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Energy decay for a viscoelastic Kirchhoff plate equation with a delay term

Boundary Value Problems20162016:174

https://doi.org/10.1186/s13661-016-0682-8

  • Received: 8 July 2016
  • Accepted: 15 September 2016
  • Published:

Abstract

A nonlinear viscoelastic Kirchhoff plate equation with a time delay term in the internal feedback is considered. Under suitable assumptions, we establish the general rates of energy decay of the initial and boundary value problem by using the energy perturbation method.

Keywords

  • general decay
  • Kirchhoff plate
  • delay feedbacks

MSC

  • 35L75
  • 35B40
  • 35B35

1 Introduction

In this paper, we are concerned with the following nonlinear viscoelastic Kirchhoff plate equation with a time delay term in the internal feedback:
$$\begin{aligned} &u_{tt}(x,t)+\Delta^{2}u(x,t)- \operatorname{div} F\bigl(\nabla u(x,t)\bigr)-\sigma(t) \int^{t}_{0}g(t-s)\Delta^{2} u(s)\,ds+ \mu_{1}|u_{t}|^{m-1}u_{t}(x,t) \\ &\quad{}+\mu_{2}\bigl|u_{t}(x,t-\tau)\bigr|^{m-1}u_{t}(x,t- \tau)=0, \end{aligned}$$
(1.1)
where \(\Omega\subseteq {\mathbb{R}}^{n} \) (\(n\geq1\)) is a bounded domain with smooth boundary Ω. The function \(u=u(x,t)\) is the transverse displacement of a plate filament, and \(\sigma(t)\) and \(g(t)\) are positive functions defined on \(\mathbb{R}^{+}\). \(\mu_{1}\), \(\mu_{2}\) are positive constants and \(\tau>0\) represents the time delay.
To equation (1.1), we add the following initial conditions:
$$\begin{aligned} \left \{ \textstyle\begin{array}{@{}l} u(x,0)=u_{0}(x),\qquad u_{t}(x,0)=u_{1}(x), \quad x\in\Omega, \\ u_{t}(x,t-\tau)=f_{0}(x,t-\tau), \quad x\in\Omega, t\in(0,\tau), \end{array}\displaystyle \right . \end{aligned}$$
(1.2)
and the support boundary conditions
$$\begin{aligned} u=\Delta u=0 \quad \mbox{on } \partial\Omega\times\mathbb{R}^{+}. \end{aligned}$$
(1.3)
In 1950, Woinowsky-Krieger [1] introduced the one-dimensional nonlinear equation of vibration of beams
$$u_{tt}+\alpha u_{xxxx}- \biggl(\beta+\gamma \int^{L}_{0}|u_{x}|^{2}\,dx \biggr)u_{xx}=0, $$
where L is the length of the beam and \(\alpha,\beta,\gamma\) are positive physical constants. Since then many mathematicians studied the related model in one dimension and higher dimensions. The main results are mainly concerned with global existence, stability, and long-time dynamics, and many results may be found in the literature. It has been stabilized by means of different controls, for example, internal damping, boundary controls, dynamic boundary conditions, distributed damping and heat damping, and so on. See, for example, Brito [2], Cavalcanti et al. [35], Jorge Silva and Ma [6], Ma [7], Ma and Narciso [8], Oliveira and Lima [9], Park [10], Patcheu [11], Munõz Rivera [12, 13], Yang [14, 15], and the references therein. We would like here to mention the work of Andrade et al. [16]. In this paper the authors studied a viscoelastic plate equation with p-Laplacian and memory terms with strong damping,
$$\begin{aligned} u_{tt}+\Delta^{2}u-\Delta_{p}u+ \int^{t}_{0}g(t-s)\Delta u(s)\,ds-\Delta u_{t}+f(u)=0, \end{aligned}$$
(1.4)
and proved the existence of weak solutions by using Faedo-Galerkin approximations to the IBVP of (1.4). In addition, they obtained the uniqueness of strong solutions and the exponential stability of solutions to (1.4) under some suitable conditions on the memory kernel g and a forcing term f. For \(\sigma(t)>0\), Messaoudi [17] considered the following viscoelastic wave equation:
$$u_{tt}-\Delta u+\sigma(t) \int^{t}_{0}g(t-\tau)\Delta u(\tau)\,d\tau=0. $$
Under some assumptions on the relaxation function g and the potential σ, the author established a general decay property which depends on the behavior of σ and g. Jorge Silva et al. [18] studied the following viscoelastic Kirchhoff plate equation:
$$u_{tt}-\sigma(t)\Delta u_{tt}+\Delta^{2}u- \operatorname{div}F(\nabla u)- \int^{t}_{0}g(t-s)\Delta^{2} u(s)\,ds=0, $$
and they mainly proved the global well-posedness of the solution for \(\sigma(t)=1\) and similarly the result holds for \(\sigma(t)=0\). Moreover, the authors established the general rates of energy decay of the system for \(\sigma\in[0,\infty)\). For more results on viscoelastic equations, we can refer to Berrimi and Messaoudi [19], Messaoudi [20], Messaoudi and Tartar [21, 22], Tatar [23], and the references therein.
In recent years, there has been published much work concerning the wave equation with time delay effects and the delay effects often appear in many practical problems. In Nicaise and Pignotti [24], the authors studied a wave equation with time delay,
$$\begin{aligned} u_{tt}-\Delta u+\mu_{1}u_{t}+\mu_{2}u_{t}(t- \tau)=0, \end{aligned}$$
and established stability results under the assumption \(0<\mu_{2}<\mu_{1}\). In [25], Kirane and Said-Houari studied a viscoelastic wave equation with a delay term in internal feedbacks, and they proved the global well-posedness of the IBVP to the equation by using some suitable assumptions on the relaxation function and some restriction on the parameters \(\mu_{1}\) and \(\mu_{2}\). Furthermore, under the assumption \(\mu_{2}\leq\mu_{1}\), they obtained a general decay result of the total energy to the system. Dai and Yang [26] improved the results in [25] under weaker conditions. For the plate equation with time delay term, Park [27] considered
$$\begin{aligned} u_{tt}+\Delta^{2}u-M \bigl(\|\nabla u\|^{2} \bigr) \Delta u+\sigma(t) \int^{t}_{0}g(t-s)\Delta u(s)\,ds+a_{0}u_{t}+a_{1}u_{t}(t- \tau)=0, \end{aligned}$$
which can be regarded as an extensive weak viscoelastic plate equation with a linear time delay term. The author obtained a general decay result of energy by using suitable energy and Lyapunov functionals. In [28], one of the present authors investigated an extensible plate equation with a weak viscoelastic term and a time delay term in the internal feedback,
$$\begin{aligned} u_{tt}+\Delta^{2}u-M \bigl(\|\nabla u\|^{2} \bigr) \Delta u- \int^{t}_{0}g(t-s)\Delta u(s)\,ds+ \mu_{1}u_{t}+\mu_{2}u_{t}(t-\tau)=0, \end{aligned}$$
and established the global well-posedness of the initial and boundary value problem by using the Faedo-Galerkin approximations and some energy estimates. Moreover, the author proved a general rate result of energy decay when the weight of the delay is less than the weight of the damping. Recently, Yang [29] studied a viscoelastic plate equation with a linear time delay term
$$\begin{aligned} u_{tt}+\Delta^{2}u- \int^{t}_{0}g(t-s)\Delta^{2}u(s)\,ds+ \mu_{1}u_{t}+\mu _{2}u_{t}(t-\tau)=0. \end{aligned}$$
The author obtained the global well-posedness of the IBVP to the equation and established the decay property of energy for either \(0<|\mu_{2}|<\mu_{1}\) or \(\mu_{1}=0\), \(0<|\mu_{2}|<a\), and \(\zeta_{2}>\zeta_{0}\), but one needs more assumptions on the kernel g. For more some results concerning the different boundary conditions under an appropriate assumption between \(\mu_{1}\) and \(\mu_{2}\), one can refer to Datko et al. [30], Kafini et al. [31], Nicaise and Pignotti [32], Nicaise et al. [33], Nicaise and Valein [34], and the references therein.

