leq\mu_{1}$ by using the Faedo-Galerkin approximations
together with some energy estimates. Provided that
$g:\mathbb{R}^{+}\to\mathbb{R}^{+}$ is a $C^{1}$ function satisfying $g(0)>0$
and (1.2), and there exists a positive non-increasing differentiable function
$\zeta(t)$ such that

$g^{\prime}(t)\leq-\zeta(t)g(t)\text{ for }t\geq 0\text{ and
}\int_{0}^{+\infty}\zeta(t)dt=+\infty,$ (1.5)

by establishing suitable Lyapunov functionals, they also obtained the
corresponding exponential stability for $\mu_{2}<\mu_{1}$ and for
$\mu_{2}=\mu_{1}$, respectively. Subsequently, Dai and Yang [7] proved an
existence result of problem (1.4) without restrictions of
$\mu_{1},~{}\mu_{2}>0$ and $\mu_{2}\leq\mu_{1}$. Making use of the
viscoelasticity term controls the delay term, they also proved an energy decay
result for problem (1.4) in the case $\mu_{1}=0$ provided that
$g:\mathbb{R}^{+}\to\mathbb{R}^{+}$ is a $C^{1}$ function satisfying $g(0)>0$
and (1.2), and there exists a positive constant $\zeta$ such that

$g(t)\leq-\zeta g(t)\text{ for }t>0.$ (1.6)

Liu [15] generalized the results obtained by Kirane and Said-Houari [11]. That
is, by the similar method in [11], they established a general energy decay
result for problem (1.4) with $\tau(t)$ instead of $\tau$. In the absence of
the source term $b|u|^{p-2}u$ in problem (1.1), Wu [21] proved an energy decay
by the similar method in [11], and generalized the results to the time-varying
delay in [22]. There are many papers concerning with the stability of
viscoelastic equations with time delay, the interested readers may refer to
[3, 8, 16] and the reference therein. However, the relaxation function $g$ are
mainly limited to satisfying among the three conditions, which are (1.5),
(1.6) and that $g:\mathbb{R}^{+}\to\mathbb{R}^{+}$ is a differentiable
function satisfying $g(0)>0$ and (1.2), and there exists a positive function
$G\in C^{1}(\mathbb{R}^{+})$ and $G$ is linear or strictly increasing and
strictly convex $C^{2}$ function on $(0,r]$, $r<1$, with
$G(0)=G^{\prime}(0)=0$, such that

$g^{\prime}(t)\leq-G(g(t))\text{ for }t>0.$

Until recently, Chellaoua and Boukhatem [5] generalized the previous
conditions that the relaxation function $g$ satisfied, specifically
investigated the following second-order abstract viscoelastic equation in
Hilbert spaces

$\begin{cases}u_{tt}+Au-\int_{0}^{\infty}g(s)Bu(t-s)ds+\mu_{1}u_{t}(t)+\mu_{2}u_{t}(t-\tau)=0&t>0,\\\
u_{t}(t-\tau)=f_{0}(t-\tau)&t\in(0,\tau),\\\
u(-t)=u_{0}(t),~{}u_{t}(0)=u_{1}&t\geq 0,\end{cases}$

where $A:D(A)\to H$ and $B:D(B)\to H$ are a self-adjoint linear positive
operator with domains $D(A)\subset D(B)\subset H$ such that the embeddings are
dense and compact. They established an explicit and general decay results of
the energy solution by introducing a suitable Lyapunov functional and some
properties of the convex functions under the condition $(\mathbf{H_{1}})$.
Chellaoua and Boukhatem also addressed the stability results for the following
second-order abstract viscoelastic equation in Hilbert spaces with time-
varying delay in [6]

$\begin{cases}u_{tt}+Au-\int_{0}^{t}g(t-s)Bu(s)ds+\mu_{1}u_{t}(t)+\mu_{2}u_{t}(t-\tau(t))=0&t>0,\\\
u_{t}(t-\tau(0))=f_{0}(t-\tau(0))&t\in(0,\tau(0)),\\\
u(0)=u_{0},~{}u_{t}(0)=u_{1}&t\geq 0,\end{cases}$

under the condition $(\mathbf{H_{1}})$. It is worth pointing out that Mustafa
[17] first proposed the condition $(\mathbf{H_{1}})$ to study the decay rates
for the following initial boundary value problem

$\begin{cases}u_{tt}-\Delta u+\int_{0}^{t}g(t-s)\Delta u(s)ds=0&\text{in
}\Omega\times(0,\infty),\\\ u=0&\text{on }\partial\Omega\times(0,\infty),\\\
u(x,0)=u_{0}(x),~{}u_{t}(x,0)=u_{1}(x)&x\in\Omega.\end{cases}$

After that, many authors popularized the method used by Mustafa in [17]. The
readers may see the references [2, 9, 10, 18, 19] to get more details.

Motivated by the above works, we are committed to considering the stability of
problem (1.1) when the relaxation function $g$ satisfies the condition
$(\mathbf{H_{1}})$. To the best of our knowledge, there is no decay result for
problem (1.1) where the relaxation functions satisfy $(\mathbf{H_{1}})$,
although Wu [23] has investigated problem (1.1) and proved the blow-up result
with nonpositive and positive initial energy. With minimal conditions on the
relaxation function $g$, we establish a general and optimal energy decay rates
of problem (1.1) in Theorem 3.3. Our proof is based on the multiplier method
and the similar arguments in [5, 17] but it is different from before since the
presence of $\Delta u_{tt}$, the time delay and the force source
$b|u|^{p-2}u$. The outline of this paper is as follows: In Section 2, we give
some preliminary results. Section 3 is used to present the energy decay and
its proof. In Section 4, we give the possible generalizations.

##  2\. Preliminaries

Throughout this paper, we denote by $\|\cdot\|_{p}$ and $\|\nabla\cdot\|_{2}$
the norm on $L^{p}(\Omega)$ with $1\leq p\leq\infty$ and $H_{0}^{1}(\Omega)$,
respectively. Let $\lambda_{1}$ be the first eigenvalue of the following
boundary value problem

$\begin{cases}-\Delta\psi=\lambda\psi&\quad\text{for }x\in\Omega,\\\
\psi=0&\quad\text{for }x\in\partial\Omega.\end{cases}$

The symbol $c_{s}$ is the optimal embedding constant of
$H_{0}^{1}(\Omega)\hookrightarrow L^{p}(\Omega)$.

In order to the completeness of results, in what follows, we state the known
results in [23]. Let us introduce the new variable

$z(x,\kappa,t)=u_{t}(x,t-\tau\kappa)\quad\text{ for
}x\in\Omega,~{}\kappa\in(0,1),$

then problem (1.1) is equivalent to

$\begin{cases}|u_{t}|^{\rho}u_{tt}-\Delta u-\Delta
u_{tt}+\int_{0}^{t}g(t-s)\Delta u(s)ds\\\
\quad+\mu_{1}u_{t}(x,t)+\mu_{2}z(x,1,t)=b|u|^{p-2}u&(x,t)\in\Omega\times(0,\infty),\\\
\tau
z_{t}(x,\kappa,t)+z_{\kappa}(x,\kappa,t)=0&(x,t)\in\Omega\times(0,\infty),~{}\kappa\in(0,1),\\\
z(x,0,t)=u_{t}(x,t)&(x,t)\in\Omega\times(0,\infty),\\\
z(x,\kappa,0)=f_{0}(x,-\tau\kappa)&x\in\Omega,\\\
u_{t}(x,t-\tau)=f_{0}(x,t-\tau)&(x,t)\in\Omega\times(0,\tau),\\\
u(x,0)=u_{0}(x),~{}u_{t}(x,0)=u_{1}(x)&x\in\Omega,\\\
u(x,t)=0&(x,t)\in\partial\Omega\times[0,\infty).\end{cases}$ (2.1)

######  Theorem 2.1 (Theorem 2.3 in [23]).

Suppose that $|\mu_{2}|\leq\mu_{1}$, $(\mathbf{H_{1}})$ and $(\mathbf{H_{2}})$
hold. Assume that $u_{0},~{}u_{1}\in H_{0}^{1}(\Omega)$ and $f_{0}\in
L^{2}(\Omega\times(0,1))$, then there exists a unique solution $(u,z)$ of
problem (2.1) satisfying

$u,u_{t}\in C([0,T);H_{0}^{1}(\Omega)),\quad z\in
C([0,T);L^{2}(\Omega\times(0,1))),$

for $T>0.$

Define the energy functional of problem (2.1) as follows

$\begin{split}E(t)&=\frac{1}{\rho+2}\|u_{t}\|_{\rho+2}^{\rho+2}+\frac{1}{2}\Big{(}1-\int_{0}^{t}g(s)ds\Big{)}\|\nabla
u\|_{2}^{2}+\frac{1}{2}(g\circ\nabla u)(t)\\\ &\quad+\frac{1}{2}\|\nabla
u_{t}\|_{2}^{2}+\frac{\xi}{2}\int_{\Omega}\int_{0}^{1}z^{2}(x,\kappa,t)d\kappa
dx-\frac{b}{p}\|u\|_{p}^{p},\end{split}$ (2.2)

here $\xi$ is a positive constant so that

$\tau|\mu_{2}|\leq\xi\leq\tau(2\mu_{1}-|\mu_{2}|)$ (2.3)

and

$(g\circ u)(t)=\int_{0}^{t}g(t-s)\|u(s)-u(t)\|_{2}^{2}ds.$

######  Lemma 2.2 (Lemma 3.1 in [23]).

$E(t)$ is a non-increasing function and

$\begin{split}E^{\prime}(t)&\leq-\omega(\|u_{t}\|_{2}^{2}+\|z(x,1,t)\|_{2}^{2})+\frac{1}{2}(g^{\prime}\circ\nabla
u)(t)-\frac{1}{2}g(t)\|\nabla u\|_{2}^{2}\\\
&\leq-\omega(\|u_{t}\|_{2}^{2}+\|z(x,1,t)\|_{2}^{2})\leq 0\quad\text{ for all
}t\geq 0,\end{split}$ (2.4)

where
$\omega=\min\Big{\\{}\mu_{1}-\frac{\xi}{2\tau}-\frac{|\mu_{2}|}{2},\frac{\xi}{2\tau}-\frac{|\mu_{2}|}{2}\Big{\\}}\geq
0.$

######  Lemma 2.3.

If $u$ is a solution for problem $(\ref{2.1})$ and

$E(0)<E_{1}=\frac{p-2}{2p}\sigma_{1}^{2},\quad l\|\nabla
u_{0}\|_{2}^{2}<\sigma_{1}^{2},$

here $\sigma_{1}=b^{-\frac{1}{p-2}}B_{1}^{-\frac{p}{p-2}}$,
$B_{1}=\frac{c_{s}^{p}}{l^{\frac{p}{2}}}$, then there exists a positive
constant $\sigma_{2}$ satisfying $0<\sigma_{2}<\sigma_{1}$ such that

$l\|\nabla u\|_{2}^{2}+(g\circ\nabla u)(t)\leq\sigma_{2}^{2}\quad\text{ for
all }t\geq 0.$ (2.5)

###### Proof.

Taking the combination of (2.2) and (1.2) with the embedding
$H_{0}^{1}(\Omega)\hookrightarrow L^{p}(\Omega)$, one has

