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Global existence and exponential decay of solutions for generalized coupled non-degenerate Kirchhoff system with a time varying delay term

Abstract

The paper studies a system of nonlinear viscoelastic Kirchhoff system with a time varying delay and general coupling terms. We prove the global existence of solutions in a bounded domain using the energy and Faedo–Galerkin methods with respect to the condition on the parameters in the coupling terms together with the weight condition as regards the delay terms in the feedback and the delay speed. Furthermore, we construct some convex function properties, and we prove the uniform stability estimate.

Introduction

The Kirchhoff equation belongs to the famous wave equation’s models describing the transverse vibration of a string fixed in its ends. It has been introduced in 1876 by Kirchhoff [8] and it is more general than the D’Alembert equation. In one dimensional space it takes the following form:

$$ \frac{\partial ^{2}u}{\partial t^{2}}- \biggl(\frac{P_{0}}{\rho h}+ \frac{E}{2 L \rho } \int _{0}^{L} \biggl\vert \frac{\partial u}{\partial x}(x, t) \biggr\vert ^{2}\,dx\biggr) \frac{\partial ^{2}u}{\partial x^{2}}=0, $$
(1.1)

where the function \(u(x,t)\) is the vertical displacement at the space coordinate x, varying in the segment \([0,L]\) and over time \(t>0\), ρ is the mass density, h is the area of the cross section of the string, \(P_{0}\) is the initial tension on the string, L is the length of the string and E is the Young modulus of the material. The nonlinear coefficient

$$ C(t)= \int _{0}^{L} \biggl\vert \frac{\partial u}{\partial x}(x, t) \biggr\vert ^{2} \,dx$$

is obtained by the variation of the tension during the deformation of the string. When we do not have an initial tension (i.e. \(P_{0}=0\)), we call that a degenerate case as opposed to the non-degenerate case.

In this paper, we are interested in studying, in \(\mathcal{A}=\varOmega \times (0,\infty )\), the following coupled viscoelastic Kirchhoff system:

$$\begin{aligned} \textstyle\begin{cases} \vert u_{t} \vert ^{l}u_{tt}-M( \Vert \nabla u \Vert ^{2})\Delta u -\Delta u_{tt}+\int _{0}^{t}h_{1}(t-s) \Delta u(s)\,ds-\mu _{1}\Delta u_{t}(x,t-\tau (t))+f_{1}(u,v) \\ \quad =0\quad \text{in } \varOmega \times \mathopen]0,+\infty \mathclose[, \\ \vert v_{t} \vert ^{l}v_{tt}-M( \Vert \nabla v \Vert ^{2})\Delta v -\Delta v_{tt}+\int _{0}^{t}h_{2}(t-s) \Delta v(s)\,ds-\mu _{2}\Delta v_{t}(x,t-\tau (t))+f_{2}(u,v) \\ \quad =0 \quad \text{in } \varOmega \times \mathopen]0,+\infty \mathclose[, \\ u(x,t)=v(x,t)=0 \quad \text{on } \varGamma \times \mathopen]0,+\infty \mathclose[, \\ (u(x,0),v(x,0))=(u_{0}(x),v_{0}(x)), \qquad (u_{t}(x,0),v_{t}(x,0))=(u_{1}(x),v_{1}(x)) \quad \text{in } \varOmega , \\ (u_{t}(x,t-\tau (0)),v_{t}(x,t-\tau (0)))=(f_{0}(x,t-\tau (0)),g_{0}(x,t- \tau (0))) \quad \text{in }\varOmega \times \mathopen]0,\tau (0)\mathclose[, \end{cases}\displaystyle \end{aligned}$$
(1.2)

in which Ω is an n dimensional bounded domain of \(\mathbb{R}^{n}\) and we have a smooth boundary Γ, \(l>0\), \(\mu _{1}\) and \(\mu _{2}\) are positive real constants, \(h_{1}\) and \(h_{2}\) are positive functions with exponential decay, and \(\tau (t)\) is a positive time varying delay. In addition the initial condition \((u_{0},v_{0},u_{1},v_{1},f_{0},g_{0})\) will be specified in their function space later. M is a smooth function defined by

$$ \begin{aligned} M:\quad& \mathbb{R}_{+}\longrightarrow \mathbb{R}, _{+} \\ &r\longmapsto M(r)=a+br^{\gamma }, \end{aligned}$$

with \(a,b>0\), and \(\gamma \geq 1\). \(f_{1}\) and \(f_{2}\) are two functions taking a particular form that we will make precise later.

The problem (1.2) is a description of axially moving viscoelastic strings composed of two different materials (like the wires of electricity) that are nonhomogeneous and which will be of influence on its moving, specially on the acceleration. From the mathematical point of view, this influence is represented by \(\vert w_{t} \vert ^{l}w^{{\prime \prime }}\), where \(\vert w_{t} \vert ^{l}\) is the material density, varying the velocity. A lot of work has been published with this term, for example see [11] and [14], where we find different results about the global existence and nonexistence of solutions and the decay of energy.

In recent years, the study of wave equations with delay has become an active area and with different forms of delay (constant [7], switching [5], varying in time [12], distributed [6]). The delay appears in modeling of a lot of domains, like the physical, chemical, biological and engineering domains. It is introduced when we have a time lag between an action on a system and a response of the system to this action. Furthermore, a delay can be small enough in feedback yet can destabilize a system [10], or improve the performance of the system [17].

In the absence of delay, Cavalcanti et al. [3] studied the following viscoelastic wave equations with strong damping:

$$ \begin{aligned} \vert u_{t} \vert ^{l}u_{tt}- \Delta u + \int _{0}^{t}h_{1}(t-s) \Delta u(s) \,ds-\mu \Delta u_{t}(x,t)=0, \quad \text{in } \varOmega \times \mathopen]0,+\infty \mathclose[. \end{aligned}$$

They used the Fadeo–Galerkin method to prove the global existence of a solution; also an explicit decay rate of the energy has been given provided \(m>0\).

In the other hand, in the same case and for \(l=0\), Raslan et al. [16] and El-Sayed et al. [4] have studied coupled equal width wave equations with strong damping, as they were looking for the new exact solution.

The problem treated in [2] has the following form:

$$ \begin{aligned} u_{tt}-\Delta u +\mu _{1}\sigma (t) g_{1} \bigl(u_{t}(x,t) \bigr)+ \mu _{2}\sigma (t) g_{2} \bigl(u_{t} \bigl(x,t-\tau (t) \bigr) \bigr)=0,\quad \text{in } \varOmega \times \mathopen]0,+\infty \mathclose[. \end{aligned}$$

Under the assumptions set on \(g_{1}\), \(g_{2}\), σ and τ, the authors have gotten the global existence of a solution and the decay rate of the energy.

Recently, Mezouar and Boulaaras [13] have studied the viscoelastic non-degenerate Kirchhoff equation with varying delay term in the internal feedback.

In the present paper, we extend our recently published paper in [13] for a coupled system (1.2). The famous technique of using the presence of a delay in the PDE problem is to set a new variable defined by a velocity dependent on the delay, which will give us a new problem equivalent to our studied problem; but the last one is a coupled system without delay. After this, we can prove the existence of global solutions in suitable Sobolev spaces by combining the energy method with the Fadeo–Galerkin procedure and under the choice of a suitable Lyapunov functional, we establish an exponential decay result.

The outline of the paper is as follows: In the second section, some hypotheses related to the problem are given and we state our main result. Then in the third section, the global existence of weak solutions is proven. Finally, in the fourth section, we give the uniform energy decay.

Preliminaries and assumptions

Similar to that [12], we present the new variables

$$ z_{1}(x,\rho ,t)=u_{t} \bigl(x,t-\rho \tau (t) \bigr),\quad x \in \varOmega , \rho \in (0,1), t>0, $$

and

$$ z_{2}(x,\rho ,t)=v_{t} \bigl(x,t-\rho \tau (t) \bigr),\quad x \in \varOmega , \rho \in (0,1), t>0. $$

Then we have

$$ \tau (t)z^{\prime }_{1}(x,\rho ,t)+ \bigl(1-\rho \tau ^{\prime }(t) \bigr) \frac{\partial }{\partial \rho }z_{1}(x,\rho ,t)=0,\quad \text{in } \varOmega \times (0,1) \times (0,+\infty ). $$
(1.3)

In the same way, we have

$$ \tau (t)z^{\prime }_{2}(x,\rho ,t)+ \bigl(1-\rho \tau ^{\prime }(t) \bigr) \frac{\partial }{\partial \rho }z_{2}(x,\rho ,t)=0,\quad \text{in } \varOmega \times (0,1) \times (0,+\infty ). $$
(1.4)

Therefore, problem (1.2) is equivalent to

$$\begin{aligned} \textstyle\begin{cases} \vert u_{t} \vert ^{l}u_{tt} -M( \Vert \nabla u \Vert ^{2})\Delta u -\Delta u_{tt}+\int _{0}^{t}h_{1}(t-s) \Delta u(s)\,ds- \mu _{1}\Delta z_{1}(x,1,t)+f_{1}(u,v)=0 \\ \quad \text{in $\varOmega \times \mathopen]0,+\infty \mathclose[$,} \\ \vert v_{t} \vert ^{l}v_{tt}-M( \Vert \nabla v \Vert ^{2})\Delta v -\Delta v_{tt}+\int _{0}^{t}h_{2}(t-s) \Delta v(s)\,ds- \mu _{2}\Delta z_{2}(x,1,t)+f_{2}(u,v)=0 \\ \quad \text{in $\varOmega \times \mathopen]0,+\infty \mathclose[$,} \\ \tau (t)z^{\prime }_{1}(x,\rho ,t)+(1-\rho \tau ^{\prime }(t)) \frac{\partial }{\partial \rho }z_{1}(x,\rho ,t)=0,\quad \text{in } \varOmega \times (0,1) \times (0,+\infty ), \\ \tau (t)z^{\prime }_{2}(x,\rho ,t)+(1-\rho \tau ^{\prime }(t)) \frac{\partial }{\partial \rho }z_{2}(x,\rho ,t)=0,\quad \text{in } \varOmega \times (0,1) \times (0,+\infty ), \\ u(x,t)=v(x,t)=0, \quad \text{on }\partial \varOmega \times \mathopen[0,\infty \mathclose[, \\ (z_{1}(x,0,t),z_{2}(x,0,t))=(u_{t}(x,t),v_{t}(x,t)), \quad \text{on } \varOmega \times \mathopen]0,\infty \mathclose[, \\ (u(x,0),v(x,0))=(u_{0}(x),v_{0}(x)),\qquad (u_{t}(x,0),v_{t}(x,0))=(u_{1}(x),v_{1}(x)), \quad \text{in }\varOmega , \\ (z_{1}(x,\rho ,0),z_{2}(x,\rho ,0))=(f_{0}(x,-\rho \tau (0)),g_{0}(x,- \rho \tau (0)) ) , \quad \text{in }\varOmega \times \mathopen]0,1\mathclose[.\end{cases}\displaystyle \end{aligned}$$
(1.5)

Throughout this work and for simplifying our formulas, we will adopt the notation \(z_{i}\), u and v instead of \(z_{i}(x,\rho ,t)\), \(u(x,t)\) and \(v(x,t)\), except if that makes things inconvenient.

In order to demonstrate the main result in this paper, a few assumptions are needed.

(A-1):

Consider that \(0< l\leq \gamma \) verifies

$$ \textstyle\begin{cases} \gamma \leq \frac{2}{n-2}&\text{in the case } n > 2, \\ \gamma < \infty & \text{in the case } n\leq 2. \end{cases} $$
(A-2):

As regards the relaxation functions \(h_{i}:{\mathbb{R}_{+}}\rightarrow {\mathbb{R}_{+}}\) we see that they are bounded \(C^{1}\) functions such that

$$ a- \int _{0}^{\infty }h_{i}(s)\,ds\geq k>0. $$

We assume also that there exist some positive constants \(\zeta _{i} \) verifying

$$ h_{i}^{\prime }(t)\leq -\zeta _{i} h_{i}(t) $$

for \(i=1,2\).

(A-3):

We have \(\tau \in C^{2}([0,T],[\tau _{0},\tau _{1}])\) a positive function, where

$$ \tau ^{\prime }(t)\leq d< 1,\quad \forall t\in [0,T]. $$
(A-4):

\(f_{1}(u,v)=\alpha v+b_{1} \vert v \vert ^{q+1} \vert u \vert ^{p-1}u\) and \(f_{2}(u,v)=\alpha u+b_{2} \vert u \vert ^{p+1} \vert v \vert ^{q-1}v\) where \(\alpha >0\), \(b_{1}=(p+1)(p+q)\), \(b_{2}=(q+1)(p+q)\) such that p and q are conjugate (i.e. \(\frac{1}{p}+\frac{1}{q}=1\)), \(p,q<\gamma -\frac{1}{2}\) and satisfy

$$ 2\leq p,q\leq \textstyle\begin{cases} \sqrt{\frac{n}{2(n-2)}}& \text{if } n> 2, \\ +\infty &\text{if } n\leq 2. \end{cases} $$

The energy related to the system solution of (1.5) is defined as follows:

$$ \begin{aligned}[b] E(t)&=\frac{1}{l+2} \bigl( \Vert u_{t} \Vert ^{l+2}_{l+2}+ \Vert v_{t} \Vert ^{l+2}_{l+2} \bigr)+ \frac{b}{2(\gamma +1)} \bigl( \Vert \nabla u \Vert ^{2(\gamma +1)}+ \Vert \nabla v \Vert ^{2(\gamma +1)} \bigr) \\ &\quad{} +\frac{1}{2} \biggl(a- \int _{0}^{t}h_{1}(s)\,ds \biggr) \Vert \nabla u \Vert ^{2}+ \frac{1}{2} \biggl(a- \int _{0}^{t}h_{2}(s)\,ds \biggr) \Vert \nabla v \Vert ^{2} \\ &\quad{} +\frac{1}{2} \bigl( \Vert \nabla u_{t} \Vert ^{2}+ \Vert \nabla v_{t} \Vert ^{2} \bigr) \\ &\quad{}+\frac{1}{2}(h_{1}\mathbin{o}\nabla u) (t)+ \frac{1}{2}(h_{2}\mathbin{o}\nabla v) (t)+ \xi \tau (t) \int _{0}^{1} \bigl( \Vert \nabla z_{1} \Vert ^{2}+ \Vert \nabla z_{2} \Vert ^{2} \bigr) \,d\rho \\ &\quad{}+\alpha \int _{\varOmega }uv\,dx+(p+q) \int _{\varOmega } \vert u \vert ^{p+1} \vert v \vert ^{q+1} \,dx, \end{aligned} $$
(1.6)

where ξ is a positive constant such that

$$ \frac{\max \{\mu _{1},\mu _{2}\}}{2(1-d)}< \xi $$
(1.7)

and

$$ (h_{i}\mathbin{o}w) (t)= \int _{0}^{t}h_{i}(t-s) \bigl\Vert w( \cdot ,t)-w(\cdot ,s) \bigr\Vert ^{2}\,ds,\quad \text{for }i=1,2. $$

Lemma 1.1

(Sobolev–Poincaré’s inequality)

Let q be a number with

$$ 2\leq q< +\infty \quad (n=1,2)\quad \textit{or}\quad 2\leq q\leq 2n/(n-2)\quad (n \geq 3). $$

Then there exists a constant \(C_{s}=C_{s}(\varOmega ,q) \) such that

$$ \Vert u \Vert _{q}\leq C_{s} \Vert \nabla u \Vert _{2} \quad \textit{for } u\in H^{1}_{0}( \varOmega ). $$

We present the following lemma.

