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Asymptotic behavior for a viscoelastic Kirchhoff equation with distributed delay and Balakrishnan–Taylor damping

Abstract

A nonlinear viscoelastic Kirchhoff-type equation with Balakrishnan–Taylor damping and distributed delay is studied. By the energy method we establish the general decay rate under suitable hypothesis.

Introduction

Let \(\mathcal{H}=\Omega \times (\tau _{1}, \tau _{2})\times (0, \infty )\), in the present work, we consider the following Kirchhoff equation:

$$\begin{aligned} \textstyle\begin{cases} \vert u_{t} \vert ^{p} u_{tt}- (\zeta _{0}+\zeta _{1} \Vert \nabla u \Vert ^{2}_{2}+\sigma (\nabla u,\nabla u_{t})_{L^{2}(\Omega )} ) \Delta u(t)-\Delta u_{tt}(t) \\ \quad {}+\alpha (t) \int _{0}^{t}h(t-\varrho )\Delta u(\varrho )\,d \varrho +\beta _{1} \vert u_{t}(t) \vert ^{m-2} u_{t}(t) \\ \quad {}+ \int _{\tau _{1}}^{\tau _{2}} \vert \beta _{2}(s) \vert \vert u_{t}(t-s) \vert ^{m-2} u_{t}(t-s)\,ds=0, \\ u( x,0) =u_{0}( x), \qquad u_{t}( x,0) =u_{1}( x), \quad \text{in } \Omega, \\ u_{t}( x,-t) =f_{0}( x,t), \quad \text{in } \Omega \times (0, \tau _{2}), \\ u( x,t) =0, \quad \text{in } \partial \Omega \times (0, \infty ), \end{cases}\displaystyle \end{aligned}$$
(1.1)

where \(\Omega \in \mathbb{R}^{N}\) is a bounded domain with sufficiently smooth boundary Ω. \(\zeta _{0}\), \(\zeta _{1}\), σ, \(\beta _{1}\) are positive constants, \(p\geq 0\) for \(N=1,2\), and \(0\leq p\leq \frac{4}{N-2}\) for \(N\geq 3\), and \(m\geq 1\) for \(N=1,2\), and \(1< m\leq \frac{N+2}{N-2}\) for \(N\geq 3\). \(\tau _{1}<\tau _{2}\) are nonnegative constants such that \(\beta _{2} : [\tau _{1}, \tau _{2}] \rightarrow \mathbb{R}\) represents distributive time delay, h, α are positive functions.

Physically, the relationship between the stress and strain history in the beam inspired by Boltzmann theory is called viscoelastic damping term, where the kernel of the term of memory is the function h. See [46, 911, 1318, 22, 29, 31, 32, 34, 35]. It has been studied by many authors, especially in Kirchhoff’s equations (see [8, 10, 1921, 2326, 30, 33]).

In [2], Balakrishnan and Taylor proposed a new model of damping called the Balakrishnan–Taylor damping, as it relates to the span problem and the plate equation. For more depth, here are some papers that focused on the study of this damping: [2, 3, 8, 10, 16, 18, 20, 27, 35].

The effect of the delay often appears in many applications and practical problems and turns a lot of systems into different problems worth studying. Recently, the stability and the asymptotic behavior of evolution systems with time delay, especially the distributed delay effect, have been studied by many authors. See [7, 1012, 14, 28].

Based on all of the above, we believe that the combination of these terms of damping (memory term, Balakrishnan–Taylor damping, and the distributed delay) in one particular problem with the addition of \(\alpha (t)\) to the term of memory and the distributed delay term (\(\int _{\tau _{1}}^{\tau _{2}}\vert \beta _{2}(s)\vert \vert u_{t}(t-s) \vert ^{m-2} u_{t}(t-s)\,ds\)) constitutes a new problem worthy of study and research, different from the above that we will try to shed light on.

Our paper is divided into several sections: in the next section we lay down the hypotheses, concepts, and lemmas we need, and in the last section we prove our main result.

Preliminaries

For studying our problem, in this section we will need some materials.

Firstly, we introduce the following hypotheses for \(\beta _{2}\), h, and α:

  1. (A1)

    \(h,\alpha :\mathbb{R}_{+}\rightarrow \mathbb{R}_{+}\) are nonincreasing \(C^{1}\) functions satisfying

    $$ \begin{gathered} h(t)>0, \qquad \alpha (t)>0, \qquad l_{0}= \int _{0}^{\infty }h(\varrho ) \,d\varrho < \infty , \\ \zeta _{0}-2\alpha (t) \int _{0}^{t }h( \varrho ) \,d\varrho \geq l>0, \end{gathered} $$
    (2.1)

    where \(l=1-l_{0}\).

  2. (A2)

    \(\exists \vartheta : \mathbb{R}_{+}\rightarrow \mathbb{R}_{+}\) is a nonincreasing \(C^{1}\) function satisfying

    $$ \vartheta ( t) h ( t )+h^{\prime }( t)\leq 0, \quad t\geq 0 \quad \text{and} \quad \lim_{t\rightarrow \infty } \frac{-\alpha '(t)}{\vartheta (t)\alpha (t)}=0. $$
    (2.2)
  3. (A3)

    \(\beta _{2}:[\tau _{1}, \tau _{2}]\rightarrow \mathbb{R}\) is a bounded function satisfying

    $$\begin{aligned} \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \beta _{2}(s) \bigr\vert \,ds< \beta _{1}. \end{aligned}$$
    (2.3)

Let us introduce

$$ (h\circ \psi ) (t):= \int _{\Omega } \int _{0}^{t}h(t-\varrho ) \bigl\vert \psi (t)- \psi (\varrho ) \bigr\vert ^{2}\,d\varrho \,dx $$

and

$$ M(t):= \bigl(\zeta _{0}+\zeta _{1} \Vert \nabla u \Vert ^{2}_{2}+\sigma \bigl( \nabla u(t),\nabla u_{t}(t) \bigr)_{L^{2}(\Omega )} \bigr). $$

Lemma 2.1

(Sobolev–Poincare inequality [1])

Let \(2\leq q<\infty\) (\(n=1,2\)) or \(2\leq q<\frac{2n}{n-2}\) (\(n\geq 3\)). Then \(\exists c_{*}=c(\Omega ,q)>0\) such that

$$ \Vert u \Vert _{q}\leq c_{*} \Vert \nabla u \Vert _{2}, \quad \forall u\in H^{1}_{0}(\Omega ). $$

As in [28], we take the following new variables:

$$ y(x, \rho , s, t)=u_{t}(x, t-s\rho ) $$

which satisfy

$$ \textstyle\begin{cases} s y_{t}(x, \rho , s, t)+y_{\rho }(x, \rho , s, t)=0, \\ y(x, 0, s, t)=u_{t}(x, t). \end{cases} $$
(2.4)

So, problem (1.1) can be written as

$$ \textstyle\begin{cases} \vert u_{t} \vert ^{p} u_{tt}- (\zeta _{0}+\zeta _{1} \Vert \nabla u \Vert ^{2}_{2}+\sigma (\nabla u,\nabla u_{t})_{L^{2}(\Omega )} ) \Delta u(t)+\alpha (t) \int _{0}^{t}h(t-\varrho )\Delta u( \varrho )\,d\varrho \\ \quad {}-\Delta u_{tt}(t)+\beta _{1} \vert u_{t}(t) \vert ^{m-2} u_{t}(t)+ \int _{\tau _{1}}^{\tau _{2}} \vert \beta _{2}(s) \vert \vert y(x,1,s,t) \vert ^{m-2} y(x,1,s,t)\,ds=0, \\ s y_{t}(x, \rho , s, t)+y_{\rho }(x, \rho , s, t)=0, \\ u( x,0) =u_{0}( x), \qquad u_{t}( x,0) =u_{1}( x), \quad \text{in } \Omega, \\ y(x,\rho ,s,0)=f_{0}(x,\rho s), \quad \text{in } \Omega \times (0,1)\times (0, \tau _{2}), \\ u( x,t) =0, \quad \text{in } \partial \Omega \times (0, \infty ), \end{cases} $$
(2.5)

where

$$ (x, \rho , s, t)\in \Omega \times (0,1)\times (\tau _{1},\tau _{2}) \times (0,\infty ). $$

Now, we give the energy functional.

