1 Introduction

Let Ω be an open bounded set of \({\mathbb{R}}^{2}\) with a sufficiently smooth boundary \(\Gamma =\Gamma _{0} \cup \Gamma _{1} \), \(\Gamma _{0}\) and \(\Gamma _{1} \) are closed and disjoint. Denote by \(\nu = (\nu _{1} , \nu _{2} )\) the external unit normal to Γ, and by \(\eta = ( - \nu _{2} , \nu _{1} )\) the unitary tangent positively oriented on Γ. In this paper we consider the following von Karman system with memory:

$$\begin{aligned}& w_{tt} -k\Delta w_{tt} + \Delta ^{2} w - \int _{0}^{t} h(t-s) \Delta ^{2} w (s)\,d s = [ w, v ] \quad \text{in} \ \Omega \times (0 ,\infty ), \end{aligned}$$
(1.1)
$$\begin{aligned}& \Delta ^{2} v = -[w, w] \quad \text{in} \ \Omega \times (0 ,\infty ), \end{aligned}$$
(1.2)
$$\begin{aligned}& v = \frac{\partial v }{\partial \nu} = 0 \quad \text{on} \ \Gamma \times (0, \infty ), \end{aligned}$$
(1.3)
$$\begin{aligned}& w = \frac{\partial w }{\partial \nu} = 0 \quad \text{on} \ \Gamma _{0} \times (0, \infty ), \end{aligned}$$
(1.4)
$$\begin{aligned}& {\mathcal {B}}_{1} w - {\mathcal {B}}_{1} \biggl\{ \int _{0}^{t} h(t-s ) w(s )\,d s \biggr\} =0 \quad \text{on} \ \Gamma _{1} \times (0 ,\infty ), \end{aligned}$$
(1.5)
$$\begin{aligned}& {\mathcal {B}}_{2} w- k\frac{\partial w_{tt}}{\partial \nu}- { \mathcal {B}}_{2} \biggl\{ \int _{0}^{t} h(t-s) w(s )\,d s \biggr\} =0 \quad \text{on} \ \Gamma _{1} \times (0, \infty ), \end{aligned}$$
(1.6)
$$\begin{aligned}& w( x,y,0) = w_{0} (x,y) , \qquad w_{t} (x,y,0) = w_{1} (x,y) \quad \text{in} \ \Omega , \end{aligned}$$
(1.7)

where the function h satisfies some conditions to be specified later and von Karman bracket is given by

$$ [w, v] = w_{xx}v_{yy} - 2 w_{xy} v_{xy}+ w_{yy}v_{xx} . $$

Here

$$\begin{aligned} {\mathcal {B}}_{1} w = \Delta w + (1-\mu ) B_{1} w \quad \text{and}\quad {\mathcal {B}}_{2} w = \frac{ \partial \Delta w}{\partial \nu } + (1- \mu ) B_{2} w, \end{aligned}$$

where constant \(\mu ( 0< \mu < \frac{1}{2})\) is Poisson’s ratio and

$$\begin{aligned} B_{1} w = 2 \nu _{1} \nu _{2} w_{xy} - \nu _{1}^{2} w_{yy} - \nu _{2}^{2} w_{xx}, \qquad B_{2} w = \frac{\partial }{\partial \eta}\bigl[\bigl(\nu _{1}^{2} - \nu _{2}^{2} \bigr)w_{xy} + \nu _{1} \nu _{2} ( w_{yy}-w_{xx})\bigr]. \end{aligned}$$

The equations describe small vibrations of a thin plate of uniform thickness. The second term in (1.1) represents rotational inertia.

Munoz Rivera and Menzala [2] discussed the exponential decay of the energy for problem (1.1)–(1.7) under the usual condition

$$\begin{aligned} -c_{0} h(t) \leq h' (t) \leq -c_{1} h(t) , \quad 0 \leq h'' (t) \leq c_{2} h(t) \end{aligned}$$
(1.8)

for some \(c_{i}\), \(i=0,1,2\). Moreover, they showed that when the kernel h decays polynomially, the energy also decays with the same rate. Raposo and Santos [3] generalized the decay result of [2]. They investigated the general decay of the solutions for problem (1.1)–(1.7) under a more general condition on h such as

$$\begin{aligned} h'(t) \leq - \xi (t) h(t), \qquad \xi (t) >0, \qquad \xi '(t) \leq 0 , \quad \forall t \geq 0, \end{aligned}$$
(1.9)

where ξ is a nonincreasing and positive function. Kang [4] proved that the solutions for problem (1.1)–(1.7) decay exponentially to zero as time goes to infinity in case

$$ h'(t) +\gamma h(t) \geq 0 , \qquad \bigl[h'(t)+ \gamma h(t)\bigr]e^{\alpha t} \in L^{1} (0, \infty ) ,\quad \forall t \geq 0, $$

for some \(\gamma , \alpha >0\). Lately, Kang [5] improved the decay result of [3] without imposing any restrictive assumptions on the behavior of the relaxation function at infinity. The author considered the general stability result for problem (1.1)–(1.7) under a relaxation function satisfying

$$\begin{aligned} h'(t) \leq -H \bigl( h(t) \bigr), \end{aligned}$$
(1.10)

where H is a nonnegative function, with \(H(0) =0\), and H is linear or strictly increasing and strictly convex on \((0, r]\) for some \(r >0\). Recently, Balegh et al. [6] studied the general decay rate of the energy for problem (1.1)–(1.7) with nonlinear boundary delay term. The relaxation function h satisfies

$$\begin{aligned} h'(t) \leq -\xi (t) H \bigl( h(t) \bigr), \end{aligned}$$
(1.11)

where ξ is a positive nonincreasing differentiable function and H satisfies the same conditions as (1.10) for some \(0< r <1\).

For the case \(h=0\) in (1.1) with nonlinear boundary dissipation, Horn and Lasiecka [7] and Bradley and Lasiecka [8] proved the uniform decay rates for the solution when t goes to infinity.

Moreover, Cavalcanti et al. [9] considered the following problem (1.1) with the rotational inertia coefficient \(k=0\):

$$\begin{aligned} \textstyle\begin{cases} u_{tt} + \Delta ^{2} u -{\int _{0}^{t} h(t-s) \Delta ^{2} u (s)\,d s} = [ u, v ] & \text{in} \ \Omega \times (0 ,\infty ), \\ \Delta ^{2} v = -[u, u] & \text{in} \ \Omega \times (0 ,\infty ), \\ u = \frac{\partial u }{\partial \nu} = 0, \qquad v = \frac{\partial v }{\partial \nu} = 0 & \text{on} \ \Gamma \times (0, \infty ), \end{cases}\displaystyle \end{aligned}$$
(1.12)

where the relaxation kernel h satisfies (1.10) and H is a positive, strictly increasing, and convex function with \(H(0)=0\). The rotational inertia ensures the regularity of solutions that is needed in the estimates. They proved the global existence of weak and regular solutions and provided sharp and general decay rate estimates without accounting for regularizing effects of rotational inertia by using the method introduced in [10]. Park [11] established an arbitrary rate of decay for problem (1.12) using the assumptions on the relaxation function due to Tatar [1].