Equation (1.1) is a Kirchhoff plate equation with a memory term and a nonlinear time delay term in the internal feedback. To the best of our knowledge, the general rate of energy decay for system (1.1)-(1.3) were not previously considered. So the main objective of the present work is to establish the stability of initial boundary value problem (1.1)-(1.3).

The outline of this paper is as follows. In Section 2, we give some preparations for our consideration and our main results. In Section 3, we establish the general decay result of the energy by using energy perturbation method.

2 Assumptions and main results

We first introduce the following Hilbert spaces:
$$\begin{aligned} V_{0}=L^{2}(\Omega),\qquad V_{1}=H^{1}_{0}( \Omega),\qquad V_{2}=H^{2}(\Omega)\cap H^{1}_{0}( \Omega), \end{aligned}$$
with norms
$$\begin{aligned} \|u\|_{V_{0}}=\|u\|, \qquad\|u\|_{V_{1}}=\|\nabla u\|_{2} \quad\mbox{and}\quad \|u\|_{V_{2}}=\|\Delta u\|, \end{aligned}$$
respectively. The notation \(\|\cdot\|_{p}\) denotes the \(L^{p}\)-norm, and \((\cdot,\cdot)\) is the \(L^{2}\)-inner product. In particular, we write \(\|\cdot\|\) instead of \(\|\cdot\|_{2}\) when \(p=2\). The constants \(\lambda_{0},\lambda_{1},\lambda_{2},\lambda>0\) represent the embedding constants
$$\begin{aligned} \lambda_{0}\|u\|^{2}\leq\|\nabla u\|^{2},\qquad \lambda_{1}\|u\|^{2}\leq\|\Delta u\|^{2},\qquad \lambda_{2}\|\nabla u\|^{2}\leq\|\Delta u\|^{2}, \qquad \lambda=\frac{1}{\lambda_{1}}+\frac{1}{\lambda_{2}}, \end{aligned}$$
for \(u\in V_{2}\).

For the relaxation function g and the potential σ, we assume

(A1) \(g,\sigma:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}\) are nonincreasing differentiable functions satisfying
$$ \begin{aligned} & g(0)>0,\qquad l_{0}= \int^{\infty}_{0}g(s)\,ds>\infty, \qquad\sigma(t)>0,\\ & 1-2 \sigma(t) \int^{t}_{0}g(s)\,ds\geq l>0,\quad \mbox{for } t\geq0, \end{aligned} $$
(2.1)
with \(l=1-l_{0}\), and there exists a nonincreasing differentiable function \(\zeta:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}\) satisfying
$$\begin{aligned} \zeta(t)>0,\quad g'(t)\leq-\zeta(t)g(t) \quad\mbox{for } t \geq0, \lim_{t\rightarrow\infty}\frac{-\sigma'(t)}{\zeta(t)\sigma (t)}=0. \end{aligned}$$
(2.2)
(A2) The constant m satisfies
$$\begin{aligned} m\geq1 \quad\mbox{if } n=1,2,\qquad 1\leq m\leq\frac{n+2}{n-2} \quad\mbox{if } n \geq3. \end{aligned}$$
(2.3)
(A3) The function \(F:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}\) is a \(C^{1}\)-vector field given by \(F=(F_{1},\ldots,F_{n})\) satisfying for every \(j=1,2,\ldots,n\),
$$\begin{aligned} \bigl|\nabla F_{j}(u)\bigr|\leq k_{j} \bigl(1+|u|^{\frac{p_{j}-1}{2}} \bigr),\quad \forall u\in \mathbb{R}^{n}, \end{aligned}$$
(2.4)
where \(k_{j}\) are positive constants and the constants \(p_{j}\) satisfy
$$\begin{aligned} p_{j}\geq1 \quad\mbox{if } n=1,2,\qquad 1\leq p_{j}\leq \frac{n+2}{n-2}\quad \mbox{if } n\geq3. \end{aligned}$$
(2.5)
Moreover, the function F is a conservative vector field with \(F=\nabla f\), where \(f:\mathbb{R}^{n}\rightarrow\mathbb{R}\) is a real valued function satisfying
$$\begin{aligned} 0\leq f(u)\leq F(u)u+\alpha l|u|^{2}, \quad\forall u\in \mathbb{R}^{n}, \end{aligned}$$
(2.6)
where \(\alpha\in[0,\mu)\) with \(\mu=\lambda_{2}\frac{1-2\sigma(t)\int^{t}_{0}g(s)\,ds}{2l}\).

The vector field F satisfying a condition like (2.4) possesses an interesting property. One can find the detailed proof in [18].

Remark 2.1

Let \(F:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}\) is a \(C^{1}\)-vector field given by \(F=(F_{1},\ldots,F_{n})\). If there exist positive constants \(k_{1},\ldots,k_{n}\) and \(q_{1},\ldots,q_{n}\) such that, for every \(j=1,\ldots,n\),
$$\begin{aligned} \bigl|\nabla F_{j}(u)\bigr|\leq k_{j}\bigl(1+|u|^{q_{j}} \bigr),\quad \forall u\in\mathbb{R}^{n}. \end{aligned}$$
Then there exists a positive constant \(K=K(k_{j},q_{j},n), j=1,\ldots,n\), such that, for all \(x,y\in\mathbb{R}^{n}\),
$$\begin{aligned} \bigl|F(x)-F(y)\bigr|\leq K\sum^{n}_{j=1} \bigl(1+|x|^{q_{j}}+|y|^{q_{j}}\bigr)|x-y|. \end{aligned}$$
In particular, we have
$$\begin{aligned} \bigl|F(x)\bigr|\leq \bigl|F(0)\bigr|+K\sum^{n}_{j=1} \bigl(1+|x|^{q_{j}}\bigr)|x|,\quad \forall x\in\mathbb{R}^{n}. \end{aligned}$$
(2.7)
Now we give some estimates related to the convolution operator. By direct calculations, we shall see below that
$$\begin{aligned}& \begin{aligned}[b] &\sigma(t) (g\ast u,u_{t})\\ &\quad=-\frac{\sigma(t)}{2}g(t)\bigl\| u(t) \bigr\| ^{2}-\frac{d}{dt} \biggl[\frac {\sigma(t)}{2}(g\circ u) (t)- \frac{\sigma(t)}{2} \biggl( \int^{t}_{0}g(s)\,ds \biggr)\bigl\| u(t)\bigr\| ^{2} \biggr]\\ &\qquad{}+\frac{\sigma(t)}{2}\bigl(g'\circ u\bigr) (t)+\frac{\sigma'(t)}{2}(g \circ u) (t)-\frac{\sigma'(t)}{2} \int^{t}_{0}g(s)\,ds\bigl\| u(t)\bigr\| ^{2}, \end{aligned} \end{aligned}$$
(2.8)
$$\begin{aligned}& (g\ast u,u)\leq 2 \biggl( \int^{t}_{0}g(s)\,ds \biggr)\bigl\| u(t)\bigr\| ^{2}+ \frac {1}{4}(g\circ u) (t), \end{aligned}$$
(2.9)
where
$$\begin{aligned} (g\ast u) (t)= \int^{t}_{0}g(t-s)u(s)\,ds, \qquad(g\circ u) (t)= \int^{t}_{0}g(t-s)\bigl\| u(t)-u(s)\bigr\| ^{2}\,ds. \end{aligned}$$
Motivated by [32, 34], we introduce the following new dependent variable to deal with the delay feedback term:
$$\begin{aligned} z(x,\rho,t)=u_{t}(x,t-\tau\rho),\quad x\in\Omega, \rho \in(0,1), t>0, \end{aligned}$$
(2.10)
which gives us
$$\begin{aligned} \tau z_{t}(x,\rho,t)+z_{\rho}(x,\rho,t)=0,\quad \mbox{in } \Omega\times(0,1)\times(0,\infty). \end{aligned}$$
(2.11)
Thus, problem (1.1)-(1.3) is equivalent to
$$ \textstyle\begin{cases} u_{tt}+\Delta^{2}u-\operatorname{div}F(\nabla u)-\sigma(t)\int ^{t}_{0}g(t-s)\Delta^{2} u(s)\,ds\\ \quad{} +\mu_{1}|u_{t}|^{m-1}u_{t}+\mu _{2}|z(x,1,t)|^{m-1}z(x,1,t)=0,\\ \tau z_{t}(x,\rho,t)+z_{\rho}(x,\rho,t)=0, \end{cases} $$
(2.12)
where \(x\in\Omega,\rho\in(0,1)\) and \(t>0\), and the initial and boundary conditions are
$$\begin{aligned} \left \{ \textstyle\begin{array}{@{}l} u(x,0)=u_{0},\qquad u_{t}(x,0)=u_{1},\quad x\in\Omega\\ z(x,\rho,0)=f_{0}(x,-\rho\tau), \quad (x,t)\in\Omega\times(0,\tau),\\ u=\Delta u=0, \quad \mbox{on } \partial\Omega\times\mathbb{R}^{+},\\ z(x,0,t)=u_{t}(x,t),\quad x\in\Omega, t>0. \end{array}\displaystyle \right . \end{aligned}$$
(2.13)
Let ξ be a positive constant satisfying
$$\begin{aligned} \tau\frac{\mu_{2}m}{m+1}< \xi< \tau\frac{(m+1)\mu_{1}-\mu _{2}}{m+1}. \end{aligned}$$
(2.14)
Now we define the weak solutions of (1.1)-(1.3): for given initial data \((u_{0},u_{1})\in V_{2}\times V_{0} \), we say that a function \(U=(u,u_{t})\in C(\mathbb{R}^{+},V_{2}\times V_{0})\) is a weak solution to the problem (1.1)-(1.3) if \(U(0)=(u_{0},u_{1})\) and
$$\begin{aligned} &(u_{tt},\omega)+(\Delta u,\Delta\omega)+\bigl(F(\nabla u),\nabla \omega\bigr)-\sigma(t) \int^{t}_{0}g(t-s) \bigl(\Delta u(s),\Delta\omega \bigr)\,ds\\ &\quad{} +\mu_{1}\bigl(|u_{t}|^{m-1}u_{t},\omega \bigr)+\mu _{2}\bigl(\bigl|u_{t}(t-\tau)\bigr|^{m-1}u_{t}(t- \tau),\omega\bigr)=0, \end{aligned}$$
for all \(\omega\in V_{2}\).