Lemma 2.2

The energy functional E, defined by

$$\begin{aligned} E( t) =&\frac{1}{p+2} \Vert u_{t} \Vert _{p+2}^{p+2}+ \frac{1}{2} \biggl( \zeta _{0}-\alpha (t) \int _{0}^{t}h(\varrho )\,d\varrho \biggr) \bigl\Vert \nabla u(t) \bigr\Vert _{2}^{2} \\ &{}+\frac{1}{2} \bigl\Vert \nabla u_{t}(t) \bigr\Vert _{2}^{2}+ \frac{\zeta _{1}}{4} \bigl\Vert \nabla u(t) \bigr\Vert _{2}^{4}+ \frac{\alpha (t)}{2}(h\circ \nabla u) (t) \\ &{}+\frac{m-1}{m} \int _{0}^{1} \int _{ \tau _{1}}^{\tau _{2}}s \bigl\vert \beta _{2}(s) \bigr\vert \bigl\Vert y(x,\rho ,s,t) \bigr\Vert ^{m}_{m} \,ds \,d\rho , \end{aligned}$$
(2.6)

satisfies

$$\begin{aligned} E^{\prime } ( t ) \leq &-\eta _{0} \bigl\Vert u_{t}(t) \bigr\Vert ^{m}_{m}+ \frac{\alpha (t)}{2}\bigl(h^{\prime }\circ \nabla u\bigr) (t) \\ &{}-\frac{\alpha '(t)}{2} \biggl( \int _{0}^{t}h(\varrho )\,d\varrho \biggr) \bigl\Vert \nabla u(t) \bigr\Vert ^{2}_{2}- \frac{\sigma }{4} \biggl(\frac{d}{dt} \bigl\{ \bigl\Vert \nabla u(t) \bigr\Vert ^{2}_{2} \bigr\} \biggr)^{2}, \end{aligned}$$
(2.7)

where \(\eta _{0}=\beta _{1}-\int _{\tau _{1}}^{\tau _{2}}\vert \beta _{2}(s) \vert \,ds>0\).

Proof

Taking the inner product of (2.5)1 with \(u_{t}\), then integrating over Ω, we find

$$\begin{aligned} &\bigl( \vert u_{t} \vert ^{p} u_{tt}(t),u_{t}(t) \bigr)_{L^{2}(\Omega )}-\bigl(M(t) \Delta u(t),u_{t}(t) \bigr)_{L^{2}(\Omega )}-\bigl(\Delta u_{tt}(t),u_{t}(t) \bigr)_{L^{2}( \Omega )} \\ &\quad {}+\biggl(\alpha (t) \int _{0}^{t}h(t-\varrho )\Delta u(\varrho )\,d \varrho ,u_{t}(t)\biggr)_{L^{2}( \Omega )}+\beta _{1}\bigl( \vert u_{t} \vert ^{m-2}u_{t},u_{t} \bigr)_{L^{2}( \Omega )} \\ &\quad {}+ \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \beta _{2}(s) \bigr\vert \bigl( \bigl\vert y(x,1,s,t) \bigr\vert ^{m-2} y(x,1,s,t),u_{t}(t)\bigr)_{L^{2}(\Omega )}\,ds=0. \end{aligned}$$
(2.8)

By computation, integration by parts, and the last condition in (2.5), we get

$$\begin{aligned}& \begin{aligned}[b] \bigl( \vert u_{t} \vert ^{p} u_{tt}(t),u_{t}(t) \bigr)_{L^{2}(\Omega )}&= \int _{ \Omega }u_{t}(t). \vert u_{t} \vert ^{p} u_{tt}(t)\,dx \\ &=\frac{1}{p+2}\frac{d}{dt} \bigl( \bigl\Vert u_{t}(t) \bigr\Vert ^{p+2}_{p+2} \bigr),\end{aligned} \end{aligned}$$
(2.9)
$$\begin{aligned}& \begin{aligned}[b] -\bigl(\Delta u_{tt}(t),u_{t}(t)\bigr)_{L^{2}(\Omega )}&= \int _{\Omega }\nabla u_{t}(t) \nabla u_{tt}(t) \,dx \\ &=\frac{1}{2}\frac{d}{dt} \bigl( \bigl\Vert \nabla u_{t}(t) \bigr\Vert ^{2}_{2} \bigr). \end{aligned} \end{aligned}$$
(2.10)

By integration by parts, we find

$$\begin{aligned} &{-}\bigl(M(t)\Delta u(t),u_{t}(t)\bigr)_{L^{2}(\Omega )} \\ &\quad =-\bigl( \bigl(\zeta _{0}+\zeta _{1} \Vert \nabla u \Vert ^{2}_{2}+\sigma \bigl( \nabla u(t),\nabla u_{t}(t)\bigr)_{L^{2}(\Omega )} \bigr)\Delta u(t),u_{t}(t) \bigr)_{L^{2}( \Omega )} \\ &\quad = \bigl(\zeta _{0}+\zeta _{1} \Vert \nabla u \Vert ^{2}_{2}+\sigma \bigl( \nabla u(t),\nabla u_{t}(t)\bigr)_{L^{2}(\Omega )} \bigr) \int _{\Omega } \nabla u(t).\nabla u_{t}(t)\,dx \\ &\quad = \bigl(\zeta _{0}+\zeta _{1} \Vert \nabla u \Vert ^{2}_{2}+\sigma \bigl( \nabla u(t),\nabla u_{t}(t)\bigr)_{L^{2}(\Omega )} \bigr)\frac{d}{dt} \biggl\{ \int _{\Omega } \bigl\vert \nabla u(t) \bigr\vert ^{2} \,dx \biggr\} \\ &\quad =\frac{d}{dt} \biggl\{ \frac{1}{2} \biggl(\zeta _{0}+ \frac{\zeta _{1}}{2} \Vert \nabla u \Vert ^{2}_{2} \biggr) \bigl\Vert \nabla u(t) \bigr\Vert ^{2}_{2} \biggr\} +\frac{\sigma }{4}\frac{d}{dt} \bigl\{ \bigl\Vert \nabla u(t) \bigr\Vert ^{2}_{2} \bigr\} ^{2}, \end{aligned}$$
(2.11)

and we have

$$\begin{aligned} &\biggl( \int _{0}^{t}h(t-\varrho )\Delta u(\varrho )\,d \varrho ,u_{t}(t)\biggr)_{L^{2}( \Omega )} \\ &\quad = \int _{0}^{t}h(t-\varrho ) \bigl(\Delta u(\varrho ),u_{t}(t)\bigr)_{L^{2}( \Omega )}\,d\varrho \\ &\quad =- \int _{0}^{t}h(t-\varrho ) \biggl[ \int _{\Omega }\nabla u(x,\varrho ) \nabla u(x,t)\,dx \biggr]\,d \varrho , \end{aligned}$$
(2.12)

and

$$ -\nabla u(x,\varrho ).\nabla u(x,t)=\frac{1}{2}\frac{d}{dt} \bigl\{ \bigl\vert \nabla u(x,\varrho )-\nabla u(x,t) (t) \bigr\vert ^{2} \bigr\} - \frac{1}{2}\frac{d}{dt} \bigl\{ \bigl\vert \nabla u(x,t) \bigr\vert ^{2} \bigr\} , $$
(2.13)

then

$$\begin{aligned} &{-} \int _{0}^{t}h(t-\varrho ) \bigl(\nabla u(\varrho ), \nabla u_{t}(t)\bigr)_{L^{2}( \Omega )}\,d\varrho \\ &\quad =- \int _{0}^{t}h(t-\varrho ) \int _{\Omega } \biggl[\frac{1}{2} \frac{d}{dt} \bigl\{ \bigl\vert \nabla u(x,\varrho )-\nabla u(x,t) \bigr\vert ^{2} \bigr\} \biggr]\,dx \,ds. \\ &\qquad {}- \int _{0}^{t}h(t-\varrho ) \int _{\Omega } \biggl[\frac{1}{2} \frac{d}{dt} \bigl\{ \bigl\vert \nabla u(x,t) \bigr\vert ^{2} \bigr\} \biggr]dxd \varrho \\ &\quad =\frac{1}{2} \int _{0}^{t}h(t-\varrho ) \biggl[\frac{d}{dt} \biggl\{ \int _{\Omega } \bigl\vert \nabla u(x,t)-\nabla u(x,\varrho ) \bigr\vert ^{2}\,dx \biggr\} \biggr]\,d\varrho \\ &\qquad {}-\frac{1}{2} \int _{0}^{t}h(t-\varrho ) \biggl[\frac{d}{dt} \bigl\{ \bigl\Vert \nabla u(x,t) \bigr\Vert _{2}^{2} \bigr\} \biggr]\,dx \,d\varrho . \end{aligned}$$
(2.14)