$\begin{split}E(t)&\geq\frac{l}{2}\|\nabla
u\|_{2}^{2}+\frac{1}{2}(g\circ\nabla
u)(t)-\frac{bB_{1}^{p}}{p}\Big{(}l^{\frac{1}{2}}\|\nabla u\|_{2}\Big{)}^{p}\\\
&\geq F\Big{(}\sqrt{l\|\nabla u\|_{2}^{2}+(g\circ\nabla
u)(t)}\Big{)},\end{split}$ (2.6)

where $F(x)=\frac{1}{2}x^{2}-\frac{bB_{1}^{p}}{p}x^{p}\text{ for }x>0.$ We
know that $F$ is strictly increasing in $(0,\sigma_{1})$, strictly decreasing
in $(\sigma_{1},\infty)$, and $F$ has a maximum at $\sigma_{1}$ with the
maximum value $E_{1}$. Since $E(0)<E_{1}$, there exists a
$\sigma_{2}<\sigma_{1}$ such that $F(\sigma_{2})=E(0)$. Set
$\sigma_{0}:=\sqrt{l\|\nabla u_{0}\|_{2}^{2}}$, recall (2.6), then
$F(\sigma_{0})\leq E(0)=F(\sigma_{2})$, which implies
$\sigma_{0}\leq\sigma_{2}$ since the given condition
$\sigma_{0}^{2}<\sigma_{1}^{2}$. In order to complete the proof of (2.5), we
suppose by contradiction that for some $t^{0}>0$,

$\sigma(t^{0})=\sqrt{l\|\nabla u(t_{0})\|_{2}^{2}+(g\circ\nabla
u)(t_{0})}>\sigma_{2}.$

The continuity of $\sqrt{l\|\nabla u\|_{2}^{2}+(g\circ\nabla u)(t)}$
illustrates that we may choose $t^{0}$ such that
$\sigma_{1}>\sigma(t^{0})>\sigma_{2}$, then we have

$E(0)=F(\sigma_{2})<F(\sigma(t^{0}))\leq E(t^{0}).$

This is a contradiction since Lemma 2.2. ∎

######  Lemma 2.4.

Under all the conditions of Lemma 2.3, there exists a positive constant
$\mathcal{D}$ such that for all $t\geq 0$,

$\|u\|_{p}^{p}\leq\mathcal{D}E(t)\leq\mathcal{D}E(0),$ (2.7)
$\begin{split}&\frac{1}{\rho+2}\|u_{t}\|_{\rho+2}^{\rho+2}+\frac{1}{2}\Big{(}1-\int_{0}^{t}g(s)ds\Big{)}\|\nabla
u\|_{2}^{2}+\frac{1}{2}(g\circ\nabla u)(t)+\frac{1}{2}\|\nabla
u_{t}\|_{2}^{2}\\\
&\quad+\frac{\xi}{2}\int_{\Omega}\int_{0}^{1}z^{2}(x,\kappa,t)d\kappa
dx\leq\mathcal{D}E(t)\leq\mathcal{D}E(0).\end{split}$ (2.8)

###### Proof.

Using the embedding $H_{0}^{1}(\Omega)\hookrightarrow L^{p}(\Omega)$, (2.2)
and (2.5), we have

$\begin{split}\frac{b}{p}\|u\|_{p}^{p}&\leq\frac{bB_{1}^{p}}{p}\Big{(}l^{\frac{1}{2}}\|\nabla
u\|_{2}\Big{)}^{p}\leq\frac{bB_{1}^{p}}{p}\Big{(}l\|\nabla
u\|_{2}^{2}+(g\circ\nabla u)(t)\Big{)}^{\frac{p-2}{2}}[l\|\nabla
u\|_{2}^{2}+(g\circ\nabla u)(t)]\\\ &\leq\frac{2bB_{1}^{p}}{p}\Big{(}l\|\nabla
u\|_{2}^{2}+(g\circ\nabla
u)(t)\Big{)}^{\frac{p-2}{2}}\Big{(}E(t)+\frac{b}{p}\|u\|_{p}^{p}\Big{)}\\\
&\leq\frac{2bB_{1}^{p}}{p}\sigma_{2}^{p-2}\Big{(}E(t)+\frac{b}{p}\|u\|_{p}^{p}\Big{)},\end{split}$

which yields (2.7) with

$\mathcal{D}=\frac{2pB_{1}^{p}\sigma_{2}^{p-2}}{p-2bB_{1}^{p}\sigma_{2}^{p-2}}>0.$
(2.9)

One has (2.8) by combining (2.7) with (2.2), ∎

######  Lemma 2.5 (Lemma 4.1 in [2]).

For $u\in H_{0}^{1}(\Omega)$, we have for all $t\geq 0$,

$\int_{\Omega}\Big{(}\int_{0}^{t}g(t-s)(\nabla u(s)-\nabla
u(t))ds\Big{)}^{2}dx\leq C_{\alpha}(h_{\alpha}\circ\nabla u)(t)$ (2.10)

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

$C_{\alpha}=\int_{0}^{\infty}\frac{g^{2}(s)}{\alpha
g(s)-g^{\prime}(s)}ds\text{ and }h_{\alpha}(t)=\alpha g(t)-g^{\prime}(t).$
(2.11)

Let us follow from the proof of Lemma 4.1 in [2], we have in fact

$\int_{\Omega}\Big{(}\int_{0}^{t}g(t-s)(u(s)-u(t))ds\Big{)}^{2}dx\leq
C_{\alpha}(h_{\alpha}\circ u)(t).$ (2.12)

######  Lemma 2.6 (Lemma 2.2 in [6]).

There exist positive constants $\gamma$ and $t_{1}$ such that

$g^{\prime}(t)\leq-\gamma g(t)\quad\text{for }t\in[0,t_{1}].$ (2.13)

######  Lemma 2.7.

Let $u$ be a solution of problem (2.1), then the functional

$I_{1}(t)=\frac{1}{\rho+1}\int_{\Omega}|u_{t}|^{\rho}u_{t}udx+\int_{\Omega}\nabla
u_{t}\nabla udx,$ (2.14)

satisfies, for $\varepsilon>0$ and for all $t\geq 0$,

$\begin{split}I_{1}^{\prime}(t)&\leq\frac{1}{\rho+1}\|u_{t}\|_{\rho+2}^{\rho+2}-\Big{[}l-\Big{(}1+\frac{\mu_{1}}{\lambda_{1}}+\frac{\mu_{2}}{\lambda_{1}}\Big{)}\varepsilon\Big{]}\|\nabla
u\|_{2}^{2}+\frac{1}{4\varepsilon}C_{\alpha}(h_{\alpha}\circ\nabla u)(t)\\\
&\quad+\frac{|\mu_{2}|}{4\varepsilon}\|z(x,1,t)\|_{2}^{2}+b\|u\|_{p}^{p}+\|\nabla
u_{t}\|_{2}^{2}+\frac{\mu_{1}}{4\varepsilon}\|u_{t}\|_{2}^{2}.\end{split}$
(2.15)

###### Proof.

Multiplying the first identity in problem (2.1) by $u$, integrating on $x$
over $\Omega$, and then using integration by parts, we give

$\begin{split}&\int_{\Omega}|u_{t}|^{\rho}u_{tt}udx+\int_{\Omega}\nabla
u_{tt}\nabla udx\\\ &\quad=-\Big{(}1-\int_{0}^{t}g(s)ds\Big{)}\|\nabla
u\|_{2}^{2}+\int_{\Omega}\nabla u(t)\int_{0}^{t}g(t-s)\nabla(u(s)-u(t))ds\\\
&\quad\quad-\int_{\Omega}\mu_{1}u_{t}(x,t)udx-\int_{\Omega}\mu_{2}z(x,1,t)udx+b\|u\|_{p}^{p}.\end{split}$
(2.16)

Differentiating (2.14) on $t$, and using (2.16), one has

$\begin{split}I_{1}^{\prime}(t)&=\frac{1}{\rho+1}\|u_{t}\|_{\rho+2}^{\rho+2}+\int_{\Omega}|u_{t}|^{\rho}u_{tt}udx+\int_{\Omega}\nabla
u_{tt}\nabla udx+\|\nabla u_{t}\|_{2}^{2}\\\
&=\frac{1}{\rho+1}\|u_{t}\|_{\rho+2}^{\rho+2}-\Big{(}1-\int_{0}^{t}g(s)ds\Big{)}\|\nabla
u\|_{2}^{2}\\\ &\quad+\int_{\Omega}\nabla
u(t)\int_{0}^{t}g(t-s)\nabla(u(s)-u(t))ds\\\
&\quad-\int_{\Omega}\mu_{1}u_{t}(x,t)udx-\int_{\Omega}\mu_{2}z(x,1,t)udx+b\|u\|_{p}^{p}+\|\nabla
u_{t}\|_{2}^{2}.\end{split}$ (2.17)

Applying Cauchy’s inequality with $\varepsilon>0$ and
$\lambda_{1}\|u\|_{2}^{2}\leq\|\nabla u\|_{2}^{2}$, it follows that

$\begin{split}-\int_{\Omega}\mu_{1}u_{t}(x,t)udx&\leq\frac{\mu_{1}}{4\varepsilon}\|u_{t}\|_{2}^{2}+\mu_{1}\varepsilon\|u\|_{2}^{2}\leq\frac{\mu_{1}}{4\varepsilon}\|u_{t}\|_{2}^{2}+\frac{\mu_{1}\varepsilon}{\lambda_{1}}\|\nabla
u\|_{2}^{2},\end{split}$ (2.18)
$-\int_{\Omega}\mu_{2}z(x,1,t)udx\leq\frac{|\mu_{2}|}{4\varepsilon}\|z(x,1,t)\|_{2}^{2}+\frac{|\mu_{2}|\varepsilon}{\lambda_{1}}\|\nabla
u\|_{2}^{2}.$ (2.19)

It follows from Cauchy’s inequality with $\varepsilon>0$ and (2.10) that

$\int_{\Omega}\nabla u(t)\int_{0}^{t}g(t-s)(\nabla u(s)-\nabla
u(t))ds\leq\varepsilon\|\nabla
u\|_{2}^{2}+\frac{1}{4\varepsilon}C_{\alpha}(h\circ\nabla u)(t).$ (2.20)

Inserting (2.18)-(2.20) into (2.17), we obtain (2.15). ∎

######  Lemma 2.8.

Under all the conditions of Lemma 2.3, let $u$ be a solution of problem (2.1),
then the functional

$I_{2}(t)=\int_{\Omega}\Big{(}\Delta
u_{t}-\frac{1}{\rho+1}|u_{t}|^{\rho}u_{t}\Big{)}\int_{0}^{t}g(t-s)(u(t)-u(s))dsdx,$
(2.21)

satisfies, for $\delta>0$ and for all $t\geq 0$,

$\begin{split}I_{2}^{\prime}(t)&\leq B_{1}\|\nabla
u\|_{2}^{2}+B_{2}(h_{\alpha}\circ\nabla
u)(t)+\Big{[}B_{3}-\int_{0}^{t}g(s)ds\Big{]}\|\nabla u_{t}\|_{2}^{2}\\\
&\quad+\delta\|z(x,1,t)\|_{2}^{2}-\int_{0}^{t}g(s)ds\cdot\frac{1}{\rho+1}\|u_{t}\|_{\rho+2}^{\rho+2},\end{split}$
(2.22)

here $B_{1}$, $B_{2}$ and $B_{3}$ are positive constants depending on $\delta$
shown in (2.31).