When \(k=h=0\) in (1.1) with nonlinear boundary dissipation, Favini et al. [12] and Horn and Lasiecka [13] proved global existence, uniqueness, and regularity of solutions and uniform decay rates of weak solutions, respectively.

For the case \(k=h=0\) in (1.1) with memory-type boundary condition, Feng and Soufyane [14] obtained an optimal explicit and general energy decay result. For more results on von Karman plate equation with memory-type boundary condition, we refer to [15, 16].

On the other hand, for the viscoelastic wave equation, Cavalcanti et al. [17] proved exponential and polynomial decay under the usual condition (1.8). Later, this assumption was relaxed by several authors [18–20]. Messaoudi [21] considered general stability for the viscoelastic equation

$$\begin{aligned} u_{tt} -\Delta u+ \int _{0}^{t} h(t-s)\Delta u(s)\,d s =0 \quad \text{in} \ \Omega \times (0, \infty ) , \end{aligned}$$
(1.13)

where the relaxation function h satisfies

$$\begin{aligned} h'(t) \leq - \xi (t) h(t), \qquad \frac{ \vert \xi '(t) \vert }{ \vert \xi (t) \vert } \leq k_{0}, \qquad \xi (t) >0, \qquad \xi '(t) \leq 0 , \quad \forall t \geq 0. \end{aligned}$$
(1.14)

Tatar [22] investigated polynomial asymptotic stability of solutions for problem (1.13) under the condition

$$\begin{aligned} h'(t) \leq 0 \quad \text{for almost all} \ t>0. \end{aligned}$$
(1.15)

Moreover, Tatar [1] established an arbitrary decay rate for problem (1.13) with assumptions as follows:

$$\begin{aligned} \int _{0}^{\infty }h(s) \gamma (s)\,ds < +\infty , \end{aligned}$$
(1.16)

where a nondecreasing function \(\gamma (t) >0\) such that \(\frac{\gamma '(t)}{\gamma (t)}=\eta (t)\) is a decreasing function. As for problem of decay of the solutions for a viscoelastic system under condition (1.16), we also refer the reader to [11, 23] and the references therein. Later, Mustafa and Messaoudi [24] showed a general decay rate result for problem (1.13) with condition (1.10) on a relaxation function. The stability of the solutions to a viscoelastic system under condition (1.9) was studied in [25–28] and the references therein.

Motivated by these works, we study an arbitrary decay of solutions for problem (1.1)–(1.7) for relaxation functions satisfying condition (1.16). This result improves earlier ones concerning exponential and polynomial decay for problem (1.1)–(1.7).

The plan of the paper is as follows: in Sect. 2, we prepare some notation and material needed for our work. In Sect. 3, we show an arbitrary decay result of the solutions for problem (1.1)–(1.7).

2 Preliminaries

We define

$$ V=\bigl\{ w \in H^{1} (\Omega ) ; w =0 \text{ on } \Gamma _{0} \bigr\} ,\qquad W= \biggl\{ w \in H^{2} (\Omega ) ; w = \frac{\partial w }{\partial \nu} =0 \text{ on } \Gamma _{0} \biggr\} . $$

Integration by parts formula yields

$$\begin{aligned} \bigl( \Delta ^{2} w , v \bigr)= a(w, v) + ({ \mathcal {B}}_{2} w ,v )_{\Gamma}- \biggl({\mathcal {B}}_{1} w , \frac{\partial v }{\partial \nu} \biggr)_{ \Gamma}, \end{aligned}$$
(2.1)

where the bilinear symmetric form \(a(w, v)\) is given by

$$ a(w, v)= \int _{\Omega} \bigl\{ w_{xx} v_{xx} +w_{yy}v_{yy} +\mu (w_{xx}v_{yy}+w_{yy}v_{xx})+ 2 ( 1 -\mu ) w_{xy} v_{xy} \bigr\} \,d\Omega , $$

where \(d\Omega =dx\,dy\). Because \(\Gamma _{0} \neq \emptyset \), we see that for \(c_{0}>0\) and \({c_{1}}>0\),

$$\begin{aligned} c_{0} \Vert w \Vert ^{2}_{H^{2} (\Omega )} \leq a(w, w) \leq {c_{1}} \Vert w \Vert ^{2}_{H^{2} (\Omega )}. \end{aligned}$$
(2.2)

The Sobolev imbedding theorem implies that for positive constants \(C_{p} \) and \(C_{s}\),

$$\begin{aligned} \Vert w \Vert ^{2} \leq C_{p} a(w, w) , \qquad \Vert \nabla w \Vert ^{2} \leq C_{s} a(w, w) , \quad \forall w \in W . \end{aligned}$$
(2.3)

By the symmetry of \(a(\cdot ,\cdot )\), we get that for any \(w\in C^{1} (0, T; H^{2}(\Omega ))\),

$$\begin{aligned} a(h*w, w_{t} ) =&-\frac{1}{2}h(t) a(w, w) +\frac{1}{2}\bigl(h' \square \partial ^{2} w \bigr) (t) \\ &{}-\frac{1}{2} \frac{d}{dt} \biggl\{ \bigl(h \square \partial ^{2} w\bigr) (t) - \biggl( \int _{0}^{t} h(s)\,ds \biggr) a(w, w) \biggr\} , \end{aligned}$$
(2.4)

where

$$\begin{aligned}& (h*w) (t) := \int _{0}^{t} h(t- s) w (s )\,d s , \\& \bigl(h \square \partial ^{2} w\bigr) (t) := \int _{0}^{t} h(t-s) a\bigl(w(\cdot , t)-w( \cdot , s),w(\cdot , t)-w(\cdot , s)\bigr)\,ds . \end{aligned}$$

We introduce relative results of the Airy stress function and von Karman bracket.

Lemma 2.1

([2, 29])

Let \(w, u \in H^{2}(\Omega )\) and \(v \in H^{2}_{0} (\Omega )\). Then

$$\begin{aligned} \int _{\Omega }w[v, u]\,d\Omega = \int _{\Omega }v [w, u]\,d \Omega . \end{aligned}$$
(2.5)

Lemma 2.2

([12])

If \(w, v \in H^{2}(\Omega )\), then \([w, v] \in L^{2}(\Omega )\) and satisfies

$$\begin{aligned} \Vert v \Vert _{W^{2, \infty}(\Omega )} \leq c \Vert w \Vert _{H^{2} (\Omega )}^{2} \quad \textit{and}\quad \bigl\Vert [w, v] \bigr\Vert \leq c \Vert w \Vert _{H^{2}(\Omega )} \Vert v \Vert _{W^{2, \infty}( \Omega )} , \end{aligned}$$
(2.6)

where \(c>0\).

