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


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.


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

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

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.


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

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

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

  2. (A2)

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

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

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

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

Let us introduce

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


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

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

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

So, problem (1.1) can be written as

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


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

Now, we give the energy functional.

Lemma 2.2

The energy functional E, defined by

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


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

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


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

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

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

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

By integration by parts, we find

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

and we have

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


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


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

By (2.1), we get

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


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

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

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

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

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

By Young’s inequality, we have

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

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

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


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


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

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 \(\varepsilon >0\),

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


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

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

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

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


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

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

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

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 \(\delta >0\),

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


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}$$

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

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


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

Similarly, we have

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

By exploiting the Sobolev embedding, we have

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


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

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

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


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

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

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

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


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

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

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

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

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

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

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

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

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


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

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

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

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

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}$$

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

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


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

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 \(k_{i}\), \(i=1,\ldots,6\),

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

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

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


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

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


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

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

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

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

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

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

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

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

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

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

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

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

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

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  1. 1.

    Adams, R., Fourier, J.: Sobolev Space. Academic Press, New York (2003)

    Google Scholar 

  2. 2.

    Balakrishnan, A.V., Taylor, L.W.: Distributed parameter nonlinear damping models for flight structures. In: Proceedings (Damping 89), Flight Dynamics Lab and Air Force Wright Aeronautical Labs. WPAFB, Washington (1989)

    Google Scholar 

  3. 3.

    Bass, R.W., Zes, D.: Spillover nonlinearity, and flexible structures. In: The Fourth NASA Workshop on Computational Control of Flexible Aerospace Systems. Washington: NASA Conference Publication 10065, pp. 1–14 (1991)

    Google Scholar 

  4. 4.

    Bathory, M., Bulíček, M., Málek, J.: Large data existence theory for three-dimensional unsteady flows of rate-type viscoelastic fluids with stress diffusion. Adv. Nonlinear Anal. 10(1), 501–521 (2021)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Benbernou, S., Gala, S., Ragusa, M.A.: On the regularity criteria for the 3D magnetohydrodynamic equations via two components in terms of BMO space. Math. Methods Appl. Sci. 37(15), 2320–2325 (2014)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Bland, D.R.: The Theory of Linear Viscoelasticity. Courier Dover Publications, Mineola (2016)

    MATH  Google Scholar 

  7. 7.

    Boulaaras, S., Choucha, A., Ouchenane, D., Cherif, B.: Blow up of solutions of two singular nonlinear viscoelastic equations with general source and localized frictional damping terms. Adv. Differ. Equ. 2020, 310 (2020)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Boulaaras, S., Draifia, A., Zennir, Kh.: General decay of nonlinear viscoelastic Kirchhoff equation with Balakrishnan-Taylor damping and logarithmic nonlinearity. Math. Methods Appl. Sci. 42, 4795–4814 (2019)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Cavalcanti, M., Domingos Cavalcanti, V., Lasiecka, I., Webler, C.: Intrinsic decay rates for the energy of a nonlinear viscoelastic equation modeling the vibrations of thin rods with variable density. Adv. Nonlinear Anal. 6(2), 121–145 (2017)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Choucha, A., Boulaaras, S., Ouchenane, D., Beloul, S.: General decay of nonlinear viscoelastic Kirchhoff equation with Balakrishnan-Taylor damping, logarithmic nonlinearity and distributed delay terms. Math. Methods Appl. Sci. 44(7), 5436–5457 (2021).

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Choucha, A., Boulaaras, S.M., Ouchenane, D., Cherif, B.B., Abdalla, M.: Exponential stability of swelling porous elastic with a viscoelastic damping and distributed delay term. J. Funct. Spaces 2021, Article ID 5581634 (2021).

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Choucha, A., Ouchenane, D., Boulaaras, S.: Well posedness and stability result for a thermoelastic laminated Timoshenko beam with distributed delay term. Math. Methods Appl. Sci. 43(17), 9983–10004 (2020).

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Choucha, A., Ouchenane, D., Boulaaras, S.: Blow-up of a nonlinear viscoelastic wave equation with distributed delay combined with strong damping and source terms. J. Nonlinear Funct. Anal. 2020, Article ID 31 (2020).

    Article  Google Scholar 

  14. 14.

    Choucha, A., Ouchenane, D., Zennir, Kh., Feng, B.: Global well-posedness and exponential stability results of a class of Bresse-Timoshenko-type systems with distributed delay term. Math. Methods Appl. Sci., 1–26 (2020).

    Article  Google Scholar 

  15. 15.