###### Proof.

Differentiating (2.21) on $t$, and using the first identity in problem (2.1)
and integration by parts, we have

$\begin{split}I_{2}^{\prime}(t)&=\int_{\Omega}\Big{(}\Delta
u_{tt}-|u_{t}|^{\rho}u_{tt}\Big{)}\int_{0}^{t}g(t-s)(u(t)-u(s))dsdx\\\
&\quad+\int_{\Omega}\Big{(}\Delta
u_{t}-\frac{1}{\rho+1}|u_{t}|^{\rho}u_{t}\Big{)}\int_{0}^{t}g_{t}(t-s)(u(t)-u(s))dsdx\\\
&\quad+\int_{\Omega}\Big{(}\Delta
u_{t}-\frac{1}{\rho+1}|u_{t}|^{\rho}u_{t}\Big{)}\int_{0}^{t}g(t-s)u_{t}(t)dsdx\\\
&=\int_{\Omega}\nabla u(t)\int_{0}^{t}g(t-s)\nabla(u(t)-u(s))ds\\\
&\quad-\int_{\Omega}\int_{0}^{t}g(t-s)\nabla
u(s)ds\int_{0}^{t}g(t-s)\nabla(u(t)-u(s))dsdx\\\
&\quad+\int_{\Omega}\mu_{1}u_{t}(x,t)\int_{0}^{t}g(t-s)(u(t)-u(s))dsdx\\\
&\quad+\int_{\Omega}\mu_{2}z(x,1,t)\int_{0}^{t}g(t-s)(u(t)-u(s))dsdx\\\
&\quad-b\int_{\Omega}|u|^{p-2}u\int_{0}^{t}g(t-s)(u(t)-u(s))dsdx\\\
&\quad-\int_{\Omega}\nabla u_{t}\int_{0}^{t}g_{t}(t-s)\nabla(u(t)-u(s))dsdx\\\
&\quad-\int_{\Omega}\frac{1}{\rho+1}|u_{t}|^{\rho}u_{t}\int_{0}^{t}g_{t}(t-s)(u(t)-u(s))dsdx\\\
&\quad-\int_{0}^{t}g(s)ds\cdot\|\nabla
u_{t}\|_{2}^{2}-\int_{0}^{t}g(s)ds\cdot\frac{1}{\rho+1}\|u_{t}\|_{\rho+2}^{\rho+2}\\\
&=J_{1}+J_{2}+\cdots+J_{8}+J_{9}.\end{split}$ (2.23)

It is direct from Cauchy’s inequality with $\delta>0$ and (2.10) that

$J_{1}\leq\delta\|\nabla
u\|_{2}^{2}+\frac{1}{4\delta}C_{\alpha}(h_{\alpha}\circ\nabla u)(t),$ (2.24)
$\begin{split}J_{2}&\leq\delta\int_{\Omega}\Big{(}\int_{0}^{t}g(t-s)|\nabla
u(s)-\nabla u(t)|+|\nabla u(t)|ds\Big{)}^{2}dx\\\
&\quad+\frac{1}{4\delta}\int_{\Omega}\Big{(}\int_{0}^{t}g(t-s)|\nabla
u(s)-\nabla u(t)|ds\Big{)}^{2}dx\\\ &\leq
2\delta\int_{\Omega}\Big{(}\int_{0}^{t}g(t-s)|\nabla u(s)-\nabla
u(t)|ds\Big{)}^{2}dx+2\delta(1-l)^{2}\|\nabla u(t)\|_{2}^{2}\\\
&\quad+\frac{1}{4\delta}\int_{\Omega}\Big{(}\int_{0}^{t}g(t-s)|\nabla
u(s)-\nabla u(t)|ds\Big{)}^{2}dx\\\
&\leq\Big{(}2\delta+\frac{1}{4\delta}\Big{)}C_{\alpha}(h_{\alpha}\circ\nabla
u)(t)+2\delta(1-l)^{2}\|\nabla u\|_{2}^{2}.\end{split}$ (2.25)

Cauchy’s inequality with $\delta>0$ and (2.12) yield

$\begin{split}J_{3}\leq\delta\|u_{t}\|_{2}^{2}+\frac{\mu_{1}^{2}}{4\delta}C_{\alpha}(h_{\alpha}\circ
u)(t)\leq\frac{\delta}{\lambda_{1}}\|\nabla
u_{t}\|_{2}^{2}+\frac{\mu_{1}^{2}}{4\delta\lambda_{1}}C_{\alpha}(h_{\alpha}\circ\nabla
u)(t),\end{split}$ (2.26)
$\begin{split}J_{4}&\leq\delta\|z(x,1,t)\|_{2}^{2}+\frac{\mu_{2}^{2}}{4\delta\lambda_{1}}C_{\alpha}(h_{\alpha}\circ\nabla
u)(t).\end{split}$ (2.27)

It follows from Cauchy’s inequality with $\delta>0$, (2.12), the embedding
$H_{0}^{1}(\Omega)\hookrightarrow L^{2(p-1)}(\Omega)$ and (2.8) that

$\begin{split}J_{5}&\leq
b\delta\|u\|_{2(p-1)}^{2(p-1)}+\frac{b}{4\delta}\int_{\Omega}\Big{(}\int_{0}^{t}g(t-s)(u(t)-u(s))ds\Big{)}^{2}dx\\\
&\leq
b\delta\|u\|_{2(p-1)}^{2(p-1)}+\frac{b}{4\delta}C_{\alpha}(h_{\alpha}\circ
u)(t)\\\ &\leq b\delta
c_{s}^{2(p-1)}\Big{(}\frac{2\mathcal{D}}{l}E(0)\Big{)}^{p-2}\|\nabla
u\|_{2}^{2}+\frac{b}{4\delta\lambda_{1}}C_{\alpha}(h_{\alpha}\circ\nabla
u)(t).\end{split}$ (2.28)

Recalling the definition of $g^{\prime}(t)$ in (2.11), and using Cauchy’s
inequality with $\delta>0$, (2.10) and Hölder’s inequality, one obtains

$\begin{split}J_{6}&=-\int_{\Omega}\nabla u_{t}\int_{0}^{t}\alpha
g(t-s)\nabla(u(t)-u(s))dsdx\\\ &\quad+\int_{\Omega}\nabla
u_{t}\int_{0}^{t}h_{\alpha}(t-s)\nabla(u(t)-u(s))dsdx\\\ &\leq\delta\|\nabla
u_{t}\|_{2}^{2}+\frac{\alpha^{2}}{4\delta}\int_{\Omega}\Big{(}\int_{0}^{t}g(t-s)\nabla(u(t)-u(s))ds\Big{)}^{2}dx\\\
&\quad+\delta\|\nabla
u_{t}\|_{2}^{2}+\frac{1}{4\delta}\int_{\Omega}\Big{(}\int_{0}^{t}h_{\alpha}(t-s)\nabla(u(t)-u(s))ds\Big{)}^{2}dx\\\
&\leq 2\delta\|\nabla
u_{t}\|_{2}^{2}+\frac{\alpha^{2}}{4\delta}C_{\alpha}(h_{\alpha}\circ\nabla
u)(t)\\\
&\quad+\frac{1}{4\delta}\int_{0}^{t}h_{\alpha}(s)ds\int_{0}^{t}h_{\alpha}(t-s)\|\nabla(u(t)-u(s))\|_{2}^{2}ds\\\
&\leq 2\delta\|\nabla
u_{t}\|_{2}^{2}+\Big{(}\frac{\alpha^{2}}{4\delta}C_{\alpha}+\frac{\alpha(1-l)+g(0)}{4\delta}\Big{)}(h_{\alpha}\circ\nabla
u)(t).\end{split}$ (2.29)

Similarly, we get

$\begin{split}J_{7}&=-\int_{\Omega}\frac{1}{\rho+1}|u_{t}|^{\rho}u_{t}\int_{0}^{t}\alpha
g(t-s)(u(t)-u(s))dsdx\\\
&\quad+\int_{\Omega}\frac{1}{\rho+1}|u_{t}|^{\rho}u_{t}\int_{0}^{t}h_{\alpha}(t-s)(u(t)-u(s))dsdx\\\
&\leq\frac{\delta}{\rho+1}\|u_{t}\|_{2(\rho+1)}^{2(\rho+1)}+\frac{\alpha^{2}}{4(\rho+1)\delta}\int_{\Omega}\Big{(}\int_{0}^{t}g(t-s)(u(t)-u(s))ds\Big{)}^{2}dx\\\
&\quad+\frac{\delta}{\rho+1}\|u_{t}\|_{2(\rho+1)}^{2(\rho+1)}+\frac{1}{4(\rho+1)\delta}\int_{\Omega}\Big{(}\int_{0}^{t}h_{\alpha}(t-s)(u(t)-u(s))ds\Big{)}^{2}dx\\\
&\leq\frac{2\delta}{\rho+1}\|u_{t}\|_{2(\rho+1)}^{2(\rho+1)}+\frac{\alpha^{2}}{4(\rho+1)\delta}C_{\alpha}(h_{\alpha}\circ
u)(t)\\\
&\quad+\frac{1}{4(\rho+1)\delta}\int_{0}^{t}h_{\alpha}(s)ds\int_{0}^{t}h_{\alpha}(t-s)\|u(t)-u(s)\|_{2}^{2}ds\\\
&\leq\frac{2\delta}{\rho+1}c_{s}^{2(\rho+1)}\Big{(}2\mathcal{D}E(0)\Big{)}^{\frac{\rho}{2}}\|\nabla
u_{t}\|_{2}^{2}\\\
&\quad+\Big{(}\frac{\alpha^{2}}{4(\rho+1)\delta}C_{\alpha}+\frac{\alpha(1-l)+g(0)}{4(\rho+1)\delta}\Big{)}\frac{1}{\lambda_{1}}(h_{\alpha}\circ\nabla
u)(t).\end{split}$ (2.30)

Inserting (2.25)-(2.30) into (2.23), one has

$\begin{split}I_{2}^{\prime}(t)&\leq B_{1}\|\nabla
u\|_{2}^{2}+B_{2}(h_{\alpha}\circ\nabla
u)(t)+\Big{[}B_{3}-\int_{0}^{t}g(s)ds\Big{]}\|\nabla u_{t}\|_{2}^{2}\\\
&\quad+\delta\|z(x,1,t)\|_{2}^{2}-\int_{0}^{t}g(s)ds\cdot\frac{1}{\rho+1}\|u_{t}\|_{\rho+2}^{\rho+2},\end{split}$

with

$\begin{cases}B_{1}=\delta+2\delta(1-l)^{2}+b\delta
c_{s}^{2(p-1)}\Big{(}\frac{2\mathcal{D}}{l}E(0)\Big{)}^{p-2};\\\
B_{2}=\Big{[}\frac{1}{2\delta}+2\delta+\frac{\mu_{1}^{2}}{4\delta\lambda_{1}}+\frac{\mu_{2}^{2}}{4\delta\lambda_{1}}+\frac{b}{4\delta\lambda_{1}}+\frac{\alpha^{2}}{4\delta}+\frac{\alpha^{2}}{4(\rho+1)\delta\lambda_{1}}\Big{]}C_{\alpha}\\\
\quad\quad\quad+\frac{\alpha(1-l)+g(0)}{4\delta}+\frac{\alpha(1-l)+g(0)}{4(\rho+1)\delta\lambda_{1}};\\\
B_{3}=\frac{\delta}{\lambda_{1}}+2\delta+\frac{2\delta}{\rho+1}c_{s}^{2(\rho+1)}\Big{(}2\mathcal{D}E(0)\Big{)}^{\frac{\rho}{2}}.\end{cases}$
(2.31)

∎

######  Lemma 2.9.

The functional

$I_{3}(t)=\int_{0}^{1}e^{-2\tau\kappa}\|z(x,\kappa,t)\|_{2}^{2}d\kappa$ (2.32)

satisfies for all $t\geq 0$,

$I_{3}^{\prime}(t)=-2I_{3}(t)+\frac{1}{\tau}\|u_{t}\|_{2}^{2}-\frac{e^{-2\tau}}{\tau}\|z(x,1,t)\|_{2}^{2}.$
(2.33)

###### Proof.

Differentiating (2.32) on $t$, and using the second identity in (2.1), it is
direct that