Lemma 1.2

[15] Forh, φ\(C^{1}\)-real functions, we have

$$ \begin{aligned}[b] & \frac{d}{dt} \biggl[(h\mathbin{o}\varphi ) (t)- \biggl( \int _{0}^{t}h(s)\,ds\biggr) \Vert \varphi \Vert ^{2} \biggr] \\ &\quad =(h^{\prime 2}-2 \int _{\varOmega } \int _{0}^{t}h(t-s) \varphi (s)\varphi _{t} (t) \,ds\,dx\quad \forall t\geq 0. \end{aligned} $$
(1.8)

Lemma 1.3

Let\((u,v,z_{1},z_{2})\)be a solution of the problem (1.5). Then the energy functional defined by (1.6) satisfies

$$ \begin{aligned}[b] E^{\prime }(t)&\leq -\beta \bigl( \bigl\Vert \nabla z_{1}(x,1,t) \bigr\Vert ^{2}+ \bigl\Vert \nabla z_{2}(x,1,t) \bigr\Vert ^{2} \bigr)+\lambda \bigl( \Vert \nabla u_{t} \Vert ^{2}+ \Vert \nabla v_{t} \Vert ^{2} \bigr) \\ &\quad{} + \frac{1}{2} \bigl[ \bigl(h_{1}^{\prime }o\nabla u \bigr) (t)+ \bigl(h_{2}^{\prime }o \nabla v \bigr) (t) \bigr], \end{aligned} $$
(1.9)

where\(\lambda =\xi +\frac{\mu }{2} \), \(\beta =\xi (1-d)-\frac{\mu }{2}\)and\(\mu =\max \{\mu _{1},\mu _{2}\}\)are positive.

Proof

After the multiplication of the first equation in (1.5) by \(u_{t}\) followed by integration of the result by parts over Ω, we get

$$ \begin{aligned}[b] & \frac{d}{dt} \biggl[\frac{1}{l+2} \Vert u_{t} \Vert _{l+2}^{l+2}+ \frac{b}{2(\gamma +1)} \Vert \nabla u \Vert ^{2(\gamma +1)}+\frac{1}{2}a \Vert \nabla u \Vert ^{2}+\frac{1}{2} \Vert \nabla u_{t} \Vert ^{2} \biggr] \\ &\quad{} - \int _{ \varOmega } \int _{0}^{t}h_{1}(t-s)\nabla u(s)\nabla u_{t}(t)\,ds\,dx\\ & \quad{}+\mu _{1} \int _{\varOmega }\nabla u_{t}\nabla z_{1}(x,1,t) \,dx+\alpha \int _{\varOmega }vu_{t}\,dx+b_{1} \int _{\varOmega } \vert v \vert ^{q+1} \vert u \vert ^{p-1}uu_{t} \,dx=0. \end{aligned} $$
(1.10)

Using (1.8) and (1.10) leads to

$$ \begin{aligned}[b] & \frac{d}{dt} \biggl[\frac{1}{l+2} \Vert u_{t} \Vert _{l+2}^{l+2}+ \frac{b}{2(\gamma +1)} \Vert \nabla u \Vert ^{2(\gamma +1)}+\frac{1}{2} \biggl(a- \int _{0}^{t}h_{1}(s)\,ds \biggr) \Vert \nabla u \Vert ^{2} \\ &\quad{} +\frac{1}{2} \Vert \nabla u_{t} \Vert ^{2}+\frac{1}{2}(h_{1}o\nabla u) (t) \biggr] \\ & \quad{}+\frac{1}{2}h_{1}(t) \Vert \nabla u \Vert ^{2}-\frac{1}{2} \bigl(h_{1}^{ \prime }o\nabla u \bigr) (t)+\mu _{1} \int _{\varOmega }\nabla u_{t}\nabla z_{1}(x,1,t) \,dx\\ &\quad{} +\alpha \int _{\varOmega }vu_{t}\,dx+b_{1} \int _{\varOmega } \vert v \vert ^{q+1} \vert u \vert ^{p-1}uu_{t} \,dx=0. \end{aligned} $$
(1.11)

Similarly by multiplying the second equation in (1.5) by \(v_{t}\), integrating over Ω and using integration by parts, we get

$$ \begin{aligned}[b] & \frac{d}{dt} \biggl[\frac{1}{l+2} \Vert v_{t} \Vert _{l+2}^{l+2}+ \frac{b}{2(\gamma +1)} \Vert \nabla v \Vert ^{2(\gamma +1)}+\frac{1}{2} \biggl(a- \int _{0}^{t}h_{2}(s)\,ds \biggr) \Vert \nabla v \Vert ^{2} \\ &\quad{} +\frac{1}{2} \Vert \nabla v_{t} \Vert ^{2}+\frac{1}{2}(h_{2}o\nabla v) (t) \biggr] \\ & \quad{}+\frac{1}{2}h_{2}(t) \Vert \nabla v \Vert ^{2}-\frac{1}{2} \bigl(h_{2}^{ \prime }o\nabla v \bigr) (t)+\mu _{2} \int _{\varOmega }\nabla v_{t}\nabla z_{2}(x,1,t) \,dx+\alpha \int _{\varOmega }uv_{t}\,dx\\ &\quad{} +b_{2} \int _{\varOmega } \vert u \vert ^{p+1} \vert v \vert ^{q-1}vv_{t} \,dx=0. \end{aligned} $$
(1.12)

Multiplying the third equation in (1.5) by \(\xi \Delta z_{1}\) and integrating the result over \(\varOmega \times (0,1)\), we obtain

$$ \xi \tau (t) \int _{\varOmega } \int _{0}^{1}z_{1}^{\prime }\Delta z_{1} \,d\rho \,dx=-\xi \int _{\varOmega \times (0,1)} \bigl(1-\rho \tau ^{\prime }(t) \bigr) \frac{\partial }{\partial \rho }z_{1}\Delta z_{1}\,d\rho \,dx. $$

Consequently,

$$ \begin{aligned}[b] \frac{d}{dt} \biggl(\xi \tau (t) \int _{0}^{1} \Vert \nabla z_{1} \Vert ^{2}\,d\rho \biggr)&=\xi \int _{\varOmega \times (0,1)} \biggl[\tau ^{\prime }(t) \vert \nabla z_{1} \vert ^{2}-\rho \tau ^{\prime }(t)) \frac{\partial }{\partial \rho } \vert \nabla z_{1} \vert ^{2} \biggr] \,d\rho \,dx\\ &=-\xi \int _{0}^{1}\frac{\partial }{\partial \rho } \bigl( \bigl(1-\rho \tau ^{\prime }(t) \bigr) \Vert \nabla z_{1} \Vert ^{2} \bigr)\,d\rho \\ &=\xi \bigl[ \Vert \nabla u_{t} \Vert ^{2}-\xi \bigl(1- \tau ^{\prime }(t) \bigr) \bigl\Vert \nabla z_{1}(x,1,t) \bigr\Vert ^{2} \bigr]. \end{aligned} $$
(1.13)

Similarly we get

$$ \frac{d}{dt} \biggl(\xi \tau (t) \int _{0}^{1} \Vert \nabla z_{2} \Vert ^{2} \,d\rho \biggr)=\xi \bigl[ \Vert \nabla v_{t} \Vert ^{2}- \bigl(1-\tau ^{\prime }(t) \bigr) \bigl\Vert \nabla z_{2}(x,1,t) \bigr\Vert ^{2} \bigr]. $$
(1.14)

Combining (1.11)–(1.14), taking the derivation of energy leads to

$$ \begin{aligned} E^{\prime }(t)&=\xi \bigl[ \Vert \nabla u_{t} \Vert ^{2}+ \Vert \nabla v_{t} \Vert ^{2}- \bigl(1-\tau ^{\prime }(t) \bigr) \bigl( \bigl\Vert \nabla z_{1}(x,1,t) \bigr\Vert ^{2}+ \bigl\Vert \nabla z_{2}(x,1,t) \bigr\Vert ^{2} \bigr) \bigr] \\ & \quad{}-\frac{1}{2} \bigl[h_{1}(t) \Vert \nabla u \Vert ^{2}+h_{2}(t) \Vert \nabla v \Vert ^{2} \bigr] \\ & \quad{}+\frac{1}{2} \bigl[ \bigl(h_{1}^{\prime }o\nabla u \bigr) (t)+ \bigl(h_{2}^{\prime }o\nabla v \bigr) (t) \bigr]- \mu _{1} \int _{\varOmega }\nabla u_{t}(x,t)\nabla z_{1}(x,1,t)\,dx\\ & \quad{}-\mu _{2} \int _{\varOmega }\nabla v_{t}(x,t)\nabla z_{2}(x,1,t)\,dx. \end{aligned}$$

From \((A3)\), we find the following bound:

$$\begin{aligned} \begin{aligned}[b] E^{\prime }(t)&\leq - \biggl(\xi (1-d)- \frac{\mu _{1}}{2} \biggr) \int _{\varOmega } \bigl\vert \nabla z_{1}(x,1,t) \bigr\vert ^{2}\,dx- \biggl(\xi (1-d)- \frac{\mu _{2}}{2} \biggr) \int _{\varOmega } \bigl\vert \nabla z_{2}(x,1,t) \bigr\vert ^{2}\,dx \\ & \quad{}-\frac{1}{2}h_{1}(t) \bigl\Vert \nabla u_{t}(t) \bigr\Vert ^{2}+ \biggl(\xi + \frac{\mu _{1}}{2} \biggr) \bigl\Vert \nabla u_{t}(t) \bigr\Vert ^{2}- \frac{1}{2}h_{2}(t) \bigl\Vert \nabla v(t) \bigr\Vert ^{2} \\ &\quad{} + \biggl(\xi +\frac{\mu _{2}}{2} \biggr) \bigl\Vert \nabla v_{t}(t) \bigr\Vert ^{2} +\frac{1}{2} \bigl[ \bigl(h_{1}^{\prime }o\nabla u \bigr) (t)+ \bigl(h_{2}^{\prime }o\nabla v \bigr) (t) \bigr]. \end{aligned} \end{aligned}$$
(1.15)

Using (1.7), we complete the proof of the lemma. □

Global existence

Theorem 2.1

Let\((u_{0},v_{0})\in (H^{2}(\varOmega ) \cap H_{0}^{1}(\varOmega ))^{2}\), \((u_{1},v_{1})\in (H_{0}^{1}(\varOmega ))^{2}\)and\((f_{0},g_{0})\in (H_{0}^{1}(\varOmega , H^{1}(0,1)))^{2}\)satisfy the compatibility condition

$$ \bigl(f_{0}(\cdot ,0),g_{0}(\cdot ,0) \bigr)=(u_{1},v_{1}). $$

Assume that(A1)(A3)hold. Then the problem (1.2) admits a weak solution such that\(u,v\in L^{\infty }(0,\infty ;H^{2}(\varOmega ) \cap H_{0}^{1}(\varOmega ))\), \(u_{t},v_{t}\in L^{\infty }(0,\infty ; H_{0}^{1}(\varOmega ))\), and\(u_{tt},v_{tt}\in L^{2}(0,\infty ,H_{0}^{1}(\varOmega ))\).

Proof

As in the previous assumptions in [2] for the initial conditions \(u_{0},v_{0}\in H^{2}(\varOmega )\cap H_{0}^{1}(\varOmega )\), \(u_{1},v_{1}\in H_{0}^{1}( \varOmega )\), \(f_{0},g_{0}\in H_{0}^{1}(\varOmega ,H^{1}(0,1))\) and the basic functions, we introduce the approximate solutions \((u^{k},v^{k},z_{1}^{k},z_{2}^{k})\), \(k=1,2,3,\ldots \) , in the form

$$\begin{aligned}& u^{k}(t)=\sum_{j=1}^{k}a^{jk}(t)w^{j}, \qquad v^{k}(t)= \sum_{j=1}^{k}b^{jk}(t)w^{j}, \\& z_{1}^{k}(t)=\sum_{j=1}^{k}c^{jk}(t) \phi ^{j},\qquad z_{2}^{k}(t)=\sum _{j=1}^{k}d^{jk}(t) \phi ^{j}, \end{aligned}$$

where \(a^{jk}\), \(b^{jk}\), \(c^{jk}\) and \(d^{jk}\) (\(j=1,2,\ldots,k\)) are determined by the following ordinary differential equations:

$$\begin{aligned}& \textstyle\begin{cases} ( \vert u_{t}^{k} \vert ^{l}u_{tt}^{k},w^{j})+M( \Vert \nabla u^{k}(t) \Vert ^{2})( \nabla u^{k},\nabla w^{j})+(\nabla u_{tt}^{k},\nabla w^{j}) \\ \quad{} -\int _{0}^{t}h_{1}(t-s)(\nabla u^{k}(s),\nabla w^{j})\,ds+\mu _{1}( \nabla z_{1}^{k}(\cdot ,1)),\nabla w^{j})+(f_{1}(u^{k},v^{k}),w^{j})=0, \\ \quad\quad 1 \leq j\leq k, \\ ( \vert v_{t}^{k} \vert ^{l}v_{tt}^{k},w^{j})+M( \Vert \nabla v^{k}(t) \Vert ^{2})( \nabla v^{k},\nabla w^{j})+(\nabla v_{tt}^{k},\nabla w^{j}) \\ \quad{} -\int _{0}^{t}h_{2}(t-s)(\nabla v^{k}(s),\nabla w^{j})\,ds+\mu _{2}( \nabla z_{2}^{k}(\cdot ,1)),\nabla w^{j})+(f_{2}(u^{k},v^{k}),w^{j})=0, \\ \quad\quad 1 \leq j\leq k, \\ z_{1}^{k}(x,0,t)=u_{t}^{k}(x,t),z_{2}^{k}(x,0,t)=v_{t}^{k}(x,t),\end{cases}\displaystyle \end{aligned}$$
(2.1)
$$\begin{aligned}& u^{k}(0)=u^{k}_{0}=\sum _{j=1}^{k} \bigl(u_{0},w^{j} \bigr)w^{j}\rightarrow u_{0}, \quad \text{in } H^{2}(\varOmega ) \cap H_{0}^{1}(\varOmega ) \text{ as } k\rightarrow + \infty , \end{aligned}$$
(2.2)
$$\begin{aligned}& v^{k}(0)=v^{k}_{0}=\sum _{j=1}^{k} \bigl(v_{0},w^{j} \bigr)w^{j}\rightarrow v_{0} ,\quad \text{in } H^{2}(\varOmega ) \cap H_{0}^{1}(\varOmega ) \text{ as } k\rightarrow + \infty , \end{aligned}$$
(2.3)
$$\begin{aligned}& u^{k}_{t}(0)=u^{k}_{1}=\sum _{j=1}^{k} \bigl(u_{1},w^{j} \bigr)w^{j}\rightarrow u_{1} ,\quad \text{in } H_{0}^{1}(\varOmega ) \text{ as }k\rightarrow +\infty , \end{aligned}$$
(2.4)
$$\begin{aligned}& v_{t}^{k}(0)=v_{1}^{k}=\sum _{j=1}^{k} \bigl(v_{1},w^{j} \bigr)w^{j}\rightarrow v_{1} ,\quad \text{in } H_{0}^{1}(\varOmega )\text{ as }k\rightarrow +\infty . \end{aligned}$$
(2.5)

Also

$$\begin{aligned}& \textstyle\begin{cases} (\tau (t)\frac{\partial }{\partial t}z_{1}^{k}+(1-\rho \tau ^{\prime }(t))\frac{\partial }{\partial \rho }z_{1}^{k},\phi ^{j})=0, &1\leq j\leq k, \\ (\tau (t)\frac{\partial }{\partial t}z_{2}^{k}+(1-\rho \tau ^{\prime }(t))\frac{\partial }{\partial \rho }z_{2}^{k},\phi ^{j})=0,&\leq j\leq k,\end{cases}\displaystyle \end{aligned}$$
(2.6)
$$\begin{aligned}& z_{1}^{k}(\rho ,0)=\sum _{j=1}^{k} \bigl(f_{0},\phi ^{j} \bigr)\phi ^{j} \rightarrow f_{0},\quad \text{in } H_{0}^{1} \bigl(\varOmega ,H^{1}(0,1) \bigr) \text{ as }k\rightarrow + \infty . \end{aligned}$$
(2.7)
$$\begin{aligned}& z_{2}^{k}(\rho ,0)=\sum _{j=1}^{k} \bigl(g_{0},\phi ^{j} \bigr)\phi ^{j} \rightarrow g_{0},\quad \text{in }H_{0}^{1} \bigl(\varOmega ,H^{1}(0,1) \bigr) \text{ as }k\rightarrow + \infty . \end{aligned}$$
(2.8)

Noting that \(\frac{l}{2(l+1)}+\frac{1}{2(l+1)}+\frac{1}{2}=1\), by applying the generalized Hölder inequality, we find

$$ \bigl( \bigl\vert u^{k}_{t} \bigr\vert ^{l}u^{k}_{tt},w_{j} \bigr)= \int _{\varOmega } \bigl\vert u^{k}_{t} \bigr\vert ^{l}u^{k}_{tt}w_{j} \,dx\leq \biggl( \int _{\varOmega } \bigl\vert u^{k}_{t} \bigr\vert ^{2(l+1)}\,dx\biggr)^{\frac{l}{2(l+1)}} \bigl\Vert u^{k}_{tt} \bigr\Vert _{2(l+1)} \Vert w_{j} \Vert _{2}. $$

Since \((A1)\) holds, according to the Sobolev embedding the nonlinear terms \(( \vert u^{k}_{t} \vert ^{l}u^{k}_{tt},w_{j})\) and \(( \vert v^{k}_{t} \vert ^{l}v^{k}_{tt},w_{j})\) in (2.1) make sense (see [2]).