The following theorem is concerned with the global well-posedness of problem (2.12)-(2.13). By using the classical Faedo-Galerkin method, see, e.g., [16, 18, 28, 35], we can prove the theorem, and we omit the proof here.

Theorem 2.1

Let \(\mu_{2}\leq m\mu_{1}\), and assume the assumptions (2.1)-(2.6) hold. If the initial data \((u_{0},u_{1})\in(V_{2}\times V_{0})\), \(f_{0}\in L^{2}(\Omega\times(0,1))\), then problem (2.12)-(2.13) has a unique weak solution \((u,u_{t})\in C(0,T;V_{2}\times V_{0})\) such that, for any \(T>0\),
$$\begin{aligned} u\in L^{\infty}(0,T;V_{2}),\qquad u_{t}\in L^{\infty}(0,T;V_{0}). \end{aligned}$$
We introduce the modified energy functional to problem (2.12)-(2.13) by
$$\begin{aligned} E(t) =&\frac{1}{2}\bigl\| u_{t}(t)\bigr\| ^{2}+ \frac{1}{2} \biggl[1-\sigma(t) \biggl( \int^{t}_{0}g(s)\,ds \biggr) \biggr]\bigl\| \Delta u(t) \bigr\| ^{2}+\frac{1}{2}\sigma(t) (g\circ\Delta u) \\ &{}+\frac{\xi}{2} \int_{\Omega} \int^{1}_{0}|z|^{m+1}(x,\rho,t)\,d\rho \,dx+ \int_{\Omega}f\bigl(\nabla u(t)\bigr)\,dx. \end{aligned}$$
(2.15)

Our main result is the general decay rate of the energy, which is given by the following theorem.

Theorem 2.2

Let \(\mu_{2}< m\mu_{1}\), and assume the assumptions (2.1)-(2.6) hold. Let \((u,u_{t})\) be the weak solutions of problem (2.12)-(2.13) with the initial data \((u_{0},u_{1})\in(V_{2}\times V_{0})\), \(f_{0}\in L^{2}(\Omega\times(0,1))\). Then there exist two constants \(\beta>0\) and \(\gamma>0\) such that the energy \(E(t)\) defined by (2.15) satisfies
$$\begin{aligned} E(t)\leq\beta\exp \biggl(-\gamma \int^{t}_{0}\zeta(s)\sigma(s)\,ds \biggr), \quad\textit{for all } t\geq0. \end{aligned}$$
(2.16)

Remark 2.2

Generally speaking, the energy of problem (2.12)-(2.13) is usually defined by
$$\begin{aligned} F(t)=\frac{1}{2}\|u_{t}\|^{2}+\frac{1}{2}\| \Delta u\|^{2} +\frac{\xi}{2} \int_{\Omega}\int^{1}_{0}|z|^{m+1}\,d\rho \,dx+ \int_{\Omega}f(\nabla u)\,dx. \end{aligned}$$
From Theorem 2.2, we can also get the decay
$$\begin{aligned} F(t)\leq\beta'\exp \biggl(-\gamma \int ^{t}_{0}\zeta (s)\sigma(s)\,ds \biggr). \end{aligned}$$
(2.17)
Indeed, by (2.15) and (2.1), we have
$$\begin{aligned} E(t) =&F(t)-\frac{1}{2}\sigma(t) \int^{t}_{0}g(s)\,ds\|\Delta u\|^{2}+ \frac {1}{2}\sigma(t) (g\circ\Delta u)\\ \geq&\frac{3+l}{4}F(t), \end{aligned}$$
which, together with (2.16), implies (2.17) with \(\beta '=\frac{4\beta}{l+3}\).

3 General decay rate

In this section, we shall establish the general decay property of the solution for problem (2.12)-(2.13) in the case \(\mu_{2}< m\mu_{1}\). For this purpose we define
$$\begin{aligned} \mathcal {L}(t):=E(t)+\varepsilon_{1}\sigma(t)\Phi(t)+ \varepsilon_{2}\sigma (t)\Psi(t), \end{aligned}$$
(3.1)
where \(\varepsilon_{1}\) and \(\varepsilon_{2}\) are positive constants and
$$\begin{aligned}& \Phi(t)= \int_{\Omega}u_{t}u\,dx, \end{aligned}$$
(3.2)
$$\begin{aligned}& \Psi(t)=- \int_{\Omega}u_{t} \int^{t}_{0}g(t-s) \bigl(u(t)-u(s)\bigr)\,ds\,dx. \end{aligned}$$
(3.3)

To prove Theorem 2.2, we need the following technical lemmas.