$\begin{split}I_{3}^{\prime}(t)&=\int_{0}^{1}2e^{-2\tau\kappa}\int_{\Omega}z(x,\kappa,t)z_{t}(x,\kappa,t)dxd\kappa\\\
&=-\int_{0}^{1}\frac{2}{\tau}e^{-2\tau\kappa}\int_{\Omega}z(x,\kappa,t)z_{k}(x,\kappa,t)dxd\kappa\\\
&=-\frac{1}{\tau}\int_{\Omega}\int_{0}^{1}\Big{(}\frac{d}{d\kappa}e^{-2\tau\kappa}z^{2}(x,\kappa,t)+2\tau
e^{-2\tau\kappa}z^{2}(x,\kappa,t)\Big{)}d\kappa dx\\\
&=-2I_{3}(t)+\frac{1}{\tau}\|u_{t}\|_{2}^{2}-\frac{e^{-2\tau}}{\tau}\|z(x,1,t)\|_{2}^{2}.\end{split}$
(2.34)

∎

######  Lemma 2.10 (Lemma 3.4 in [17]).

The functional

$I_{4}(t)=\int_{0}^{t}f(t-s)\|\nabla u(s)\|_{2}^{2}ds$ (2.35)

satisfies for all $t\geq 0$,

$I_{4}^{\prime}(t)\leq 3(1-l)\|\nabla u\|_{2}^{2}-\frac{1}{2}(g\circ\nabla
u)(t),$ (2.36)

where $f(t)=\int_{t}^{\infty}g(s)ds.$

##  3\. Stability results

In this section, we will present and prove the decay results of the energy
functional $E(t)$ based on the lemmas in Section 2. To begin with, we define a
functional

$L(t)=ME(t)+\sum_{i=1}^{3}N_{i}I_{i}(t),$ (3.1)

where $M,~{}N_{1},~{}N_{2},~{}N_{3}$ are positive constants. The following
lemma is shown to illustrate that $L(t)$ is equivalent to $E(t)$.

######  Lemma 3.1.

Under all the conditions of Lemma 2.3, assume that $M$ is enough large, then
there exist two positive constants $\beta_{1}$ and $\beta_{2}$ such that

$\beta_{1}E(t)\leq L(t)\leq\beta_{2}E(t).$ (3.2)

###### Proof.

Recalling the definition of $I_{1}(t)$ in (2.14), using Young’s inequality and
Cauchy’s inequality, and then applying the embedding
$H_{0}^{1}(\Omega)\hookrightarrow L^{\rho+2}(\Omega)$ and (2.8), it is not
hard to give

$\begin{split}|I_{1}(t)|&\leq\frac{1}{\rho+2}\|u_{t}\|^{\rho+2}_{\rho+2}+\frac{1}{(\rho+1)(\rho+2)}\|u\|^{\rho+2}_{\rho+2}+\frac{1}{2}\|\nabla
u_{t}\|_{2}^{2}+\frac{1}{2}\|\nabla u\|_{2}^{2}\\\
&\leq\frac{1}{\rho+2}\|u_{t}\|^{\rho+2}_{\rho+2}+\Big{[}\frac{c_{s}^{\rho+2}}{(\rho+1)(\rho+2)}\Big{(}\frac{2\mathcal{D}}{l}E(0)\Big{)}^{\frac{\rho}{2}}+\frac{1}{2}\Big{]}\|\nabla
u\|_{2}^{2}+\frac{1}{2}\|\nabla u_{t}\|_{2}^{2}.\end{split}$ (3.3)

Recalling the definition of $I_{2}(t)$ in (2.21), using integration by parts,
Cauchy’s inequality, Young’s inequality and Hölder’s inequality, we give

$\begin{split}|I_{2}(t)|&\leq\frac{1}{2}\|\nabla
u_{t}\|_{2}^{2}+\frac{1}{2}\int_{\Omega}\Big{(}\int_{0}^{t}g(t-s)(\nabla
u(t)-\nabla u(s))ds\Big{)}^{2}dx\\\
&\quad+\frac{1}{\rho+2}\|u_{t}\|^{\rho+2}_{\rho+2}+\frac{1}{(\rho+1)(\rho+2)}\int_{\Omega}\Big{(}\int_{0}^{t}g(t-s)(u(t)-u(s))ds\Big{)}^{\rho+2}dx\\\
&\leq\frac{1}{2}\|\nabla u_{t}\|_{2}^{2}+\frac{1-l}{2}(g\circ\nabla
u)(t)+\frac{1}{\rho+2}\|u_{t}\|^{\rho+2}_{\rho+2}\\\
&\quad+\frac{1}{(\rho+1)(\rho+2)}(1-l)^{\rho+1}c_{s}^{\rho+2}\Big{(}\frac{2\mathcal{D}}{l}E(0)\Big{)}^{\frac{\rho}{2}}(g\circ\nabla
u)(t),\end{split}$ (3.4)

here we have used

$\begin{split}&\int_{\Omega}\Big{(}\int_{0}^{t}g(t-s)(u(t)-u(s))ds\Big{)}^{\rho+2}dx\\\
&\quad\leq\int_{\Omega}\Big{(}\int_{0}^{t}(g(t-s))^{\frac{\rho+1}{\rho+2}}(g(t-s))^{\frac{1}{\rho+2}}(u(t)-u(s))ds\Big{)}^{\rho+2}dx\\\
&\quad\leq\Big{(}\int_{0}^{t}g(s)ds\Big{)}^{\rho+1}\int_{0}^{t}g(t-s)\|u(t)-u(s))\|_{\rho+2}^{\rho+2}ds\\\
&\quad\leq(1-l)^{\rho+1}c_{s}^{\rho+2}\int_{0}^{t}g(t-s)\|\nabla u(t)-\nabla
u(s))\|_{2}^{\rho+2}ds\\\
&\quad\leq(1-l)^{\rho+1}c_{s}^{\rho+2}\Big{(}\frac{2\mathcal{D}}{l}E(0)\Big{)}^{\frac{\rho}{2}}(g\circ\nabla
u)(t).\end{split}$ (3.5)

Therefore, it follows from (2.8) that

$\begin{split}|L(t)-ME(t)|=|\sum_{i=1}^{3}N_{i}I_{i}(t)|\leq
CE(t),\end{split}$

here $C$ is some positive constant. ∎

######  Lemma 3.2.

Under all the conditions of Lemma 2.3, for $|\mu_{2}|<\mu_{1}$, the functional
$L(t)$ defined in (3.1) satisfies, for $t\geq t_{1}$

$\begin{split}L^{\prime}(t)&\leq-
C_{1}\|u_{t}\|_{\rho+2}^{\rho+2}-C_{2}\|\nabla u_{t}\|_{2}^{2}-4(1-l)\|\nabla
u\|_{2}^{2}\\\ &\quad+N_{1}b\|u\|_{p}^{p}+\frac{1}{4}(g\circ\nabla
u)(t)-2N_{3}\int_{0}^{1}\|z(x,\kappa,t)\|_{2}^{2}d\kappa.\end{split}$ (3.6)

where $C_{1},~{}C_{2}$ are positive constants given in (3.8).

###### Proof.

Taking the combination of (2.4), (2.15) and (2.22) with (2.33), recalling
(2.11), and applying $g_{1}=\int_{0}^{t_{1}}g(s)ds\leq\int_{0}^{t}g(s)ds$ for
$t\geq t_{1}$, one has

$\begin{split}L^{\prime}(t)&\leq-M\omega(\|u_{t}\|_{2}^{2}+\|z(x,1,t)\|_{2}^{2})+\frac{M}{2}(g^{\prime}\circ\nabla
u)(t)-\frac{M}{2}g(t)\|\nabla u\|_{2}^{2}\\\
&\quad+\frac{N_{1}}{\rho+1}\|u_{t}\|_{\rho+2}^{\rho+2}-N_{1}\Big{[}l-\Big{(}1+\frac{\mu_{1}}{\lambda_{1}}+\frac{\mu_{2}}{\lambda_{1}}\Big{)}\varepsilon\Big{]}\|\nabla
u\|_{2}^{2}+\frac{N_{1}}{4\varepsilon}C_{\alpha}(h_{\alpha}\circ\nabla
u)(t)\\\
&\quad+\frac{N_{1}|\mu_{2}|}{4\varepsilon}\|z(x,1,t)\|_{2}^{2}+N_{1}b\|u\|_{p}^{p}+N_{1}\|\nabla
u_{t}\|_{2}^{2}+N_{1}\frac{\mu_{1}}{4\varepsilon}\|u_{t}\|_{2}^{2}\\\
&\quad+N_{2}B_{1}\|\nabla u\|_{2}^{2}+N_{2}B_{2}(h_{\alpha}\circ\nabla
u)(t)+N_{2}\Big{[}B_{3}-\int_{0}^{t}g(s)ds\Big{]}\|\nabla u_{t}\|_{2}^{2}\\\
&\quad+N_{2}\delta\|z(x,1,t)\|_{2}^{2}-N_{2}\int_{0}^{t}g(s)ds\cdot\frac{1}{\rho+1}\|u_{t}\|_{\rho+2}^{\rho+2}\\\
&\quad-2N_{3}I_{3}(t)+\frac{N_{3}}{\tau}\|u_{t}\|_{2}^{2}-\frac{N_{3}e^{-2\tau}}{\tau}\|z(x,1,t)\|_{2}^{2}\\\
&\leq-C_{1}\|u_{t}\|_{\rho+2}^{\rho+2}-C_{2}\|\nabla
u_{t}\|_{2}^{2}-C_{3}\|\nabla
u\|_{2}^{2}-C_{4}\|z(x,1,t)\|_{2}^{2}-C_{5}(h_{\alpha}\circ\nabla u)(t)\\\
&\quad+N_{1}b\|u\|_{p}^{p}+\frac{\alpha M}{2}(g\circ\nabla
u)(t)-C_{6}\|u_{t}\|_{2}^{2}-2N_{3}\int_{0}^{1}\|z(x,\kappa,t)\|_{2}^{2}d\kappa\end{split}$
(3.7)

with

$\begin{cases}C_{1}=N_{2}g_{1}\cdot\frac{1}{\rho+1}-\frac{N_{1}}{\rho+1};\\\
C_{2}=N_{2}\Big{[}g_{1}-\Big{(}\frac{\delta}{\lambda_{1}}+2\delta+\frac{2\delta}{\rho+1}c_{s}^{2(\rho+1)}\Big{(}2\mathcal{D}E(0)\Big{)}^{\frac{\rho}{2}}\Big{)}\Big{]}-N_{1};\\\
C_{3}=N_{1}\Big{[}l-\Big{(}1+\frac{\mu_{1}}{\lambda_{1}}+\frac{|\mu_{2}|}{\lambda_{1}}\Big{)}\varepsilon\Big{]}-N_{2}\Big{[}\delta+2\delta(1-l)^{2}+b\delta
c_{s}^{2(p-1)}\Big{(}\frac{2\mathcal{D}}{l}E(0)\Big{)}^{p-2}\Big{]};\\\
C_{4}=\omega
M+N_{3}\frac{e^{-2\tau}}{\tau}-N_{1}\frac{|\mu_{2}|}{4\varepsilon}-N_{2}\delta;\\\
C_{5}=\frac{M}{2}-N_{1}\frac{1}{4\varepsilon}C_{\alpha}-N_{2}\Big{[}\Big{(}\frac{1}{2\delta}+2\delta+\frac{\mu_{1}^{2}}{4\delta\lambda_{1}}+\frac{\mu_{2}^{2}}{4\delta\lambda_{1}}+\frac{b}{4\delta\lambda_{1}}+\frac{\alpha^{2}}{4\delta}+\frac{\alpha^{2}}{4(\rho+1)\delta\lambda_{1}}\Big{)}C_{\alpha};\\\
\quad\quad\quad+\frac{\alpha(1-l)+g(0)}{4\delta}+\frac{\alpha(1-l)+g(0)}{4(\rho+1)\delta\lambda_{1}}\Big{]};\\\
C_{6}=\omega
M-\frac{N_{3}}{\tau}-N_{1}\frac{\mu_{1}}{4\varepsilon},\end{cases}$ (3.8)

where we have used the values of $B_{1}$, $B_{2}$ and $B_{3}$ defined in
(2.31).