A. First estimate.

Since the sequences \(u^{k}_{0}\), \(v^{k}_{0}\), \(u^{k}_{1}\), \(v^{k}_{1}\), \(z^{k}_{1}(\rho ,0)\) and \(z^{k}_{2}(\rho ,0)\) converge and from Lemma 1.3 with employing Gronwall’s lemma, we find \(C_{1}>0\) independent of k such that

$$ E^{k}(t)+\beta \int _{0}^{t} \bigl( \bigl\Vert \nabla z^{k}_{1}(x,1,s) \bigr) \bigr\Vert ^{2}+ \bigl\Vert \nabla z^{k}_{2}(x,1,s)) \bigr\Vert ^{2} )\,ds\leq C_{1}, $$
(2.9)

where

$$ \begin{aligned} E^{k}(t)&=\frac{1}{l+2} \bigl( \bigl\Vert u^{k}_{t} \bigr\Vert ^{l+2}_{l+2}+ \bigl\Vert v^{k}_{t} \bigr\Vert ^{l+2}_{l+2} \bigr)+\frac{b}{2(\gamma +1)} \bigl( \bigl\Vert \nabla u^{k} \bigr\Vert ^{2(\gamma +1)}+ \bigl\Vert \nabla v^{k} \bigr\Vert ^{2(\gamma +1)} \bigr) \\ &\quad{} +\frac{1}{2} \biggl(a- \int _{0}^{t}h_{1}(s)\,ds \biggr) \bigl\Vert \nabla u^{k} \bigr\Vert ^{2}+ \frac{1}{2} \biggl(a- \int _{0}^{t}h_{2}(s)\,ds \biggr) \bigl\Vert \nabla v^{k} \bigr\Vert ^{2} \\ &\quad{}+\frac{1}{2} \bigl( \bigl\Vert \nabla u^{k}_{t} \bigr\Vert ^{2}+ \bigl\Vert \nabla v^{k}_{t} \bigr\Vert ^{2} \bigr) \\ &\quad{}+\frac{1}{2} \bigl[ \bigl(h_{1}\mathbin{o}\nabla u^{k} \bigr) (t)+ \bigl(h_{2}\mathbin{o}\nabla v^{k} \bigr) (t) \bigr] \\ &\quad{} + \xi \tau (t) \int _{0}^{1} \bigl( \bigl\Vert \nabla z^{k}_{1}(x,\rho ,t)\bigr) \bigr\Vert ^{2}+ \bigl\Vert \nabla z^{k}_{2}(x,\rho ,t)) \bigr\Vert ^{2} )\,d\rho \\ &\quad{}+\alpha \int _{\varOmega }u^{k}v^{k}\,dx+(p+q) \int _{\varOmega } \bigl\vert u^{k} \bigr\vert ^{p+1} \bigl\vert v^{k} \bigr\vert ^{q+1} \,dx. \end{aligned}$$

Noting \((A1)\) and the estimate (2.9) yields

$$\begin{aligned}& u^{k},v^{k} \quad \text{are bounded in } L_{\mathrm{loc}}^{\infty } \bigl(0,\infty ,H_{0}^{1}(\varOmega ) \bigr), \end{aligned}$$
(2.10)
$$\begin{aligned}& u^{k}_{t},v^{k}_{t} \quad \text{are bounded in } L_{\mathrm{loc}}^{\infty } \bigl(0, \infty ,H_{0}^{1}(\varOmega ) \bigr), \end{aligned}$$
(2.11)
$$\begin{aligned}& z_{1}^{k}(x,\rho ,t),z_{2}^{k}(x,\rho ,t)\quad \text{are bounded in } L_{\mathrm{loc}}^{\infty } \bigl(0,\infty ,L^{1} \bigl(0,1,H_{0}^{1}(\varOmega ) \bigr) \bigr). \end{aligned}$$
(2.12)

B. The second estimate.

By multiplying the first side of equation (respectively, the second equation) in (2.1) by \(a_{tt}^{jk}\) (respectively, by \(b_{tt}^{jk}\)), by summing j from 1 to k, then

$$ \textstyle\begin{cases} \int _{\varOmega } \vert u_{t}^{k} \vert ^{l} \vert u_{tt}^{k} \vert ^{2}\,dx+\int _{\varOmega }M( \Vert \nabla u^{k} \Vert ^{2})\nabla u^{k}\nabla u_{tt}^{k}\,dx+ \Vert \nabla u_{tt}^{k} \Vert ^{2} \\ \quad =\int _{0}^{t}h_{1}(t-s)\int _{\varOmega } \nabla u^{k}(s)\nabla u_{tt}^{k}(t)\,dx\,ds\\ \quad\quad {}-\mu _{1}\int _{\varOmega }\nabla u_{tt}^{k}\nabla (z_{1}^{k}(x,1,t))\,dx- \int _{\varOmega }f_{1}(u^{k},v^{k})u_{tt}^{k}(t)\,dx, \\ \int _{\varOmega } \vert v_{t}^{k} \vert ^{l} \vert v_{tt}^{k} \vert ^{2}\,dx+\int _{\varOmega }M( \Vert \nabla v^{k} \Vert ^{2})\nabla v^{k}\nabla v_{tt}^{k}\,dx+ \Vert \nabla v_{tt}^{k} \Vert ^{2} \\ \quad =\int _{0}^{t}h_{2}(t-s)\int _{\varOmega } \nabla v^{k}(s)\nabla v_{tt}^{k}(t)\,dx\,ds\\ \quad \quad{}-\mu _{2}\int _{\varOmega }\nabla v_{tt}^{k}\nabla (z_{2}^{k}(x,1,t))\,dx- \int _{\varOmega }f_{2}(u^{k},v^{k})v_{tt}^{k}(t)\,dx.\end{cases} $$
(2.13)

Differentiating (2.6) with respect to t, we get

$$ \textstyle\begin{cases} ((\frac{\tau (t)}{(1-\rho \tau ^{\prime }(t))})^{\prime } \frac{\partial }{\partial t}z_{1}^{k}+\frac{\tau (t)}{(1-\rho \tau ^{\prime }(t))} \frac{\partial ^{2}}{\partial t^{2}}z_{1}^{k}+ \frac{\partial ^{2}}{\partial t\partial \rho }z_{1}^{k},\phi ^{j})=0, \\ ((\frac{\tau (t)}{(1-\rho \tau ^{\prime }(t))})^{\prime } \frac{\partial }{\partial t}z_{2}^{k}+\frac{\tau (t)}{(1-\rho \tau ^{\prime }(t))} \frac{\partial ^{2}}{\partial t^{2}}z_{2}^{k}+ \frac{\partial ^{2}}{\partial t\partial \rho }z_{2}^{k},\phi ^{j})=0.\end{cases}$$

Multiplying the first equation by \(c_{t}^{jk}\) (respectively the second equation by \(d_{t}^{jk}\)), summing over j from 1 to k, we have

$$ \textstyle\begin{cases} \frac{1}{2}(\frac{\tau (t)}{(1-\rho \tau ^{\prime }(t))})^{\prime } \Vert \frac{\partial }{\partial t}z_{1}^{k} \Vert ^{2}+\frac{1}{2} \frac{d}{dt} (\frac{\tau (t)}{(1-\rho \tau ^{\prime }(t))} \Vert \frac{\partial }{\partial t}z_{1}^{k} \Vert ^{2} )+\frac{1}{2}\frac{d}{d\rho } \Vert \frac{\partial }{\partial t}z_{1}^{k} \Vert ^{2}=0, \\ \frac{1}{2}(\frac{\tau (t)}{(1-\rho \tau ^{\prime }(t))})^{\prime } \Vert \frac{\partial }{\partial t}z_{2}^{k} \Vert ^{2}+\frac{1}{2} \frac{d}{dt} (\frac{\tau (t)}{(1-\rho \tau ^{\prime }(t))} \Vert \frac{\partial }{\partial t}z_{2}^{k} \Vert ^{2} )+\frac{1}{2}\frac{d}{d\rho } \Vert \frac{\partial }{\partial t}z_{2}^{k} \Vert ^{2}=0.\end{cases}$$

Integrating over \((0,1)\) with respect to ρ, we obtain

$$ \textstyle\begin{cases} \frac{1}{2}\int _{0}^{1}( \frac{\tau (t)}{(1-\rho \tau ^{\prime }(t))})^{\prime } \Vert \frac{\partial }{\partial t}z_{1}^{k} \Vert ^{2}\,d\rho +\frac{1}{2}\frac{d}{dt} (\int _{0}^{1} \frac{\tau (t)}{(1-\rho \tau ^{\prime }(t))} \Vert \frac{\partial }{\partial t}z_{1}^{k} \Vert ^{2}\,d\rho ) \\ \quad{}+\frac{1}{2} \Vert \frac{\partial }{\partial t}z_{1}^{k}(x,1,t) \Vert ^{2}-\frac{1}{2} \Vert u_{tt}^{k}(x,t) \Vert ^{2}=0, \\ \frac{1}{2}\int _{0}^{1}( \frac{\tau (t)}{(1-\rho \tau ^{\prime }(t))})^{\prime } \Vert \frac{\partial }{\partial t}z_{2}^{k} \Vert ^{2}\,d\rho +\frac{1}{2}\frac{d}{dt} (\int _{0}^{1} \frac{\tau (t)}{(1-\rho \tau ^{\prime }(t))} \Vert \frac{\partial }{\partial t}z_{2}^{k} \Vert ^{2}\,d\rho ) \\ \quad{}+\frac{1}{2} \Vert \frac{\partial }{\partial t}z_{2}^{k}(x,1,t) \Vert ^{2}-\frac{1}{2} \Vert v_{tt}^{k}(x,t) \Vert ^{2}=0.\end{cases} $$
(2.14)

Summing (2.13), (2.14) and as \(M(r)\geq a\), we get

$$ \textstyle\begin{cases} \int _{\varOmega } \vert u_{t}^{k} \vert ^{l} \vert u_{tt}^{k} \vert ^{2}\,dx+ \Vert \nabla u_{tt}^{k} \Vert ^{2}+\frac{1}{2}\frac{d}{dt} (\int _{0}^{1} \frac{\tau (t)}{(1-\rho \tau ^{\prime }(t))} \Vert \frac{\partial }{\partial t}z_{1}^{k} \Vert ^{2}\,d\rho )+\frac{1}{2} \Vert \frac{\partial }{\partial t}z_{1}^{k}(x,1,t) \Vert ^{2} \\ \quad \leq -a\int _{\varOmega }\nabla u^{k}\nabla u_{tt}^{k}\,dx+\frac{1}{2} \Vert u_{tt}^{k}(x,t) \Vert ^{2}-\frac{1}{2}\int _{0}^{1}( \frac{\tau (t)}{(1-\rho \tau ^{\prime }(t))})^{\prime } \Vert \frac{\partial }{\partial t}z_{1}^{k} \Vert ^{2}\,d\rho \\ \quad \quad{}+\int _{0}^{t}h_{1}(t-s)\int _{\varOmega }\nabla u^{k}(s)\nabla u_{tt}^{k}(t) \,dx\,ds-\mu _{1}\int _{\varOmega }\nabla u_{tt}^{k}\nabla z_{1}^{k}(x,1,t) \,dx\\ \quad\quad{} -\int _{\varOmega }f_{1}(u^{k},v^{k})u_{tt}^{k}(t)\,dx, \\ \int _{\varOmega } \vert v_{t}^{k} \vert ^{l} \vert v_{tt}^{k} \vert ^{2}\,dx+ \Vert \nabla v_{tt}^{k} \Vert ^{2}+\frac{1}{2}\frac{d}{dt} (\int _{0}^{1} \frac{\tau (t)}{(1-\rho \tau ^{\prime }(t))} \Vert \frac{\partial }{\partial t}z_{2}^{k} \Vert ^{2}\,d\rho )+\frac{1}{2} \Vert \frac{\partial }{\partial t}z_{2}^{k}(x,1,t) \Vert ^{2} \\ \quad \leq -a\int _{\varOmega }\nabla v^{k}\nabla v_{tt}^{k}\,dx+\frac{1}{2} \Vert v_{tt}^{k}(x,t) \Vert ^{2}-\frac{1}{2}\int _{0}^{1}( \frac{\tau (t)}{(1-\rho \tau ^{\prime }(t))})^{\prime } \Vert \frac{\partial }{\partial t}z_{2}^{k} \Vert ^{2}\,d\rho \\ \quad\quad {}+\int _{0}^{t}h_{2}(t-s)\int _{\varOmega }\nabla v^{k}(s)\nabla v_{tt}^{k}(t) \,dx\,ds-\mu _{2}\int _{\varOmega }\nabla v_{tt}^{k}\nabla z_{2}^{k}(x,1,t) \,dx\\ \quad\quad{} -\int _{\varOmega }f_{2}(u^{k},v^{k})v_{tt}^{k}(t)\,dx. \end{cases} $$
(2.15)

We estimate the right hand side of (2.15) as follows:

From the integration by parts, we have

$$ - \int _{\varOmega }f_{1} \bigl(u^{k},v^{k} \bigr)u_{tt}^{k}(t)\,dx=\alpha \int _{ \varOmega }v_{t}^{k}u_{t}^{k} \,dx-b_{1} \int _{\varOmega } \bigl\vert v^{k} \bigr\vert ^{q+1} \bigl\vert u^{k} \bigr\vert ^{p-1}u^{k}u_{tt}^{k} \,dx. $$

Using the inequality \(ab\leq \frac{1}{2}a^{2}+\frac{1}{2}b^{2}\) and Sobolev–Poincaré inequalities, we obtain

$$ \alpha \int _{\varOmega }v_{t}^{k}u_{t}^{k} \,dx\leq \frac{\alpha }{2} \bigl( \bigl\Vert v_{t}^{k} \bigr\Vert ^{2}+ \bigl\Vert u_{t}^{k} \bigr\Vert ^{2} \bigr)\leq \frac{C_{s}^{2}\alpha }{2} \bigl( \bigl\Vert \nabla v_{t}^{k} \bigr\Vert ^{2}+ \bigl\Vert \nabla u_{t}^{k} \bigr\Vert ^{2} \bigr). $$
(2.16)

On the other hand, by recalling (A-4) and Lemma 1.1 and using Young’s inequality, we get

$$ \begin{aligned}[b] & \biggl\vert \int _{\varOmega } \bigl\vert v^{k} \bigr\vert ^{q+1} \bigl\vert u^{k} \bigr\vert ^{p-1}u^{k}u_{tt}^{k} \,dx\biggr\vert \\ &\quad \leq \frac{1}{2} \int _{\varOmega } \bigl\vert v^{k} \bigr\vert ^{2(q+1)} \bigl\vert u^{k} \bigr\vert ^{2p} \,dx+\frac{1}{2} \bigl\Vert u_{tt}^{k} \bigr\Vert ^{2} \\ &\quad \leq \frac{\eta }{2} \int _{\varOmega } \bigl\vert v^{k} \bigr\vert ^{2(q+1)q}\,dx+ \frac{1}{8\eta }\int _{\varOmega } \bigl\vert u^{k} \bigr\vert ^{2p^{2}}\,dx+\frac{1}{2} \bigl\Vert u_{tt}^{k} \bigr\Vert ^{2} \\ &\quad \leq \frac{\eta }{2} \vert \varOmega \vert ^{\frac{q-1}{2q}} \bigl\Vert v^{k} \bigr\Vert _{4q^{2}}^{4(q+1)q}+ \frac{1}{8\eta } \bigl\Vert u^{k} \bigr\Vert _{2p^{2}}^{2p^{2}}+\frac{1}{2} \bigl\Vert u_{tt}^{k} \bigr\Vert ^{2} \\ &\quad \leq \frac{\eta }{2} \vert \varOmega \vert ^{\frac{q-1}{2q}}C_{s}^{4(q+1)q} \bigl\Vert \nabla v^{k} \bigr\Vert ^{4(q+1)q}+ \frac{C_{s}^{2p^{2}}}{8\eta } \bigl\Vert \nabla u^{k} \bigr\Vert ^{2p^{2}}+\frac{C_{s}^{2}}{2} \bigl\Vert \nabla u_{tt}^{k} \bigr\Vert ^{2}. \end{aligned} $$
(2.17)