As in [1], we consider the following hypotheses on the relaxation function \(h(t)\):

(H1) \(h(t)\geq 0\) for all \(t\geq 0\) and

$$\begin{aligned} 0< l:= \int _{0}^{\infty }h(s)\,ds < 1 . \end{aligned}$$
(2.7)

(H2) \(h'(t) \leq 0\) for almost all \(t>0\).

(H3) There exists a nondecreasing function \(\gamma (t)>0\) such that

$$ \frac{\gamma '(t)}{\gamma (t)}:=\eta (t) \quad \text{is a decreasing function and } \int _{0}^{\infty }h(s) \gamma (s)\,ds < +\infty . $$
(2.8)

By using Galerkin’s approximation, we get the following result for the solution (see [2]). For \((w_{0}, w_{1}) \in W \times V \), \(k>0\), and \(T>0\), system (1.1)–(1.7) has a unique weak solution. For \((w_{0}, w_{1})\) is 2-regular, the weak solution satisfies

$$\begin{aligned}& w \in C\bigl([0, T];W\cap H^{4}(\Omega )\bigr), \qquad w_{t} \in C \bigl([0, T];V\cap H^{3}(\Omega )\bigr) . \end{aligned}$$

We define the energy of problem (1.1)–(1.7) by

$$\begin{aligned}& E(t) = \frac{1}{2} \bigl\Vert w_{t}(t) \bigr\Vert ^{2}+\frac{k}{2} \bigl\Vert \nabla w_{t}(t) \bigr\Vert ^{2} +\frac{1}{2} a \bigl(w(t), w(t)\bigr) + \frac{1}{4} \Vert \Delta v \Vert ^{2}. \end{aligned}$$
(2.9)

3 Arbitrary decay of the energy

To obtain the stability of problem (1.1)–(1.7), we introduce the following notations as in [1, 30]. For every measurable set \({\mathcal {M}} \subset {\mathbb{R}}^{+}\), we denote the probability measure ĥ by

$$\begin{aligned} \hat{h} ({\mathcal {M}}) = \frac{1}{l} \int _{\mathcal {M}} h(s)\,ds . \end{aligned}$$
(3.1)

The flatness set of h is defined by

$$\begin{aligned} F_{h}=\bigl\{ s \in {\mathbb{R}}^{+} : h(s)>0 \text{ and } h'(s)=0 \bigr\} . \end{aligned}$$
(3.2)

Let \(t_{0}>0\) be a number such that \({\int _{0}^{t_{0}} h(s)\,ds:=h_{0}>0.}\) We define the modified energy by

$$\begin{aligned} {\mathcal {E}}(t) =& \frac{1}{2} \bigl\Vert w_{t}(t) \bigr\Vert ^{2}+\frac{k}{2} \bigl\Vert \nabla w_{t}(t) \bigr\Vert ^{2} +\frac{1}{4} \Vert \Delta v \Vert ^{2}\\ &{} +\frac{1}{2} \biggl(1- \int _{0}^{t} h(s)\,ds \biggr) a\bigl(w(t), w(t) \bigr) + \frac{1}{2}\bigl( h \square \partial ^{2} w\bigr) (t). \end{aligned}$$

Multiplying (1.1) by \(w_{t} (t) \) and using (2.4), we have

$$\begin{aligned} {\mathcal {E}}'(t) =-\frac{1}{2}h(t) a \bigl(w(t), w(t)\bigr) +\frac{1}{2}\bigl(h' \square \partial ^{2} w \bigr) (t). \end{aligned}$$
(3.3)

From (2.7) one sees that

$$\begin{aligned} E(t) \leq \frac{1}{1-l} {\mathcal {E}}(t), \quad \forall t\geq 0. \end{aligned}$$
(3.4)

First, we define the standard functionals

$$\begin{aligned}& \Phi (t) = \int _{\Omega }w_{t}(t) w (t)\,d\Omega + k \int _{\Omega } \nabla w_{t}(t) \nabla w(t)\,d \Omega ,\\& \Psi (t) = \int _{\Omega }\bigl(k \Delta w_{t}(t) - w_{t}(t) \bigr) \int _{0}^{t} h(t-s) \bigl(w(t)-w(s)\bigr)\,d s \,d \Omega , \end{aligned}$$

and the new one

$$ \Xi (t) = \int _{0}^{t} G_{\gamma}(t-s) a \bigl(w(s), w(s)\bigr)\,ds , $$

where

$$ G_{\gamma}(t) = \gamma (t)^{-1} \int _{t}^{\infty }h(s) \gamma (s)\,ds . $$

Now let us define the perturbed modified energy by

$$\begin{aligned} {\mathcal {F}}(t) = M {\mathcal {E}}(t) +\xi _{1} \Phi (t) + \xi _{2} \Psi (t)+ \xi _{3}\Xi (t), \end{aligned}$$
(3.5)

where M and \(\xi _{i}(i=1,2,3)\) are positive constants to be specified later. Using the methods presented in [1, 4, 5], we get the following lemmas.

Lemma 3.1

Assume that (H1) holds. Then, for \(M>0\) large, there exist \(\alpha _{0}>0\) and \(\alpha _{1}>0\) such that

$$\begin{aligned} \alpha _{0} \bigl({\mathcal {E}}(t)+\Xi (t) \bigr) \leq {\mathcal {F}}(t) \leq \alpha _{1} \bigl( {\mathcal {E}}(t)+\Xi (t) \bigr), \quad \forall t \geq 0. \end{aligned}$$
(3.6)

Proof

From Young’s inequality, (2.3), and (2.7), we obtain

$$\begin{aligned} \bigl\vert \Phi (t) \bigr\vert \leq \frac{1}{2} \bigl\Vert w_{t}(t) \bigr\Vert ^{2} + \frac{k}{2} \bigl\Vert \nabla w_{t}(t) \bigr\Vert ^{2} + \frac{C_{p} +C_{s} k}{2} a \bigl(w(t),w(t)\bigr) \leq C_{1} {\mathcal {E}}(t) \end{aligned}$$
(3.7)

and

$$\begin{aligned} \bigl\vert \Psi (t) \bigr\vert \leq \frac{1}{2} \bigl\Vert w_{t}(t) \bigr\Vert ^{2} + \frac{k}{2} \bigl\Vert \nabla w_{t}(t) \bigr\Vert ^{2} + \frac{(C_{p} + C_{s} k) l}{2} \bigl(h \square \partial ^{2} w \bigr) (t)\leq C_{2} {\mathcal {E}}(t), \end{aligned}$$
(3.8)

where \(C_{1}=\max \{ 1, \frac{C_{p} +C_{s} k}{1-l} \}\) and \(C_{2}=\max \{ 1, (C_{p} +C_{s} k )l\}\). By (3.7) and (3.8), we find that

$$\begin{aligned} \bigl\vert F(t)- M {\mathcal {E}}(t)-\xi _{3} \Xi (t) \bigr\vert \leq C_{3} {\mathcal {E}}(t), \end{aligned}$$

where \(C_{3}=\xi _{1} C_{1} +\xi _{2}C_{2} \). Setting \(\alpha _{0}=\min \{ M-C_{3}, \xi _{3} \} \), \(\alpha _{1} =\max \{ M+C_{3}, \xi _{3}\}\) and taking \(M>0\) large, we complete the proof of Lemma 3.1. □