Next, we choose $\delta$ such that

$\begin{split}\delta&<\Bigg{\\{}\frac{lg_{1}}{16\Big{[}1+2(1-l)^{2}+bc_{s}^{2(p-1)}\Big{(}\frac{2\mathcal{D}}{l}E(0)\Big{)}^{p-2}\Big{]}},\frac{lg_{1}}{1024(1-l)^{2}},\\\
&\quad\quad\frac{5g_{1}}{8\Big{(}\frac{1}{\lambda_{1}}+2+\frac{2}{\rho+1}c_{s}^{2(\rho+1)}\Big{(}2\mathcal{D}E(0)\Big{)}^{\frac{\rho}{2}}\Big{)}}\Bigg{\\}}.\end{split}$

Let us choose $N_{1}=\frac{3}{8}g_{1}N_{2}$, then

$C_{1}=N_{2}g_{1}\cdot\frac{1}{\rho+1}-\frac{3}{8}g_{1}N_{2}\frac{1}{\rho+1}=\frac{5}{8}g_{1}N_{2}\frac{1}{\rho+1}>0,\quad
C_{2}>0.$

Let us fix

$\varepsilon=\frac{3l}{4}\frac{1}{1+\frac{\mu_{1}}{\lambda_{1}}+\frac{|\mu_{2}|}{\lambda_{1}}},$

then

$C_{3}=\frac{N_{1}l}{4}-N_{2}\Big{[}\delta+2\delta(1-l)^{2}+b\delta
c_{s}^{2(p-1)}\Big{(}\frac{2\mathcal{D}}{l}E(0)\Big{)}^{p-2}\Big{]}>\frac{l}{32}g_{1}N_{2}>0.$

By taking $N_{2}=\frac{1}{8\delta(1-l)}$, we get

$C_{3}>\frac{l}{32}g_{1}N_{2}=\frac{lg_{1}}{256\delta(1-l)}>4(1-l).$

Since $g^{\prime}(s)\leq 0$, one has $\frac{\alpha g^{2}(s)}{\alpha
g(s)-g^{\prime}(s)}\leq g(s)$, further we get

$\lim_{\alpha\to 0^{+}}\alpha C_{\alpha}=\lim_{\alpha\to
0^{+}}\int_{0}^{\infty}\frac{\alpha g^{2}(s)}{\alpha g(s)-g^{\prime}(s)}ds=0.$

Thus, there exists $0<\alpha_{0}<1$ so that if $\alpha<\alpha_{0}$, then

$\alpha
C_{\alpha}<\frac{1}{8\Big{[}N_{2}\Big{(}\frac{1}{2\delta}+2\delta+\frac{\mu_{1}^{2}}{4\delta\lambda_{1}}+\frac{\mu_{2}^{2}}{4\delta\lambda_{1}}+\frac{b}{4\delta\lambda_{1}}+\frac{\alpha^{2}}{4\delta}+\frac{\alpha^{2}}{4(\rho+1)\delta\lambda_{1}}\Big{)}+N_{1}\frac{1}{4\varepsilon}\Big{]}}.$

Let us choose $M$ sufficiently large such that for $\alpha=\frac{1}{2M}$,

$C_{5}=\frac{M}{4}-N_{2}\Big{[}\frac{\alpha(1-l)+g(0)}{4\delta}+\frac{\alpha(1-l)+g(0)}{4(\rho+1)\delta\lambda_{1}}\Big{]}>0,$
$C_{4}=\omega
M+N_{3}\frac{e^{-2\tau}}{\tau}-N_{1}\frac{|\mu_{2}|}{4\varepsilon}-N_{2}\delta>0,$
$C_{6}=\omega M-\frac{N_{3}}{\tau}-N_{1}\frac{\mu_{1}}{4\varepsilon}>0,$

here we have used $\omega>0$ given in Lemma 2.2 since $|\mu_{2}|<\mu_{1}$.

Based on the above discussion, one has from (3.7)

$\begin{split}L^{\prime}(t)&\leq-
C_{1}\|u_{t}\|_{\rho+2}^{\rho+2}-C_{2}\|\nabla u_{t}\|_{2}^{2}-4(1-l)\|\nabla
u\|_{2}^{2}\\\ &\quad+N_{1}b\|u\|_{p}^{p}+\frac{1}{4}(g\circ\nabla
u)(t)-2N_{3}\int_{0}^{1}\|z(x,\kappa,t)\|_{2}^{2}d\kappa.\end{split}$

∎

Now, we give the following stability results.

######  Theorem 3.3.

For $|\mu_{2}|<\mu_{1}$, suppose that $(\mathbf{H_{1}}),~{}(\mathbf{H_{2}})$
hold and

$E(0)<E_{1},\quad l\|\nabla u_{0}\|_{2}^{2}<\sigma_{1}^{2},$

then there exist positive constants $k_{1},k_{2},k_{3}$ and $k_{4}$ such that
the solution of problem (1.1) satisfies for all $t\geq t_{1}$,

$E(t)\leq\begin{cases}k_{1}e^{-k_{2}\int_{t_{1}}^{t}\zeta(s)ds}&\quad\text{for
}G\text{ is linear};\\\
k_{4}G_{1}^{-1}\Big{(}k_{3}\int_{t_{1}}^{t}\zeta(s)ds\Big{)}&\quad\text{for
}G\text{ is nonlinear},\end{cases}$

where $E_{1}$ and $\sigma_{1}$ are shown in Lemma 2.3,
$G_{1}(t)=\int_{t}^{r}\frac{1}{sG^{\prime}(s)}ds$ is strictly decreasing and
convex in $(0,r]$ with $\lim_{t\to 0}G_{1}(t)=+\infty.$

###### Proof.

Using (2.13) and (2.4), one has, for $t\geq t_{1}$,

$\begin{split}&\int_{0}^{t_{1}}g(s)\|\nabla u(t)-\nabla u(t-s)\|_{2}^{2}ds\\\
&\quad\leq-\frac{1}{\gamma}\int_{0}^{t_{1}}g^{\prime}(s)\|\nabla u(t)-\nabla
u(t-s)\|_{2}^{2}ds\leq-cE^{\prime}(t)\end{split}$ (3.9)

here $c$ is used to denote a generic positive constant throughout this proof.
Define a functional $F(t)$ that is obviously equivalent to $E(t)$ as follows

$F(t)=L(t)+cE(t),$

then based on (3.6), (2.2), (3.9), for some $m>0$ and for any $t\geq t_{1}$,
we have

$\begin{split}F^{\prime}(t)&\leq-
C_{1}\|u_{t}\|_{\rho+2}^{\rho+2}-C_{2}\|\nabla u_{t}\|_{2}^{2}-4(1-l)\|\nabla
u\|_{2}^{2}\\\ &\quad+N_{1}b\|u\|_{p}^{p}+\frac{1}{4}(g\circ\nabla
u)(t)-2N_{3}\int_{0}^{1}\|z(x,\kappa,t)\|_{2}^{2}d\kappa+cE^{\prime}(t)\\\
&\leq-mE(t)-\Big{(}\frac{bc}{p}-N_{1}b\Big{)}\|u\|_{p}^{p}+c(g\circ\nabla
u)(t)+cE^{\prime}(t)\\\ &\leq-mE(t)+c\int_{t_{1}}^{t}g(s)\|\nabla u(t)-\nabla
u(t-s)\|_{2}^{2}ds,\end{split}$ (3.10)

here we have chosen $N_{1}$ so small that $\frac{bc}{p}-N_{1}b>0$.

In what follows, we will discuss in two cases.

Case 1: $G$ is linear Multiplying (3.10) by $\zeta(t)$, using
$(\mathbf{H_{1}})$ and (2.4), one gives

$\begin{split}\zeta(t)F^{\prime}(t)&\leq-m\zeta(t)E(t)+c\zeta(t)\int_{t_{1}}^{t}g(s)\|\nabla
u(t)-\nabla u(t-s)\|_{2}^{2}ds\\\
&\leq-m\zeta(t)E(t)-c\int_{t_{1}}^{t}g^{\prime}(s)\|\nabla u(t)-\nabla
u(t-s)\|_{2}^{2}ds\\\ &\leq-m\zeta(t)E(t)-cE^{\prime}(t),\end{split}$

which implies

$(\zeta(t)F(t)+cE(t))^{\prime}\leq-m\zeta(t)E(t)\quad\text{for }t\geq t_{1}.$

Integrating the above inequality over $(t_{1},t)$, and using the fact that
$\zeta(t)F(t)+cE(t)$ is equivalent to $E(t)$, one has

$E(t)\leq k_{1}e^{-k_{2}\int_{t_{1}}^{t}\zeta(s)ds}\quad\text{for }t\geq
t_{1},$

here $k_{1}$ and $k_{2}$ are constants.

Case 2: $G$ is nonlinear Define a functional

$H(t)=L(t)+I_{4}(t).$

Taking the combination of Lemma 3.1 and the non-negativity of $E(t)$ obtained
by Lemma 2.4 with the definition of $I_{4}(t)$ in (2.35) , it is not difficult
to get the non-negativity of $H(t)$. It follows from (3.6) and (2.36) that for
some $m_{1}>0$ and $t\geq t_{1}$,

$\begin{split}H^{\prime}(t)&=L^{\prime}(t)+I^{\prime}_{4}(t)\\\ &\leq-
C_{1}\|u_{t}\|_{\rho+2}^{\rho+2}-C_{2}\|\nabla u_{t}\|_{2}^{2}-(1-l)\|\nabla
u\|_{2}^{2}\\\ &\quad+N_{1}b\|u\|_{p}^{p}-\frac{1}{4}(g\circ\nabla
u)(t)-2N_{3}\int_{0}^{1}\|z(x,\kappa,t)\|_{2}^{2}d\kappa\leq-
m_{1}E(t).\end{split}$

Integrating the above inequality over $(t_{1},t)$ yields

$m_{1}\int_{t_{1}}^{t}E(s)ds\leq H(t_{1})-H(t)\leq H(t_{1}),$

which implies

$\int_{0}^{\infty}E(s)ds<+\infty.$ (3.11)

Define

$\lambda(t)=p\int_{t_{1}}^{t}\|\nabla u(t)-\nabla u(t-s)\|_{2}^{2}ds,$

by using (2.8), then we give

$\begin{split}\lambda(t)&\leq 2p\int_{0}^{t}\Big{(}\|\nabla
u(t)\|_{2}^{2}+\|\nabla u(t-s)\|_{2}^{2}\Big{)}ds\\\
&\leq\frac{8p\mathcal{D}}{l}\int_{0}^{t}\Big{(}E(t)+E(t-s)\Big{)}ds\leq\frac{16p\mathcal{D}}{l}\int_{0}^{t}E(t-s)ds\\\
&=\frac{16p\mathcal{D}}{l}\int_{0}^{t}E(s)ds\leq\int_{0}^{\infty}E(s)ds<+\infty.\end{split}$

Thus, we can choose $p$ so small that for $t\geq t_{1}$,

$\lambda(t)<1.$ (3.12)

It is direct that

$G(\theta z)\leq\theta G(z)\quad\text{for }0\leq\theta\leq 1\text{ and
}z\in(0,r],$ (3.13)

since $G$ is strictly convex on $(0,r]$ and $G(0)=0$. Based on (1.3), (3.12),
(3.13) and Jensen’s inequality, one gives