    Coleman, B.D., Noll, W.: Foundations of linear viscoelasticity. Rev. Mod. Phys. 33(2), 239 (1961)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Feng, B., Soufyane, A.: Existence and decay rates for a coupled Balakrishnan-Taylor viscoelastic system with dynamic boundary conditions. Math. Models Methods Appl. Sci. 43, 3375–3391 (2020)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Gala, S., Galakhov, E., Ragusa, M.A., Salieva, O.: Beale-Kato-Majda regularity criterion of smooth solutions for the Hall-MHD equations with zero viscosity. Bull. Braz. Math. Soc. (2021).

    Article  Google Scholar 

  18. 18.

    Gheraibia, B., Boumaza, N.: General decay result of solution for viscoelastic wave equation with Balakrishnan-Taylor damping and a delay term. Z. Angew. Math. Phys. 71, 198 (2020).

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Kirchhoff, G.: Vorlesungen uber Mechanik. Tauber, Leipzig (1883)

    MATH  Google Scholar 

  20. 20.

    Liu, W., Zhu, B., Li, G., Wang, D.: General decay for a viscoelastic Kirchhoff equation with Balakrishnan-Taylor damping, dynamic boundary conditions and a time-varying delay term. Evol. Equ. Control Theory 6, 239–260 (2017)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Makvand Chaharlang, M.M., Razani, A.: Two weak solutions for some Kirchhoff-type problem with Neumann boundary condition. Georgian Math. J. 28(3), 429–438 (2021)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Mesloub, F., Boulaaras, S.: General decay for a viscoelastic problem with not necessarily decreasing kernel. J. Appl. Math. Comput. 58, 647–665 (2018).

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Mezouar, N., Boulaaras, S.: Global existence and decay of solutions for a class of viscoelastic Kirchhoff equation. Bull. Malays. Math. Sci. Soc. 43, 725–755 (2020)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Mezouar, N., Boulaaras, S.: Global existence and exponential decay of solutions for generalized coupled non-degenerate Kirchhoff system with a time varying delay term. Bound. Value Probl. (2020).

    MathSciNet  Article  MATH  Google Scholar 

  25. 25.

    Mingqi, X., Radulescu, V.D., Zhang, B.: Nonlocal Kirchhoff diffusion problems: local existence and blow-up of solutions. Nonlinearity 31(7), 3228–3250 (2018)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Mingqi, X., Radulescu, V.D., Zhang, B.: Nonlocal Kirchhoff problems with singular exponential nonlinearity. Appl. Math. Optim. 84(1), 915–954 (2021)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Mu, C., Ma, J.: On a system of nonlinear wave equations with Balakrishnan-Taylor damping. Z. Angew. Math. Phys. 65, 91–113 (2014)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Nicaise, A.S., Pignotti, C.: Stabilization of the wave equation with boundary or internal distributed delay. Differ. Integral Equ. 21(9–10), 935–958 (2008)

    MathSciNet  MATH  Google Scholar 

  29. 29.

    Ouchenane, D., Boulaaras, S., Mesloub, F.: General decay for a viscoelastic problem with not necessarily decreasing kernel. Appl. Anal. 98(44), 1–17 (2018).

    MathSciNet  Article  MATH  Google Scholar 

  30. 30.

    Polidoro, S., Ragusa, M.A.: Sobolev-Morrey spaces related to an ultraparabolic equation. Manuscr. Math. 96(3), 371–392 (1998)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Razani, A.: Shock waves in gas dynamics. Surv. Math. Appl., 2, 59–89 (2007)

    MathSciNet  MATH  Google Scholar 

  32. 32.

    Razani, A.: Subsonic detonation waves in porous media. Phys. Scr. 94, 085209, 6 pages (2019).

    Article  Google Scholar 

  33. 33.

    Tian, L., Cheng, Z.: A triangular plate bending element based on discrete Kirchhoff theory with simple explicit expression. Mathematics 9(11), 1181 (2021)

    Article  Google Scholar 

  34. 34.

    Wu, Y., Qiao, Z.H., Hamdani, M.K., Kou, B.Y., Yang, L.B.: A lass of variable-order fractional p(.)-Kirchhoff-type systems. J. Funct. Spaces 2021, 5558074 (2021)

    MATH  Google Scholar 

  35. 35.

    Zarai, A., Tatar, N.: Global existence and polynomial decay for a problem with Balakrishnan-Taylor damping. Arch. Math. 46, 157–176 (2010)

    MathSciNet  MATH  Google Scholar 

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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).

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


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