By (2.1), we get

$$\begin{aligned} &\frac{1}{2} \int _{0}^{t}h(t-\varrho ) \biggl[\frac{d}{dt} \biggl\{ \int _{ \Omega } \bigl\vert \nabla u(x,t)-\nabla u(x,\varrho ) \bigr\vert ^{2}\,dx \biggr\} \biggr]\,d\varrho \\ &\quad =\frac{1}{2}\frac{d}{dt} \biggl\{ \int _{0}^{t}h(t-\varrho ) \biggl[ \int _{\Omega } \bigl\vert \nabla u(x,t)-\nabla u(x,\varrho ) \bigr\vert ^{2}\,dx \biggr] \biggr\} \,d\varrho \\ &\qquad {}-\frac{1}{2} \int _{0}^{t}h'(t-\varrho ) \biggl[ \int _{\Omega } \bigl\vert \nabla u(x,t)-\nabla u(x,\varrho ) \bigr\vert ^{2}\,dx \biggr]\,d\varrho \\ &\quad =\frac{1}{2}\frac{d}{dt}(h\circ \nabla u) (t)- \frac{1}{2}\bigl(h'\circ \nabla u\bigr) (t), \end{aligned}$$
(2.15)

and

$$\begin{aligned} &{-}\frac{1}{2} \int _{0}^{t}h(t-\varrho ) \biggl[\frac{d}{dt} \bigl\{ \bigl\Vert \nabla u(t) \bigr\Vert _{2}^{2} \bigr\} \biggr]\,dx \,d\varrho \\ &\quad =-\frac{1}{2} \biggl( \int _{0}^{t}h(t-\varrho )\,d\varrho \biggr) \biggl( \frac{d}{dt} \bigl\{ \bigl\Vert \nabla u(t) \bigr\Vert _{2}^{2} \bigr\} \biggr)\,dx \\ &\quad =-\frac{1}{2} \biggl( \int _{0}^{t}h(\varrho )\,d\varrho \biggr) \biggl( \frac{d}{dt} \bigl\{ \bigl\Vert \nabla u(t) \bigr\Vert _{2}^{2} \bigr\} \biggr)\,dx \\ &\quad =-\frac{1}{2}\frac{d}{dt} \biggl\{ \biggl( \int _{0}^{t}h(\varrho )\,d \varrho \biggr) \bigl\Vert \nabla u(t) \bigr\Vert _{2}^{2} \biggr\} + \frac{1}{2}h(t) \bigl\Vert \nabla u(t) \bigr\Vert _{2}^{2}. \end{aligned}$$
(2.16)

Inserting (2.15) and (2.16) into (2.14) gives

$$\begin{aligned} & \biggl(\alpha (t) \int _{0}^{t}h(t-\varrho )\Delta u(\varrho )\,d \varrho , u_{t}(t) \biggr)_{L^{2}(\Omega )} \\ &\quad =\frac{d}{dt} \biggl\{ \frac{\alpha (t)}{2}(h\circ \nabla u) (t) - \frac{\alpha (t)}{2} \biggl( \int _{0}^{t}h(\varrho )\,d\varrho \biggr) \bigl\Vert \nabla u(t) \bigr\Vert _{2}^{2} \biggr\} \\ &\qquad {}-\frac{\alpha (t)}{2}\bigl(h'\circ \nabla u\bigr) (t)+ \frac{\alpha (t)}{2}h(t) \bigl\Vert \nabla u(t) \bigr\Vert _{2}^{2} \\ &\qquad {}-\frac{\alpha '(t)}{2}(h\circ \nabla u) (t)+\frac{\alpha '(t)}{2} \biggl( \int _{0}^{t}h(\varrho )\,d\varrho \biggr) \bigl\Vert \nabla u(t) \bigr\Vert _{2}^{2}. \end{aligned}$$
(2.17)

Now, multiplying equation (2.5)2 by \(- y\vert \beta _{2}(s)\vert \), integrating over \(\Omega \times (0, 1)\times (\tau _{1}, \tau _{2})\), and using (2.4)2, we get

$$\begin{aligned} &\frac{d}{dt }\frac{m-1}{m} \int _{\Omega } \int _{0}^{1} \int _{\tau _{1}}^{ \tau _{2}}s \bigl\vert \beta _{2}(s) \bigr\vert . \bigl\vert y(x,\rho ,s,t) \bigr\vert ^{m}\,ds \,d \rho \,dx \\ &\quad =-(m-1) \int _{\Omega } \int _{0}^{1} \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \beta _{2}(s) \bigr\vert . \vert y \vert ^{m-1} y_{\rho }\,ds \,d\rho \,dx \\ &\quad =-\frac{m-1}{m} \int _{\Omega } \int _{0}^{1} \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \beta _{2}( \varrho ) \bigr\vert \frac{d}{d\rho } \bigl\vert y(x,\rho ,s,t) \bigr\vert ^{m}\,ds \,d\rho \,dx \\ &\quad =\frac{m-1}{m} \int _{\Omega } \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \beta _{2}(s) \bigr\vert \bigl( \bigl\vert y( x, 0 , s, t) \bigr\vert ^{m} - \bigl\vert y(x, 1, s, t) \bigr\vert ^{m} \bigr)\,ds \,dx \\ &\quad =\frac{m-1}{m} \biggl( \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \beta _{2}(s) \bigr\vert \,ds \biggr) \int _{\Omega } \bigl\vert u_{t}(t) \bigr\vert ^{m}\,dx \\ &\qquad {}-\frac{m-1}{m} \int _{\Omega } \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \beta _{2}(s) \bigr\vert . \bigl\vert y ( x, 1, s, t ) \bigr\vert ^{m} \,ds \,dx \\ &\quad =\frac{m-1}{m} \biggl( \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \beta _{2}(s) \bigr\vert \,ds \biggr) \bigl\Vert u_{t}(t) \bigr\Vert _{m}^{m} \\ &\qquad {}-\frac{m-1}{m} \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \beta _{2}(s) \bigr\vert \bigl\Vert y ( x, 1, s, t ) \bigr\Vert _{m}^{m} \,ds . \end{aligned}$$
(2.18)

By Young’s inequality, we have

$$\begin{aligned} & \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \beta _{2}(s) \bigr\vert \bigl( \bigl\vert y(x,1,s,t) \bigr\vert ^{m-2}y(x,1,s,t),u_{t}(t) \bigr)_{L^{2}(\Omega )}\,ds \\ &\quad \leq \frac{1}{m} \biggl( \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \beta _{2}(s) \bigr\vert \,ds \biggr) \bigl\Vert u_{t}(t) \bigr\Vert _{m}^{m}+\frac{m-1}{m} \int _{\tau _{1}}^{ \tau _{2}} \bigl\vert \beta _{2}(s) \bigr\vert \bigl\Vert y ( x, 1, s, t ) \bigr\Vert _{m}^{m} \,ds. \end{aligned}$$
(2.19)

By inserting (2.9)–(2.11) and (2.17)–(2.19) into (2.8), we find (2.6) and (2.7).

Hence, by (2.2), we get the function E is nonincreasing \(\forall t\geq t_{1}\). This completes of the proof. □

Now we state the local existence of problem (2.5), whose proof can be found in [23, 24].

Theorem 2.3

Suppose that (2.1)(2.3) are satisfied. Then, for any \(u_{0},u_{1}\in H^{1}_{0}(\Omega )\cap L^{2}(\Omega )\), and \(f_{0}\in L^{2}(\Omega ,(0,1),(\tau _{1},\tau _{2}))\), there exists a weak solution u of problem (2.5) such that

$$\begin{aligned} &u\in C\bigl(]0,T[,H^{1}_{0}(\Omega )\bigr)\cap C^{1}\bigl(]0,T[,L^{2}(\Omega )\bigr), \\ &u_{t}\in C\bigl(]0,T[,H^{1}_{0}(\Omega )\bigr)\cap L^{2}\bigl(]0,T[,L^{2}\bigl(\Omega ,(0,1),( \tau _{1},\tau _{2})\bigr)\bigr). \end{aligned}$$

General decay

In this section, we state and prove the asymptotic behavior of system (2.5). For this goal, we set

$$\begin{aligned} \Psi (t) :=&\frac{1}{p+1} \int _{\Omega } u(t) \vert u _{t} \vert ^{p} u _{t}(t) \,dx+\frac{\sigma }{4} \bigl\Vert \nabla u(t) \bigr\Vert _{2}^{4}+ \int _{\Omega } \nabla u(t)\nabla u_{t}(t)\,dx \end{aligned}$$
(3.1)

and

$$\begin{aligned} \Phi (t) :=& \int _{\Omega } \biggl(\Delta u_{t}-\frac{1}{p+1} \vert u _{t} \vert ^{p}u_{t} \biggr) \int _{0}^{t}h(t-\varrho ) \bigl(u(t)-u(\varrho ) \bigr)\,d \varrho \,dx, \end{aligned}$$
(3.2)

and

$$\begin{aligned} \Theta (t) :=& \int _{0}^{1} \int _{\tau _{1}}^{\tau _{2}}s e^{-\rho s} \bigl\vert \beta _{2}(s) \bigr\vert . \bigl\Vert y(x,\rho ,s,t) \bigr\Vert _{m}^{m} \,ds \,d \rho . \end{aligned}$$
(3.3)