Lemma 3.2

Assume that (H1)–(H3) hold. Then, for each \(t_{0} > 0\) and all measurable sets \({\mathcal {M}}\) and \({\mathcal {N}}\) with \({\mathcal {M}}={\mathbb{R}}^{+} \setminus {\mathcal {N}}\), it is satisfied that

$$\begin{aligned} {\mathcal {F}}'(t) \leq& \bigl\{ \xi _{1} +\xi _{2} (\delta _{2}- h_{0} ) \bigr\} \bigl\Vert w_{t}(t) \bigr\Vert ^{2} + k \bigl\{ \xi _{1} +\xi _{2} (\delta _{2}- g_{0} ) \bigr\} \bigl\Vert \nabla w_{t}(t) \bigr\Vert ^{2} \\ &{} + \biggl[ \xi _{2} \biggl\{ ( 1- h_{0} ) \biggl( \delta _{1} + \frac{3l\hat{h}({\mathcal {N}})}{2} \biggr) +\delta _{3} C_{*} E^{2}(0) \biggr\} \\ &{}-\xi _{1} \biggl(1-\frac{l}{2} \biggr) +\xi _{3} G_{\gamma}(0) \biggr] a\bigl(w(t), w(t)\bigr) \\ & {} + \xi _{2}l \biggl( \frac{ 1-h_{0} }{4\delta _{1}} + 1+ \frac{1}{\delta _{1}} +\frac{C_{p}}{2\delta _{3}} \biggr) \int _{{ \mathcal {M}}_{t}} h(t-s) a\bigl(w(t)-w(s), w(t)-w(s) \bigr)\,ds \\ & {} +\xi _{2} l \hat{h}({\mathcal {N}}) \biggl( 1+ \delta _{1} + \frac{C_{p}}{2\delta _{3}} \biggr) \int _{{\mathcal {N}}_{t}} h(t-s) a\bigl(w(t)-w(s), w(t)-w(s) \bigr)\,ds \\ & {} - \frac{\xi _{1}}{2} \bigl( h \square \partial ^{2} w\bigr) (t) +\frac{\xi _{2}( 1- h_{0} )}{2} \int _{{\mathcal {N}}_{t}} h(t-s) a\bigl(w(s), w(s)\bigr)\,ds \\ &{} + \biggl( \frac{M}{2}- \frac{\xi _{2}h(0)(C_{s} h +C_{p})}{4\delta _{2}} \biggr) \bigl( h' \square \partial ^{2} w\bigr) (t) \\ & {} + \biggl( \frac{\xi _{1}}{2} -\xi _{3} \biggr) \int _{0}^{t} h(t-s)a\bigl(w(s), w(s)\bigr)\,ds \\ &{} -\xi _{3}\eta (t) \Xi (t)-\xi _{1} \Vert \Delta v \Vert ^{2} , \quad \forall t \geq t_{0} , \end{aligned}$$
(3.9)

where \(C_{*}\) is a positive constant.

Proof

From (1.1)–(1.6), (2.1), (2.5), and (2.7), we have

$$\begin{aligned} \Phi '(t) =& \bigl\Vert w_{t}(t) \bigr\Vert ^{2} +k \bigl\Vert \nabla w_{t}(t) \bigr\Vert ^{2} -a\bigl(w(t), w(t)\bigr) + \frac{1}{2} \biggl( \int _{0}^{t} h(s)\,ds \biggr) a\bigl(w(t), w(t) \bigr) \\ & {}+ \frac{1}{2} \int _{0}^{t} h(t-s) a\bigl(w(s), w(s)\bigr)\,ds - \frac{1}{2} \bigl(h \square \partial ^{2} w \bigr) (t) - \Vert \Delta v \Vert ^{2} \\ \leq& \bigl\Vert w_{t}(t) \bigr\Vert ^{2} +k \bigl\Vert \nabla w_{t}(t) \bigr\Vert ^{2} - \biggl(1- \frac{l}{2} \biggr) a\bigl(w(t), w(t) \bigr)\\ & {}+ \frac{1}{2} \int _{0}^{t} h(t-s) a\bigl(w(s), w(s)\bigr)\,ds - \frac{1}{2} \bigl(h \square \partial ^{2} w \bigr) (t) - \Vert \Delta v \Vert ^{2}. \end{aligned}$$
(3.10)

Similarly, we conclude that

$$\begin{aligned} \Psi '(t) =& \biggl(1- \int _{0}^{t} h(s)\,ds \biggr) \int _{0}^{t} h(t-s)a \bigl( w(t)-w(s), w(t) \bigr)\,ds \\ & {} + \int _{0}^{t} h(t-s)a \biggl( w(t)-w(s), \int _{0}^{t} h(t-\tau ) \bigl( w(t)-w( \tau ) \bigr)\,d \tau \biggr)\,ds \\ &{} - k \int _{0}^{t} h'(t-s) \bigl( \nabla w(t)-\nabla w(s), \nabla w_{t}(t)\bigr)\,ds \\ &{} - \int _{0}^{t} h'(t-s) \bigl(w(t)-w(s), w_{t}(t)\bigr)\,d s \\ &{} - \int _{0}^{t} h(t-s) \bigl( w(t)-w(s), [w, v] \bigr)\,ds - \biggl( \int _{0}^{t} h(s)\,ds \biggr) \bigl\Vert w_{t}(t) \bigr\Vert ^{2} \\ &{}- k \biggl( \int _{0}^{t} h(s)\,ds \biggr) \bigl\Vert \nabla w_{t}(t) \bigr\Vert ^{2} \\ :=& \biggl( 1- \int _{0}^{t} h(s)\,ds \biggr) I_{1}+I_{2}+\cdots +I_{5} \\ &{} - \biggl( \int _{0}^{t} h(s)\,ds \biggr) \bigl\Vert w_{t}(t) \bigr\Vert ^{2}-k \biggl( \int _{0}^{t} h(s)\,ds \biggr) \bigl\Vert \nabla w_{t}(t) \bigr\Vert ^{2}. \end{aligned}$$
(3.11)