Hence from summing (2.16) and (2.17) we deduce that

$$ \begin{aligned}[b] \biggl\vert - \int _{\varOmega }f_{1} \bigl(u^{k},v^{k} \bigr)u_{tt}^{k}(t)\,dx\biggr\vert &\leq \frac{C_{s}^{2}\alpha }{2} \bigl( \bigl\Vert \nabla v_{t}^{k} \bigr\Vert ^{2}+ \bigl\Vert \nabla u_{t}^{k} \bigr\Vert ^{2} \bigr) \\ & \quad{}+\frac{b_{1}\eta }{2} \vert \varOmega \vert ^{\frac{q-1}{2q}}C_{s}^{4(q+1)q} \bigl\Vert \nabla v^{k} \bigr\Vert ^{4(q+1)q} \\ & \quad{}+\frac{b_{1}C_{s}^{2p^{2}}}{8\eta } \bigl\Vert \nabla u^{k} \bigr\Vert ^{2p^{2}}+ \frac{b_{1}C_{s}^{2}}{2} \bigl\Vert \nabla u_{tt}^{k} \bigr\Vert ^{2}. \end{aligned} $$
(2.18)

Similarly

$$ \begin{aligned}[b] \biggl\vert - \int _{\varOmega }f_{2} \bigl(u^{k},v^{k} \bigr)v_{tt}^{k}(t)\,dx\biggr\vert &\leq \frac{C_{s}^{2}\alpha }{2} \bigl( \bigl\Vert \nabla v_{t}^{k} \bigr\Vert ^{2}+ \bigl\Vert \nabla u_{t}^{k} \bigr\Vert ^{2} \bigr) \\ & \quad{}+\frac{b_{2}\eta }{2} \vert \varOmega \vert ^{\frac{p-1}{2p}}C_{s}^{4(p+1)p} \bigl\Vert \nabla u^{k} \bigr\Vert ^{4(p+1)p} \\ & \quad{}+\frac{b_{2}C_{s}^{2p^{2}}}{8\eta } \bigl\Vert \nabla v^{k} \bigr\Vert ^{2q^{2}}+ \frac{b_{2}C_{s}^{2}}{2} \bigl\Vert \nabla v_{tt}^{k} \bigr\Vert ^{2}. \end{aligned}$$
(2.19)

Also by Young’s inequality, we get

$$ \textstyle\begin{cases} \vert \int _{\varOmega }a\nabla u^{k}\nabla u_{tt}^{k}\,dx\vert \leq \eta \Vert \nabla u_{tt}^{k} \Vert ^{2}+\frac{a^{2}}{4\eta } \Vert \nabla u^{k} \Vert ^{2}, \\ \vert \int _{\varOmega }a\nabla v^{k}\nabla v_{tt}^{k}\,dx\vert \leq \eta \Vert \nabla v_{tt}^{k} \Vert ^{2}+\frac{a^{2}}{4\eta } \Vert \nabla v^{k} \Vert ^{2}. \end{cases}$$
(2.20)

We have

$$ \begin{aligned}[b] & \biggl\vert \int _{0}^{t}h_{1}(t-s) \int _{\varOmega }\nabla u^{k}(s) \nabla u_{tt}^{k}(t)\,dx\,ds\biggr\vert \\ &\quad \leq \eta \bigl\Vert \nabla u_{tt}^{k} \bigr\Vert ^{2}+\frac{1}{4\eta } \int _{\varOmega } \biggl( \int _{0}^{t}h_{1}(t-s)\nabla u^{k}(s) \,ds\biggr)^{2}\,dx\\ &\quad \leq \eta \bigl\Vert \nabla u_{tt}^{k} \bigr\Vert ^{2}+\frac{1}{4\eta } \int _{0}^{t}h_{1}(s) \,ds\int _{\varOmega } \int _{0}^{t}h_{1}(t-s) \bigl\vert \nabla u^{k}(s) \bigr\vert ^{2}\,ds\,dx\\ &\quad \leq \eta \bigl\Vert \nabla u_{tt}^{k} \bigr\Vert ^{2}+\frac{a-k}{4\eta } \int _{0}^{t}h_{1}(t-s) \bigl\Vert \nabla u^{k}(s) \bigr\Vert ^{2}\,ds\\ &\quad \leq \eta \bigl\Vert \nabla u_{tt}^{k} \bigr\Vert ^{2}+ \frac{(a-k)h_{1}(0)}{4\eta }\int _{0}^{t} \bigl\Vert \nabla u^{k}(s) \bigr\Vert ^{2}\,ds. \end{aligned} $$
(2.21)

Similarly

$$ \begin{aligned}[b] & \biggl\vert \int _{0}^{t}h_{2}(t-s) \int _{\varOmega }\nabla v^{k}(s)\nabla v_{tt}^{k}(t) \,dx\,ds\biggr\vert \\ &\quad \leq \eta \bigl\Vert \nabla v_{tt}^{k} \bigr\Vert ^{2}+ \frac{(a-k)h_{2}(0)}{4\eta } \int _{0}^{t} \bigl\Vert \nabla v^{k}(s) \bigr\Vert ^{2}\,ds\end{aligned} $$
(2.22)

and

$$ \textstyle\begin{cases} \vert \mu _{1}\int _{\varOmega }\nabla u_{tt}^{k}\nabla z_{1}^{k}(x,1,t) \,dx\vert \leq \eta \mu _{1}^{2} \Vert \nabla u_{tt}^{k} \Vert ^{2}+ \frac{1}{4\eta } \Vert \nabla z_{1}^{k}(x,1,t) \Vert ^{2}, \\ \vert \mu _{2}\int _{\varOmega }\nabla v_{tt}^{k}\nabla z_{2}^{k}(x,1,t) \,dx\vert \leq \eta \mu _{2}^{2} \Vert \nabla v_{tt}^{k} \Vert ^{2}+ \frac{1}{4\eta } \Vert \nabla z_{2}^{k}(x,1,t) \Vert ^{2}. \end{cases} $$
(2.23)

Taking into account (2.18)–(2.23) into (2.15) yields

$$ \textstyle\begin{cases} \int _{\varOmega } \vert u_{t}^{k} \vert ^{l} \vert u_{tt}^{k} \vert ^{2}\,dx+ \Vert \nabla u_{tt}^{k} \Vert ^{2}+\frac{1}{2}\frac{d}{dt} (\int _{0}^{1} \frac{\tau (t)}{1-\rho \tau ^{\prime }(t)} \Vert \frac{\partial }{\partial t}z_{1}^{k} \Vert ^{2}\,d\rho ) +\frac{1}{2} \Vert \frac{\partial }{\partial t}z_{1}^{k}(x,1,t) \Vert ^{2} \\ \quad \leq (\eta (\mu _{1}^{2}+2)+\frac{C_{s}^{2}}{2}) \Vert \nabla u_{tt}^{k} \Vert ^{2} \\ \quad\quad {}+\frac{a^{2}}{4\eta } \Vert \nabla u^{k} \Vert ^{2}+ \frac{C_{s}^{2}\alpha }{2}( \Vert \nabla v_{t}^{k} \Vert ^{2}+ \Vert \nabla u_{t}^{k} \Vert ^{2})+ \frac{b_{1}\eta }{2} \vert \varOmega \vert ^{\frac{q-1}{2q}}C_{s}^{4(q+1)q} \Vert \nabla v^{k} \Vert ^{4(q+1)q} \\ \quad\quad {}+\frac{b_{1}C_{s}^{2p^{2}}}{8\eta } \Vert \nabla u^{k} \Vert ^{2p^{2}}+ \frac{b_{1}C_{s}^{2}}{2} \Vert \nabla u_{tt}^{k} \Vert ^{2} \\ \quad\quad {}+\frac{1}{4\eta } \Vert \nabla z_{1}^{k}(x,1,t) \Vert ^{2}+ \frac{1}{4\eta }(a-k)h_{1}(0)\int _{0}^{t} \Vert \nabla u^{k}(s) \Vert ^{2}\,ds- \frac{1}{2}\int _{0}^{1} ( \frac{\tau (t)}{1-\rho \tau ^{\prime }(t)} )^{\prime } \Vert \frac{\partial }{\partial t}z_{1}^{k} \Vert ^{2}\,d\rho , \\ \int _{\varOmega } \vert v_{t}^{k} \vert ^{l} \vert v_{tt}^{k} \vert ^{2}\,dx+ \Vert \nabla v_{tt}^{k} \Vert ^{2}+\frac{1}{2}\frac{d}{dt} (\int _{0}^{1} \frac{\tau (t)}{1-\rho \tau ^{\prime }(t)} \Vert \frac{\partial }{\partial t}z_{2}^{k} \Vert ^{2}\,d\rho ) +\frac{1}{2} \Vert \frac{\partial }{\partial t}z_{2}^{k}(x,1,t) \Vert ^{2} \\ \quad \leq (\eta (\mu _{2}^{2}+2)+\frac{C_{s}^{2}}{2}) \Vert \nabla v_{tt}^{k} \Vert ^{2} \\ \quad \quad{}+\frac{a^{2}}{4\eta } \Vert \nabla v^{k} \Vert ^{2}+ \frac{C_{s}^{2}\alpha }{2}( \Vert \nabla v_{t}^{k} \Vert ^{2}+ \Vert \nabla u_{t}^{k} \Vert ^{2})+ \frac{b_{2}\eta }{2} \vert \varOmega \vert ^{\frac{p-1}{2p}}C_{s}^{4(p+1)p} \Vert \nabla u^{k} \Vert ^{4(p+1)p} \\ \quad \quad{}+\frac{b_{2}C_{s}^{2p^{2}}}{8\eta } \Vert \nabla v^{k} \Vert ^{2q^{2}}+ \frac{b_{2}C_{s}^{2}}{2} \Vert \nabla v_{tt}^{k} \Vert ^{2} \\ \quad\quad {}+\frac{1}{4\eta } \Vert \nabla z_{2}^{k}(x,1,t) \Vert ^{2}+ \frac{1}{4\eta }(a-k)h_{2}(0)\int _{0}^{t} \Vert \nabla v^{k}(s) \Vert ^{2}\,ds\\ \quad \quad{}-\frac{1}{2}\int _{0}^{1} ( \frac{\tau (t)}{1-\rho \tau ^{\prime }(t)} )^{\prime } \Vert \frac{\partial }{\partial t}z_{2}^{k} \Vert ^{2}\,d\rho .\end{cases}$$

By using (A3) and taking the first estimate (2.9) into account, we infer

$$ \textstyle\begin{cases} \int _{\varOmega } \vert u_{t}^{k} \vert ^{l} \vert u_{tt}^{k} \vert ^{2}\,dx+(1-(\eta (\mu _{1}^{2}+2)+\frac{(1+b_{1})C_{s}^{2}}{2})) \Vert \nabla u_{tt}^{k} \Vert ^{2} \\ \quad{} + \frac{1}{2}\frac{d}{dt} (\int _{0}^{1} \frac{\tau (t)}{1-\rho \tau ^{\prime }(t)} \Vert \frac{\partial }{\partial t}z_{1}^{k} \Vert ^{2}\,d\rho ) \\ \quad{}+\frac{1}{2} \Vert \frac{\partial }{\partial t}z_{1}^{k}(x,1,t) \Vert ^{2} \leq C_{2}+\frac{1}{4\eta }(a-k_{1})h_{1}(0)C_{1}T, \\ \int _{\varOmega } \vert v_{t}^{k} \vert ^{l} \vert v_{tt}^{k} \vert ^{2}\,dx+(1-(\eta (\mu _{2}^{2}+2)+\frac{(1+b_{2})C_{s}^{2}}{2})) \Vert \nabla v_{tt}^{k} \Vert ^{2} \\ \quad{} + \frac{1}{2}\frac{d}{dt} (\int _{0}^{1} \frac{\tau (t)}{1-\rho \tau ^{\prime }(t)} \Vert \frac{\partial }{\partial t}z_{2}^{k} \Vert ^{2}\,d\rho ) \\ \quad{}+\frac{1}{2} \Vert \frac{\partial }{\partial t}z_{2}^{k}(x,1,t) \Vert ^{2} \leq C_{2}+\frac{1}{4\eta }(a-k_{2})h_{2}(0)C_{1}T,\end{cases} $$
(2.24)

where \(C_{2}\) is a positive constant that depends on η, α, a, \(C_{s}\), \(\vert \varOmega \vert \), \(b_{1}\), \(b_{2}\), p, q, \(C_{1}\) for \(i=1,2\).

Integrating (2.24) over (0,t) we obtain

$$ \textstyle\begin{cases} \int _{0}^{t}\int _{\varOmega } \vert u_{t}^{k} \vert ^{l} \vert u_{tt}^{k} \vert ^{2}\,dx\,dt+(1-( \eta (\mu _{1}^{2}+2)+\frac{(1+b_{1})C_{s}^{2}}{2}))\int _{0}^{t} \Vert \nabla u_{tt}^{k}(s) \Vert ^{2}\,ds\\ \quad{}+\int _{0}^{1}\frac{\tau (t)}{1-\rho \tau ^{\prime }(t)} \Vert \frac{\partial }{\partial t}z_{1}^{k} \Vert ^{2}\,d\rho +\frac{1}{2} \int _{0}^{1} \Vert \frac{\partial }{\partial t}z_{1}^{k}(x,1,t) \Vert ^{2}\,dt\leq (C_{2}+ \frac{1}{4\eta }(a-k)h_{1}(0)C_{1}T)T, \\ \int _{0}^{t}\int _{\varOmega } \vert v_{t}^{k} \vert ^{l} \vert v_{tt}^{k} \vert ^{2}\,dx\,dt+(1-( \eta (\mu _{2}^{2}+2)+\frac{(1+b_{2})C_{s}^{2}}{2}))\int _{0}^{t} \Vert \nabla v_{tt}^{k}(s) \Vert ^{2}\,ds\\ \quad{}+\int _{0}^{1}\frac{\tau (t)}{1-\rho \tau ^{\prime }(t)} \Vert \frac{\partial }{\partial t}z_{2}^{k} \Vert ^{2}\,d\rho +\frac{1}{2} \int _{0}^{1} \Vert \frac{\partial }{\partial t}z_{2}^{k}(x,1,t) \Vert ^{2}\,dt\leq (C_{2}+ \frac{1}{4\eta }(a-k)h_{2}(0)C_{1}T)T. \end{cases}$$

For a suitable \(\eta >0\) such that \(1-(\eta (\mu _{i}^{2}+2)+\frac{(1+b_{i})C_{s}^{2}}{2})>0\) for \(i=1,2\), we obtain the second estimate

$$ \begin{aligned}[b] &\int _{0}^{t} \bigl( \bigl\Vert \nabla u_{tt}^{k}(s) \bigr\Vert ^{2}+ \bigl\Vert \nabla v_{tt}^{k}(s) \bigr\Vert ^{2} \bigr) \,ds\\ &\quad{} + \int _{0}^{1} \frac{\tau (t)}{1-\rho \tau ^{\prime }(t)} \biggl( \biggl\Vert \frac{\partial }{\partial t}z_{1}^{k} \biggr\Vert ^{2}+ \biggl\Vert \frac{\partial }{\partial t}z_{2}^{k} \biggr\Vert ^{2} \biggr)\,d\rho \leq C_{3}. \end{aligned} $$
(2.25)