$\begin{split}I(t)&=\frac{1}{p\lambda(t)}\int_{t_{1}}^{t}\lambda(t)(-g^{\prime}(s))p\|\nabla
u(t)-\nabla u(t-s)\|_{2}^{2}ds\\\
&\geq\frac{1}{p\lambda(t)}\int_{t_{1}}^{t}\lambda(t)\zeta(s)G(g(s))p\|\nabla
u(t)-\nabla u(t-s)\|_{2}^{2}ds\\\
&\geq\frac{\zeta(t)}{p\lambda(t)}\int_{t_{1}}^{t}\overline{G}(\lambda(t)g(s))p\|\nabla
u(t)-\nabla u(t-s)\|_{2}^{2}ds\\\
&\geq\frac{\zeta(t)}{p}\overline{G}\Big{(}p\int_{t_{1}}^{t}g(s)\|\nabla
u(t)-\nabla u(t-s)\|_{2}^{2}ds\Big{)},\end{split}$

which yields

$\int_{t_{1}}^{t}g(s)\|\nabla u(t)-\nabla
u(t-s)\|_{2}^{2}ds\leq\frac{1}{p}\overline{G}^{-1}\Big{(}\frac{pI(t)}{\zeta(t)}\Big{)},$
(3.14)

where $G$ has an extension $\overline{G}$ which is a strictly increasing and
strictly convex $C^{2}$ function on $(0,+\infty)$ as in Remark 2.1 [6].
Therefore, (3.10) becomes

$F^{\prime}(t)\leq-
mE(t)+\frac{c}{p}\overline{G}^{-1}\Big{(}\frac{pI(t)}{\zeta(t)}\Big{)}.$
(3.15)

Let us define the functional

$F_{1}(t)=\overline{G}^{\prime}\Big{(}\frac{r_{1}E(t)}{E(0)}\Big{)}F(t)+E(t)$

with $0<r_{1}<r,$ then $F_{1}$ is equivalent to $E$ and

$\begin{split}F_{1}^{\prime}(t)&=\frac{r_{1}E^{\prime}(t)}{E(0)}\overline{G}^{\prime\prime}\Big{(}\frac{r_{1}E(t)}{E(0)}\Big{)}F(t)+\overline{G}^{\prime}\Big{(}\frac{r_{1}E(t)}{E(0)}\Big{)}F^{\prime}(t)+E^{\prime}(t)\\\
&\leq-
mE(t)\overline{G}^{\prime}\Big{(}\frac{r_{1}E(t)}{E(0)}\Big{)}+\frac{c}{p}\overline{G}^{-1}\Big{(}\frac{pI(t)}{\zeta(t)}\Big{)}\overline{G}^{\prime}\Big{(}\frac{r_{1}E(t)}{E(0)}\Big{)}+E^{\prime}(t)\end{split}$
(3.16)

by using (3.15), (2.4), $G^{\prime}>0$ and $G^{\prime\prime}>0$. Let
$\overline{G}^{*}$ be the convex conjugate of $G$ in the sense of Young in
[1], which is given by

$\overline{G}^{*}(s)=s(\overline{G}^{\prime})^{-1}(s)-\overline{G}\Big{[}(\overline{G}^{\prime})^{-1}(s)\Big{]}$
(3.17)

and it satisfies the following Young’s inequality

$AB\leq\overline{G}^{*}(A)+\overline{G}(B).$ (3.18)

Choosing

$A=\overline{G}^{\prime}\Big{(}\frac{r_{1}E(t)}{E(0)}\Big{)}\text{ and
}B=\overline{G}^{-1}\Big{(}\frac{pI(t)}{\zeta(t)}\Big{)},$

then using (3.18), (3.17) and the non-negativity of $\overline{G},$ (3.16)
becomes

$\begin{split}F_{1}^{\prime}(t)&\leq-
mE(t)\overline{G}^{\prime}\Big{(}\frac{r_{1}E(t)}{E(0)}\Big{)}+\frac{c}{p}\overline{G}^{-1}\Big{(}\frac{pI(t)}{\zeta(t)}\Big{)}\overline{G}^{\prime}\Big{(}\frac{r_{1}E(t)}{E(0)}\Big{)}+E^{\prime}(t)\\\
&\leq-
mE(t)\overline{G}^{\prime}\Big{(}\frac{r_{1}E(t)}{E(0)}\Big{)}+\frac{c}{p}\overline{G}^{*}\Big{(}\overline{G}^{\prime}\Big{(}\frac{r_{1}E(t)}{E(0)}\Big{)}\Big{)}+c\frac{I(t)}{\zeta(t)}+E^{\prime}(t)\\\
&\leq-
mE(t)\overline{G}^{\prime}\Big{(}\frac{r_{1}E(t)}{E(0)}\Big{)}+\frac{c}{p}\frac{r_{1}E(t)}{E(0)}\overline{G}^{\prime}\Big{(}\frac{r_{1}E(t)}{E(0)}\Big{)}\\\
&\quad-\overline{G}\Big{(}\frac{r_{1}E(t)}{E(0)}\Big{)}+c\frac{I(t)}{\zeta(t)}+E^{\prime}(t)\\\
&\leq-
mE(t)\overline{G}^{\prime}\Big{(}\frac{r_{1}E(t)}{E(0)}\Big{)}+\frac{c}{p}\frac{r_{1}E(t)}{E(0)}\overline{G}^{\prime}\Big{(}\frac{r_{1}E(t)}{E(0)}\Big{)}+c\frac{I(t)}{\zeta(t)}+E^{\prime}(t).\end{split}$
(3.19)

Note that $\eqref{01}$ implies

$I(t)\leq\int_{t_{1}}^{t}-g^{\prime}(s)\|\nabla u(t)-\nabla
u(t-s)\|_{2}^{2}ds\leq-2E^{\prime}(t),$

then one has

$\zeta(t)F_{1}^{\prime}(t)\leq-m\zeta(t)E(t)G^{\prime}\Big{(}\frac{r_{1}E(t)}{E(0)}\Big{)}+\frac{c}{p}\zeta(t)\frac{r_{1}E(t)}{E(0)}G^{\prime}\Big{(}\frac{r_{1}E(t)}{E(0)}\Big{)}-cE^{\prime}(t)$
(3.20)

by multiplying (3.19) by $\zeta(t)$ and by using the fact

$\overline{G}^{\prime}\Big{(}\frac{r_{1}E(t)}{E(0)}\Big{)}=G^{\prime}\Big{(}\frac{r_{1}E(t)}{E(0)}\Big{)}.$

Define the functional $F_{2}(t)=\zeta(t)F_{1}(t)+cE(t)$ which is equivalent to
$E(t)$, which means

$\gamma_{1}F_{2}(t)\leq E(t)\leq\gamma_{2}F_{2}(t)$ (3.21)

for some $\gamma_{1}$ and $\gamma_{2}$. Under a suitable choice of $r_{1}$ and
for a positive constant $k$, we have

$F_{2}^{\prime}(t)\leq-k\zeta(t)\frac{E(t)}{E(0)}G^{\prime}\Big{(}\frac{r_{1}E(t)}{E(0)}\Big{)}=-k\zeta(t)G_{2}\Big{(}\frac{E(t)}{E(0)}\Big{)}$
(3.22)

with $G_{2}(t)=tG^{\prime}(r_{1}t)$. Obviously, $G_{2}$ and $G_{2}^{\prime}$
are positive in $(0,1]$ since

$G_{2}^{\prime}(t)=G^{\prime}(r_{1}t)+r_{1}t^{2}G^{\prime\prime}(r_{1}t)$

and the convexity of $G$ in $(0,r].$ (3.22) and (3.21) imply

$\Big{(}\frac{\gamma_{1}F_{2}(t)}{E(0)}\Big{)}^{\prime}\leq-k\zeta(t)\frac{\gamma_{1}}{E(0)}G_{2}\Big{(}\frac{E(t)}{E(0)}\Big{)}\leq-
k_{3}\zeta(t)G_{2}\Big{(}\frac{\gamma_{1}F_{2}(t)}{E(0)}\Big{)}$ (3.23)

with $k_{3}=k\frac{\gamma_{1}}{E(0)}$. Setting
$R(t)=\frac{\gamma_{1}F_{2}(t)}{E(0)}$, and then integrating (3.23) over
$(t_{1},t)$, one has

$\int_{t_{1}}^{t}-\frac{R^{\prime}(s)}{G_{2}(R(s))}ds\geq\int_{t_{1}}^{t}k_{3}\zeta(s)ds.$

Since $r_{1}R(t_{1})<r$, we have

$G_{1}(r_{1}R(t))=\int_{r_{1}R(t)}^{r_{1}R(t_{1})}\frac{1}{sG^{\prime}(s)}ds\geq
k_{3}\int_{t_{1}}^{t}\zeta(s)ds.$

It is noted that $G_{1}$ is strictly decreasing function on $(0,r]$ and
$\lim_{t\to 0}G_{1}(t)=+\infty$ in Theorem 3.3, then

$R(t)\leq\frac{1}{r_{1}}G_{1}^{-1}\Big{(}k_{3}\int_{t_{1}}^{t}\zeta(s)ds\Big{)}.$

Since $R(t)$ is equivalent to $E(t)$, further one obtains

$E(t)\leq k_{4}G_{1}^{-1}\Big{(}k_{3}\int_{t_{1}}^{t}\zeta(s)ds\Big{)}$

with $k_{4}=\frac{1}{r_{1}}.$

This completes the proof of this theorem. ∎

##  4\. Possible generalizations

Our results can be generalized to the following initial boundary value problem
with the time-varying delay $\tau(t)$

$\begin{cases}|u_{t}|^{\rho}u_{tt}-\Delta u-\Delta
u_{tt}+\int_{0}^{t}g(t-s)\Delta u(s)ds\\\
\quad+\mu_{1}u_{t}(x,t)+\mu_{2}u_{t}(x,t-\tau(t))=b|u|^{p-2}u&(x,t)\in\Omega\times(0,\infty),\\\
u_{t}(x,t-\tau(0))=f_{0}(x,t-\tau(0))&(x,t)\in\Omega\times(0,\tau(0)),\\\
u(x,0)=u_{0}(x),~{}u_{t}(x,0)=u_{1}(x)&x\in\Omega,\\\
u(x,t)=0&(x,t)\in\partial\Omega\times[0,\infty),\end{cases}$

equipped with the following assumptions in addition to $(\mathbf{H_{1}})$ and
$(\mathbf{H_{2}})$:

$(A_{1})$ the function $\tau\in W^{2,\infty}([0,T])$, for all $T>0$ such that

$0<\tau_{0}\leq\tau(t)\leq\tau_{1}\quad\text{for all }t>0,$
$\tau^{\prime}(t)\leq d<1\quad\text{for all }t>0,$

where $\tau_{0}$ and $\tau_{1}$ are positive numbers.

$(A_{2})$ the coefficients of delay and dissipation satisfy

$|\mu_{2}|\leq\frac{2(1-d)}{2-d}\mu_{1}.$

The above assumptions in fact are given in [6]. To be more precise, we need to
define the new energy functional

$\begin{split}E(t)&=\frac{1}{\rho+2}\|u_{t}\|_{\rho+2}^{\rho+2}+\frac{1}{2}\Big{(}1-\int_{0}^{t}g(s)ds\Big{)}\|\nabla
u\|_{2}^{2}+\frac{1}{2}(g\circ\nabla u)(t)\\\ &\quad+\frac{1}{2}\|\nabla
u_{t}\|_{2}^{2}+\frac{\xi}{2}\tau(t)\int_{\Omega}\int_{0}^{1}z^{2}(x,\kappa,t)d\kappa
dx-\frac{b}{p}\|u\|_{p}^{p},\end{split}$

where $\xi$ satisfies

$\frac{|\mu_{2}|}{1-d}\leq\xi\leq 2\mu_{1}-|\mu_{2}|.$

We also need to replace $I_{3}(t)$ in Lemma 2.9 by

$I_{3}(t)=\int_{0}^{1}e^{-2\tau(t)\kappa}\|z(x,\kappa,t)\|_{2}^{2}d\kappa,$

further, one has

$I_{3}^{\prime}(t)\leq-2I_{3}(t)-\frac{(1-d)e^{-2\tau_{1}}}{\tau_{1}}\|z(x,1,t)\|_{2}^{2}+\frac{1}{\tau_{0}}\|u_{t}\|_{2}^{2}$

as shown in Lemma 2.8 of [6]. Based on the above changes, we believe that it
is possible to get the similar results in Theorem 3.3 by following the steps
in this paper.