For all measurable sets \({\mathcal {M}}\) and \({\mathcal {N}}\) such that \({\mathcal {M}}={\mathbb{R}}^{+} \setminus {\mathcal {N}}\), using Young’s inequality, (2.7), and (3.1), we obtain that for \(\delta _{1}>0\),

$$\begin{aligned} I_{1} =& \int _{{\mathcal {M}}_{t}} h(t-s)a \bigl( w(t)-w(s), w(t) \bigr)\,ds + \biggl( \int _{{\mathcal {N}}_{t}} h(s)\,ds \biggr) a \bigl( w(t), w(t) \bigr) \\ & {}- \int _{{ \mathcal {N}}_{t}} h(t-s)a \bigl( w(s), w(t) \bigr)\,ds \\ \leq &\biggl( \delta _{1} + \frac{3l\hat{h}({\mathcal {N}})}{2} \biggr) a\bigl(w(t), w(t)\bigr) + \frac{l}{4\delta _{1}} \int _{{\mathcal {M}}_{t}} h(t-s)a \bigl(w(t)-w(s), w(t)-w(s) \bigr)\,ds \\ & {} + \frac{1}{2} \int _{{\mathcal {N}}_{t}} h(t-s) a \bigl( w(s), w(s) \bigr)\,ds , \end{aligned}$$
(3.12)

where \({\mathcal {M}}_{t}:={\mathcal {M}}\cap [0, t]\) and \({\mathcal {N}}_{t}:={\mathcal {N}}\cap [0, t]\). Similarly, we have that for \(\delta _{1}>0\),

$$\begin{aligned} I_{2} =& a\biggl( \int _{0}^{t} h(t-s) \bigl( w(t)-w(s)\bigr)\,ds , \int _{0}^{t} h(t-s) \bigl( w(t)-w(s)\bigr)\,ds \biggr) \\ =& a\biggl( \int _{{\mathcal {M}}_{t}} h(t-s) \bigl( w(t)-w(s)\bigr)\,ds, \int _{{ \mathcal {M}}_{t}} h(t-s) \bigl( w(t)-w(s)\bigr)\,ds \biggr) \\ & {}+ 2 a\biggl( \int _{{\mathcal {M}}_{t}} h(t-s) \bigl( w(t)-w(s)\bigr)\,ds, \int _{{\mathcal {N}}_{t}} h(t-s) \bigl( w(t)-w(s)\bigr)\,ds \biggr) \\ & {}+ a\biggl( \int _{{\mathcal {N}}_{t}} h(t-s) \bigl( w(t)-w(s)\bigr)\,ds, \int _{{\mathcal {N}}_{t}} h(t-s) \bigl( w(t)-w(s)\bigr)\,ds\biggr) \\ \leq &\biggl(1+\frac{1}{\delta _{1}} \biggr)l \int _{{\mathcal {M}}_{t}} h(t-s) a\bigl( w(t)-w(s) , w(t)-w(s) \bigr)\,ds \\ & {}+ (1+\delta _{1} )l\hat{h}({\mathcal {N}}) \int _{{\mathcal {N}}_{t}} h(t-s) a\bigl( w(t)-w(s) , w(t)-w(s) \bigr)\,ds . \end{aligned}$$
(3.13)

Applying Young’s inequality and (2.3), we get that for \(\delta _{2}>0\),

$$\begin{aligned}& \begin{aligned}[b] \vert I_{3} \vert &\leq k\delta _{2} \bigl\Vert \nabla w_{t}(t) \bigr\Vert ^{2} + \frac{k}{4\delta _{2}} \int _{\Omega } \biggl( \int _{0}^{t} h' (t-s) \bigl\vert \nabla w(t)-\nabla w(s) \bigr\vert \,d s \biggr)^{2}\,d \Omega \\ & \leq k\delta _{2} \bigl\Vert \nabla w_{t}(t) \bigr\Vert ^{2} - \frac{ h(0)C_{s} k }{4\delta _{2} } \bigl(h' \square \partial ^{2} w\bigr) (t) , \end{aligned} \end{aligned}$$
(3.14)
$$\begin{aligned}& \vert I_{4} \vert \leq \delta _{2} \bigl\Vert w_{t}(t) \bigr\Vert ^{2}- \frac{h(0)C_{p}}{4 \delta _{2}}\bigl( h' \square \partial ^{2} w \bigr) (t). \end{aligned}$$
(3.15)

By Young’s inequality, we find that for \(\delta _{3}>0\),

$$\begin{aligned} & \vert I_{5} \vert \leq \delta _{3} \bigl\Vert [w,v] \bigr\Vert ^{2}+\frac{1}{4\delta _{3}} \biggl\Vert \int _{0}^{t} h(t-s) \bigl( w(t)-w(s)\bigr)\,ds \biggr\Vert ^{2} . \end{aligned}$$
(3.16)

Using (2.2), (2.6), (2.9), (3.4) and the fact \({\mathcal {E}}(t) \leq {\mathcal {E}}(0)=E(0)\), we see that

$$\begin{aligned} \bigl\Vert [w,v] \bigr\Vert ^{2} \leq& c^{4} \bigl\Vert w(t) \bigr\Vert ^{2}_{H^{2}(\Omega )} \bigl\Vert w(t) \bigr\Vert ^{4}_{H^{2}( \Omega )} \leq \frac{ c^{4} }{c_{0}}a\bigl(w(t),w(t)\bigr) \biggl(\frac{2}{c_{0}} E(t) \biggr)^{2} \\ \leq& \frac{ c^{4} }{c_{0}}a\bigl(w(t),w(t)\bigr) \biggl(\frac{2}{c_{0}(1-l)} { \mathcal {E}}(t) \biggr)^{2} \leq C_{*} E^{2}(0) a\bigl(w(t), w(t)\bigr) , \end{aligned}$$

where \(C_{*}= \frac{4c^{4}}{c_{0}^{3} (1-l)^{2}}\). From Young’s inequality, (2.3), and (3.1), we obtain

$$\begin{aligned}& \biggl\Vert \int _{0}^{t} h(t-s) \bigl( w(t)-w(s)\bigr)\,ds \biggr\Vert ^{2}\\& \quad = \biggl\Vert \int _{{\mathcal {M}}_{t}} h(t-s) \bigl( w(t)-w(s)\bigr)\,ds + \int _{{ \mathcal {N}}_{t}} h(t-s) \bigl( w(t)-w(s)\bigr)\,ds \biggr\Vert ^{2} \\& \quad \leq 2l \int _{{\mathcal {M}}_{t}} h(t-s) \bigl\Vert w(t)-w(s) \bigr\Vert ^{2}\,ds + 2l \hat{h}({\mathcal {N}}) \int _{{\mathcal {N}}_{t}} h(t-s) \bigl\Vert w(t)-w(s) \bigr\Vert ^{2}\,ds \\& \quad \leq 2l C_{p} \int _{{\mathcal {M}}_{t}} h(t-s) a\bigl( w(t)-w(s), w(t)-w(s) \bigr)\,ds \\& \qquad {}+ 2l \hat{h}({\mathcal {N}})C_{p} \int _{{\mathcal {N}}_{t}} h(t-s) a\bigl( w(t)-w(s), w(t)-w(s)\bigr)\,ds . \end{aligned}$$