We observe from the estimate (2.9) and (2.25) that there exist subsequences \((u^{m})\) of \((u^{k})\) and \((v^{m})\) of \((v^{k})\) such that

$$\begin{aligned}& \bigl(u^{m},v^{m} \bigr)\rightharpoonup (u,v)\quad \text{weakly star in }L^{\infty } \bigl(0,T,H_{0}^{1}( \varOmega ) \bigr), \end{aligned}$$
(2.26)
$$\begin{aligned}& \bigl(u_{t}^{m},v_{t}^{m} \bigr) \rightharpoonup (u_{t},v_{t})\quad \text{weakly star in } L^{\infty } \bigl(0,T,H_{0}^{1}(\varOmega ) \bigr), \end{aligned}$$
(2.27)
$$\begin{aligned}& \bigl(u_{tt}^{m},v_{tt}^{m} \bigr) \rightharpoonup (u_{tt},v_{tt}) \quad \text{weakly in}L^{2} \bigl(0,T,H_{0}^{1}(\varOmega ) \bigr), \end{aligned}$$
(2.28)
$$\begin{aligned}& \bigl(z_{1}^{m},z_{1}^{m} \bigr) \rightharpoonup (z_{1},z_{2})\quad \text{weakly star in }L^{\infty } \bigl(0,T,H_{0}^{1} \bigl(\varOmega ,L^{2}(0,1) \bigr) \bigr), \end{aligned}$$
(2.29)
$$\begin{aligned}& \biggl(\frac{\partial }{\partial t}z_{1}^{m},\frac{\partial }{\partial t}z_{2}^{m} \biggr)\rightharpoonup \biggl(\frac{\partial }{\partial t}z_{1}, \frac{\partial }{\partial t}z_{2} \biggr)\quad \text{weakly star in }L^{\infty } \bigl(0,T,L^{2} \bigl( \varOmega \times (0,1) \bigr) \bigr). \end{aligned}$$
(2.30)

In the following, we will treat the nonlinear term. From the first estimate (2.9) and Lemma 1.1, we deduce

$$ \begin{aligned} \bigl\Vert \bigl\vert u_{t}^{k} \bigr\vert ^{l}u_{t}^{k} \bigr\Vert _{L^{2}(0,T,L^{2}( \varOmega )}&= \int _{0}^{T} \bigl\Vert u_{t}^{k} \bigr\Vert _{2(l+1)}^{2(l+1)}\,dt\\ &\leq C_{s}^{2(l+1)} \int _{0}^{T} \bigl\Vert \nabla u_{t}^{k} \bigr\Vert _{2}^{2(l+1)} \,dt\leq C_{4} ,\end{aligned}$$

where \(C_{4}\) depends only on \(C_{s}\), \(C_{1}\), Tl.

On the other hand, from the Aubin–Lions theorem (see Lions [9]), we deduce that there exists a subsequence of \((u^{m})\), still denoted by \((u^{m})\), such that

$$ u_{t}^{m}\rightarrow u_{t}\quad \text{strongly in }L^{2} \bigl(0,T,L^{2}( \varOmega ) \bigr) , $$
(2.31)

which implies

$$ u_{t}^{m}\rightarrow u_{t}\quad \text{almost everywhere in } \mathcal{A}. $$
(2.32)

Hence

$$ \bigl\vert u_{t}^{m} \bigr\vert ^{l}u_{t}^{m} \rightarrow \vert u_{t} \vert ^{l}u_{t}\quad \text{almost everywhere in }\mathcal{A}, $$
(2.33)

where \(\mathcal{A}=\varOmega \times (0,T)\). Thus, using (2.31), (2.33) and the Lions lemma, we derive

$$ \bigl\vert u_{t}^{m} \bigr\vert ^{l}u_{t}^{m} \rightharpoonup \vert u_{t} \vert ^{l}u_{t} \quad \text{weakly in } L^{2} \bigl(0,T,L^{2}(\varOmega ) \bigr) ;$$
(2.34)

similarly

$$ \bigl\vert v_{t}^{m} \bigr\vert ^{l}v_{t}^{m} \rightharpoonup \vert v_{t} \vert ^{l}v_{t} \quad \text{weakly in } L^{2} \bigl(0,T,L^{2}(\varOmega ) \bigr) $$
(2.35)

and

$$ \bigl(z_{1}^{m},z_{2}^{m} \bigr) \rightarrow (z_{1},z_{2})\quad \text{strongly in } L^{2} \bigl(0,T,L^{2}(\varOmega ) \bigr), $$

which implies \((z_{1}^{m},z_{2}^{m})\rightarrow (z_{1},z_{2})\) almost everywhere in \(\mathcal{A}\).

The sequences \((u^{m})\) and \((v^{m})\) satisfy

$$ f_{1} \bigl(u^{m},v^{m} \bigr)\rightarrow f_{1}(u,v)\quad \text{strongly in }L^{ \infty } \bigl(0,T,L^{2}(\varOmega ) \bigr) $$
(2.36)

and

$$ f_{2} \bigl(u^{m},v^{m} \bigr)\rightarrow f_{2}(u,v)\quad \text{strongly in }L^{ \infty } \bigl(0,T,L^{2}(\varOmega ) \bigr); $$
(2.37)

we have

$$ \bigl\Vert f_{1} \bigl(u^{m},v^{m} \bigr)-f_{1}(u,v) \bigr\Vert ^{2}= \int _{\varOmega } \bigl\vert \bigl\vert v^{k} \bigr\vert ^{q+1} \bigl\vert u^{k} \bigr\vert ^{p}u^{k}- \vert v \vert ^{q+1} \vert u \vert ^{p}u \vert \bigr\vert ^{2}\,dx. $$

As we add and subtract \(\vert v^{k} \vert ^{q+1} \vert u \vert ^{p}u\) to the previous formula, we obtain

$$\begin{aligned} \begin{aligned}[b] \bigl\Vert f_{1} \bigl(u^{m},v^{m} \bigr)-f_{1}(u,v) \bigr\Vert ^{2}&\leq \int _{\varOmega } \big\vert \bigl\vert v^{k} \bigr\vert ^{q+1} \bigl\vert \bigl\vert u^{k} \bigr\vert ^{p}u^{k}- \vert u \vert ^{p}u \bigr\vert + \vert u \vert ^{p+1} \big\vert \bigl\vert v^{k} \bigr\vert ^{q+1}- \bigl\vert v \bigl\vert ^{q+1} \bigr\vert \bigr\vert ^{2}\,dx\\ &\leq 2 \biggl[ \int _{\varOmega } \bigl\vert v^{k} \bigr\vert ^{2(q+1)} \bigl\vert \bigl\vert u^{k} \bigr\vert ^{p}u^{k}- \vert u \vert ^{p}u \bigr\vert ^{2} \,dx\\ &\quad{} + \int _{\varOmega } \vert u \vert ^{2(p+1)} \bigl\vert \bigl\vert v^{k} \bigr\vert ^{q+1}- \vert v \vert ^{q+1} \bigr\vert ^{2}\,dx\biggr]. \end{aligned} \end{aligned}$$
(2.38)

We use the following elementary inequalities:

$$\begin{aligned}& \bigl\vert \vert a \vert ^{k}- \vert b \vert ^{k} \bigr\vert \leq C \vert a-b \vert \bigl( \vert a \vert ^{k-1}+ \vert b \vert ^{k-1} \bigr), \\& \bigl\vert \vert a \vert ^{k}a- \vert b \vert ^{k}b \bigr\vert \leq C \vert a-b \vert \bigl( \vert a \vert ^{k}+ \vert b \vert ^{k} \bigr), \end{aligned}$$

and

$$ (a+b)^{2}\leq 2 \bigl(a^{2}+b^{2} \bigr), $$

for some constant C, \(\forall k\geq 1\) and \(\forall a,b\in \mathbb{R}\). Hence (2.38) becomes

$$ \begin{aligned}[b] \bigl\Vert f_{1} \bigl(u^{m},v^{m} \bigr)-f_{1}(u,v) \bigr\Vert ^{2}\leq {}&4C \biggl[ \int _{ \varOmega } \bigl\vert v^{k} \bigr\vert ^{2(q+1)} \bigl\vert u^{k}-u \bigr\vert ^{2} \bigl( \bigl\vert u^{k} \bigr\vert ^{2p}+ \vert u \vert ^{2p} \bigr)\,dx\\ &{}+ \int _{\varOmega } \vert u \vert ^{2(p+1)} \bigl\vert v^{k}-v \bigr\vert ^{2} \bigl( \bigl\vert v^{k} \bigr\vert ^{2q}+ \vert v \vert ^{2q} \bigr)\,dx\biggr].\end{aligned} $$
(2.39)

The typical term in the above formula can be estimated as follows.

Noting that \(\frac{l}{2p}+\frac{1}{2q}+\frac{1}{2}=1\), by applying the generalized Hölder inequality, we find

$$ \begin{aligned}[b] & \int _{\varOmega } \bigl\vert v^{k} \bigr\vert ^{2(q+1)} \bigl\vert u^{k}-u \bigr\vert ^{2} \bigl\vert u^{k} \bigr\vert ^{2p}\,dx\\ &\quad \leq \biggl( \int _{\varOmega } \bigl\vert v^{k} \bigr\vert ^{4(q+1)}\,dx\biggr)^{\frac{1}{2}} \biggl( \int _{\varOmega } \bigl\vert u^{k}-u \bigr\vert ^{4q} \,dx\biggr)^{\frac{1}{2q}} \biggl( \int _{\varOmega } \bigl\vert u^{k} \bigr\vert ^{4p^{2}}\,dx\biggr)^{\frac{1}{2p}}. \end{aligned} $$
(2.40)

Recalling \((A4)\), Lemma 1.1 and (2.9), we get

$$ \int _{\varOmega } \bigl\vert v^{k} \bigr\vert ^{2(q+1)} \bigl\vert u^{k}-u \bigr\vert ^{2} \bigl\vert u^{k} \bigr\vert ^{2p}\,dx\leq C \bigl\Vert \nabla \bigl(u^{k}-u \bigr) \bigr\Vert ^{2} .$$
(2.41)

Hence (2.39) yields

$$ \bigl\Vert f_{1} \bigl(u^{m},v^{m} \bigr)-f_{1}(u,v) \bigr\Vert ^{2}\leq C \bigl[ \bigl\Vert \nabla \bigl(u^{k}-u \bigr) \bigr\Vert ^{2}+ \bigl\Vert \nabla \bigl(v^{k}-v \bigr) \bigr\Vert ^{2} \bigr] .$$
(2.42)

As \((u^{m})\), \((v^{m})\) are Cauchy sequences in \(L^{\infty }(0,T,H_{0}^{1}(\varOmega ))\) (we prove it as in [1]) then we deduce (2.36). Similarly we get the convergence (2.37).

By multiplying (2.1) and (2.6) by \(\theta (t)\in \mathcal{D}(0,T)\) and by integrating over \((0,T)\), it follows that

$$ \textstyle\begin{cases} -\frac{1}{l+1}\int _{0}^{T}( \vert u_{t}^{k}(t) \vert ^{l}u_{t}^{k}(t),w^{j}) \theta ^{\prime }(t)\,dt+\int _{0}^{T}M( \Vert \nabla u^{k}(t) \Vert ^{2})( \nabla u^{k}(t),\nabla w^{j})\theta (t)\,dt\\ \quad{}+\int _{0}^{T}(\nabla u_{tt}^{k},\nabla w^{j})\theta (t)\,dt-\int _{0}^{T} \int _{0}^{t}h_{1}(t-s)(\nabla u^{k}(s),\nabla w^{j})\theta (t)\,ds\,dt\\ \quad{}+\mu _{1}\int _{0}^{T}(\nabla z_{1}^{k}(\cdot ,1)),\nabla w^{j})\theta (t) \,dt+\int _{0}^{T}(f_{1}(u^{k},v^{k}),w^{j})\theta (t)\,dt=0, \\ -\frac{1}{l+1}\int _{0}^{T}( \vert v_{t}^{k}(t) \vert ^{l}v_{t}^{k}(t),w^{j}) \theta ^{\prime }(t)\,dt+\int _{0}^{T}M( \Vert \nabla v^{k}(t) \Vert ^{2})( \nabla v^{k}(t),\nabla w^{j})\theta (t)\,dt\\ \quad{}+\int _{0}^{T}(\nabla v_{tt}^{k},\nabla w^{j})\theta (t)\,dt-\int _{0}^{T} \int _{0}^{t}h_{2}(t-s)(\nabla v^{k}(s),\nabla w^{j})\theta (t)\,ds\,dt\\ \quad{}+\mu _{2}\int _{0}^{T}(\nabla z_{2}^{k}(\cdot ,1)),w^{j})\theta (t)\,dt+ \int _{0}^{T}(f_{2}(u^{k},v^{k}),w^{j})\theta (t)\,dt=0, \\ \int _{0}^{T}\int _{0}^{1}\int _{\varOmega }(\tau (t) \frac{\partial }{\partial t}z_{1}^{k}+(1-\rho \tau ^{\prime }(t)) \frac{\partial }{\partial \rho }z_{1}^{k})\phi ^{j}\theta (t)\,dx\,d\rho \,dt=0, \\ \int _{0}^{T}\int _{0}^{1}\int _{\varOmega }(\tau (t) \frac{\partial }{\partial t}z_{2}^{k}+(1-\rho \tau ^{\prime }(t)) \frac{\partial }{\partial \rho }z_{2}^{k})\phi ^{j}\theta (t)\,dx\,d\rho \,dt=0,\end{cases} $$
(2.43)

for all \(j=1,\ldots,k\).

The convergence of (2.26)–(2.30), (2.35), (2.34), (2.36) and (2.37) is sufficient to pass to the limit in (2.43). This completes the proof of the theorem. □

Exponential decay rate

In order to make precise the asymptotic behavior of our solutions, we introduce some functionality to determine a suitable Lyapunov functional equivalent to E.

Theorem 3.1

Assume that(A1)(A3)hold. Then for every\(t_{0}>0\)there exist positive constantsKand\(c^{\prime }\)such that the energy defined by (1.6) obeys the following decay:

$$ E(t)\leq K e^{-c^{\prime }t},\quad \forall t\geq t_{0}. $$
(3.1)

Lemma 3.2

Along a solution of the problem (1.5) the functional

$$ I(t)=\tau (t) \int _{0}^{1}e^{-2\tau (t)\rho } \bigl( \Vert \nabla z_{1} \Vert ^{2}+ \Vert \nabla z_{2} \Vert ^{2} \bigr)\,d\rho $$
(3.2)

satisfies the following estimates:

$$\begin{aligned}& \bigl\vert I(t) \bigr\vert \leq \frac{1}{\xi }E(t), \end{aligned}$$
(3.3)
$$\begin{aligned}& \begin{aligned}[b] I'(t)&\leq -2\tau (t)e^{-2\tau _{1}} \int _{0}^{1} \bigl( \Vert \nabla z_{1} \Vert ^{2}+ \Vert \nabla z_{2} \Vert ^{2} \bigr)\,d\rho + \Vert \nabla u_{t} \Vert ^{2}+ \Vert \nabla v_{t} \Vert ^{2} \\ &\quad{}-(1-d)e^{-2\tau _{1}} \bigl( \bigl\Vert \nabla z_{1}(x,1,t) \bigr\Vert ^{2}+ \bigl\Vert \nabla z_{2}(x,1,t) \bigr\Vert ^{2} \bigr). \end{aligned} \end{aligned}$$
(3.4)

Proof

(ii) A direct derivation of (3.2) gives

$$ I^{\prime }(t)= \int _{0}^{1} \bigl[\tau ^{\prime -2\tau (t)\rho } \bigl( \Vert \nabla z_{1} \Vert ^{2}+ \Vert \nabla z_{2} \Vert ^{2} \bigr) +\tau (t)e^{-2\tau (t)\rho } \bigl( \bigl\Vert \nabla z^{\prime }_{1} \bigr\Vert ^{2}+ \bigl\Vert \nabla z^{\prime }_{2} \bigr\Vert ^{2} \bigr) \bigr]\,d\rho . $$