First, since the function h is positive and continuous, for all \(t_{0}>0\), we have

$$\begin{aligned} \int _{0}^{t}h(\varrho )\,d\varrho \geq \int _{0}^{t_{0}}h(\varrho )\,d \varrho :=h_{0}, \quad \forall t\geq t_{0}. \end{aligned}$$

Lemma 3.1

The functional \(\Psi (t)\) defined in (3.1) satisfies, for any \(\varepsilon >0\),

$$\begin{aligned} \Psi '(t) \leq &\frac{1}{p+1} \Vert u_{t} \Vert _{p+2}^{p+2}-\bigl(l- \varepsilon (c_{1}+c_{2}) \bigr) \Vert \nabla u \Vert _{2}^{2}-\zeta _{1} \Vert \nabla u \Vert _{2}^{4}+\frac{\alpha (t)}{4}(h\circ \nabla u) (t) \\ &{}+ \Vert \nabla u_{t} \Vert _{2}^{2}+c( \varepsilon ) \biggl( \Vert u_{t} \Vert _{m}^{m}+ \int _{\tau _{1}}^{\tau _{2}}\bigl\vert \beta _{2}(s)\bigr| \bigl\Vert y(x,1,s,t) \bigr\Vert _{m}^{m}\,ds \biggr). \end{aligned}$$
(3.4)

Proof

A differentiation of (3.1) and using (2.5)1 give

$$\begin{aligned} \Psi '(t) =&\frac{1}{p+1} \Vert u_{t} \Vert _{p+2}^{p+2}+ \int _{\Omega } \vert u_{t} \vert ^{p}u_{tt}u \,dx+\sigma \Vert \nabla u \Vert _{2}^{2} \int _{\Omega }\nabla u_{t}\nabla u\,dx \\ &{}+ \int _{\Omega }\nabla u(t)\nabla u_{tt}(t)\,dx+ \Vert \nabla u_{t} \Vert _{2}^{2} \\ =&\frac{1}{p+1} \Vert u_{t} \Vert _{p+2}^{p+2}- \zeta _{0} \Vert \nabla u \Vert _{2}^{2}-\zeta _{1} \Vert \nabla u \Vert _{2}^{4} \underbrace{-\beta _{1} \int _{\Omega } \vert u_{t} \vert ^{m-2} u_{t}u\,dx}_{J_{1}} \\ &{}+ \underbrace{\alpha (t) \int _{\Omega }\nabla u(t) \int _{0}^{t}h(t-\varrho )\nabla u(\varrho )\,d \varrho \,dx}_{J_{2}}+ \Vert \nabla u_{t} \Vert _{2}^{2} \\ &{}- \underbrace{ \int _{\Omega } \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \beta _{2}(s) \bigr\vert \bigl\vert y(x,1,s,t) \bigr\vert ^{m-2} y(x,1,s,t).u \,ds \,dx}_{J_{3}}. \end{aligned}$$
(3.5)

We estimate the last three terms of the RHS of (3.5). Applying Hölder’s, Sobolev–Poincare, and Young’s inequalities, (2.1) and (2.6), we find

$$\begin{aligned} J_{1} \leq &\varepsilon \beta _{1}^{m} \Vert u \Vert _{m}^{m}+c( \varepsilon ) \Vert u_{t} \Vert _{m}^{m} \\ \leq &\varepsilon \beta _{1}^{m}c_{p}^{m} \Vert \nabla u \Vert _{2}^{m}+c( \varepsilon ) \Vert u_{t} \Vert _{m}^{m} \\ \leq &\varepsilon \beta _{1}^{m}c_{p}^{m} \biggl(\frac{E(0)}{l} \biggr)^{(m-2)/2} \Vert \nabla u \Vert _{2}^{2}+c(\varepsilon ) \Vert u_{t} \Vert _{m}^{m} \\ \leq &\varepsilon c_{1} \Vert \nabla u \Vert _{2}^{2}+c( \varepsilon ) \Vert u_{t} \Vert _{m}^{m} \end{aligned}$$
(3.6)

and

$$\begin{aligned} J_{2} \leq &2\alpha (t) \biggl( \int _{0}^{t}h(\varrho )\,d\varrho \biggr) \Vert \nabla u \Vert _{2}^{2}+\frac{\alpha (t)}{4}(h\circ \nabla u) (t) \\ \leq &(\zeta _{0}-l) \Vert \nabla u \Vert _{2}^{2}+ \frac{\alpha (t)}{4}(h\circ \nabla u) (t). \end{aligned}$$
(3.7)

Similar to \(J_{1}\), we have

$$\begin{aligned} J_{3} \leq &\varepsilon c_{2} \Vert \nabla u \Vert _{2}^{2}+c( \varepsilon ) \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \beta _{2}(s) \bigr\vert . \bigl\Vert y(x,1,s,t) \bigr\Vert _{m}^{m} \,ds. \end{aligned}$$
(3.8)

Combining (3.6)–(3.8) and (3.5), we get

$$\begin{aligned} \Psi '(t) \leq &\frac{1}{p+1} \Vert u_{t} \Vert _{p+2}^{p+2}-\bigl(l- \varepsilon (c_{1}+c_{2}) \bigr) \Vert \nabla u \Vert _{2}^{2}-\zeta _{1} \Vert \nabla u \Vert _{2}^{4}+ \Vert \nabla u_{t} \Vert _{2}^{2} \\ &{}+\frac{\alpha (t)}{4}(h\circ \nabla u) (t)+c(\varepsilon ) \biggl( \Vert u_{t} \Vert _{m}^{m}+ \int _{\tau _{1}}^{\tau _{2}}\bigl\vert \beta _{2}(s)\bigr| \bigl\Vert y(x,1,s,t) \bigr\Vert _{m}^{m}\,ds \biggr). \end{aligned}$$

 □

Lemma 3.2

The functional \(\Phi (t)\) defined in (3.2) satisfies, for any \(\delta >0\),

$$\begin{aligned} \Phi '(t) \leq &-\frac{1}{p+1} \biggl( \int _{0}^{t}h(\varrho )\,d\varrho \biggr) \Vert u_{t} \Vert _{p+2}^{p+2}+\delta \bigl(\zeta _{0}+2h_{0}^{2} \alpha (t) \bigr) \Vert \nabla u \Vert _{2}^{2} \\ &{}+\zeta _{1}\delta \Vert \nabla u \Vert _{2}^{4}+ \delta \frac{\sigma E(0)}{l} \biggl(\frac{1}{2}\frac{d}{dt} \Vert \nabla u \Vert _{2}^{2} \biggr)^{2} \\ &{}+ \biggl(c(\delta )+\biggl(2\delta +\frac{1}{4\delta }\biggr)c\alpha (t) \biggr) (h \circ \nabla u) (t) \\ &{}+c(\delta ) \biggl( \Vert u_{t} \Vert _{m}^{m}+ \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \beta _{2}(\varrho )\bigr| \bigl\Vert y(x,1,s,t) \bigr\Vert _{m}^{m}\,ds \biggr) \\ &{}+ \biggl(\delta _{1}\bigl(1+c\bigl(E(0)\bigr)^{p}\bigr)- \int _{0}^{t}h(\varrho )\,d\varrho \biggr) \Vert \nabla u_{t} \Vert _{2}^{2} \\ &{}- \biggl(\frac{h(0)c_{p}^{2}}{4\delta _{1}}+c(\delta _{1}) \biggr) \bigl(h' \circ \nabla u\bigr) (t). \end{aligned}$$
(3.9)