Inserting these estimates into (3.16), we have

$$\begin{aligned} \vert I_{5} \vert \leq& \delta _{3} C_{*} E^{2}(0) a\bigl(w(t), w(t)\bigr) + \frac{l C_{p}}{2\delta _{3}} \int _{{\mathcal {M}}_{t}} h(t-s) a\bigl( w(t)-w(s), w(t)-w(s) \bigr)\,ds \\ & {}+ \frac{l \hat{h}({\mathcal {N}})C_{p}}{2\delta _{3}} \int _{{ \mathcal {N}}_{t}} h(t-s) a\bigl( w(t)-w(s), w(t)-w(s)\bigr)\,ds . \end{aligned}$$
(3.17)

Substituting (3.12)–(3.15) and (3.17) into (3.11), we arrive at

$$\begin{aligned} \Psi '(t) \leq& k \biggl( \delta _{2}- \int _{0}^{t} h(s)\,ds \biggr) \bigl\Vert \nabla w_{t}(t) \bigr\Vert ^{2} + \biggl( \delta _{2}- \int _{0}^{t} h(s)\,ds \biggr) \bigl\Vert w_{t}(t) \bigr\Vert ^{2} \\ & {} + \biggl\{ \biggl( 1- \int _{0}^{t} h(s)\,ds \biggr) \biggl( \delta _{1} + \frac{3l\hat{h}({\mathcal {N}})}{2} \biggr) +\delta _{3} C_{*} E^{2}(0) \biggr\} a\bigl(w(t), w(t)\bigr) \\ & {} +l \biggl\{ \biggl( 1- \int _{0}^{t} h(s)\,ds \biggr) \frac{1}{4\delta _{1}} + 1+\frac{1}{\delta _{1}} + \frac{C_{p}}{2\delta _{3}} \biggr\} \\ & {} \times \int _{{\mathcal {M}}_{t}} h(t-s) a\bigl( w(t)-w(s), w(t)-w(s) \bigr)\,ds \\ & {} + l \hat{h}({\mathcal {N}}) \biggl( 1+\delta _{1} + \frac{C_{p}}{2\delta _{3}} \biggr) \int _{{\mathcal {N}}_{t}} h(t-s) a\bigl( w(t)-w(s), w(t)-w(s) \bigr)\,ds \\ & {} +\frac{1}{2} \biggl( 1- \int _{0}^{t} h(s)\,ds \biggr) \int _{{ \mathcal {N}}_{t}} h(t-s) a\bigl(w(s), w(s)\bigr)\,ds \\ & {} - \frac{h(0)(C_{s} k +C_{p})}{4\delta _{2}}\bigl( h' \square \partial ^{2} w \bigr) (t) . \end{aligned}$$
(3.18)

A differentiation of \(\Xi (t)\) yields

$$\begin{aligned} \Xi '(t) =& G_{\gamma}(0)a\bigl(w(t), w(t) \bigr) - \int _{0}^{t} \frac{\gamma '(t-s)}{\gamma (t-s)} G_{\gamma}(t-s) a\bigl(w(s), w(s)\bigr)\,ds\\ &{} - \int _{0}^{t} h(t-s)a\bigl(w(s), w(s)\bigr)\,ds \\ \leq &G_{\gamma}(0)a\bigl(w(t), w(t)\bigr) - \eta (t) \Xi (t) - \int _{0}^{t} h(t-s)a\bigl(w(s), w(s)\bigr)\,ds, \end{aligned}$$
(3.19)

where we have used the fact that \(\frac{\gamma '(t)}{\gamma (t)} =\eta (t)\) is a nonincreasing function. Since h is positive, we get \(\int _{0}^{t} h(s)\,ds \geq h_{0}\) for all \(t\geq t_{0}\), and combining (3.3), (3.5), (3.10), (3.18), and (3.19), we obtain the desired estimate (3.9). □

Now, we are ready to prove the following arbitrary decay result.

Theorem 3.1

Assume that (H1)–(H3), \(E(0)<\frac{l}{\sqrt{C_{*}C_{p}}}\), and \(\hat{h}({F}_{h})<\frac{1}{8}\) hold. If \(h_{0}> \frac{3l}{8-l}\) and \(G_{\gamma}(0)< \frac{(8-l)h_{0} -3l}{16}\), then there exist positive constants \(t_{0}\), ω, and C such that

$$ E(t) \leq \frac{C}{ \gamma (t)^{\omega}} \quad \textit{for } t \geq t_{0} . $$

Proof

As in [1, 30], we introduce the sets

$$ {\mathcal {M}}_{n}=\bigl\{ s\in {\mathbb{R}}^{+} : n h'(s)+h(s) \leq 0\bigr\} \quad \text{and} \quad {\mathcal {N}}_{n} ={\mathbb{R}}^{+} \setminus {\mathcal {M}}_{n} , \quad n\in{\mathbb{N}}. $$

Observe that

$$ \bigcup_{n=1}^{\infty }{\mathcal {M}}_{n} = {\mathbb{R}}^{+} \setminus \{ { F}_{h} \cup {N}_{h} \}, $$

where \({N}_{h}\) is the null set where \(h'\) is not defined and \({F}_{h}\) is given in (3.2). Because \({\mathcal {N}}_{n +1} \subset {\mathcal {N}}_{n}\) for all n and \(\cap _{n=1}^{\infty }{\mathcal {N}}_{n} = { F}_{h} \cup{N}_{h}\), we have

$$\begin{aligned} \lim_{n\to \infty} \hat{h}({\mathcal {N}}_{n} )=\hat{h}({ F}_{h}) . \end{aligned}$$
(3.20)

Choosing \({\mathcal {M}}= {\mathcal {M}}_{n}\), \({\mathcal {N}}={\mathcal {N}}_{n}\), and \(\delta _{3} =\frac{(2-l)\xi _{1}}{4C_{*}E^{2}(0) \xi _{2}}\) in (3.9), we find that