Recalling (1.3)–(1.4)

$$ \begin{aligned} I^{\prime }(t)&= \int _{0}^{1} \biggl[\tau ^{\prime -2\tau (t) \rho } \bigl( \Vert \nabla z_{1} \Vert ^{2}+ \Vert \nabla z_{2} \Vert ^{2} \bigr) \\ &\quad{} -\tau (t)e^{-2\tau (t) \rho } \bigl(1- \rho \tau ^{\prime }(t) \bigr)\frac{\partial }{\partial \rho } \bigl( \Vert \nabla z_{1} \Vert ^{2}+ \Vert \nabla z_{2} \Vert ^{2} \bigr) \biggr]\,d\rho \\ &=- \int _{0}^{1} \biggl[\frac{\partial }{\partial \rho } \bigl(e^{-2\tau (t) \rho } \bigl(1-\tau ^{\prime }(t)\rho \bigr) \bigl( \Vert \nabla z_{1} \Vert ^{2}+ \Vert \nabla z_{2} \Vert ^{2} \bigr) \bigr) \\ &\quad{}-2\tau (t) e^{-2\tau (t)\rho } \bigl( \Vert \nabla z_{1} \Vert ^{2}+ \Vert \nabla z_{2} \Vert ^{2} \bigr) \biggr]\,d\rho \\ &= \Vert \nabla u_{t} \Vert ^{2}+ \Vert \nabla v_{t} \Vert ^{2}-e^{-2\tau (t)} \bigl(1-\tau ^{ \prime }(t) \bigr) \bigl( \bigl\Vert \nabla z_{1}(x,1,t) \bigr\Vert ^{2}+ \bigl\Vert \nabla z_{2}(x,1,t) \bigr\Vert ^{2} \bigr)-2I(t). \end{aligned}$$

Because the exponential function \(e^{-2\rho \tau (t)}\) decreases on \((0,1)\times (\tau _{0},\tau _{1}) \) and from \((A3)\), we get the results of this lemma. □

Lemma 3.3

Along a solution of the problem (1.5) the functional

$$ \phi (t)=\frac{1}{l+1} \int _{\varOmega } \bigl( \vert u_{t} \vert ^{l}u_{t} u+ \vert v_{t} \vert ^{l}v_{t} v \bigr)\,dx+\int _{\varOmega }\nabla u_{t}\nabla u\,dx+ \int _{\varOmega }\nabla v_{t} \nabla v\,dx$$

verifies the estimates

$$ \begin{aligned}[b] \bigl\vert \phi (t) \bigr\vert &\leq \frac{1}{l+2} \bigl( \Vert u_{t} \Vert ^{l+2}_{l+2}+ \Vert v_{t} \Vert ^{l+2}_{l+2} \bigr)+ \biggl(\frac{(l+1)^{-1}}{l+2}c_{s}^{l+2}+\frac{c}{2} \biggr) \bigl( \Vert \nabla u \Vert ^{l+2}+ \Vert \nabla v \Vert ^{l+2} \bigr) \\ &\quad{} +\frac{1}{2} \bigl( \Vert \nabla u_{t} \Vert ^{2}+ \Vert \nabla v_{t} \Vert ^{2} \bigr) \end{aligned} $$
(3.5)

and

$$ \begin{aligned}[b] \phi ^{\prime }(t)&\leq \frac{1}{l+1} \bigl( \Vert u_{t} \Vert _{l+2}^{l+2}+ \Vert v_{t} \Vert _{l+2}^{l+2} \bigr) \\ &\quad{} + \biggl(\eta (a-k+1)-k+ \biggl(\frac{b_{1}+b_{2}}{2}+\alpha \biggr)C^{2}_{s} \biggr) \bigl( \Vert \nabla u \Vert ^{2}+ \Vert \nabla v \Vert ^{2} \bigr) \\ &\quad{}+\frac{1}{4\eta } \bigl[(h_{1}o\nabla u) (t)+ (h_{2}o\nabla v) (t) \bigr] + \frac{\mu ^{2}_{1}}{4\eta } \bigl\Vert \nabla z_{1}(x,1,t) \bigr\Vert ^{2}+\frac{\mu ^{2}_{2}}{4\eta } \bigl\Vert \nabla z_{2}(x,1,t) \bigr\Vert ^{2} \\ &\quad{} + \Vert \nabla u_{t} \Vert ^{2}+ \Vert \nabla v_{t} \Vert ^{2}. \end{aligned}$$
(3.6)

Proof

(i) Applying Young’s inequality, Sobolev–Poincaré’s inequality and \(L^{l+2}\hookrightarrow L^{2}\), we find

$$\begin{aligned} \bigl\vert \phi (t) \bigr\vert &\leq \frac{1}{l+2} \Vert u_{t} \Vert ^{l+2}_{l+2}+ \frac{(l+1)^{-1}}{l+2} \Vert u \Vert _{l+2}^{l+2}+ \frac{1}{l+2} \Vert v_{t} \Vert ^{l+2}_{l+2}+ \frac{(l+1)^{-1}}{l+2} \Vert v \Vert _{l+2}^{l+2} \\ &\quad{}+\frac{1}{2} \Vert \nabla u_{t} \Vert ^{2}+\frac{1}{2} \Vert \nabla u \Vert ^{2}+ \frac{1}{2} \Vert \nabla v_{t} \Vert ^{2}+ \frac{1}{2} \Vert \nabla v \Vert ^{2} \\ &\leq \frac{1}{l+2} \Vert u_{t} \Vert ^{l+2}_{l+2}+ \frac{(l+1)^{-1}}{l+2}c_{s}^{l+2} \Vert \nabla u \Vert ^{l+2}+\frac{1}{l+2} \Vert v_{t} \Vert ^{l+2}_{l+2}+ \frac{(l+1)^{-1}}{l+2}c_{s}^{l+2} \Vert \nabla v \Vert ^{l+2} \\ &\quad{}+\frac{1}{2} \Vert \nabla u_{t} \Vert ^{2}+\frac{1}{2} \Vert \nabla u \Vert ^{2}+ \frac{1}{2} \Vert \nabla v_{t} \Vert ^{2}+ \frac{1}{2} \Vert \nabla v \Vert ^{2} \\ &\leq \frac{1}{l+2} \bigl( \Vert u_{t} \Vert ^{l+2}_{l+2}+ \Vert v_{t} \Vert ^{l+2}_{l+2} \bigr)+ \biggl( \frac{(l+1)^{-1}}{l+2}c_{s}^{l+2}+ \frac{c}{2} \biggr) \bigl( \Vert \nabla u \Vert ^{l+2}+ \Vert \nabla v \Vert ^{l+2} \bigr) \\ &\quad{} +\frac{1}{2} \bigl( \Vert \nabla u_{t} \Vert ^{2}+ \Vert \nabla v_{t} \Vert ^{2} \bigr). \end{aligned}$$

(ii) Taking a direct derivation of (3.2) and replacing \(\vert u_{t} \vert ^{l}u_{tt}\), \(\vert v_{t} \vert ^{l}v_{tt}\) from the first and seconde equations of (1.5) give

$$\begin{aligned} \begin{aligned}[b] \phi ^{\prime }(t)&=\frac{1}{l+1} \int _{\varOmega } \bigl( \vert u_{t} \vert ^{l}u_{t} \bigr)^{ \prime }u\,dx+ \frac{1}{l+1} \int _{\varOmega } \vert u_{t} \vert ^{l+2} \,dx\\ & \quad{}+\frac{1}{l+1} \int _{\varOmega } \bigl( \vert v_{t} \vert ^{l}v_{t} \bigr)^{\prime }v\,dx+ \frac{1}{l+1}\int _{\varOmega } \vert v_{t} \vert ^{l+2} \,dx\\ & \quad{}+ \int _{\varOmega }\nabla u_{tt}\nabla u\,dx+ \int _{\varOmega }\nabla u_{t} \nabla u_{t} \,dx+ \int _{\varOmega }\nabla v_{tt}\nabla v\,dx+ \int _{ \varOmega }\nabla v_{t}\nabla v_{t} \,dx\\ &= \int _{\varOmega } \bigl[ \vert u_{t} \vert ^{l}u_{tt} \bigr]u\,dx+\frac{1}{l+1} \Vert u_{t} \Vert _{l+2}^{l+2}+ \int _{\varOmega } \bigl[ \vert v_{t} \vert ^{l}v_{tt} \bigr]v\,dx+\frac{1}{l+1} \Vert v_{t} \Vert _{l+2}^{l+2} \\ & \quad{}- \int _{\varOmega }\Delta u_{tt}u\,dx+ \Vert \nabla u_{t} \Vert ^{2}- \int _{\varOmega }\Delta v_{tt}u\,dx+ \Vert \nabla v_{t} \Vert ^{2} \\ &=\frac{1}{l+1} \bigl( \Vert u_{t} \Vert _{l+2}^{l+2}+ \Vert v_{t} \Vert _{l+2}^{l+2} \bigr)+ \int _{\varOmega } \bigl[ \vert u_{t} \vert ^{l}u_{tt}-\Delta u_{tt} \bigr]u\,dx+\int _{\varOmega } \bigl[ \vert v_{t} \vert ^{l}v_{tt} \\ & \quad{}-\Delta v_{tt} \bigr]v\,dx+ \Vert \nabla u_{t} \Vert ^{2}+ \Vert \nabla v_{t} \Vert ^{2} \\ &=\frac{1}{l+1} \bigl( \Vert u_{t} \Vert _{l+2}^{l+2}+ \Vert v_{t} \Vert _{l+2}^{l+2} \bigr)+ \int _{\varOmega } \biggl[-f_{1}(u,v)+M \bigl( \Vert \nabla u \Vert ^{2} \bigr)\Delta u \\ & \quad{}- \int _{0}^{t}h_{1}(t-s)\Delta u(s) \,ds+\mu _{1}\Delta z_{1}(x,1,t) \biggr]u \,dx\\ & \quad{}+ \int _{\varOmega } \biggl[-f_{2}(u,v)+M \bigl( \Vert \nabla v \Vert ^{2} \bigr)\Delta v- \int _{0}^{t}h_{2}(t-s)\Delta v(s) \,ds+\mu _{2}\Delta z_{2}(x,1,t) \biggr]v\,dx\\ & \quad{}+ \Vert \nabla u_{t} \Vert ^{2}+ \Vert \nabla v_{t} \Vert ^{2} \\ &=\frac{1}{l+1} \bigl( \Vert u_{t} \Vert _{l+2}^{l+2}+ \Vert v_{t} \Vert _{l+2}^{l+2} \bigr)-M \bigl( \Vert \nabla u \Vert ^{2} \bigr) \Vert \nabla u \Vert ^{2} \\ & \quad{}+ \int _{\varOmega }\nabla u(t) \int _{0}^{t}h_{1}(t-s)\nabla u(s) \,ds\,dx- \mu _{1} \int _{\varOmega }\nabla z_{1}(x,1,t)\nabla u\,dx\\ & \quad{}-M \bigl( \Vert \nabla v \Vert ^{2} \bigr) \Vert \nabla v \Vert ^{2}+ \int _{\varOmega } \nabla v(t) \int _{0}^{t}h_{2}(t-s)\nabla v(s) \,ds\,dx\\ & \quad{}-\mu _{2} \int _{\varOmega }\nabla z_{2}(x,1,t)\nabla v\,dx+ \Vert \nabla u_{t} \Vert ^{2}+ \Vert \nabla v_{t} \Vert ^{2} \\ & \quad{}-(b_{1}+b_{2}) \int _{\varOmega } \vert v \vert ^{q+1} \vert u \vert ^{p+1}\,dx-2\alpha \int _{ \varOmega }uv\,dx. \end{aligned} \end{aligned}$$
(3.7)

As \(M(r)\geq a\) and making use of Young’s inequality we obtain

$$\begin{aligned} \phi ^{\prime }(t)&\leq \frac{1}{l+1} \bigl( \Vert u_{t} \Vert _{l+2}^{l+2}+ \Vert v_{t} \Vert _{l+2}^{l+2} \bigr)-a \Vert \nabla u \Vert ^{2}+ \int _{ \varOmega }\nabla u(t)\int _{0}^{t}h_{1}(t-s)\nabla u(s) \,ds\,dx \\ & \quad{}+\frac{\mu _{1}^{2}}{4\eta } \bigl\Vert \nabla z_{1}(x,1,t) \bigr\Vert ^{2}+ \eta \Vert \nabla u \Vert ^{2} \\ & \quad{}-a \Vert \nabla v \Vert ^{2}+ \int _{\varOmega }\nabla v(t) \int _{0}^{t}h_{2}(t-s) \nabla v(s) \,ds\,dx \\ & \quad{}+\frac{\mu _{2}^{2}}{4\eta } \bigl\Vert \nabla z_{2}(x,1,t) \bigr\Vert ^{2}+ \eta \Vert \nabla v \Vert ^{2}+ \Vert \nabla u_{t} \Vert ^{2}+ \Vert \nabla v_{t} \Vert ^{2} \\ & \quad{}-(b_{1}+b_{2}) \int _{\varOmega } \vert v \vert ^{q+1} \vert u \vert ^{p+1}\,dx-2\alpha \int _{ \varOmega }uv\,dx. \end{aligned}$$
(3.8)

By use of Young’s inequality, the third term in the right side is estimated as follows:

$$ \begin{aligned} \int _{\varOmega }\nabla u(t) \int _{0}^{t}h_{1}(t-s)\nabla u(s) \,ds\,dx&\leq \int _{0}^{t}h(t-s) \int _{\varOmega } \bigl\vert \nabla u(t) \bigl( \nabla u(s)-\nabla u(t) \bigr) \bigr\vert \,dx\,ds\\ & \quad{}+ \bigl\Vert \nabla u(t) \bigr\Vert ^{2} \int _{0}^{t}h_{1}(t-s)\,ds\\ &\leq (1+\eta ) \bigl\Vert \nabla u(t) \bigr\Vert ^{2} \int _{0}^{t}h_{1}(s)\,ds\\ &\quad{} + \frac{1}{4\eta } \int _{0}^{t}h_{1}(t-s) \bigl\Vert \nabla u(s)- \nabla u(t)) \bigr\Vert ^{2} \,ds\\ &\leq (1+\eta ) (a-k) \bigl\Vert \nabla u(t) \bigr\Vert ^{2}+ \frac{1}{4\eta }(h_{1}o\nabla u) (t). \end{aligned}$$

Similarly

$$ \int _{\varOmega }\nabla v(t) \int _{0}^{t}h_{2}(t-s)\nabla v(s) \,ds\,dx\leq (1+\eta ) (a-k) \bigl\Vert \nabla v(t) \bigr\Vert ^{2}+\frac{1}{4\eta }(h_{2}o \nabla v) (t), $$

and from \((A4)\)

$$ -(b_{1}+b_{2}) \int _{\varOmega } \vert v \vert ^{q+1} \vert u \vert ^{p+1}\,dx-2\alpha \int _{ \varOmega }uv\,dx\leq \biggl(\frac{b_{1}+b_{2}}{2}+\alpha \biggr)C_{s}^{2} \bigl( \Vert \nabla v \Vert ^{2}+ \Vert \nabla u \Vert ^{2} \bigr). $$