Lemma 3.1

Under the assumptions in Theorem 2.2, the modified energy functional defined by (2.15) satisfies there exist two positive constants \(c_{1}\) and \(c_{2}\) such that, for any \(t\geq0\),
$$\begin{aligned} E'(t) \leq&-c_{1}\|u_{t} \|^{m+1}_{m+1}-c_{2}\bigl\| z(x,1,t)\bigr\| ^{m+1}_{m+1} +\frac{\sigma(t)}{2}\bigl(g'\circ\Delta u\bigr) \\ &{}-\frac{\sigma'(t)}{2} \int^{t}_{0}g(s)\,ds\|\Delta u\|^{2}. \end{aligned}$$
(3.4)

Proof

First the direct calculation yields
$$\begin{aligned} \int_{\Omega}F(\nabla u)\cdot\nabla u \,dx= \int_{\Omega}\nabla f(\nabla u)\cdot\nabla u \,dx=\frac{d}{dt} \int_{\Omega}f(\nabla u)\,dx. \end{aligned}$$
(3.5)
Multiplying the first equation in (2.12) by \(u_{t}\), integrating the result over Ω, and using integration by parts, (2.8) and (3.5), we can obtain
$$\begin{aligned} &\frac{d}{dt} \biggl[\frac{1}{2}\|u_{t} \|^{2}+\frac{1}{2}\|\Delta u\|^{2}+\frac{\sigma(t)}{2}(g \circ\Delta)-\frac{\sigma(t)}{2} \biggl( \int^{t}_{0}g(s)\,ds \biggr)\|\Delta u \|^{2}+ \int_{\Omega}f(\nabla u)\,dx \biggr] \\ &\quad{} +\mu_{1}\|u_{t}\|^{m+1}_{m+1} + \mu_{2} \int_{\Omega}\bigl|z(x,1,t)\bigr|^{m-1}z(x,1,t)u_{t}\,dx+ \frac{\sigma (t)}{2}g(t)\|\Delta u\|^{2} \\ &\qquad{} -\frac{\sigma(t)}{2}\bigl(g'\circ\Delta u\bigr)- \frac{\sigma'(t)}{2}(g\circ\Delta u)+\frac{\sigma'(t)}{2} \int^{t}_{0}g(s)\,ds\|\Delta u\|^{2}=0. \end{aligned}$$
(3.6)
Multiplying the second equation in (2.12) by ξz and integrating the result over \(\Omega\times(0,1)\), we have
$$\begin{aligned} &\xi\frac{d}{dt} \int_{\Omega}\int^{1}_{0}\bigl|z(x,\rho,t)\bigr|^{m-1}z(x,\rho ,t) \,d\rho\,dx \\ &\quad= -\frac{\xi}{\tau(m+1)} \int_{\Omega}\int^{1}_{0}\frac{\partial }{\partial \rho}\bigl|z(x, \rho,t)\bigr|^{m+1}\,d\rho \,dx \\ &\quad=-\frac{\xi}{\tau(m+1)} \int_{\Omega}\bigl(\bigl|z(x,1,t)\bigr|^{m+1}-\bigl|z(x,0,t)\bigr|^{m+1} \bigr)\,dx \\ &\quad=-\frac{\xi}{\tau} \int_{\Omega}\bigl|z(x,1,t)\bigr|^{m+1}\,dx+\frac{\xi }{\tau} \int_{\Omega}|u_{t}|^{m+1}\,dx. \end{aligned}$$
(3.7)
By using Young’s inequality, we get
$$\begin{aligned} &\mu_{2}\biggl\vert \int_{\Omega}\bigl|z(x,1,t)\bigr|^{m-1}z(x,1,t)u_{t}\,dx \biggr\vert \\ &\quad\leq \frac{\mu_{2}m}{m+1} \int_{\Omega}\bigl|z(x,1,t)\bigr|^{m+1}\,dx+\frac{\mu_{2}}{m+1} \int_{\Omega}|u_{t}|^{m+1}\,dx, \end{aligned}$$
which, together with (3.6)-(3.7), gives us
$$\begin{aligned} E'(t) \leq&- \biggl(\mu_{1}-\frac{\xi}{2\tau}- \frac{\mu _{2}}{m+1} \biggr)\|u_{t}\|^{m+1}_{m+1}- \biggl(\frac{\xi}{2\tau}-\frac {\mu_{2} m}{m+1} \biggr)\bigl\| z(x,1,t) \bigr\| ^{m+1}_{m+1}\\ &{}+\frac{\sigma(t)}{2}\bigl(g'\circ\Delta u\bigr)-\frac{\sigma'(t)}{2} \int^{t}_{0}g(s)\,ds\|\Delta u\|^{2}. \end{aligned}$$
By using condition (2.14), we get
$$\begin{aligned} c_{1}:=\mu_{1}-\frac{\xi}{2\tau}-\frac{\mu_{2}}{m+1}>0 \quad \mbox{and}\quad c_{2}:=\frac{\xi}{\tau}-\frac{\mu_{2} m}{m+1}>0, \end{aligned}$$
which implies the desired inequality (3.4). The proof is now complete. □

Lemma 3.2

Under the assumptions in Theorem 2.2, for the functional \(\Phi(t)\) defined in (3.2) there exists a positive constant \(c_{3}\) such that, for any \(t\geq0\),
$$\begin{aligned} \Phi'(t) \leq&\|u_{t}\|^{2}-c_{3} \|\Delta u\|^{2}+C_{\varepsilon}\|u_{t}\|^{m+1}_{m+1}+C_{\varepsilon}\bigl\| z(x,1,t)\bigr\| ^{m+1}_{m+1}+\frac{\sigma(t)}{4}(g\circ\Delta u), \end{aligned}$$
(3.8)
where \(C_{\varepsilon}>0\) is a constant depending for any \(\varepsilon>0\).

Proof

By using the first equation of (2.12), we obtain
$$\begin{aligned} \frac{d}{dt}\Phi(t) =& \int_{\Omega}u_{tt}u\,dx+\|u_{t} \|^{2} \\ =&\|u_{t}\|^{2}+ \int_{\Omega} \biggl(-\Delta^{2}u+\operatorname{div}F( \nabla u)+\sigma(t) \int^{t}_{0}g(t-s)\Delta^{2} u(s)\,ds \\ &{}-\mu_{1}|u_{t}|^{m_{1}}u_{t}- \mu_{2}\bigl|z(x,1,t)\bigr|^{m-1}z(x,1,t) \biggr)\cdot u\,dx \\ =&\|u_{t}\|^{2}-\|\Delta u\|^{2}- \int_{\Omega}F(\nabla u)\nabla u\,dx-\mu_{2} \int_{\Omega}\bigl|z(x,1,t)\bigr|^{m-1}z(x,1,t)u\,dx \\ &{}+\sigma(t) \int_{\Omega}\int^{t}_{0}g(t-s)\Delta u(s)\,ds\cdot\Delta u\,dx- \mu_{1} \int_{\Omega}|u_{t}|^{m-1}u_{t}u\,dx. \end{aligned}$$
(3.9)
Using Young’s inequality, the embedding theorem, and (2.15), we know that, for any \(\varepsilon>0\),
$$\begin{aligned} \biggl\vert \mu_{1} \int_{\Omega}|u_{t}|^{m-1}u_{t}u\,dx \biggr\vert \leq& \int _{\Omega}|u_{t}|^{m} |u|\,dx \\ \leq&\varepsilon\|u\|^{m+1}_{m+1}+C_{\varepsilon}\|u_{t}\| ^{m+1}_{m+1} \\ \leq&C\varepsilon\|\nabla u\|^{m+1}+C_{\varepsilon}\|u_{t} \| ^{m+1}_{m+1} \\ \leq&\frac{C\varepsilon}{\lambda_{2}} \biggl(\frac{2E(0)}{l} \biggr)^{m-1}\|\Delta u\|^{2}+C_{\varepsilon}\|u_{t}\|^{m+1}_{m+1}, \end{aligned}$$
(3.10)
where the constant \(C>0\) is the embedding constant.
Similarly we get
$$\begin{aligned} \biggl\vert \mu_{2} \int_{\Omega}\bigl|z(x,1,t)\bigr|^{m-1}z(x,1,t)u\,dx\biggr\vert \leq \frac{C\varepsilon}{\lambda_{2}} \biggl(\frac{2E(0)}{l} \biggr)^{m-1}\| \Delta u \|^{2}+C_{\varepsilon}\bigl\| z(x,1,t)\bigr\| ^{m+1}_{m+1}. \end{aligned}$$
(3.11)
We infer from (2.6) that
$$\begin{aligned} - \int_{\Omega}F(\nabla u)\nabla u\,dx\leq\alpha l \int_{\Omega}|\nabla u|^{2}\,dx\leq\frac{\alpha l}{\lambda_{2}}\| \Delta u\|^{2}. \end{aligned}$$
(3.12)
Combining (2.11) and (3.10)-(3.12) with (3.9), we can get
$$\begin{aligned} \Phi'(t) \leq&\|u_{t}\|^{2}- \biggl[ \biggl(1-2\sigma(t) \int ^{t}_{0}g(s)\,ds \biggr)-\frac{2C\varepsilon}{\lambda_{2}} \biggl(\frac {2E(0)}{l} \biggr)^{m-1}-\frac{\alpha l}{\lambda_{2}} \biggr]\|\Delta u\|^{2} \\ &{}+C_{\varepsilon}\|u_{t}\|^{m+1}_{m+1}+C_{\varepsilon}\bigl\| z(x,1,t)\bigr\| ^{m+1}_{m+1}+\frac{\sigma(t)}{4}(g\circ\Delta u). \end{aligned}$$
(3.13)
Due to (2.1) and (2.6) and choosing \(\varepsilon>0\) small enough, we know that
$$\begin{aligned} c_{3}:= \biggl(1-2\sigma(t) \int^{t}_{0}g(s)\,ds \biggr)-\frac{2C\varepsilon }{\lambda_{2}} \biggl(\frac{2E(0)}{l} \biggr)^{m-1}-\frac{\alpha l}{\lambda_{2}}>0, \end{aligned}$$
which, together with (3.13), give us (3.8). The proof is hence complete. □