$$\begin{aligned} {\mathcal {F}}'(t) \leq &\bigl\{ \xi _{1} +\xi _{2} (\delta _{2}- h_{0} ) \bigr\} \bigl\Vert w_{t}(t) \bigr\Vert ^{2} + k \bigl\{ \xi _{1} +\xi _{2} (\delta _{2}- h_{0} ) \bigr\} \bigl\Vert \nabla w_{t}(t) \bigr\Vert ^{2} \\ & {}+ \biggl\{ \xi _{2} ( 1- h_{0} ) \biggl( \delta _{1} + \frac{3l\hat{h}({\mathcal {N}}_{n})}{2} \biggr) -\frac{\xi _{1}}{2} \biggl(1- \frac{l}{2} \biggr) +\xi _{3} G_{\gamma}(0) \biggr\} a\bigl(w(t),w(t)\bigr) \\ & {}+ \biggl\{ \xi _{2}l \biggl( \frac{ 1-h_{0} }{4\delta _{1}} + 1+ \frac{1}{\delta _{1}} + \frac{2C_{*}C_{p}E^{2}(0)\xi _{2}}{(2-l)\xi _{1}} \biggr) -\frac{1}{n} \biggl( \frac{M}{2}-\frac{\xi _{2}h(0)(C_{s} k +C_{p})}{4\delta _{2}} \biggr) \biggr\} \\ & {} \times \int _{{\mathcal {M}}_{nt}} h(t-s) a\bigl( w(t)-w(s), w(t)-w(s) \bigr)\,ds \\ & {}+ \biggl\{ \xi _{2} l \hat{h}({\mathcal {N}}_{n}) \biggl( 1+\delta _{1} + \frac{2C_{*}C_{p}E^{2}(0)\xi _{2}}{(2-l)\xi _{1}} \biggr) - \frac{\xi _{1}}{2} \biggr\} \bigl( h \square \partial ^{2} w\bigr) (t) -\xi _{3} \eta (t) \Xi (t) \\ & {} + \biggl\{ \frac{\xi _{2}( 1- h_{0} )}{2} + \frac{\xi _{1}}{2} -\xi _{3} \biggr\} \int _{0}^{t} h(t-s)a\bigl(w(s), w(s)\bigr)\,ds \\ & {}-\xi _{1} \Vert \Delta v \Vert ^{2} , \quad \forall t \geq t_{0} , \end{aligned}$$
(3.21)

where \({\mathcal {M}}_{nt}={\mathcal {M}}_{n}\cap [0, t]\). For small \(0<\varepsilon < h_{0}\), by taking \(\xi _{1} =(h_{0}-\varepsilon )\xi _{2}\), (3.21) yields

$$\begin{aligned} {\mathcal {F}}'(t) \leq& \xi _{2} (\delta _{2}- \varepsilon ) \bigl\Vert w_{t}(t) \bigr\Vert ^{2} + k\xi _{2} ( \delta _{2}- \varepsilon ) \bigl\Vert \nabla w_{t}(t) \bigr\Vert ^{2} \\ & {}+ \biggl\{ \xi _{2} ( 1- h_{0} ) \biggl( \delta _{1} + \frac{3l\hat{h}({\mathcal {N}}_{n})}{2} \biggr) -(h_{0}- \varepsilon )\xi _{2} \bigl(\beta +(1-\beta ) \bigr) \biggl( \frac{2-l}{4} \biggr)\\ & {} +\xi _{3} G_{ \gamma}(0) \biggr\} a\bigl(w(t),w(t)\bigr) \\ & {}+ \biggl\{ \xi _{2}l \biggl( \frac{ 1-h_{0} }{4\delta _{1}} + 1+ \frac{1}{\delta _{1}} + \frac{2C_{*}C_{p}E^{2}(0)}{(2-l)(h_{0}-\varepsilon )} \biggr) - \frac{1}{n} \biggl( \frac{M}{2}- \frac{\xi _{2}h(0)(C_{s} k +C_{p})}{4\delta _{2}} \biggr) \biggr\} \\ &{} \times \int _{{\mathcal {M}}_{nt}} h(t-s) a\bigl( w(t)-w(s),w(t)-w(s) \bigr)\,ds \\ & {}+ \xi _{2} \biggl\{ l \hat{h}({\mathcal {B}}_{n}) \biggl( 1+\delta _{1} + \frac{2C_{*}C_{p}E^{2}(0)}{(2-l)(h_{0}-\varepsilon )} \biggr) - \frac{h_{0}-\varepsilon}{2} \biggr\} \bigl( h \square \partial ^{2} w\bigr) (t) - \xi _{3}\eta (t) \Xi (t) \\ & {} + \biggl\{ \frac{\xi _{2}( 1- \varepsilon )}{2} -\xi _{3} \biggr\} \int _{0}^{t} h(t-s)a\bigl(w(s), w(s)\bigr)\,ds \\ & {} -(h_{0}-\varepsilon )\xi _{2} \Vert \Delta v \Vert ^{2} , \quad \forall t \geq t_{0} , \end{aligned}$$
(3.22)

where \(\beta =\frac{3l(1-h_{0})}{4(2-l)h_{0}}\). From (3.20) and \(\hat{h}({ F}_{h})<\frac{1}{8}\), there exists \(n_{0} \in { N}\) large such that

$$\begin{aligned} \hat{h}({\mathcal {N}}_{n})< \frac{1}{8} \end{aligned}$$
(3.23)

for \(n\geq n_{0}\). By (3.23), we get that for \(n\geq n_{0} \),

$$\begin{aligned} ( 1- h_{0} ) \biggl( \frac{3l\hat{h}({\mathcal {N}}_{n})}{2} \biggr) < \beta h_{0} \biggl(\frac{2-l}{4} \biggr) . \end{aligned}$$

Then we can take a constant \(\varepsilon _{1}>0\) such that

$$\begin{aligned} ( 1- h_{0} ) \biggl( \frac{3l\hat{h}({\mathcal {N}}_{n})}{2} \biggr) < \beta (h_{0}-\varepsilon ) \biggl(\frac{2-l}{4} \biggr) \quad \text{for } n \geq n_{0}\ \text{and}\ 0< \varepsilon \leq \varepsilon _{1} . \end{aligned}$$
(3.24)

Because \(l=\int _{0}^{\infty }h(s)\,ds\) and \(E(0)< \frac{l}{\sqrt{C_{*}C_{p}}}\), there exists \(t_{1}>0\) large such that

$$ \frac{l}{2} < h_{0} \quad \text{and} \quad \sqrt{C_{*}C_{p}} E(0) < h_{0} < l \quad \text{for } t_{0} \geq t_{1}, $$

and then there exists a positive constant \(\varepsilon _{2}>0\) with \(\varepsilon _{2} \leq \varepsilon _{1}\) small such that

$$\begin{aligned} \frac{l}{2} < h_{0}-\varepsilon \quad \text{and} \quad \sqrt{C_{*}C_{p}} E(0) < h_{0}-\varepsilon < l\quad \text{for } t_{0} \geq t_{1}\ \text{and}\ 0< \varepsilon \leq \varepsilon _{2}. \end{aligned}$$
(3.25)

By (3.23) and (3.25), we have that for \(t_{0}\geq t_{1}\), \(\ n\geq n_{0} \), and \(0<\varepsilon \leq \varepsilon _{2}\),

$$\begin{aligned} l \hat{h}({\mathcal {N}}_{n}) \biggl( 1+ \frac{2C_{*}C_{p}E^{2}(0)}{(2-l)(h_{0}-\varepsilon )} \biggr) - \frac{h_{0}-\varepsilon}{2} < & l \hat{h}({\mathcal {N}}_{n}) +\hat{h}({ \mathcal {N}}_{n}) \frac{2C_{*}C_{p}E^{2}(0)}{h_{0}-\varepsilon}- \frac{h_{0}-\varepsilon}{2} \\ < & \frac{l}{8}- \frac{h_{0}-\varepsilon}{4} < 0. \end{aligned}$$
(3.26)