Proof

A differentiation of (3.2) and using (2.5)1 give

$$\begin{aligned} \Phi '(t) =& \int _{\Omega } \bigl(\Delta u_{tt}-u_{tt} \vert u _{t} \vert ^{p} \bigr) \int _{0}^{t}h(t-\varrho ) \bigl(u(t)-u(\varrho ) \bigr)\,d\varrho \,dx \\ &{}+ \int _{\Omega } \biggl(\Delta u_{t}-\frac{1}{p+1} \vert u _{t} \vert ^{p} u_{t} \biggr) \int _{0}^{t}h'(t-\varrho ) \bigl(u(t)-u(\varrho )\bigr)\,d\varrho \,dx \\ &{}-\frac{1}{p+1} \biggl( \int _{0}^{t}h(\varrho )\,d\varrho \biggr) \Vert u_{t} \Vert _{p+2}^{p+2}- \biggl( \int _{0}^{t}h(\varrho )\,d\varrho \biggr) \Vert \nabla u_{t} \Vert _{2}^{2} \\ =&\underbrace{\bigl(\zeta _{0}+\zeta _{1} \Vert \nabla u \Vert _{2}^{2}\bigr) \int _{\Omega }\nabla u \int _{0}^{t}h(t-\varrho ) \bigl(\nabla u(t)-\nabla u(\varrho )\bigr)\,d\varrho \,dx}_{J_{1}} \\ &\underbrace{{}+\sigma \int _{\Omega }\nabla u\nabla u_{t}\,dx. \int _{\Omega }\nabla u \int _{0}^{t}h(t-\varrho ) \bigl(\nabla u(t)-\nabla u(\varrho )\bigr)\,d\varrho \,dx}_{J_{2}} \\ &\underbrace{{}-\alpha (t) \int _{\Omega } \biggl( \int _{0}^{t}h(t-\varrho )\nabla u(\varrho )\,d \varrho \biggr). \biggl( \int _{0}^{t}h(t-\varrho ) \bigl(\nabla u(t)-\nabla u(\varrho )\bigr)\,d\varrho \biggr)\,dx}_{J_{3}} \\ &\underbrace{-\beta _{1} \int _{\Omega } \vert u_{t} \vert ^{m-2} u_{t} \biggl( \int _{0}^{t}h(t-\varrho ) \bigl(u(t)-u(\varrho ) \bigr)\,d\varrho \biggr) \,dx}_{J_{4}} \\ &{}- \int _{\Omega } \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \beta _{2}(s) \bigr\vert \bigl\vert y(x,1,s,t) \bigr\vert ^{m-2} y(x,1,s,t) \\ &\underbrace{{}\times \biggl( \int _{0}^{t}h(t-\varrho ) \bigl( u(t)- u(\varrho ) \bigr)\,d\varrho \biggr) \,ds \,dx}_{J_{5}} \\ &\underbrace{{}-\frac{1}{p+1} \int _{\Omega } \vert u _{t} \vert ^{p} u_{t} \int _{0}^{t}h'(t-\varrho ) \bigl(u(t)-u(\varrho )\bigr)\,d\varrho \,dx}_{J_{6}} \\ &\underbrace{{}- \int _{\Omega }\nabla u _{t} \int _{0}^{t}h'(t-\varrho ) \bigl( \nabla u(t)-\nabla u(\varrho )\bigr)\,d\varrho \,dx}_{J_{7}} \\ &{}-\frac{1}{p+1} \biggl( \int _{0}^{t}h(\varrho )\,d\varrho \biggr) \Vert u_{t} \Vert _{p+2}^{p+2}- \biggl( \int _{0}^{t}h(\varrho )\,d\varrho \biggr) \Vert \nabla u_{t} \Vert _{2}^{2}. \end{aligned}$$
(3.10)

By estimating the terms \(J_{i}\), \(i=1,\ldots,7\), of the RHS of (3.10), exploiting Hölder’s, Sobolev–Poincare, and Young’s inequalities, (2.1) and (2.6), we find

$$\begin{aligned} \vert J_{1} \vert \leq &\bigl(\zeta _{0}+\zeta _{1} \Vert \nabla u \Vert _{2}^{2}\bigr) \biggl(\delta \Vert \nabla u \Vert _{2}^{2}+ \frac{c}{4\delta }(h\circ \nabla u) (t) \biggr) \\ \leq &\delta \zeta _{0} \Vert \nabla u \Vert _{2}^{2}+\delta \zeta _{1} \Vert \nabla u \Vert _{2}^{4}+ \biggl(\frac{\zeta _{0}c}{4\delta }+ \frac{\zeta _{1}cE(0)}{4l\delta } \biggr) (h\circ \nabla u) (t) \end{aligned}$$
(3.11)

and

$$\begin{aligned}& \begin{aligned}[b] J_{2}&\leq \delta \sigma \biggl( \int _{\Omega }\nabla u\nabla u_{t}\,dx \biggr)^{2} \Vert \nabla u \Vert _{2}^{2}+ \frac{\sigma c}{4\delta }(h \circ \nabla u) (t) \\ &\leq \delta \frac{\sigma E(0)}{l} \biggl(\frac{1}{2}\frac{d}{dt} \Vert \nabla u \Vert _{2}^{2} \biggr)^{2}+ \frac{\sigma c}{4\delta }(h \circ \nabla u) (t), \end{aligned} \end{aligned}$$
(3.12)
$$\begin{aligned}& \begin{aligned}[b] \vert J_{3} \vert &\leq \delta \alpha (t) \int _{\Omega } \biggl( \int _{0}^{t}h(t- \varrho ) \bigl( \bigl\vert \nabla u(t)-\nabla u(\varrho ) \bigr\vert -\nabla \bigl\vert u(t) \bigr\vert \bigr)\,d\varrho \biggr)^{2}\,dx \\ &\quad {}+\frac{1}{4\delta }\alpha (t) \int _{\Omega } \biggl( \int _{0}^{t}h(t- \varrho ) \bigl(\nabla u(t)-\nabla u(\varrho )\bigr)\,d\varrho \biggr)^{2}\,dx \\ &\leq 2\delta h_{0}^{2}\alpha (t) \Vert \nabla u \Vert _{2}^{2}+ \biggl(2 \delta +\frac{1}{4\delta } \biggr)c\alpha (t) (h\circ \nabla u) (t), \end{aligned} \end{aligned}$$
(3.13)
$$\begin{aligned}& \begin{aligned}[b] \vert J_{4} \vert &\leq c(\delta ) \Vert u_{t} \Vert _{m}^{m}+\delta \beta _{1}^{m} \int _{\Omega } \biggl( \int _{0}^{t}h(t-\varrho ) \bigl(u(t)- u( \varrho ) \bigr)\,d\varrho \biggr)^{m}\,dx \\ &\leq c(\delta ) \Vert u_{t} \Vert _{m}^{m}+ \delta \beta _{1}^{m}c_{p}^{m} \int _{0}^{t}h(t-\varrho ) \bigl\Vert \nabla u(t)- \nabla u(\varrho ) \bigr\Vert ^{m}_{2} \,d\varrho \\ &\leq c(\delta ) \Vert u_{t} \Vert _{m}^{m}+ \delta \biggl(\beta _{1}^{m}c_{p}^{m} \biggl( \frac{E(0)}{l}\biggr)^{(m-2)/2} \biggr) (h\circ \nabla u) (t) \\ &\leq c(\delta ) \Vert u_{t} \Vert _{m}^{m}+ \delta c_{3}(h\circ \nabla u) (t). \end{aligned} \end{aligned}$$
(3.14)

Similarly, we have

$$\begin{aligned} \vert J_{5} \vert \leq &c(\delta ) \bigl\Vert y(x,1,s,t) \bigr\Vert _{m}^{m}+ \delta c_{4}(h\circ \nabla u) (t). \end{aligned}$$
(3.15)

By exploiting the Sobolev embedding, we have

$$\begin{aligned} \vert J_{6} \vert \leq &\frac{1}{p+1} \biggl(\delta _{1} \Vert u_{t} \Vert _{2(p+1)}^{2(p+1)}+ \frac{c}{\delta _{1}} \int _{\Omega } \int _{0}^{t}\bigl(-h'(t- \varrho ) \bigr) \bigl\vert u(t)- u(\varrho ) \bigr\vert ^{2}\,d\varrho \,dx \biggr) \\ \leq &c\delta _{1}\bigl(E(0)\bigr)^{p} \Vert \nabla u_{t} \Vert _{2}^{2}-c( \delta _{1}) \bigl(h'\circ \nabla u\bigr) (t) \end{aligned}$$
(3.16)

and

$$\begin{aligned} \vert J_{7} \vert \leq &\delta _{1} \Vert \nabla u_{t} \Vert _{2}^{2}- \frac{h(0)}{4\delta _{1}} \bigl(h'\circ \nabla u\bigr) (t). \end{aligned}$$
(3.17)