Thus, (3.6) is valid. □

Lemma 3.4

Along a solution of the problem (1.5) the functional

$$\begin{aligned} \psi (t) =& \int _{\varOmega } \biggl(\Delta u_{t}-\frac{1}{l+1} \vert u_{t} \vert ^{l}u_{t} \biggr) \int _{0}^{t}h_{1}(t-s) \bigl(u(t)-u(s) \bigr)\,ds\,dx+ \int _{\varOmega } \biggl( \Delta v_{t} \\ &{}-\frac{1}{l+1} \vert v_{t} \vert ^{l}v_{t} \biggr) \int _{0}^{t}h_{2}(t-s) \bigl(v(t)-v(s) \bigr) \,ds\,dx \end{aligned}$$

satisfies the estimates

$$\begin{aligned} \bigl\vert \psi (t) \bigr\vert &\leq \frac{1}{2} \bigl( \Vert \nabla u_{t} \Vert ^{2}+ \Vert \nabla v_{t} \Vert ^{2} \bigr)+\frac{1}{2}(a-k) \biggl(1 \\ & \quad{}+\frac{(l+1)^{-1}}{(l+2)}(a-k)^{l}c_{s}^{l+2} \biggr){} \bigl[ (h_{1}o \nabla u) (t)+(h_{2}o\nabla v) (t) \bigr] \\ & \quad{}+\frac{(l+1)^{-1}}{(l+2)}(a-k)^{l+2}c_{s}^{l+2}2^{2l+1} \bigl( \Vert \nabla u \Vert ^{2(l+1)}+ \Vert \nabla v \Vert ^{2(l+1)} \bigr) \\ & \quad{}+\frac{1}{l+2} \bigl( \Vert u_{t} \Vert _{l+2}^{l+2}+ \Vert v_{t} \Vert _{l+2}^{l+2} \bigr) \end{aligned}$$
(3.9)

and

$$\begin{aligned} \psi ^{\prime }(t) &\leq \delta {} \biggl[ (a-k)+ \frac{(l+1)^{-1}}{(l+2)} \bigl(h_{1}(0) \bigr)^{l+2}c_{s}^{l+2}2^{2(l+1)}+b_{2} \frac{c_{s}^{4(p+1)}}{2} \\ & {} +\frac{c_{s}^{4p}}{2}b_{1} \biggr]M \bigl( \Vert \nabla u \Vert ^{2} \bigr) \Vert \nabla u \Vert ^{2} \\ & {} + \biggl(2\delta (a-k)^{2}+\frac{\alpha c_{s}^{2}}{2} \biggr) \Vert \nabla u \Vert ^{2}+ \biggl(\frac{M( \Vert \nabla u \Vert ^{2})}{4\delta } \\ & {} + \biggl(2\delta +\frac{1}{3\delta }+\frac{\alpha c_{s}^{2}}{2} \biggr) (a-k) \biggr)(h_{1}o\nabla u) (t) \\ & {} -\frac{h_{1}(0)}{4\delta } \biggl(1+\frac{(l+1)^{-1}}{(l+2)} \bigl(h_{1}(0) \bigr)^{l}c_{s}^{l+2} \biggr) \bigl(h_{1}^{\prime }o\nabla u \bigr) (t) \\ & {} + \biggl(\delta - \int _{0}^{t}h_{1}(s)\,ds\biggr) \Vert \nabla u_{t} \Vert ^{2}+ \mu _{1}^{2} \delta \bigl\Vert \nabla z_{1}(x,1,t) \bigr\Vert ^{2} \\ & {} +\frac{1}{l+1} \biggl(1- \int _{0}^{t}h_{1}(s)\,ds\biggr) \Vert u_{t} \Vert _{l+2}^{l+2} \\ & {} +\delta {} \biggl[ (a-k)+\frac{(l+1)^{-1}}{(l+2)} \bigl(h_{2}(0) \bigr)^{l+2}c_{s}^{l+2}2^{2(l+1)}+b_{1} \frac{c_{s}^{4(q+1)}}{2} \\ & {} +\frac{c_{s}^{4q}}{2}b_{2} \biggr]M \bigl( \Vert \nabla v \Vert ^{2} \bigr) \Vert \nabla v \Vert ^{2} \\ & {} + \biggl(2\delta (a-k)^{2}+\frac{\alpha c_{s}^{2}}{2} \biggr) \Vert \nabla v \Vert ^{2}+ \biggl(\frac{M( \Vert \nabla v \Vert ^{2})}{4\delta } \\ & {} + \biggl(2\delta +\frac{1}{3\delta }+\frac{\alpha c_{s}^{2}}{2} \biggr) (a-k) \biggr)(h_{2}o\nabla v) (t) \\ & {} -\frac{h_{2}(0)}{4\delta } \biggl(1+\frac{(l+1)^{-1}}{(l+2)} \bigl(h_{2}(0) \bigr)^{l}c_{s}^{l+2} \biggr) \bigl(h_{2}^{\prime }o\nabla v \bigr) (t) \\ & {} + \biggl(\delta - \int _{0}^{t}h_{2}(s)\,ds\biggr) \Vert \nabla v_{t} \Vert ^{2}+ \mu _{2}^{2} \delta \bigl\Vert \nabla z_{2}(x,1,t) \bigr\Vert ^{2} \\ & {} +\frac{1}{l+1} \biggl(1- \int _{0}^{t}h_{2}(s)\,ds\biggr) \Vert v_{t} \Vert _{l+2}^{l+2}, \end{aligned}$$
(3.10)

where\(\delta >0\)and\(c_{s}\)is the Sobolev embedding constant.

Proof

We have

$$\begin{aligned} \psi (t) =&- \int _{\varOmega }\nabla u_{t} \int _{0}^{t}h_{1}(t-s) \bigl(\nabla u(t)- \nabla u(s) \bigr)\,ds\,dx\\ &{}- \int _{\varOmega }\frac{1}{l+1} \vert u_{t} \vert ^{l}u_{t} \int _{0}^{t}h_{1}(t-s) \bigl(u(t)-u(s) \bigr)\,ds\,dx\\ &{} - \int _{\varOmega }\nabla v_{t} \int _{0}^{t}h_{2}(t-s) \bigl(\nabla v(t)- \nabla v(s) \bigr)\,ds\,dx\\ &{}- \int _{ \varOmega }\frac{1}{l+1} \vert v_{t} \vert ^{l}v_{t} \int _{0}^{t}h_{2}(t-s) \bigl(v(t)-v(s) \bigr)\,ds\,dx. \end{aligned}$$

We use Young’s inequality with the conjugate exponents \(p^{\prime }=\frac{l+2}{l+1}\) and \(q^{\prime }=l+2\), then the second term in the right hand side can be estimated as

$$ \begin{aligned}[b] & \biggl\vert - \int _{\varOmega }\frac{1}{l+1} \vert u_{t} \vert ^{l}u_{t} \int _{0}^{t}h_{1}(t-s) \bigl(u(t)-u(s) \bigr)\,ds\,dx\biggr\vert \\ &\quad \leq \frac{1}{l+1} \biggl\vert \int _{\varOmega } \bigl( \vert u_{t} \vert ^{l}u_{t} \bigr) \biggl( \int _{0}^{t}h_{1}(t-s) \bigl(u(t)-u(s) \bigr) \,ds\biggr)\,dx\biggr\vert \\ &\quad \leq \frac{1}{l+1} \biggl[\frac{1}{p^{\prime }} \int _{\varOmega } \bigl\vert \vert u_{t} \vert ^{l}u_{t} \bigr\vert ^{p^{\prime }}\,dx+ \frac{1}{q^{\prime }} \int _{ \varOmega } \biggl\vert \int _{0}^{t}h_{1}(t-s) \bigl(u(t)-u(s) \bigr)\,ds\biggr\vert ^{q^{\prime }}\,dx\biggr] \\ &\quad \leq \frac{1}{l+1} \biggl[\frac{1}{p^{\prime }} \int _{\varOmega } \bigl( \vert u_{t} \vert ^{l+1} \bigr)^{p^{ \prime }}\,dx+\frac{1}{q^{\prime }} \int _{\varOmega } \biggl( \int _{0}^{t}h_{1}(t-s) \bigl\vert u(t)-u(s) \bigr\vert \,ds\biggr)^{q^{\prime }}\,dx\biggr] \\ &\quad \leq \frac{1}{l+2} \Vert u_{t} \Vert _{l+2}^{l+2} \\ &\quad\quad{} + \frac{(l+1)^{-1}}{l+2} \int _{\varOmega } \biggl[ \int _{0}^{t} \bigl(h_{1}(t-s) \bigr)^{\frac{l+1}{l+2}} \bigl( \bigl(h_{1}(t-s) \bigr)^{\frac{1}{l+2}} \bigl\vert u(t)-u(s) \bigr\vert \bigr)\,ds\biggr]^{l+2}\,dx. \end{aligned} $$
(3.11)

We get by using Hölder’s inequality

$$\begin{aligned} \begin{aligned}[b] & \int _{\varOmega } \biggl[ \int _{0}^{t} \bigl(h_{1}(t-s) \bigr)^{ \frac{l+1}{l+2}} \bigl( \bigl(h_{1}(t-s) \bigr)^{\frac{1}{l+2}} \bigl\vert u(t)-u(s) \bigr\vert \bigr)\,ds\biggr]^{l+2}\,dx\\ &\quad \leq \int _{\varOmega } \biggl[ \biggl( \int _{0}^{t} \bigl( \bigl(h_{1}(t-s) \bigr)^{ \frac{l+1}{l+2}} \bigr)^{p^{\prime }}\,ds\biggr)^{\frac{1}{p^{\prime }}} \biggl( \int _{0}^{t} \bigl( \bigl(h_{1}(t-s) \bigr)^{\frac{1}{l+2}} \bigl\vert u(t)-u(s) \bigr\vert \bigr)^{q^{\prime }} \,ds\biggr)^{ \frac{1}{q^{\prime }}} \biggr]^{l+2}\,dx\\ &\quad \leq \int _{\varOmega } \biggl[ \biggl( \int _{0}^{t}h_{1}(t-s)\,ds\biggr)^{ \frac{l+1}{l+2}} \biggl( \int _{0}^{t}h_{1}(t-s) \bigl\vert u(t)-u(s) \bigr\vert ^{l+2}\,ds\biggr)^{ \frac{1}{l+2}} \biggr]^{l+2}\,dx\\ &\quad \leq \biggl( \int _{0}^{t}h_{1}(t-s)\,ds\biggr)^{l+1} \int _{0}^{t}h_{1}(t-s) \bigl\Vert u(t)-u(s) \bigr\Vert _{l+2}^{l+2}\,ds\\ &\quad \leq (a-k)^{l+1}c_{s}^{l+2} \int _{0}^{t}\sqrt{h_{1}(t-s)} \sqrt{h_{1}(t-s)} \bigl\Vert \nabla u(t)-\nabla u(s) \bigr\Vert ^{l+1} \bigl\Vert \nabla u(t)-\nabla u(s) \bigr\Vert \,ds\\ &\quad \leq (a-k)^{l+1}c_{s}^{l+2} \biggl( \frac{1}{2} \int _{0}^{t}h_{1}(t-s) \bigl\Vert \nabla u(t)-\nabla u(s) \bigr\Vert ^{2l+2}\,ds\\ &\quad\quad{} +\frac{1}{2} \int _{0}^{t}h_{1}(t-s) \bigl\Vert \nabla u(t)-\nabla u(s) \bigr\Vert ^{2}\,ds\biggr) \\ &\quad \leq (a-k)^{l+1}c_{s}^{l+2} \biggl( \frac{1}{2} \int _{0}^{t}h_{1}(t-s) \bigl\Vert 2 \nabla u(t) \bigr\Vert ^{2l+2}\,ds+\frac{1}{2}(h_{1}o \nabla u) (t) \biggr) \\ &\quad \leq (a-k)^{l+1}c_{s}^{l+2} \biggl(2^{2l+1}(a-k) \bigl\Vert \nabla u(t) \bigr\Vert ^{2(l+1)}+\frac{1}{2}(h_{1}o \nabla u) (t) \biggr). \end{aligned} \end{aligned}$$
(3.12)

Combining (3.12) with (3.11) we obtain

$$ \begin{aligned}[b] &\biggl\vert - \int _{\varOmega }\frac{1}{l+1} \vert u_{t} \vert ^{l}u_{t} \int _{0}^{t}h_{1}(t-s) \bigl(u(t)-u(s) \bigr)\,ds\,dx\biggr\vert \\ &\quad \leq \frac{1}{l+2} \Vert u_{t} \Vert _{l+2}^{l+2} \\ &\quad \quad{}+\frac{(l+1)^{-1}}{l+2} \biggl[(a-k)^{l+1}c_{s}^{l+2} \biggl(2^{2l+1}(a-k) \bigl\Vert \nabla u(t) \bigr\Vert ^{2(l+1)}+\frac{1}{2}(h_{1}o\nabla u) (t) \biggr) \biggr]. \end{aligned} $$
(3.13)

In the same way, we get

$$ \begin{aligned}[b] & \biggl\vert - \int _{\varOmega }\nabla u_{t} \int _{0}^{t}h_{1}(t-s) \bigl( \nabla u(t)-\nabla u(s) \bigr)\,ds\,dx\biggr\vert \\ &\quad \leq \frac{1}{2} \Vert \nabla u_{t} \Vert ^{2}+ \frac{1}{2}\int _{\varOmega } \biggl( \int _{0}^{t}h_{1}(t-s) \bigl\vert \nabla u(t)-\nabla u(s) \bigr\vert \,ds\biggr)^{2} \,dx\\ &\quad \leq \frac{1}{2} \Vert \nabla u_{t} \Vert ^{2}+ \frac{1}{2}(a-k) (h_{1}o \nabla u) (t). \end{aligned} $$
(3.14)

Similarly

$$\begin{aligned} \textstyle\begin{cases} \vert -\int _{\varOmega }\frac{1}{l+1} \vert v_{t} \vert ^{l}v_{t}\int _{0}^{t}h_{2}(t-s)(v(t)-v(s)) \,ds\,dx\vert \leq \frac{1}{l+2} \Vert v_{t} \Vert ^{l+2} \\ \quad{}+\frac{(l+1)^{-1}}{l+2} [(a-k)^{l+1}c_{s}^{l+2} (2^{2l+1}(a-k) \Vert \nabla v(t) \Vert ^{2(l+1)}+\frac{1}{2}(h_{2}o\nabla v)(t) ) ] \\ \vert -\int _{\varOmega }\nabla v_{t}\int _{0}^{t}h(t-s)(\nabla v(t)- \nabla v(s))\,ds\,dx\vert \leq \frac{1}{2} \Vert \nabla v_{t} \Vert ^{2}+ \frac{1}{2}(a-k)(h_{2}o\nabla v)(t).\end{cases}\displaystyle \end{aligned}$$
(3.15)

Combining (3.13),(3.14) and (3.15), we deduce (i).

(ii) We use the Leibnitz formula and the first and second equations of (1.5) to find