Then, from (3.24) and (3.26), we can choose \(\delta _{1}>0\) small enough such that for \(t_{0}\geq t_{1}\), \(n\geq n_{0}\), and \(0<\varepsilon \leq \varepsilon _{2} \),

$$\begin{aligned} & ( 1- h_{0} ) \biggl( \delta _{1} + \frac{3l\hat{h}({\mathcal {N}}_{n})}{2} \biggr) -\beta (h_{0}- \varepsilon ) \biggl( \frac{2-l}{4} \biggr) < 0 , \end{aligned}$$
(3.27)
$$\begin{aligned} & l \hat{h}({\mathcal {N}}_{n}) \biggl( 1+\delta _{1}+ \frac{2C_{*}C_{p}E^{2}(0)}{(2-l)(h_{0}-\varepsilon )} \biggr) - \frac{h_{0}-\varepsilon}{2}< 0 . \end{aligned}$$
(3.28)

From the fact \(\frac{3l}{8-l} < h_{0} <l\), we see that \(1-\beta =\frac{(8-l)h_{0}-3l}{4(2-l)h_{0}}>0\). Once \(n_{0}\), \(\varepsilon _{2}\), and \(t_{1}\) are fixed, we choose \(n=n_{0}\), \(\varepsilon =\varepsilon _{2}\), and \(t_{0}=t_{1}\). Next we take \(\xi _{2}\) and \(\xi _{3}\) satisfying

$$\begin{aligned} \frac{\xi _{2}}{2} < \xi _{3} < \frac{(8-l)h_{0}-3l}{32G_{\gamma }(0)} \xi _{2} . \end{aligned}$$
(3.29)

This is possible if \(G_{\gamma }(0) < \frac{(8-l)h_{0} -3l}{16} \). Using (3.25) and (3.29), we obtain

$$\begin{aligned} \frac{\xi _{2}(1-\varepsilon )}{2} -\xi _{3} < 0 \end{aligned}$$
(3.30)

and

$$\begin{aligned} \xi _{3} G_{\gamma}(0) -\xi _{2}(1-\beta ) (h_{0} -\varepsilon ) \biggl( \frac{2-l}{4} \biggr) < \frac{(8-l)h_{0}-3l}{16} \biggl(\frac{1}{2}- \frac{h_{0}-\varepsilon}{h_{0}} \biggr) \xi _{2} < 0 . \end{aligned}$$
(3.31)

Finally, we select \(\delta _{2}>0\) small enough and \(M>0\) large enough so that

$$\begin{aligned} \delta _{2}-\varepsilon < 0 \end{aligned}$$
(3.32)

and

$$\begin{aligned} \xi _{2}l \biggl( \frac{ 1-h_{0} }{4\delta _{1}} + 1+ \frac{1}{\delta _{1}} + \frac{2C_{*}C_{p}E^{2}(0)}{(2-l)(h_{0}-\varepsilon )} \biggr) - \frac{1}{n} \biggl( \frac{M}{2}- \frac{\xi _{2}h(0)(C_{s} k +C_{p})}{4\delta _{2}} \biggr) < 0, \end{aligned}$$
(3.33)

respectively. Combining (3.22), (3.27), (3.28), (3.30)–(3.32), and (3.33), we deduce that

$$\begin{aligned} {\mathcal {F}}' (t) \leq -C_{4}{\mathcal {E}}(t) -\xi _{3}\eta (t) \Xi (t), \quad t\geq t_{0}, \end{aligned}$$

for some positive constant \(C_{4}\). Using the fact that \(\eta (t)\) is decreasing and Lemma 3.1, we find that

$$\begin{aligned} {\mathcal {F}}' (t) \leq& -C_{4} \frac{\eta (t)}{\eta (t_{0})}{\mathcal {E}}(t) -\xi _{3}\eta (t) \Xi (t)\leq -C_{5} \eta (t) \bigl( {\mathcal {E}}(t) + \Xi (t) \bigr) \\ \leq& -\omega \eta (t){\mathcal {F}}(t) , \quad t\geq t_{0}, \end{aligned}$$
(3.34)

where \(C_{5}=\min \{ \frac{C_{4}}{\eta (t_{0})} , \xi _{3} \}\) and \(\omega =\frac{C_{5}}{\alpha _{1}}\). From (2.8), (3.6), and (3.34), we conclude that

$$\begin{aligned} \alpha _{0} \bigl( {\mathcal {E}}(t) +\Xi (t) \bigr) \leq& {\mathcal {F}}(t) \leq {\mathcal {F}}(t_{0}) e^{-\omega \int _{t_{0}}^{t} \eta (s)\,ds } ={ \mathcal {F}}(t_{0})e^{-\omega \int _{t_{0}}^{t} \frac{\gamma '(s)}{\gamma (s)}\,ds } \\ =& {\mathcal {F}}(t_{0})\gamma (t_{0})^{ \omega}\gamma (t)^{-\omega} , \quad t\geq t_{0}. \end{aligned}$$

By the fact \(\Xi (t)\geq 0\) and (3.4), we infer that

$$ E(t) \leq \frac{C}{\gamma (t)^{\omega}} , \quad t \geq t_{0} , $$

where \(C= \frac{{\mathcal {F}}(t_{0})\gamma (t_{0})^{\omega}}{\alpha _{0} (1-l)} \). □

Remark

We give some examples to illustrate the decay of energy given by Theorem 3.1 (see [1, 11]).

(1) \(\gamma (t) =e^{\alpha t}\), \(\alpha >0\), gives \(\eta (t)=\alpha \) and \(E(t) \leq \frac{C}{e^{\omega \alpha t}}\) for some positive constants C and ω.

(2) \(\gamma (t) =(1+t)^{\alpha}\), \(\alpha >0\), leads to \(\eta (t)=\alpha (1+t)^{-1}\) and \(E(t) \leq \frac{C}{(1+t)^{\omega}}\) for some positive constants C and ω.

4 Conclusions

In this paper, we study the von Karman plate model with long range memory. Our result is obtained without imposing the usual relation between the relaxation function h and its derivative. Assume that (H1)–(H3), \(E(0)<\frac{l}{\sqrt{C_{*}C_{p}}}\), and \(\hat{h}({F}_{h})<\frac{1}{8}\) hold. If \(h_{0}> \frac{3l}{8-l}\) and \(G_{\gamma}(0)< \frac{(8-l)h_{0} -3l}{16}\), then there exist positive constants \(t_{0}\), ω, and C such that

$$ E(t) \leq \frac{C}{ \gamma (t)^{\omega}} \quad \text{for } t \geq t_{0} . $$