According to (3.11)–(3.17) and (3.10), we get (3.9). □

Lemma 3.3

The functional \(\Theta (t)\) defined in (3.3) satisfies

$$\begin{aligned} \Theta '(t) \leq &-\eta _{1} \int _{0}^{1} \int _{\tau _{1}}^{\tau _{2}} s \bigl\vert \beta _{2}(s) \bigr\vert . \bigl\Vert y(x,\rho ,s,t) \bigr\Vert _{m}^{m} \,ds \,d \rho \\ &{}-\eta _{1} \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \beta _{2}(s) \bigr\vert . \bigl\Vert y ( x, 1, s, t ) \bigr\Vert _{m}^{m} \,ds +\beta _{1} \bigl\Vert u_{t}(t) \bigr\Vert _{m}^{m}. \end{aligned}$$
(3.18)

Proof

Differentiating \(\Theta (t)\) and using (2.5)2 give

$$\begin{aligned} \Theta '( t) =&-m \int _{\Omega } \int _{0}^{1} \int _{\tau _{1}}^{\tau _{2}} e^{-s\rho } \bigl\vert \beta _{2}(s) \bigr\vert . \vert y \vert ^{m-1} y_{\rho } ( x, \rho , s, t ) \,ds \,d\rho \,dx \\ =&- \int _{\Omega } \int _{0}^{1} \int _{\tau _{1}}^{\tau _{2}} s e^{-s \rho } \bigl\vert \beta _{2}(s) \bigr\vert . \bigl\vert y(x,\rho ,s,t) \bigr\vert ^{m} \,ds \,d \rho \,dx \\ &{}- \int _{\Omega } \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \beta _{2}(s) \bigr\vert \bigl[e^{-s} \bigl\vert y ( x, 1, s, t ) \bigr\vert ^{m}- \bigl\vert y ( x, 0, s, t ) \bigr\vert ^{m} \bigr] \,ds \,dx. \end{aligned}$$

Applying \(y(x, 0, s, t)=u_{t}(x, t)\) and \(e^{-s}\leq e^{-s\rho }\leq 1\) for any \(0<\rho <1\) and setting \(\eta _{1}=e^{-\tau _{2}}\), we obtain

$$\begin{aligned} \Theta '( t) \leq &-\eta _{1} \int _{\Omega } \int _{0}^{1} \int _{\tau _{1}}^{ \tau _{2}} s \bigl\vert \beta _{2}(s) \bigr\vert . \bigl\vert y(x,\rho ,s,t) \bigr\vert ^{m} \,ds \,d\rho \,dx \\ &{}-\eta _{1} \int _{\Omega } \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \beta _{2}(s) \bigr\vert \bigl\vert y(x,1,s,t) \bigr\vert ^{m} \,ds \,dx + \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \beta _{2}(s) \bigr\vert \,ds \int _{\Omega } \vert u_{t} \vert ^{m}(t) \,dx, \end{aligned}$$

using (2.3), we find (3.18). □

Now, we introduce the functional

$$\begin{aligned} \mathcal{G}(t) :=&E(t)+\varepsilon _{1}\alpha (t)\Psi (t)+ \varepsilon _{2}\alpha (t)\Phi (t)+\varepsilon _{3}\alpha (t)\Theta (t) \end{aligned}$$
(3.19)

for some positive constants \(\varepsilon _{i}\), \(i=1,2,3\), to be determined.

Lemma 3.4

There exist \(\mu _{1},\mu _{2}>0\) such that

$$\begin{aligned} \mu _{1}E(t)\leq \mathcal{G}(t)\leq \mu _{2}E(t). \end{aligned}$$
(3.20)

Proof

From (3.1), by using Hölder’s inequality (for \(q_{1}=\frac{p+2}{p+1}\), \(q_{2}=p+2\)), Young’s, and Poincare inequalities (for \(\kappa >0\)), and \(\Vert u_{t}\Vert ^{p}_{p+2}\leq [(p+2)E(0)]^{\frac{p}{(p+2)}}\), we find

$$\begin{aligned} \Psi (t) \leq &\frac{1}{p+1} \bigl\Vert u_{t}(t) \bigr\Vert ^{p+1}_{p+2} \bigl\Vert u(t) \bigr\Vert _{p+2}+ \frac{1}{2} \bigl( \bigl\Vert \nabla u_{t}(t) \bigr\Vert _{2}^{2}+ \bigl\Vert \nabla u(t) \bigr\Vert _{2}^{2} \bigr) \\ \leq &\frac{\kappa }{2(p+1)^{2}} \bigl\Vert u_{t}(t) \bigr\Vert ^{2(p+1)}_{p+2}+ \frac{1}{2\kappa } \bigl\Vert u(t) \bigr\Vert ^{2}_{p+2} \\ &{}+\frac{1}{2} \bigl( \bigl\Vert \nabla u_{t}(t) \bigr\Vert _{2}^{2}+ \bigl\Vert \nabla u(t) \bigr\Vert _{2}^{2} \bigr) \\ \leq &\frac{\kappa }{2(p+1)^{2}} \bigl\Vert u_{t}(t) \bigr\Vert ^{p}_{p+2} \bigl\Vert u_{t}(t) \bigr\Vert ^{p+2}_{p+2}+\frac{1}{2\kappa } \bigl\Vert u(t) \bigr\Vert ^{2}_{p+2} \\ &{}+\frac{1}{2} \bigl( \bigl\Vert \nabla u_{t}(t) \bigr\Vert _{2}^{2}+ \bigl\Vert \nabla u(t) \bigr\Vert _{2}^{2} \bigr) \\ \leq &\frac{\kappa [(p+2)E(0)]^{\frac{p}{(p+2)}}}{2(p+1)^{2}} \bigl\Vert u_{t}(t) \bigr\Vert ^{p+2}_{p+2}+c(\kappa ) \bigl\Vert \nabla u(t) \bigr\Vert ^{2}_{2}+ \frac{1}{2} \bigl\Vert \nabla u_{t}(t) \bigr\Vert _{2}^{2}, \end{aligned}$$
(3.21)

where \(c(\kappa )=(\frac{C_{0}}{2\kappa }+\frac{1}{2})\).

According to (3.21) and from (3.2)–(3.3), we get

$$\begin{aligned} \bigl\vert \mathcal{G}(t)-E(t) \bigr\vert \leq &\varepsilon _{1} \bigl\vert \alpha (t) \bigr\vert \biggl(\frac{\kappa [(p+2)E(0)]^{\frac{p}{(p+2)}}}{2(p+1)^{2}} \bigl\Vert u_{t}(t) \bigr\Vert ^{p+2}_{p+2}+c(\kappa ) \bigl\Vert \nabla u(t) \bigr\Vert ^{2}_{2} \biggr) \\ &{}+(\varepsilon _{1}+\varepsilon _{2})\frac{ \vert \alpha (t) \vert }{2} \bigl\Vert \nabla u_{t}(t) \bigr\Vert _{2}^{2}+ \varepsilon _{1}\sigma \frac{ \vert \alpha (t) \vert }{4} \bigl\Vert \nabla u(t) \bigr\Vert _{2}^{4} \\ &{}+\varepsilon _{2}\frac{ \vert \alpha (t) \vert }{2(p+1)} \bigl\Vert u_{t}(t) \bigr\Vert _{2(p+1)}^{2(p+1)}+\varepsilon _{2} \frac{ \vert \alpha (t) \vert (\zeta _{0}-l)c(p)}{2}(h\circ \nabla u) (t) \\ &{}+\varepsilon _{3} \bigl\vert \alpha (t) \bigr\vert \int _{0}^{1} \int _{\tau _{1}}^{ \tau _{2}}s e^{-\rho s} \bigl\vert \beta _{2}(s) \bigr\vert . \bigl\Vert y(x,\rho ,s,t) \bigr\Vert _{m}^{m} \,ds \,d\rho , \end{aligned}$$
(3.22)

where \(c(p)=(\frac{c_{p}}{p+1}+1)\).