In addition, we have a conjecture that our results can also be extended to the
following initial boundary value problem:

$\begin{cases}|u_{t}|^{\rho}u_{tt}-\Delta u-\Delta
u_{tt}+\int_{0}^{t}g(t-s)\Delta u(s)ds\\\
\quad+\mu_{1}|u_{t}(x,t)|^{m-2}u_{t}(x,t)\\\
\quad+\mu_{2}|u_{t}(x,t-\tau)|^{m-2}u_{t}(x,t-\tau)=b|u|^{p-2}u&(x,t)\in\Omega\times(0,\infty),\\\
u_{t}(x,t-\tau)=f_{0}(x,t-\tau)&(x,t)\in\Omega\times(0,\tau),\\\
u(x,0)=u_{0}(x),~{}u_{t}(x,0)=u_{1}(x)&x\in\Omega,\\\
u(x,t)=0&(x,t)\in\partial\Omega\times[0,\infty).\end{cases}$

### Acknowledgements

The author would like to express her sincere gratitude to Professor Wenjie Gao
and Professor Bin Guo in Jilin University for their support and constant
encouragement.

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# Theory of moiré magnetism in twisted bilayer $\alpha$-RuCl3

Muhammad Akram  Department of Physics, Arizona State University, Tempe, AZ
85287, USA  Jesse Kapeghian  Department of Physics, Arizona State University,
Tempe, AZ 85287, USA  Jyotirish Das  Department of Physics, Arizona State
University, Tempe, AZ 85287, USA  Roser Valentí  Institut für Theoretische
Physik, Goethe-Universität Frankfurt, 60438 Frankfurt am Main, Germany  Antia
S. Botana  Department of Physics, Arizona State University, Tempe, AZ 85287,
USA<EMAIL_ADDRESS>Onur Erten  Department of Physics, Arizona State
University, Tempe, AZ 85287, USA<EMAIL_ADDRESS>
###### Abstract

Twisted heterostructures of van der Waals materials have received much
attention for their many remarkable properties. Here, we present a
comprehensive theory of the long-range ordered magnetic phases of twisted
bilayer $\alpha$-RuCl3 via a combination of first-principles calculations and
atomistic simulations. While a monolayer exhibits zigzag antiferromagnetic
order with three possible ordering wave vectors, a rich phase diagram is
obtained for moiré superlattices as a function of interlayer exchange and
twist angle. For large twist angles, each layer spontaneously picks a single
zigzag ordering wave vector, whereas, for small twist angles, the ground state
involves a combination of all three wave vectors in a complex hexagonal domain
structure. This multi-domain order minimizes the interlayer energy while
enduring the energy cost due to the domain wall formation. Our results
indicate that magnetic frustration due to stacking-dependent interlayer
exchange in moiré superlattices can be used to tune the magnetic ground state
and enhance quantum fluctuations in $\alpha$-RuCl3.

KEYWORDS: 2D vdW magnets, $\alpha$-RuCl3, moiré patterns.

$\alpha$-RuCl3 is a van der Waals (vdW) magnet that has attracted a lot of
attention in recent years as a promising candidate for a Kitaev quantum spin
liquid 1. However, its zigzag antiferromagnetic (AFM) order below TN $\sim$ 7
K 2, 3, 4, 5, 6 indicates deviations from the Kitaev model. The specific
nature of these deviations has been theoretically and experimentally
intensively scrutinized, with many works pointing to additional large
anisotropic couplings beyond the Kitaev interaction and long-range exchange
likely stabilizing magnetic order 7, 8, 9, 10, 11, 12. Under application of an
in-plane magnetic field the zigzag order is suppressed leading to a field-
induced ‘disordered’ phase, whose nature is presently still under intensive
debate, specially due to initial reports of the observation of half-quantized
thermal Hall conductivity 13. This behavior would correspond to a chiral
Kitaev spin liquid state 1. However, these results have not been corroborated
so far 14, 15, 16. Alternatively, there have been promising routes to
investigate the emergence of novel phases beyond the zigzag AFM order by
considering such approaches as chemical doping 17, strain fields 18, 19, 20,
21 and graphene substrates 22, 23, 24, 25, 26, 27, 28, 29. In the present
work, we explore yet a new route to modify the magnetism in $\alpha$-RuCl3, by
exploring the properties of twisted bilayers of $\alpha$-RuCl3.

Moiré superlattices of van der Waals materials have surfaced as new tunable
quantum platforms for the realization of emergent phases on every front
including graphene30, semiconductors31, and superconductors32. While moiré
engineering of electronic phases has been studied extensively, research in
moiré superlattices comprised of magnetic materials is at its early stages. A
range of novel non-coplanar phases have been predicted in moiré vdW magnets33,
34, 35, 36, 37, 38, 39, 40, 41, 42, 43 and some of these phases have been
observed experimentally44, 45, 46. Given that monolayers of $\alpha$-RuCl3 can
be isolated by exfoliation methods47, 48 and heterostructures with other vdW
materials such as graphene can be constructed25, moiré engineering in
$\alpha$-RuCl3 can be explored as a means to tune its magnetism.

Here, we study the long-range ordered magnetic phases of twisted bilayer
$\alpha$-RuCl3 by a combination of first-principles calculations and atomistic
simulations. Our main results are as follows: (i) we obtain the stacking
dependent interlayer exchange within the moiré unit cell and show that the two
layers are coupled antiferromagnetically for different stacking orders. (ii)
Among the three inequivalent ordering wave vectors ($q_{i}$, $i=1,~{}2,~{}3$)
for the zigzag AFM order (see Fig. 2), the two layers spontaneously pick
different $q$’s on each layer for large twist angles. This single-$q$ phase
($1q-1q$) has no domain wall but the interlayer exchange energy averages to
almost zero. (iii) For small angles or large moiré periodicity, the ground
state incorporates all of the three $q$’s in a complex domain structure
($3q-3q$). These domains resemble a hexagonal shape and minimize the
interlayer exchange. However, these domains are separated by domain walls that
cost energy. (iv) Comparing the interlayer exchange and domain wall energy, we
obtain an analytical formula for the transition between these two phases which
agrees well with the atomistic simulations. (v) In the vicinity of the
($1q-1q$) and ($3q-3q$) phase boundary, we obtain an intermediate phase with
two $q$’s on each layer ($2q-2q$) that has both domain walls and finite
interlayer exchange energy.

We start by introducing the spin Hamiltonian which describes the magnetic
properties of twisted bilayer $\alpha$-RuCl3. For simplicity we limit
ourselves to a minimal set of exchange parameters, which have proven to
provide a reasonable description of several experimental observations in
$\alpha$-RuCl3 9:

$\mathcal{H}=\mathcal{H}_{intra}^{1}+\mathcal{H}_{intra}^{2}+\mathcal{H}_{inter},$
(1)

where $\mathcal{H}_{intra}^{1(2)}$ contains the intralayer exchange terms in
layer 1 (2) and $\mathcal{H}_{inter}$ contains the interlayer exchange,

$\displaystyle\mathcal{H}_{intra}$ $\displaystyle=$
$\displaystyle\sum_{\langle
ij\rangle_{\gamma}}\big{[}J_{1}\mathbf{S}_{i}\cdot\mathbf{S}_{j}+KS_{i}^{\gamma}S_{j}^{\gamma}+\Gamma(S_{i}^{\alpha}S_{j}^{\beta}+S_{j}^{\alpha}S_{i}^{\beta})\big{]}+J_{3}\sum_{\langle\langle\langle
ij\rangle\rangle\rangle}\mathbf{S}_{i}\cdot\mathbf{S}_{j},$ (2)
$\displaystyle\mathcal{H}_{inter}$ $\displaystyle=$
$\displaystyle\sum_{\langle ij\rangle}J_{\perp}({r}_{ij}){\bf
S}_{i}^{1}\cdot{\bf S}_{j}^{2}.$ (3)

where $\gamma=x~{},y,~{}z$ type bonds on the honeycomb lattice. $J_{1(3)}$ is
the (third) nearest-neighbor Heisenberg exchange, $K$ and $\Gamma$ are the
Kitaev and the symmetric anisotropic exchange interaction terms7, 8.
$J_{\perp}({r}_{ij})$ is the interlayer exchange coupling and ${r}_{ij}$
represents the interlayer displacement.

For the intralayer exchange parameters, we used $K=-5$ meV, $\Gamma=2.5$ meV,
$J_{1}$= -0.5 meV, and $J_{3}$ = 0.5 meV as obtained in Ref. 9. In order to
determine the interlayer exchange, we perform first-principles density
functional theory (DFT) calculations for three different bilayer stackings
(AA, AB’, and AB) shown in Fig. 1 (details on the ab initio DFT calculations
are provided in the Supporting Information). For each stacking, we extract the
energy difference ($\Delta$$E$) between two spin configurations: ferromagnetic
(FM), with FM planes coupled FM out-of-plane, and antiferromagnetic (AFM),
corresponding to FM planes coupled AFM out-of-plane. Even though the magnetic
ground state in RuCl3 is zig-zag-like, as mentioned above, we use the energy
difference between these simpler spin configurations to obtain an estimate of
the effective interlayer exchange. Importantly, for all stackings we obtain a
lower energy for an effective AFM interlayer coupling.

For the three stacking orders used, we construct an interlayer spin
hamiltonian in which the exchange coupling depends on the displacement,
$J_{\perp}(r_{ij})$ and we equate the energy difference from the ab initio DFT
alculations ($\Delta$$E$) to the corresponding effective interlayer spin
Hamiltonian. For instance, the unit cell for the AA stacking used in the
first-principles calculations has 4 interlayer bonds with $r_{ij}=0$ and 12
bonds with $r_{ij}=a/\sqrt{3}$ (see Fig. 1(a-c) for a schematic picture of the
interlayer bonds). In this manner, we obtain three equations for the three
different stacking patterns used.

$\displaystyle 4J_{\perp}(0)+12J_{\perp}(a_{0})$ $\displaystyle=$
$\displaystyle\Delta E_{AA}/2=3.2~{}{\rm meV}$ (4) $\displaystyle
8J_{\perp}(a_{0}/3)+8J_{\perp}(2a_{0}/3)$ $\displaystyle=$
$\displaystyle\Delta E_{AB^{\prime}}/2=1.6~{}{\rm meV}$ (5) $\displaystyle
2J_{\perp}(0)+18J_{\perp}(a_{0})$ $\displaystyle=$ $\displaystyle\Delta
E_{AB}/2=2.8~{}{\rm meV}$ (6)

where $a_{0}=a/\sqrt{3}$ is the bond length on the honeycomb lattice. We
implement a cut-off for the range of the exchange interaction, $J(r=2a/3)\geq
0$ and solve the Eqs. 4, 5 and 6, obtaining $J(0)=0.5$ meV,
$J(a_{0}/\sqrt{3})=0.2$ meV and $J(a_{0})=0.1$ meV. Similar values of the
interlayer exchange parameters have been reported in previous work 49, 50. To
determine a continuous $J(r_{ij})$, we fit these values to an exponential
decaying function,
$J_{\perp}(r)=J_{\perp}(0)e^{\frac{-B\sqrt{C^{2}+r^{2}}}{C}}/e^{-B}$ (as shown
in Fig. 1(d)) with $B=0.04898$ and $C=0.02943$ being the fitting parameters.