Lemma 3.3

Under the assumptions in Theorem 2.2, and for any \(\delta>0\), there exists a positive constant \(C_{\delta}\) such that the functional \(\Psi(t)\) defined in (2.3) satisfies
$$\begin{aligned} \Psi'(t) \leq&- \biggl( \int^{t}_{0}g(s)\,ds-\delta \biggr)\|u_{t} \| ^{2}+\bigl[2\delta+2\delta(1-l)^{2}\sigma(t)\bigr]\|\Delta u\|^{2}+\delta\mu_{2}\|u_{t}\|^{m+1}_{m+1} \\ &{}+\delta\mu_{2}\bigl\| z(x,1,t)\bigr\| ^{m+1}_{m+1}+C_{\delta} \bigl[1+(1-l)\sigma(t)+\bigl(E(0)\bigr)^{\frac{p-1}{2}} \bigr](g\circ\Delta u) \\ &{}-\frac{Cg(0)}{4\delta\lambda_{1}}\bigl(g'\circ\Delta u\bigr), \end{aligned}$$
(3.14)
where
$$\begin{aligned} p=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} \max_{j=1,\ldots,n}\{p_{j}\}, &\textit{if }E(0)\geq1, \\ \min_{j=1,\ldots,n}\{p_{j}\}, &\textit{if }E(0)< 1. \end{array}\displaystyle \right . \end{aligned}$$