Using the fact that \(0<\alpha (t)\leq \alpha (0)\) and \(e^{-\rho s}<1\), we find

$$\begin{aligned} \bigl\vert \mathcal{G}(t)-E(t) \bigr\vert \leq &\varepsilon _{1} \alpha (0) \biggl(\frac{\kappa [(p+2)E(0)]^{\frac{p}{(p+2)}}}{2(p+1)^{2}} \bigl\Vert u_{t}(t) \bigr\Vert ^{p+2}_{p+2}+c(\kappa ) \bigl\Vert \nabla u(t) \bigr\Vert ^{2}_{2} \biggr) \\ &{}+(\varepsilon _{1}+\varepsilon _{2})\frac{ \vert \alpha (0) \vert }{2} \bigl\Vert \nabla u_{t}(t) \bigr\Vert _{2}^{2}+ \varepsilon _{1}\sigma \frac{ \vert \alpha (0) \vert }{4} \bigl\Vert \nabla u(t) \bigr\Vert _{2}^{4} \\ &{}+\varepsilon _{2}\frac{ \vert \alpha (0) \vert c(E(0))^{p}}{2(p+1)} \bigl\Vert \nabla u_{t}(t) \bigr\Vert _{2}^{2}+\varepsilon _{2} \frac{ \vert \alpha (0) \vert (\zeta _{0}-l)c(p)}{2}(h\circ \nabla u) (t) \\ &{}+\varepsilon _{3}\alpha (0) \int _{0}^{1} \int _{\tau _{1}}^{\tau _{2}}s e^{-\rho s} \bigl\vert \beta _{2}(s) \bigr\vert . \bigl\Vert y(x,\rho ,s,t) \bigr\Vert _{m}^{m} \,ds \,d\rho \\ \leq &C(\varepsilon _{1},\varepsilon _{2},\varepsilon _{3},\kappa )E(t). \end{aligned}$$
(3.23)

We pick \(\kappa =1\) and choose \(\varepsilon _{1}\), \(\varepsilon _{2}\), and \(\varepsilon _{3}\) sufficiently small, then (3.20) follows from (3.23). □

Lemma 3.5

There exist \(k_{7},k_{8},t_{0}>0\) satisfying

$$\begin{aligned} \mathcal{G}'(t) \leq &-k_{7}\alpha (t)E(t)+k_{8} \alpha (t) (h\circ \nabla u) (t), \quad t>t_{0}. \end{aligned}$$
(3.24)

Proof

A differentiation of (3.19), using (2.7), Lemmas 3.1, 3.2, and 3.3 lead to

$$\begin{aligned} \mathcal{G}'(t) :=&E'(t)+\varepsilon _{1} \alpha '(t)\Psi (t)+ \varepsilon _{2}\alpha '(t)\Phi (t) +\varepsilon _{3}\alpha '(t) \Theta (t) \\ &{}+\varepsilon _{1}\alpha (t)\Psi '(t)+\varepsilon _{2}\alpha (t) \Phi '(t) +\varepsilon _{3} \alpha (t)\Theta '(t) . \end{aligned}$$
(3.25)

By using the fact that \(e^{-\rho s}<1\), Young’s and Sobolev–Poincare inequalities, we find

$$\begin{aligned} &\alpha '(t) \bigl(\varepsilon _{1}\Psi (t)+\varepsilon _{2}\Phi (t)+ \varepsilon _{3}\Theta (t) \bigr) \\ &\quad \leq -\alpha '(t) \biggl\{ \varepsilon _{1}C_{1} \Vert u_{t} \Vert _{p+2}^{p+2}+ \varepsilon _{1}c(\kappa ) \Vert \nabla u \Vert _{2}^{2}+ \frac{1}{2}( \varepsilon _{1}+\varepsilon _{2}C_{2}) \Vert \nabla u_{t} \Vert _{2}^{2} \\ &\qquad {}+\varepsilon _{2}C_{3}h\circ \nabla u) (t)+ \varepsilon _{3} \int _{0}^{1} \int _{\tau _{1}}^{\tau _{2}}s \bigl\vert \beta _{2}(s) \bigr\vert . \bigl\Vert y(x, \rho ,s,t) \bigr\Vert _{m}^{m} \,ds \,d\rho \biggr\} , \end{aligned}$$
(3.26)

where \(C_{1}=\frac{\kappa [(p+2)E(0)]^{p/(p+2)}}{2(p+1)^{2}}>0\), \(C_{2}=1+\frac{c(E(0))^{p}}{p+1}>0\), and \(C_{3}=\frac{(\zeta _{0}-l)c(p)}{2}>0\).

Hence, by using (2.7), Lemmas 3.1, 3.2, 3.3, and (3.26), we get

$$\begin{aligned} \mathcal{G}'(t) \leq &\alpha (t) \biggl\{ \frac{1}{p+1}( \varepsilon _{1}- \varepsilon _{2}h_{0})- \varepsilon _{1}\frac{\alpha '(t)}{\alpha (t)}C_{1} \biggr\} \Vert u_{t} \Vert _{p+2}^{p+2} \\ &{}+\alpha (t) \biggl\{ \varepsilon _{2}\delta \bigl(\zeta _{0}+2h_{0}^{2} \alpha (t)\bigr)-\varepsilon _{1}\bigl(l-\varepsilon (c_{1}+c_{2})\bigr) \\ &{} -\frac{\alpha '(t)}{2\alpha (t)}\biggl( \int _{0}^{t}h(\varrho )\,d\varrho \biggr)- \varepsilon _{1}\frac{\alpha '(t)c(\kappa )}{\alpha (t)} \biggr\} \Vert \nabla u \Vert _{2}^{2} \\ &{}+\alpha (t) \biggl\{ \varepsilon _{1}+\varepsilon _{2} \bigl[\delta _{1}\bigl(1+c\bigl(E(0)\bigr)^{p} \alpha (t) \bigr)-h_{0}\bigr]-\frac{\alpha '(t)}{2\alpha (t)}(\varepsilon _{1}+ \varepsilon _{2}C_{2}) \biggr\} \Vert \nabla u_{t} \Vert _{2}^{2} \\ &{}+\alpha (t) \{\varepsilon _{2}\zeta _{1}\delta - \varepsilon _{1} \zeta _{1} \} \Vert \nabla u \Vert _{2}^{4} \\ &{}+\alpha (t) \biggl\{ \varepsilon _{2}\delta \frac{\sigma E(0)}{l}- \frac{\sigma }{4\alpha (0)} \biggr\} \biggl(\frac{1}{2}\frac{d}{dt} \Vert \nabla u \Vert _{2}^{2} \biggr)^{2} \\ &{}+\alpha (t) \biggl\{ \varepsilon _{1}\frac{\alpha (t)}{4}+\varepsilon _{2} \biggl(c(\delta )+\biggl(2\delta +\frac{1}{4\delta }\biggr) \biggr)c\alpha (t) )- \varepsilon _{2}\frac{\alpha '(t)C_{3}}{\alpha (t)} \biggr\} (h\circ \nabla u) (t) \\ &{}+\alpha (t) \biggl\{ \frac{1}{2}-\varepsilon _{2}\biggl( \frac{h(0)c_{p}^{2}}{4\delta _{1}}+c(\delta _{1})\biggr) \biggr\} \bigl(h' \circ \nabla u\bigr) (t) \\ &{}+\alpha (t) \biggl\{ \varepsilon _{1}c(\varepsilon )+\varepsilon _{2}c( \delta )+\varepsilon _{3}\beta _{1}- \frac{\eta _{0}}{\alpha (0)} \biggr\} \Vert u_{t} \Vert _{m}^{m} \\ &{}+\alpha (t) \bigl\{ \varepsilon _{1}c(\varepsilon )+\varepsilon _{2}c( \delta )-\eta _{1}\varepsilon _{3} \bigr\} \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \beta _{2}(s)\bigr| \bigl\Vert y(x,1,s,t) \bigr\Vert _{m}^{m}\,ds \\ &{}+\alpha (t)\varepsilon _{3} \biggl\{ -\eta _{1}- \frac{\alpha '(t)}{\alpha (t)} \biggr\} \int _{0}^{1} \int _{\tau _{1}}^{ \tau _{2}} s \bigl\vert \beta _{2}(s) \bigr\vert . \bigl\Vert y(x,\rho ,s,t) \bigr\Vert _{m}^{m} \,ds \,d\rho . \end{aligned}$$
(3.27)

Next, we carefully choose our constants.

Choose δ, \(\delta _{1}\), and ε small enough such that

$$ h_{0}-\delta _{1}\bigl(1-c\bigl(E(0)\bigr)^{p} \bigr)>0, \qquad \delta < \frac{h_{0}}{4}, \qquad \frac{\delta }{(l-\varepsilon (c_{1}+c_{2}))}\bigl(\zeta _{0}+2h_{0}^{2}\bigr) \alpha (0)< \frac{1}{4}h_{0}. $$

For any fixed δ, \(\delta _{1}\), ε, we select \(\varepsilon _{1}\), \(\varepsilon _{2}\), and \(\varepsilon _{3}\) so small satisfying

$$ \frac{h_{0}}{4}\varepsilon _{2}< \varepsilon _{1}< \frac{h_{0}}{2} \varepsilon _{2} $$

and

$$\begin{aligned} &\varepsilon _{2}h_{0}-\varepsilon _{1}>0, \\ &\varepsilon _{2}\bigl[h_{0}-\delta _{1}\bigl(1-c \bigl(E(0)\bigr)^{p}\bigr)\bigr]-\varepsilon _{1}>0. \end{aligned}$$

Then, we select \(\varepsilon _{1}\), \(\varepsilon _{2}\), and \(\varepsilon _{3}\) so small that (3.20) and (3.27) remain valid, and further

$$\begin{aligned} &\zeta _{1}(\varepsilon _{1}-\varepsilon _{2} \delta )>0, \qquad \frac{\sigma }{4\alpha (0)}-\varepsilon _{2}\delta \frac{\sigma E(0)}{l}>0, \qquad \frac{1}{2}-\varepsilon _{2} \biggl( \frac{h(0)c_{p}^{2}}{4\delta }+c( \delta _{1}) \biggr)>0, \\ &\frac{\eta _{0}}{\alpha (0)}-\varepsilon _{1}c(\varepsilon )- \varepsilon _{2}c(\delta )-\varepsilon _{3}\beta _{1}>0, \qquad \eta _{1}\varepsilon _{3}-\varepsilon _{1}c( \varepsilon )- \varepsilon _{2}c(\delta )>0, \end{aligned}$$

where \(\eta _{0}=\beta _{1}-\int _{\tau _{1}}^{\tau _{2}}\vert \beta _{2}(s) \vert \,ds>0\).