Figure 1: (a-c) Schematic representation of the three different stackings used
in our first-principles DFT calculations displaying the interlayer neighbors
exchange couplings considered (for half a unit cell) as shown in Eqs. 4-6. (d)
Continuous interlayer exchange coupling function
$J_{\perp}(r)=J_{\perp}(0)e^{\frac{-B\sqrt{C^{2}+r^{2}}}{C}}/e^{-B}$ as a
function of displacement $r$. Here, we use $J_{\perp}(0)=0.5$ meV obtained
from our ab initio DFT estimation. Figure 2: Spin textures for the zigzag AFM
order for a $7\times 7$ system. There are three inequivalent ordering wave
vectors (a) $q_{1}$, (b) $q_{2}$ and (c) $q_{3}$ that are related with $C_{3}$
symmetry.  Figure 3: Moiré pattern and projection of the nearest neighbor
spins of layer-2 on the spins of layer-1 in twisted bilayer RuCl3. (a) Moiré
pattern of a twisted bilayer system. For small angle the local stackings of
the moiré unit cell can be considered as the untwisted bilayer system with one
layer shifted by displacement $r$. (b) $q_{1}$ zigzag order and the projection
of first nearest neighbour spins of layer-2 on the spins of layer-1 times an
exponential decaying function of in-plane displacement $r$ with $q_{1}$ zigzag
order in both layers of a 2$\times$2 moiré unit cell, ($L\sim a/\theta$). The
red and blue colors represent parallel and anti-parallel spins respectively.
(c) and (d) represent the same analysis as (b) for $q_{2}$ and $q_{3}$,
respectively.

For a bilayer ‘untwisted’ $\alpha$-RuCl3 with AA stacking order, the
antiferromagnetic interlayer exchange is not frustrated and the energy is
minimized when both layers have the same single-$q$ zigzag order. Due to the
three-fold rotational symmetry, there are three possible single-$q$ patterns
as shown in Fig. 2. On the contrary, for moiré superlattices, the real space
stacking within the moiré unit cell changes as a function of displacement
$\bf{R}$. For small twist angles, the local stacking can by described by a
translational shift, ${\bf r}(\bf{R})\simeq\theta\hat{z}\times{\bf R}$ as
shown in Fig. 3 (a). Due to this varying shift, the nearest neighbour spins in
two layers cannot be anti-parallel throughout the moiré unit cell when both
layers have the same single-$q$ order. To show this effect, we present in Fig.
3 (b),(c) and (d) the projections of the nearest neighbour spins of layer-2 on
the spins of layer-1 times the exponential decaying function
$e^{\frac{-B\sqrt{C^{2}+r^{2}}}{C}}/e^{-B}$ used above for $J_{\perp}$ for
$q_{1}$, $q_{2}$, and $q_{3}$ respectively. For $(1q-1q)$ zigzag order with
$q_{1}$ on each layer, the spins are parallel at $R_{0}$ with a zero shift as
shown in Fig. 3 (b). At $R_{1}$ and $R_{2}$, the translational shifts are
$a_{1}$ and $a_{2}$, respectively, and both of these shifts move a parallel
spin to the location of an anti-parallel spin. The whole pattern of varying
parallel and anti-parallel local stacking orders can be obtained in this way
for the $q_{2}$ and $q_{3}$ orders as well. Based on the local translational
shifts, the ($1q-1q$) zigzag order in a $2\times 2$ moiré unit cell can be
divided into three equal parts as shown in Fig. 3 (b), (c), and (d). In one-
third area, the spins are aligned parallel to each other and this area is
represented by red color, whereas in the second one-third area, they are
arranged anti-parallel (blue color). In the last one-third area, the spins are
both parallel and anti-parallel (white color). Therefore, the total interlayer
exchange energy averages out to zero. Even though this analysis is carried out
when both layers have the same wave vector, it holds true for different
$q_{i}$’s on each layer as well.

However, the interlayer energy can be minimized if there are multiple-$q$
zigzag patterns in each layer that are separated by domain walls. In Fig. 3
(b), (c) and (d), the white regions represent the areas where $q_{1}$, $q_{2}$
and $q_{3}$ orders cannot gain interlayer energy, respectively. These regions
are combined in a single graph in Fig. 4(a) that shows a schematic
representation of constraints on the wave vectors that minimize the interlayer
exchange energy. We implement the following color coding: the intersection of
yellow and magenta bars is represented by red diamonds where both $q_{2}$ and
$q_{3}$ orders cannot gain energy. Therefore, the red diamonds display $q_{1}$
order. Similarly, blue and green diamonds represent regions where only $q_{2}$
and $q_{3}$ can gain energy respectively. The violet, yellow and pink colors
represent regions where only $q_{1}$, $q_{2}$ and $q_{3}$ cannot gain energy.
In the white region, any of the three wave vectors can gain energy. Applying
these rules, we have constructed two examples of multi-domain structures in
Fig. 4 (b) and (c). The white hexagons in Fig. 4 (b) can be filled with any of
the three $q$ orders. The domain structure in Fig. 4(c) was predicted in Ref.
34 using a continuum model instead. Both of these configurations ((b) and (c))
fully minimize the interlayer energy and only differ by the domain wall
length.

Figure 4: (a) Schematic representation of constraints on the wave vectors that
minimize the interlayer exchange energy. Red, blue and green diamonds
represent regions where only $q_{1}$, $q_{2}$ and $q_{3}$ can gain interlayer
energy respectively. On the other hand, violet, yellow and pink colors
represent regions where $q_{1}$, $q_{2}$ and $q_{3}$ on their own cannot gain
energy. White color represent regions where all the three $q$’s can gain
interlayer energy. (b,c) Two examples of multi-domain structures that minimize
the interlayer exchange energy. Red, blue and green colors represent $q_{1}$,
$q_{2}$ and $q_{3}$ zigzag orders respectively. The white hexagons in (b)
represent regions where any of the three $q$’s is allowed. Figure 5: (a) Phase
diagram of twisted bilayer $\alpha$-RuCl3 as a function moiré period $L$ and
$J_{\perp}$. The ab initio estimate is $J_{\perp}(0)=0.5$ meV. The inset shows
the same phase diagram as a function twisted angle $\theta$ and interlayer
coupling $J_{\perp}$. Magnetization texture of (b) $(1q-1q)$ for $L=12a_{0}$
and $J_{\perp}(0)=0.2$ meV; and (c) $(3q-3q)$ for $L=36a_{0}$ and
$J_{\perp}(0)=0.6$ meV. Magnetization textures are shown for $2\times 2$ moiré
unit cells.

The competition between the domain wall and the interlayer energy leads to
interesting phases as a function of the moiré period (or twist angle) and
interlayer coupling. We obtain the corresponding phase diagram by solving the
Landau-Lifshitz-Gilbert (LLG) equation51 for both layers as shown in Fig.
5(a). For the interlayer exchange, we use the form derived from first-
principles calculations but we vary the overall amplitude, $J_{\perp}(0)$. For
small moiré period (or large twist angle), the ground state is in a ($1q-1q)$
phase as shown in in Fig. 5(b). In this phase, both layers have different
single-$q$ zigzag patterns. These zigzag patterns are deformed from perfect
order, leading to some gain in interlayer energy. The degree of deformation
increases with $J_{\perp}$. For large moiré periodicity (L) and large
$J_{\perp}$, we obtain the $(3q-3q)$ phase. In this phase, all three possible
zigzag patterns are formed as shown in Fig. 5 (c). This phase is consistent
with Fig. 4 (b) and the shape of the zigzag patterns is hexagonal. One of the
three $q$’s occupies the white hexagons of Fig. 4 (b) and it forms a larger
domain of three hexagons connecting the opposite hexagons of the same $q$.
This phase maximizes the interlayer energy and the gain in interlayer energy
in $2\times 2$ moiré unit cells is $E_{\perp}\sim J_{\perp}\times N\approx
J_{\perp}8(L/a_{0})^{2}/3$ where $N$ is the number of sites and $a_{0}$ is the
bond length. On the other hand, the cost of domain wall energy formation is
approximately $E_{DW}\sim 40L^{\prime}\times 5.6623/\sqrt{3}a_{0}$ meV where
$40L^{\prime}$ is the length of the domain walls and
$L^{\prime}=L/(4\cos(\pi/6))$ is the edge-length of the hexagons (see
Supporting Information for the derivation of $E_{DW}$). A critical moiré
period for the phase transition from ($1q-1q$) to $(3q-3q)$ can be attained by
equating $E_{DW}$ and $E_{\perp}$ leading to $L_{crit}=(14.1562\text{
meV}/J_{\perp})a_{0}$. We find that the second candidate domain structure
shown in Fig. 4 (c) has a longer domain wall length, and therefore it is not
preferred when compared to the hexagonal domain structure. The dashed line in
Fig. 5(a) represents the analytical estimate for the ($1q-1q$) to ($3q-3q$)
phase transition. For intermediate moiré periods, we obtain an intermediate
$(2q-2q)$ phase as shown in Fig. 6. In this phase, there are two kinds of
zigzag patterns giving rise to gaining a large fraction of interlayer energy.
However, it is not possible to obtain a maximum interlayer energy from double
zigzag patterns. On the contrary, such patterns have shorter domain wall
length compared to $(3q-3q)$ phase. For small $J_{\perp}$, the $(2q-2q)$ phase
exists for a range of moiré periods whereas for large J⟂ this phase exists
only for a very short range of small moiré periods and the distinction between
$(1q-1q)$ and $(2q-2q)$ phases is not evident.

Figure 6: Magnetization texture of $(2q-2q)$ for $L=30a_{0}$ and
$J_{\perp}(0)=0.2$ meV. Magnetization texture is shown for $2\times 2$ moiré
unit cells.

In conclusion, we have shown that the interplay of the stacking-dependent
interlayer exchange and twist angle can play an important role in determining
the magnetic phases of $\alpha$-RuCl3. In particular, we demonstrate that the
single-domain, $(1q-1q)$ zigzag order can be taken over by multi-$q$ patterns
in order to minimize the interlayer exchange energy. These phases appear at
small twist angles and can be used as a new route to introduce additional
frustration and tune the magnetic phases in $\alpha$-RuCl3. Interesting future
directions include estimating the magnon spectrum in multi-$q$ orders and
incorporating substrate effects.

##  1 Associated Content

Supporting Information

Structural parameters, domain wall energy estimation (pdf).

##  2 Author Information

Corresponding Author: Onur Erten – Email<EMAIL_ADDRESS>
Author Contributions: Muhammad Akram and Jyotirish Das have performed the
atomistic simulations. Jesse Kapeghian has performed the first-principles
calculations. Antia Botana and Onur Erten have conceptualized the research and
analyzed the results. All authors have contributed to writing the manuscript.

Notes: The authors declare no competing financial interest.

##  3 Acknowledgements

We thank Arun Paramekanti for fruitful discussions. OE and AB acknowledge
support from National Science Foundation Award No. DMR 2206987. MA
acknowledges support from National Science Foundation Awards No. DMR 2234352.
We acknowledge the ASU Research Computing Center for HPC resources. RV
acknowledges the DFG (German Research Foundation) for funding through the
research unit QUAST FOR 5249 (project ID: 449872909; project P4) and her
research was supported in part by the National Science Foundation under Grants
No. NSF PHY-1748958 and PHY-2309135.

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