Proof

The straightforward computation implies that
$$\begin{aligned} \Psi'(t) =&- \int_{\Omega}u_{tt}\cdot \int^{t}_{0}g(t-s) \bigl(u(t)-u(s)\bigr)\,ds\,dx \\ &{}- \int_{\Omega}u_{t} \biggl[u_{t} \int^{t}_{0}g(t-s)\,ds+ \int^{t}_{0}g'(t-s) \bigl(u(t)-u(s) \bigr)\,ds \biggr]\,dx \\ =&- \int_{\Omega}\biggl(-\Delta^{2} u+\operatorname{div}F( \nabla u)+\sigma(t) \int^{t}_{0}g(t-s)\Delta^{2} u(s)\,ds-\mu _{1}|u_{t}|^{m-1}u_{t} \\ &{}-\mu_{2}\bigl|z(x,1,t)\bigr|^{m-1}z(x,1,t) \biggr)\cdot \int ^{t}_{0}g(t-s) \bigl(u(t)-u(s)\bigr)\,ds\,dx \\ &{}- \int^{t}_{0}g(s)\,ds\|u_{t} \|^{2}- \int_{\Omega}u_{t} \int^{t}_{0}g'(t-s) \bigl(u(t)-u(s) \bigr)\,ds\,dx \\ =& \int_{\Omega}\Delta u \int^{t}_{0}g(t-s) \bigl(\Delta u(t)-\Delta u(s) \bigr)\,ds\,dx \\ &{}+ \int_{\Omega}F(\nabla u) \int^{t}_{0}g(t-s) \bigl(\nabla u(t)-\nabla u(s) \bigr)\,ds\,dx \\ &{}-\sigma(t) \int_{\Omega}\biggl( \int^{t}_{0}g(t-s)\Delta u(s)\,ds \biggr) \biggl( \int^{t}_{0}g(t-s) \bigl(\Delta u(t)-\Delta u(s) \bigr)\,ds \biggr)\,dx \\ &{}+\mu_{1} \int_{\Omega}|u_{t}|^{m-1}u_{t} \int^{t}_{0}g(t-s) \bigl(u(t)-u(s)\bigr)\,ds\,dx+\mu _{2} \int_{\Omega}\bigl|z(x,1,t)\bigr|^{m-1}z(x,1,t) \\ &{}\times \int^{t}_{0}g(t-s) \bigl(u(t)-u(s)\bigr)\,ds\,dx- \int^{t}_{0}g(s)\,ds\|u_{t}\| ^{2} \\ &{}- \int_{\Omega}u_{t} \int^{t}_{0}g'(t-s) \bigl(u(t)-u(s) \bigr)\,ds\,dx. \end{aligned}$$
(3.15)
By using Hölder’s inequality, Young’s inequality, and the embedding theorem, we can infer that, for any \(\delta>0\),
$$\begin{aligned}& \int_{\Omega}\Delta u \int^{t}_{0}g(t-s) \bigl(\Delta u(t)-\Delta u(s) \bigr)\,ds\,dx\leq \delta\|\Delta u\|^{2}+\frac{1-l}{4\delta}(g\circ \Delta u), \end{aligned}$$
(3.16)
$$\begin{aligned}& \begin{aligned}[b] &{-}\sigma(t) \int_{\Omega}\biggl( \int^{t}_{0}g(t-s)\Delta u(s)\,ds \biggr) \biggl( \int^{t}_{0}g(t-s) \bigl(\Delta u(t)-\Delta u(s) \bigr)\,ds \biggr)\,dx\\ &\quad\leq2\delta(1-l)^{2}\sigma(t)\|\Delta u\|^{2}+ \biggl(2 \delta+\frac{1}{4\delta} \biggr) (1-l)\sigma(t) (g\circ \Delta u), \end{aligned} \end{aligned}$$
(3.17)
$$\begin{aligned}& \begin{aligned}[b] &\mu_{1} \int_{\Omega}|u_{t}|^{m-1}u_{t} \int ^{t}_{0}g(t-s) \bigl(u(t)-u(s)\bigr)\,ds\,dx\\ &\quad\leq\delta \mu_{1}\|u_{t}\|^{m+1}_{m+1}+ \frac{\mu_{1}}{4\delta} \int^{t}_{0}g(t-s)\bigl\| u(t)-u(s)\bigr\| ^{m+1}_{m+1} \,ds\\ &\quad\leq\delta \mu_{1}\|u_{t}\|^{m+1}_{m+1}+ \frac{C\mu_{1}}{4\delta\lambda_{1}}(g\circ \Delta u), \end{aligned} \end{aligned}$$
(3.18)
$$\begin{aligned}& \begin{aligned}[b] &\mu_{2} \int_{\Omega}\bigl|z(x,1,t)\bigr|^{m-1}z(x,1,t) \int ^{t}_{0}g(t-s) \bigl(u(t)-u(s)\bigr)\,ds\,dx\\ &\quad\leq\delta \mu_{2}\bigl\| z(x,1,t)\bigr\| ^{m+1}_{m+1}+ \frac{C\mu_{2}}{4\delta\lambda _{1}}(g\circ\Delta u), \end{aligned} \end{aligned}$$
(3.19)
$$\begin{aligned}& \begin{aligned}[b] &{-}\int_{\Omega}u_{t} \int^{t}_{0}g'(t-s) \bigl(u(t)-u(s) \bigr)\,ds\,dx\\ &\quad\leq\delta\|u_{t}\|^{2}+\frac{1}{4\delta} \biggl( \int^{t}_{0}\bigl(-g'(t-s)\bigr)\bigl\| u(t)-u(s)\bigr\| \,ds \biggr)^{2}\\ &\quad\leq\delta\|u_{t}\|^{2}-\frac{Cg(0)}{4\delta\lambda_{1}} \bigl(g'\circ \Delta u\bigr). \end{aligned} \end{aligned}$$
(3.20)
Now we estimate the term \(\int_{\Omega}F(\nabla u)\int^{t}_{0}g(t-s)(\nabla u(t)-\nabla u(s))\,ds\,dx\) by using the method in [18]. Noting (2.5) and the embedding \(V_{2}\hookrightarrow W^{1,p_{j}+1}_{0}(\Omega)\), \(j=1,\ldots,n\), we know that there exist positive constants \(\mu_{p_{1}},\ldots,\mu_{p_{n}}\) satisfying
$$\begin{aligned} \|\nabla u\|_{p_{j}+1}\leq\mu_{p_{j}}\|\Delta u\|,\quad \forall j=1, \ldots,n. \end{aligned}$$
Using (2.7) with \(F(0)=0\), Hölder’s inequality, Young’s inequality, and the embedding theorem, we can conclude that, for any \(\delta>0\),
$$\begin{aligned} & \int_{\Omega}\bigl|F(\nabla u)\bigr|\bigl|\nabla u(t)-\nabla u(s)\bigr|\,dx \\ &\quad\leq K \int_{\Omega}\Biggl(\sum^{n}_{j=1} \bigl(1+|\nabla u|^{\frac{p_{j}-1}{2}} \bigr) \Biggr)|\nabla u|\bigl|\nabla u(t)-\nabla u(s)\bigr|\,dx \\ &\quad\leq K\sum^{n}_{j=1} \bigl(| \Omega|^{\frac{p_{j}-1}{2(p_{j}+1)}}+ \| \nabla u\|^{\frac{p_{j}-1}{2}}_{p_{j}+1} \bigr)\|\nabla u\|_{p_{j}+1}\bigl\| \nabla u(t)-\nabla u(s)\bigr\| \\ &\quad\leq \frac{K}{\lambda_{2}}\sum^{n}_{j=1} \mu_{p_{j}} \bigl(|\Omega|^{\frac {p_{j}-1}{2(p_{j}+1)}}+\|\nabla u\|^{\frac{p_{j}-1}{2}}_{p_{j}+1} \bigr)\|\Delta u\|\bigl\| \Delta u(t)-\Delta u(s)\bigr\| \\ &\quad\leq\delta\|\Delta u\|^{2}+\frac{1}{4\delta} \Biggl[ \frac{K}{\lambda_{2}}\sum^{n}_{j=1}\mu _{p_{j}} \bigl(|\Omega|^{\frac{p_{j}-1}{2(p_{j}+1)}}+\|\nabla u\|^{\frac{p_{j}-1}{2}}_{p_{j}+1} \bigr) \Biggr]^{2}\bigl\| \Delta u(t)-\Delta u(s)\bigr\| ^{2} \\ &\quad\leq\delta\|\Delta u\|^{2}+ \Biggl[\frac{2K^{2}}{\lambda_{2}^{2}} \Biggl(\sum ^{n}_{j=1}\mu _{p_{j}}| \Omega|^{\frac{p_{j}-1}{2(p_{j}+1)}} \Biggr)^{2} +\frac{2K^{2}}{\lambda_{2}^{2}} \Biggl(\sum ^{n}_{j=1}\mu_{p_{j}}^{\frac {p_{j}+1}{2}} \biggl(\frac{2}{l} \biggr)^{\frac{p_{j}-1}{4}}\bigl(E(0)\bigr)^{\frac {p_{j}-1}{4}} \Biggr)^{2} \Biggr] \\ &\qquad{} \times\frac{1}{4\delta}\bigl\| \Delta u(t)-\Delta u(s)\bigr\| ^{2} \\ &\quad:=\delta\|\Delta u\|^{2}+\frac{1}{4\delta} \bigl( \alpha_{1}+\alpha_{2}\bigl(E(0)\bigr)^{\frac {p-1}{2}} \bigr)\bigl\| \Delta u(t)-\Delta u(s)\bigr\| ^{2}, \end{aligned}$$
(3.21)
where
$$\begin{aligned} p=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} \max_{j=1,\ldots,n}\{p_{j}\},& \mbox{if }E(0)\geq1, \\ \min_{j=1,\ldots,n}\{p_{j}\},& \mbox{if } E(0)< 1, \end{array}\displaystyle \right . \end{aligned}$$
and
$$\begin{aligned}& \alpha_{1}:=\frac{2K^{2}}{\lambda_{2}^{2}} \Biggl(\sum ^{n}_{j=1}\mu _{p_{j}}|\Omega|^{\frac{p_{j}-1}{2(p_{j}+1)}} \Biggr)^{2}, \\& \alpha_{2}:=\frac{2K^{2}}{\lambda_{2}^{2}} \Biggl(\sum ^{n}_{j=1}\mu _{p_{j}}^{\frac{p_{j}+1}{2}} \biggl(\frac{2}{l} \biggr)^{\frac{p_{j}-1}{4}}\bigl(E(0)\bigr)^{\frac {p_{j}-1}{4}} \Biggr)^{2}. \end{aligned}$$
It follows from (3.21), Hölder’s inequality, and Young’s inequality that
$$\begin{aligned} & \int_{\Omega}F(\nabla u) \int^{t}_{0}g(t-s) \bigl(\nabla u(t)-\nabla u(s) \bigr)\,ds\,dx \\ &\quad\leq \int^{t}_{0}g(t-s) \biggl( \int_{\Omega}\bigl|F(\nabla u)\bigr|\bigl|\nabla u(t)-\nabla u(s)\bigr|\,dx \biggr)\,ds \\ &\quad\leq\delta\|\Delta u\|^{2}+\frac{1}{4\delta} \bigl( \alpha_{1}+\alpha_{2}\bigl(E(0)\bigr)^{\frac {p-1}{2}} \bigr) (g \circ\Delta u). \end{aligned}$$
(3.22)
Inserting (3.16)-(3.20) and (3.22) into (3.15), we obtain (3.14). The proof is therefore complete. □

Lemma 3.4

For \(\varepsilon_{1}>0\) and \(\varepsilon_{2}>0\) small enough, there exist two positive constants \(\beta_{1}\) and \(\beta_{2}\) such that
$$\begin{aligned} \beta_{1} E(t)\leq\mathcal{L}(t)\leq\beta_{2} E(t). \end{aligned}$$
(3.23)