Therefore, (3.27) becomes, for positive constants \(k_{i}\), \(i=1,\ldots,6\),

$$\begin{aligned} \mathcal{G}'(t) \leq &-\alpha (t) \biggl(k_{1}+ \varepsilon _{1} \frac{\alpha '(t)}{\alpha (t)}C_{1} \biggr) \Vert u_{t} \Vert _{p+2}^{p+2}- \alpha (t)k_{2} \Vert \nabla u \Vert _{2}^{4} \\ &{}-\alpha (t) \biggl(k_{3}+\frac{\alpha '(t)}{\alpha (t)}\biggl( \int _{0}^{t}h( \varrho )\,d\varrho \biggr)+ \varepsilon _{1} \frac{\alpha '(t)c(\kappa )}{\alpha (t)} \biggr) \Vert \nabla u \Vert _{2}^{2} \\ &{}-\alpha (t) \biggl(k_{4}+\frac{\alpha '(t)}{2\alpha (t)}(\varepsilon _{1}+ \varepsilon _{2}C_{2}) \biggr) \Vert \nabla u_{t} \Vert _{2}^{2} \\ &{}+\alpha (t) \biggl(k_{5}-\varepsilon _{2} \frac{C_{3}\alpha '(t)}{\alpha (t)} \biggr) (h\circ \nabla u) (t) \\ &{}-\alpha (t) \biggl(k_{6}+\varepsilon _{3} \frac{\alpha '(t)}{\alpha (t)} \biggr) \int _{0}^{1} \int _{\tau _{1}}^{ \tau _{2}} s \bigl\vert \beta _{2}(s) \bigr\vert . \bigl\Vert y(x,\rho ,s,t) \bigr\Vert _{m}^{m} \,ds \,d\rho . \end{aligned}$$
(3.28)

According to (2.2), \(\lim_{t\rightarrow \infty }\frac{\alpha '(t)}{\alpha (t)}=0\), we can choose \(t_{1}>t_{0}\) so that (3.28) can be written as

$$\begin{aligned} \mathcal{G}'(t) \leq &-\alpha (t) \biggl(k_{1} \Vert u_{t} \Vert _{p+2}^{p+2}+k_{2} \Vert \nabla u \Vert _{2}^{4}+k_{3} \Vert \nabla u \Vert _{2}^{2}+k_{4} \Vert \nabla u_{t} \Vert _{2}^{2}-k_{5}(h\circ \nabla u) (t) \\ &{} +k_{6} \int _{0}^{1} \int _{\tau _{1}}^{\tau _{2}} s \bigl\vert \beta _{2}(s) \bigr\vert . \bigl\Vert y(x,\rho ,s,t) \bigr\Vert _{m}^{m} \,ds \,d\rho \biggr) \\ \leq &-\alpha (t)k_{7}E(t)+\alpha (t)k_{8}(h\circ \nabla u) (t), \quad \forall t\geq t_{1}. \end{aligned}$$
(3.29)

 □

Theorem 3.6

Suppose that (2.1)(2.3) for any \((u_{0}, u_{1},f_{0})\) satisfy \(E(0) > 0\). Then the energy \(E(t)\) of (2.5) decays to zero exponentially. That is, \(\exists \lambda _{1}, \lambda _{2}>0\) such that

$$ E(t)\leq \lambda _{1} e^{-\lambda _{2}\int _{t_{1}}^{t}\alpha ( \varrho )\vartheta (\varrho )\,d\varrho }, \quad \forall t\geq t_{1}. $$
(3.30)

Proof

Multiplying (3.24) by \(\vartheta (t)\), using (2.1) and (2.7), we find

$$\begin{aligned} \vartheta (t)\mathcal{G}'(t) \leq &-k_{7}\vartheta (t) \alpha (t)E(t)+k_{8} \alpha (t)\vartheta (t) (h\circ \nabla u) (t) \\ \leq &-k_{7}\vartheta (t)\alpha (t)E(t)-k_{8}\alpha (t) \bigl(h'\circ \nabla u\bigr) (t) \\ \leq &-k_{7}\vartheta (t)\alpha (t)E(t)-k_{8} \biggl\{ 2E'(t)-\alpha '(t) \biggl( \int _{0}^{t}h(\varrho )\,d\varrho \biggr) \bigl\Vert \nabla u(t) \bigr\Vert ^{2}_{2} \biggr\} . \end{aligned}$$
(3.31)

Since \(\vartheta (t)\) is a nonincreasing function, we have

$$ \frac{d}{dt} \bigl(\vartheta (t)\mathcal{G}(t)+2k_{8}E(t) \bigr)\leq -k_{7} \vartheta (t)\alpha (t)E(t)-k_{8}\alpha '(t) \biggl( \int _{0}^{t}h( \varrho )\,d\varrho \biggr) \bigl\Vert \nabla u(t) \bigr\Vert ^{2}_{2}. $$
(3.32)

From (2.6) and (2.2) that \(l\Vert \nabla u(t)\Vert ^{2}_{2}\leq E(t)\), we find

$$\begin{aligned} \frac{d}{dt} \bigl(\vartheta (t)\mathcal{G}(t)+2k_{8}E(t) \bigr) \leq &-k_{7} \alpha (t)\vartheta (t)E(t)-k_{8}\alpha '(t) \biggl( \int _{0}^{t}h( \varrho )\,d\varrho \biggr) \bigl\Vert \nabla u(t) \bigr\Vert ^{2}_{2} \\ \leq &-k_{7}\alpha (t)\vartheta (t)E(t)-\frac{2k_{8}\alpha '(t)}{l}E(t) \\ \leq &-\alpha (t)\vartheta (t) \biggl(k_{7}+ \frac{2k_{8}l_{0}\alpha '(t)}{l\vartheta (t)\alpha (t)} \biggr)E(t). \end{aligned}$$
(3.33)

Since \(\lim_{t\rightarrow \infty } \frac{\alpha '(t)}{\vartheta (t)\alpha (t)}=0\), we can choose \(t_{1}\geq t_{0}\) such that \(k_{7}+\frac{2k_{8}l_{0}\alpha '(t)}{l\alpha (t)\vartheta (t)}>0\) for \(t\geq t_{1}\).

Finally, let

$$ \mathcal{R}(t):=\mathcal{G}(t)\vartheta (t)+2k_{8}E(t)\sim E(t). $$
(3.34)

Hence, for some \(\lambda _{2}>0\), we find

$$ \mathcal{R}'(t)\leq -\lambda _{2}\alpha (t)\vartheta (t) \mathcal{R}(t), \quad \forall t\geq t_{1}. $$
(3.35)

Integrating of (3.35) over \((t_{1}, t)\) gives

$$ \mathcal{R}(t)\leq \mathcal{R}(t_{1})e^{-\lambda _{2}\int _{t_{1}}^{t} \alpha (\varrho )\vartheta (\varrho )\,d\varrho }, \quad \forall t \geq t_{1}. $$
(3.36)

Hence, (3.30) is established by virtue of (3.34) and (3.36). The proof is complete. □

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Choucha, A., Boulaaras, S. Asymptotic behavior for a viscoelastic Kirchhoff equation with distributed delay and Balakrishnan–Taylor damping. Bound Value Probl 2021, 77 (2021). https://doi.org/10.1186/s13661-021-01555-0

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MSC

  • 35B40
  • 35L70
  • 76Exx
  • 93D20

Keywords

  • Kirchhoff equation
  • Exponential decay
  • Distributed delay term
  • Viscoelastic term
  • Energy method