Proof

Using Hölder’s inequality, Young’s inequality, and Poincaré’s inequality, we can easily get
$$\begin{aligned} \bigl|\mathcal {L}(t)-E(t)\bigr| \leq&\frac{\varepsilon_{1}}{2}\sigma(0)\|u_{t} \|^{2}+\frac {\varepsilon_{1}}{2\lambda_{1}}\sigma(0)\|\Delta u\|^{2}+ \frac{\varepsilon_{2}}{2}\sigma(0)\|u_{t}\|^{2} \\ &{}+\frac{\varepsilon_{2}}{2} \int_{\Omega}\biggl( \int ^{t}_{0}g(t-s) \bigl(u(t)-u(s)\bigr)\,ds \biggr)^{2}\,dx \\ \leq&\frac{\varepsilon_{1}+\varepsilon_{2}}{2}\sigma(0)\|u_{t}\| ^{2}+ \frac{\varepsilon_{1}}{2\lambda_{1}}\sigma(0)\|\Delta u\|^{2} +\frac{\varepsilon_{2}l_{0}}{2\lambda_{1}}\sigma(0) (g\circ\Delta u), \end{aligned}$$
which, choosing \(\varepsilon_{1}>0\) and \(\varepsilon_{2}>0\) small enough, implies (3.23). The proof is complete. □

Proof of Theorem 2.2

Combining (3.4), (3.8), and (3.14) with assumption (A1), we can obtain
$$\begin{aligned} \mathcal {L}'(t)={}&E'(t)+ \varepsilon_{1}\sigma(t)\Phi(t)+\varepsilon_{1}\sigma (t) \Phi'(t)+ \varepsilon_{2}\sigma(t)\Psi(t)+ \varepsilon_{2}\sigma(t)\Psi '(t) \\ \leq{}&-\sigma(t) \biggl(\frac{c_{1}}{\sigma(0)}-\varepsilon _{1}- \varepsilon_{2}\delta\mu_{2} \biggr)\|u_{t} \|^{m+1}_{m+1} -\sigma(t) \biggl[\varepsilon_{2} \biggl( \int^{t}_{0}g(s)\,ds-\delta \biggr)- \varepsilon_{1} \biggr]\|u_{t}\|^{2} \\ &{}-\sigma(t) \biggl(\frac{c_{2}}{\sigma(0)}-C_{\varepsilon}\varepsilon _{1}-\varepsilon_{2}\delta\mu_{2} \biggr)\bigl\| z(x,1,t) \bigr\| ^{m+1}_{m+1} +\sigma(t) \biggl(\frac{1}{2}- \varepsilon_{2}\frac{Cg(0)}{4\delta \lambda_{1}} \biggr) \bigl(g'\circ \Delta u\bigr) \\ &{}-\sigma(t) \bigl[c_{3}\varepsilon_{1}-2 \varepsilon_{2}\delta -2\varepsilon_{2}\sigma(t) (1-l)^{2} \bigr]\|\Delta u\|^{2}-\frac{1}{2} \sigma'(t) \biggl( \int^{t}_{0}g(s)\,ds \biggr)\|\Delta u \|^{2} \\ &{}+\sigma(t) \biggl[\frac{\varepsilon_{1}}{4}\sigma(t)+\varepsilon _{2}C_{\delta}\bigl[1+(1-l)\sigma(t)+\bigl(E(0)\bigr)^{\frac{p-1}{2}} \bigr] \biggr](g\circ \Delta u) \\ &{}+\varepsilon_{1}\sigma'(t) \int_{\Omega}uu_{t}\,dx+\varepsilon_{2} \sigma'(t) \int_{\Omega}u_{t} \int^{t}_{0}g(t-s) \bigl(u(s)-u(t)\bigr)\,ds\,dx. \end{aligned}$$
(3.24)
By using the Young inequality and the Poincaré inequality, we shall see that
$$\begin{aligned} &\sigma'(t) \int_{\Omega}uu_{t}\,dx+\sigma'(t) \int_{\Omega}u_{t} \int^{t}_{0}g(t-s) \bigl(u(s)-u(t)\bigr)\,ds\,dx \\ &\quad\leq-\frac{\sigma'(t)}{2\lambda_{1}}\|\Delta u\|^{2}-\sigma'(t) \|u_{t}\|^{2}-\frac{\sigma'(t)}{2\lambda_{1}} \biggl( \int ^{t}_{0}g(s)\,ds \biggr) (g\circ\Delta u). \end{aligned}$$
(3.25)
For any fixed \(t_{0}>0\), we know that, for any \(t\geq t_{0}\),
$$\begin{aligned} \int^{t}_{0}g(s)\,ds\geq \int^{t_{0}}_{0}g(s)\,ds:=g_{0}, \end{aligned}$$
which, along with (3.24)-(3.25), implies for any \(t\geq t_{0}\),
$$\begin{aligned} {L}'(t) \leq& -\sigma(t) \biggl(\frac{c_{1}}{\sigma(0)}- \varepsilon_{1}-\varepsilon _{2}\delta\mu_{2} \biggr)\|u_{t}\|^{m+1}_{m+1} -\sigma(t) \biggl[ \varepsilon_{2}(g_{0}-\delta)-\varepsilon_{1}+ \frac {\sigma'(t)}{\sigma(t)} \biggr]\|u_{t}\|^{2} \\ &{}-\sigma(t) \biggl(\frac{c_{2}}{\sigma(0)}-C_{\varepsilon}\varepsilon _{1}-\varepsilon_{2}\delta\mu_{2} \biggr)\bigl\| z(x,1,t) \bigr\| ^{m+1}_{m+1} +\sigma(t) \biggl(\frac{1}{2}- \varepsilon_{2}\frac{Cg(0)}{4\delta \lambda_{1}} \biggr) \bigl(g'\circ \Delta u\bigr) \\ &{}-\sigma(t) \biggl[c_{3}\varepsilon_{1}-2 \varepsilon_{2}\delta -2\varepsilon_{2}\sigma(t) (1-l)^{2}+\frac{1}{2\lambda_{1}}\frac{\sigma '(t)}{\sigma(t)} \biggr]\|\Delta u \|^{2} \\ &{}+\sigma(t) \biggl[\frac{\varepsilon_{1}}{4}\sigma(t)+\varepsilon _{2}C_{\delta}\bigl[1+(1-l)\sigma(t)+\bigl(E(0)\bigr)^{\frac{p-1}{2}} \bigr] -\frac{g_{0}}{2\lambda_{1}} \frac{\sigma'(t)}{\sigma(t)} \biggr](g\circ \Delta u). \end{aligned}$$
(3.26)
At this point we first choose \(0<\delta<\min\{\frac{g_{0}}{2},\frac{c_{3}g_{0}}{4[1+(1-l)^{2}]}\}\), and we get
$$\begin{aligned} g_{0}-\delta>\frac{1}{2}g_{0} \quad\mbox{and}\quad \frac{\delta}{2c_{3}}\bigl[2+2(1-l)^{2}\bigr]< \frac{1}{4}g_{0}. \end{aligned}$$
For any fixed \(\delta>0\), we take \(\varepsilon_{1}>0\) and \(\varepsilon_{2}>0\) satisfying
$$\begin{aligned} \frac{g_{0}}{4}\varepsilon_{2}< \varepsilon_{1}< \frac{g_{0}}{2}\varepsilon _{2} \end{aligned}$$
(3.27)
so small that
$$\begin{aligned}& \eta_{1}:=\varepsilon_{2}(g_{0}-\delta)- \varepsilon_{1}>0,\\& \eta_{2}:=c_{3}\varepsilon_{1}-2 \varepsilon_{2}\delta\sigma-2\varepsilon _{2} \delta(1-l)^{2}>0. \end{aligned}$$
We at last choose \(\varepsilon_{1}>0\) and \(\varepsilon_{2}>0\) small enough for (3.23) and (3.27) to remain valid, and further,
$$\begin{aligned} \frac{c_{1}}{\sigma(0)}-\varepsilon_{1}-\varepsilon_{2}\delta \mu_{2}>0,\qquad \frac{c_{2}}{\sigma(0)}-C_{\varepsilon}\varepsilon_{1}- \varepsilon _{2}\delta\mu_{2}>0, \qquad\frac{1}{2}- \varepsilon_{2}\frac{Cg(0)}{4\delta\lambda _{1}}>0. \end{aligned}$$
From this it follows that, for positive constants \(\eta_{1}\), \(\eta_{2}\), and \(\eta_{3}\),
$$\begin{aligned} \mathcal {L}'(t) \leq&-\sigma(t) \biggl(2 \eta_{1}+\frac{1}{2}\frac{\sigma '(t)}{\sigma(t)} \biggr)\|u_{t} \|^{2}-\sigma(t) \biggl(2\eta_{2}+\frac {1}{2\lambda_{1}} \frac{\sigma'(t)}{\sigma(t)} \biggr)\|\Delta u\|^{2} \\ &{}+ \biggl(\eta_{3}-\frac{g_{0}}{2\lambda_{1}}\frac{\sigma'(t)}{\sigma (t)} \biggr) \sigma(t) (g\circ\Delta u),\quad \forall t\geq t_{0}. \end{aligned}$$
(3.28)
Since \(\lim_{t\rightarrow\infty}\frac{\sigma'(t)}{\sigma(t)}=0\), we choose \(t_{1}\geq t_{0}\) and use (2.16) to get
$$\begin{aligned} \mathcal {L}'(t) \leq&-\sigma(t) \bigl( \eta_{1}\|u_{t}\|^{2}+\eta_{2}\|\Delta u \|^{2} \bigr)+\eta_{3}\sigma(t) (g\circ\Delta u) \\ \leq&-\eta_{4}\sigma(t)E(t)+\eta_{5}\sigma(t) (g\circ \Delta u),\quad \forall t_{1}\geq t_{0}, \end{aligned}$$
(3.29)
where \(\eta_{4}\) and \(\eta_{5}\) are positive constants.
Multiplying (3.29) by \(\zeta(t)\) and using (3.4), we obtain
$$\begin{aligned} \zeta(t)\mathcal {L}'(t) \leq&-\eta_{4}\zeta(t) \sigma(t)E(t)+\eta_{5}\zeta(t)\sigma (t) (g\circ\Delta u)\\ \leq&-\eta_{4}\zeta(t)\sigma(t)E(t)-\eta_{5} \biggl[2E'(t)+\sigma '(t) \biggl( \int^{t}_{0}g(s)\,ds \biggr)\|\Delta u \|^{2} \biggr], \end{aligned}$$
which, combining with (2.16), gives us for any \(t\geq t_{1}\),
$$\begin{aligned} \zeta(t)\mathcal {L}'(t)+2\eta_{5}E'(t) \leq-\sigma(t)\zeta(t) \biggl[\eta_{4}+\frac {2l\sigma'(t)}{\zeta(t)\sigma(t)} \biggl( \int^{t}_{0}g(s)\,ds \biggr) \biggr]E(t). \end{aligned}$$
(3.30)
Since \(\lim_{t\rightarrow\infty}\frac{\sigma'(t)}{\zeta(t)\sigma(t)}=0\), we can choose \(t_{2}\geq t_{1}\) so that
$$\begin{aligned} \zeta(t)\mathcal {L}'(t)+2\eta_{5}E'(t) \leq-\frac{\eta_{4}}{2}\sigma(t)\zeta(t)E(t),\quad \forall t\geq t_{2}. \end{aligned}$$
(3.31)
Let \(\mathcal{E}(t)=\zeta(t)\mathcal {L}(t)+2\eta_{5}E(t)\), then it is easy to see that \(\mathcal{E}(t)\) is equivalent to the modified energy \(E(t)\) by using (3.23), that is, there exist two positive constants \(\beta_{3}\) and \(\beta_{4}\) such that
$$\begin{aligned} \beta_{3}E(t)\leq\mathcal{E}(t)\leq \beta_{4}E(t), \end{aligned}$$
(3.32)
which, together with (3.31) and using \(\zeta'(t)\leq0\), shows that there exists a positive constant \(\gamma_{1}>0\) such that, for any \(t\geq t_{2}\),
$$\begin{aligned} \mathcal {E}'(t)\leq-\frac{\eta_{4}}{\beta_{4}}\zeta(t) \sigma(t)\mathcal {E}(t). \end{aligned}$$
(3.33)
Integrating (3.33) over \((t_{2},t)\) with respect to t, we get for any \(t\geq t_{2}\),
$$\begin{aligned} \mathcal{E}(t) \leq& \mathcal {E}(t_{2})\exp \biggl(- \frac{\eta_{4}}{\beta_{4}} \int^{t}_{t_{2}}\zeta (s)\sigma(s)\,ds \biggr)\\ \leq& \mathcal {E}(t_{2})\exp \biggl(\frac{\eta_{4}}{\beta_{4}} \int^{t_{2}}_{0}\zeta (s)\sigma(s)\,ds \biggr)\exp \biggl(-\frac{\eta_{4}}{\beta_{4}} \int ^{t}_{0}\zeta(s)\sigma(s)\,ds \biggr), \end{aligned}$$
which, using (3.32), implies for any \(t\geq t_{2}\),
$$\begin{aligned} E(t)\leq\frac{\beta_{4}}{\beta_{3}}\gamma_{1}E(t_{2}) \exp \biggl(-\frac {\eta_{4}}{\beta_{4}} \int^{t}_{0}\zeta(s)\sigma(s)\,ds \biggr). \end{aligned}$$
(3.34)
Therefore (2.16) follows by renaming the constants, and by the continuity and boundedness of \(E(t)\). The proof is hence complete. □

Remark 3.5

We illustrate several rates of energy decay through the following examples, some of which can be found in [17, 23].

Example 1

If g decays exponentially, i.e., \(\zeta(t)=a\), and \(\sigma(t)=\frac{b}{1+t}\), then (2.16) gives us
$$\begin{aligned} E(t)\leq \frac{\beta}{(1+t)^{\gamma ab}}. \end{aligned}$$

Example 2

If g decays exponentially, i.e., \(\zeta(t)=a\), and \(\sigma(t)=b\), then (2.16) gives us
$$\begin{aligned} E(t)\leq\beta e^{-ab\gamma t}. \end{aligned}$$

Example 3

When \(g(t)=ae^{-b(1+t)^{\alpha}}\) and \(\sigma(t)=\frac{1}{1+t}\) for \(a,b>0\) and \(0<\alpha<1\), then \(\zeta(t)=b\alpha(1+t)^{\alpha-1}\) satisfies (2.1)-(2.2). Estimate (2.16) takes the form
$$\begin{aligned} E(t)\leq\beta \exp \biggl(-\frac{b\alpha\gamma}{\alpha-1}(1+t)^{\alpha-1} \biggr). \end{aligned}$$

Example 4

If \(g(t)=a\exp(-b\ln^{\alpha}(1+t))\) and \(\sigma(t)=\frac{1}{\ln(1+t)}\) for \(a,b>0\) and \(\alpha>1\), we know that \(\zeta(t)=\frac{b\alpha \ln^{\alpha-1}(1+t)}{1+t}\) satisfies (2.1)-(2.2). Estimate (2.16) takes the form
$$\begin{aligned} E(t)\leq\beta \exp \biggl(-\frac{b\alpha\gamma}{\alpha-1}\ln^{\alpha-1}(1+t) \biggr). \end{aligned}$$

Declarations

Acknowledgements

This work was support by the NSFC (No. 11526164) and the Fundamental Research Funds for the Central Universities with grant number JBK160122.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Faculty of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu, 611130, P.R. China
(2)
College of Mathematics and Information Science, Beifang University of Nationalities, Yinchuan, 750021, P.R. China

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© Feng and Li 2016

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