$$\begin{aligned} \psi ^{\prime }(t)&= \int _{\varOmega } \bigl(\Delta u_{tt}- \vert u_{t} \vert ^{l}u_{tt} \bigr)\int _{0}^{t}h_{1}(t-s) \bigl(u(t)-u(s) \bigr)\,ds\,dx \\ & \quad{}+ \int _{\varOmega } \biggl(\Delta u_{t}-\frac{1}{l+1} \vert u_{t} \vert ^{l}u_{t} \biggr) \biggl(\int _{0}^{t} \bigl(h_{1}^{\prime }(t-s) \bigl(u(t)-u(s) \bigr)+h_{1}(t-s)u_{t}(t) \bigr)\,ds\biggr)\,dx \\ & \quad{}+ \int _{\varOmega } \bigl(\Delta v_{tt}- \vert v_{t} \vert ^{l}v_{tt} \bigr) \int _{0}^{t}h_{2}(t-s) \bigl(v(t)-v(s) \bigr) \,ds\,dx \\ & \quad{}+ \int _{\varOmega } \biggl(\Delta v_{t}-\frac{1}{l+1} \vert v_{t} \vert ^{l}v_{t} \biggr) \biggl(\int _{0}^{t} \bigl(h_{2}^{\prime }(t-s) \bigl(v(t)-v(s) \bigr)+h_{2}(t-s)v_{t}(t) \bigr)\,ds\biggr)\,dx \\ &= \int _{\varOmega }f_{1}(u,v) \int _{0}^{t}h_{1}(t-s) \bigl(u(t)-u(s) \bigr)\,ds\,dx \\ &\quad{} + \int _{\varOmega }M \bigl( \Vert \nabla u \Vert ^{2} \bigr) \nabla u(t) \int _{0}^{t}h_{1}(t-s) \bigl(\nabla u(t)-\nabla u(s) \bigr)\,ds\,dx \\ & \quad{}- \int _{\varOmega } \int _{0}^{t}h_{1}(t-s)\nabla u(s) \,ds\int _{0}^{t}h_{1}(t-s) \bigl( \nabla u(t)-\nabla u(s) \bigr)\,ds\,dx \\ &\quad{} +\mu _{1} \int _{\varOmega }\nabla z_{1}(x,1,t) \int _{0}^{t}h_{1}(t-s) \bigl( \nabla u(t)-\nabla u(s) \bigr)\,ds\,dx \\ &\quad{} - \int _{\varOmega }\nabla u_{t} \int _{0}^{t}h_{1}'(t-s) \bigl(\nabla u(t)- \nabla u(s) \bigr)\,ds\,dx \\ &\quad{} -\frac{1}{l+1} \int _{\varOmega } \vert u_{t} \vert ^{l}u_{t} \int _{0}^{t}h_{1}'(t-s) \bigl(u(t)-u(s) \bigr) \,ds\,dx \\ &\quad{} - \Vert \nabla u_{t} \Vert ^{2} \int _{0}^{t}h_{1}(s)\,ds- \frac{1}{l+1} \Vert u_{t} \Vert _{l+2}^{l+2} \int _{0}^{t}h_{1}(s)\,ds \\ &\quad{} + \int _{\varOmega }f_{2}(u,v) \int _{0}^{t}h_{2}(t-s) \bigl(u(t)-u(s) \bigr)\,ds\,dx \\ &\quad{} + \int _{\varOmega }M \bigl( \Vert \nabla v \Vert ^{2} \bigr) \nabla v(t) \int _{0}^{t}h_{2}(t-s) \bigl( \nabla v(t)-\nabla v(s) \bigr)\,ds\,dx \\ &\quad{} - \int _{\varOmega } \int _{0}^{t}h_{2}(t-s)\nabla v(s) \,ds\int _{0}^{t}h_{2}(t-s) \bigl( \nabla v(t)-\nabla v(s) \bigr)\,ds\,dx \\ & \quad{} +\mu _{2} \int _{\varOmega }\nabla z_{2}(x,1,t) \int _{0}^{t}h_{2}(t-s) \bigl( \nabla v(t)-\nabla v(s) \bigr)\,ds\,dx \\ & \quad{}- \int _{\varOmega }\nabla v_{t} \int _{0}^{t}h_{2}^{\prime }(t-s) \bigl( \nabla v(t)-\nabla v(s) \bigr)\,ds\,dx \\ &\quad{} -\frac{1}{l+1} \int _{\varOmega } \vert v_{t} \vert ^{l}v_{t} \int _{0}^{t}h_{2}^{\prime }(t-s) \bigl(v(t)-v(s) \bigr)\,ds\,dx \\ &\quad{} - \Vert \nabla v_{t} \Vert ^{2} \int _{0}^{t}h_{2}(s)\,ds- \frac{1}{l+1} \Vert v_{t} \Vert _{l+2}^{l+2} \int _{0}^{t}h_{2}(s)\,ds \\ &=I_{1}+I_{2}+I_{3}+I_{4}+I_{5}+I_{6}- \Vert \nabla u_{t} \Vert ^{2} \int _{0}^{t}h_{1}(s)\,ds- \frac{1}{l+1} \Vert u_{t} \Vert ^{l+2} \int _{0}^{t}h_{1}(s) \,ds \\ & \quad{}- \Vert \nabla v_{t} \Vert ^{2} \int _{0}^{t}h_{2}(s)\,ds- \frac{1}{l+1} \Vert v_{t} \Vert ^{l+2} \int _{0}^{t}h_{2}(s)\,ds, \end{aligned}$$
(3.16)

where

$$ \textstyle\begin{cases} I_{1}=\int _{\varOmega }M( \Vert \nabla u \Vert ^{2})\nabla u(t)\int _{0}^{t}h_{1}(t-s)( \nabla u(t)-\nabla u(s))\,ds\,dx\\ \hphantom{I_{1}=}{}+\int _{\varOmega }M( \Vert \nabla v \Vert ^{2})\nabla v(t)\int _{0}^{t}h_{2}(t-s)(\nabla v(t)-\nabla v(s))\,ds\,dx, \\ I_{2}=-\int _{\varOmega }\int _{0}^{t}h_{1}(t-s)\nabla u(s)\,ds\int _{0}^{t}h_{1}(t-s)( \nabla u(t)-\nabla u(s))\,ds\,dx\\ \hphantom{I_{2}=}{}-\int _{\varOmega }\int _{0}^{t}h_{2}(t-s)\nabla v(s)\,ds\int _{0}^{t}h_{2}(t-s)(\nabla v(t)-\nabla v(s))\,ds\,dx, \\ I_{3}=\mu _{1}\int _{\varOmega }\nabla z_{1}(x,1,t)\int _{0}^{t}h_{1}(t-s)( \nabla u(t)-\nabla u(s))\,ds\,dx\\ \hphantom{I_{3}=}{}+\mu _{2}\int _{\varOmega }\nabla z_{2}(x,1,t)\int _{0}^{t}h_{2}(t-s)( \nabla v(t)-\nabla v(s))\,ds\,dx,\end{cases}$$

and

$$ \textstyle\begin{cases} I_{4}=-\int _{\varOmega }\nabla u_{t}\int _{0}^{t}h_{1}^{\prime }(t-s)( \nabla u(t)-\nabla u(s))\,ds\,dx\\ \hphantom{I_{4}=}{}-\int _{\varOmega }\nabla v_{t}\int _{0}^{t}h_{2}^{\prime }(t-s)(\nabla v(t)- \nabla v(s))\,ds\,dx, \\ I_{5}=-\frac{1}{l+1}\int _{\varOmega } \vert u_{t} \vert ^{l}u_{t}\int _{0}^{t}h_{1}^{ \prime }(t-s)(u(t)-u(s))\,ds\,dx\\ \hphantom{I_{5}=}{}-\frac{1}{l+1}\int _{\varOmega } \vert v_{t} \vert ^{l}v_{t}\int _{0}^{t}h_{2}^{ \prime }(t-s)(v(t)-v(s))\,ds\,dx, \\ I_{6}=\int _{\varOmega }f_{1}(u,v)\int _{0}^{t}h_{1}(t-s)(u(t)-u(s))\,ds\,dx\\ \hphantom{I_{6}=}{}+\int _{\varOmega }f_{2}(u,v)\int _{0}^{t}h_{2}(t-s)(u(t)-u(s))\,ds\,dx.\end{cases} $$

Next we will estimate \(I_{1},\ldots,I_{6}\).

For \(I_{1}\), by applying Hölder’s and Young’s inequalities, we obtain

$$\begin{aligned} \begin{aligned}[b] \vert I_{1} \vert &\leq M \bigl( \Vert \nabla u \Vert ^{2} \bigr) \int _{\varOmega } \bigl\vert \nabla u(t) \bigr\vert \biggl( \int _{0}^{t}h_{1}(s)\,ds\biggr)^{\frac{1}{2}} \biggl( \int _{0}^{t}h_{1}(t-s) \bigl\vert \nabla u(t)-\nabla u(s) \bigr\vert ^{2}\,ds\biggr)^{ \frac{1}{2}}\,dx\\ & \quad{} +M \bigl( \Vert \nabla v \Vert ^{2} \bigr) \int _{\varOmega } \bigl\vert \nabla v(t) \bigr\vert \biggl( \int _{0}^{t}h_{2}(s) \,ds\biggr)^{\frac{1}{2}} \biggl( \int _{0}^{t}h_{2}(t-s) \bigl\vert \nabla v(t)- \nabla v(s) \bigr\vert ^{2}\,ds\biggr)^{\frac{1}{2}}\,dx\\ &\leq M \bigl( \Vert \nabla u \Vert ^{2} \bigr) \biggl[\delta \int _{\varOmega } \bigl\vert \nabla u(t) \bigr\vert ^{2}\int _{0}^{t}h_{1}(s)\,ds\,dx\\ &\quad{} +\frac{1}{4\delta } \int _{\varOmega } \int _{0}^{t}h_{1}(t-s) \bigl\vert \nabla u(t)-\nabla u(s) \bigr\vert ^{2}\,ds\,dx\biggr] \\ & \quad{}+M \bigl( \Vert \nabla v \Vert ^{2} \bigr) \biggl[\delta \int _{\varOmega } \bigl\vert \nabla v(t) \bigr\vert ^{2} \int _{0}^{t}h_{2}(s)\,ds\,dx\\ &\quad{} +\frac{1}{4\delta } \int _{\varOmega } \int _{0}^{t}h_{2}(t-s) \bigl\vert \nabla v(t)-\nabla v(s) \bigr\vert ^{2}\,ds\,dx\biggr] \\ &\leq M \bigl( \Vert \nabla u \Vert ^{2} \bigr) \biggl(\delta (a-k) \bigl\Vert \nabla u(t) \bigr\Vert ^{2}+\frac{1}{4\delta }(h_{1}o \nabla u) (t) \biggr) \\ &\quad{} +M \bigl( \Vert \nabla v \Vert ^{2} \bigr) \biggl(\delta (a-k) \bigl\Vert \nabla v(t) \bigr\Vert ^{2}+\frac{1}{4\delta }(h_{2}o \nabla v) (t) \biggr). \end{aligned} \end{aligned}$$
(3.17)

Similarly,

$$\begin{aligned}& \begin{aligned}[b] \vert I_{2} \vert &\leq \delta \int _{\varOmega } \biggl( \int _{0}^{t}h_{1}(t-s) \bigl\vert \nabla u(s) \bigr\vert \,ds\biggr)^{2}\,dx+ \frac{1}{4\delta } \int _{\varOmega } \biggl( \int _{0}^{t}h_{1}(t-s) \bigl\vert \nabla u(t)-\nabla u(s) \bigr\vert \,ds\biggr)^{2}\,dx \\ & \quad{}+\delta \int _{\varOmega } \biggl( \int _{0}^{t}h_{2}(t-s) \bigl\vert \nabla v(s) \bigr\vert \,ds\biggr)^{2}\,dx\\ &\quad{} + \frac{1}{4\delta } \int _{\varOmega } \biggl( \int _{0}^{t}h_{2}(t-s) \bigl\vert \nabla v(t)-\nabla v(s) \bigr\vert \,ds\biggr)^{2}\,dx\\ &\leq \delta \int _{\varOmega } \biggl( \int _{0}^{t}h_{1}(t-s) \bigl( \bigl\vert \nabla u(s)- \nabla u(t) \bigr\vert + \bigl\vert \nabla u(t) \bigr\vert \bigr)\,ds\biggr)^{2}\,dx\\ &\quad{} +\frac{1}{4\delta } \biggl( \int _{0}^{t}h_{1}(s) \,ds\biggr) (h_{1}o\nabla u) (t) \\ & \quad{}+\delta \int _{\varOmega } \biggl( \int _{0}^{t}h_{2}(t-s) \bigl( \bigl\vert \nabla v(s)- \nabla v(t) \bigr\vert + \bigl\vert \nabla v(t) \bigr\vert \bigr)\,ds\biggr)^{2}\,dx\\ &\quad{} +\frac{1}{4\delta } \biggl( \int _{0}^{t}h_{2}(s) \,ds\biggr) (h_{2}o\nabla v) (t) \\ &\leq 2\delta \bigl\Vert \nabla u(t) \bigr\Vert ^{2} \biggl( \int _{0}^{t}h_{1}(t)\,ds\biggr)^{2}\,dx+ \biggl(2\delta +\frac{1}{4\delta } \biggr) \biggl( \int _{0}^{t}h_{1}(s)\,ds\biggr) (h_{1}o \nabla u) (t) \\ & \quad{}+2\delta \bigl\Vert \nabla v(t) \bigr\Vert ^{2} \biggl( \int _{0}^{t}h_{2}(t)\,ds\biggr)^{2} \,dx+ \biggl(2\delta +\frac{1}{4\delta } \biggr) \biggl( \int _{0}^{t}h_{2}(s)\,ds\biggr) (h_{2}o\nabla v) (t) \\ &\leq 2\delta \bigl\Vert \nabla u(t) \bigr\Vert ^{2}(a-k)^{2}+ \biggl(2\delta + \frac{1}{4\delta } \biggr) (a-k) (h_{1}o\nabla u) (t)+2\delta \bigl\Vert \nabla v(t) \bigr\Vert ^{2}(a-k)^{2} \\ &\quad{} + \biggl(2 \delta +\frac{1}{4\delta } \biggr) (a-k) (h_{2}o\nabla v) (t), \end{aligned} \end{aligned}$$
(3.18)
$$\begin{aligned}& \begin{aligned}[b] \vert I_{3} \vert &\leq \delta \bigl(\mu _{1}^{2} \bigl\Vert \nabla z_{1}(x,1,t) \bigr\Vert ^{2}+ \mu _{2}^{2} \bigl\Vert \nabla z_{2}(x,1,t) \bigr\Vert ^{2} \bigr) \\ &\quad{} + \frac{(a-k)}{4\delta }(h_{1}o \nabla u) (t)+\frac{(a-k)}{4\delta }(h_{2}o\nabla v) (t), \end{aligned} \\& \begin{aligned} \vert I_{4} \vert &\leq \delta \int _{\varOmega } \vert \nabla u_{t} \vert ^{2}\,dx+ \frac{1}{4\delta }\int _{\varOmega } \biggl( \int _{0}^{t} \bigl\vert h_{1}^{\prime }(t-s) \bigr\vert \bigl\vert \nabla u(t)- \nabla u(s) \bigr\vert \,ds\biggr)^{2}\,dx\\ & \quad{}+\delta \int _{\varOmega } \vert \nabla v_{t} \vert ^{2}\,dx+\frac{1}{4\delta } \int _{\varOmega } \biggl( \int _{0}^{t} \bigl\vert h_{2}^{\prime }(t-s) \bigr\vert \bigl\vert \nabla v(t)- \nabla v(s) \bigr\vert \,ds\biggr)^{2}\,dx\\ &\leq \delta \Vert \nabla u_{t} \Vert ^{2}+ \frac{1}{4\delta } \int _{0}^{t} \bigl(-h_{1}^{ \prime }(t-s) \bigr)\,ds\int _{\varOmega } \int _{0}^{t}(-h_{1}^{\prime 2} \,ds\,dx\\ & \quad{}+\delta \Vert \nabla v_{t} \Vert ^{2}+ \frac{1}{4\delta } \int _{0}^{t} \bigl(-h_{2}^{ \prime }(t-s) \bigr)\,ds\int _{\varOmega } \int _{0}^{t}(-h_{2}^{\prime 2} \,ds\,dx\\ &\leq \delta \Vert \nabla u_{t} \Vert ^{2}- \frac{h_{1}(0)}{4\delta } \bigl(h_{1}^{\prime }o\nabla u \bigr) (t)+\delta \Vert \nabla v_{t} \Vert ^{2}- \frac{h_{2}(0)}{4\delta } \bigl(h_{2}^{\prime }o\nabla u \bigr) (t), \end{aligned} \end{aligned}$$
(3.19)

and using the fact that \(l\leq \gamma \)

$$\begin{aligned} \vert I_{5} \vert &\leq \frac{1}{l+2} \bigl( \Vert u_{t} \Vert _{l+2}^{l+2}+ \Vert v_{t} \Vert _{l+2}^{l+2} \bigr)+\frac{(l+1)^{-1}}{l+2} \biggl[ \bigl(h_{1}(0) \bigr)^{l+1} \int _{0}^{t} \bigl(-h_{1}^{\prime }(t-s) \bigr) \bigl\Vert u(t)-u(s) \bigr\Vert _{l+2}^{l+2} \,ds \\ & \quad{}+ \bigl(h_{2}(0) \bigr)^{l+1} \int _{0}^{t} \bigl(-h_{2}^{\prime }(t-s) \bigr) \bigl\Vert v(t)-v(s) \bigr\Vert _{l+2}^{l+2} \,ds\biggr] \\ &\leq \frac{1}{l+2} \bigl( \Vert u_{t} \Vert _{l+2}^{l+2}+ \Vert u_{t} \Vert _{l+2}^{l+2} \bigr) \\ &\quad{} + \frac{(l+1)^{-1}}{l+2}c_{s}^{l+2} \biggl[ \bigl(h_{1}(0) \bigr)^{l+1} \int _{0}^{t} \bigl(-h_{1}'(t-s) \bigr) \bigl\Vert \nabla u(t)-\nabla u(s) \bigr\Vert ^{l+2} \,ds \\ & \quad{}+ \bigl(h_{2}(0) \bigr)^{l+1} \int _{0}^{t} \bigl(-h_{2}'(t-s) \bigr) \bigl\Vert \nabla v(t)-\nabla v(s) \bigr\Vert ^{l+2} \,ds\biggr] \\ &\leq \frac{1}{l+2} \bigl( \Vert u_{t} \Vert _{l+2}^{l+2}+ \Vert u_{t} \Vert _{l+2}^{l+2} \bigr) \\ &\quad{}+\frac{(l+1)^{-1}}{l+2}c_{s}^{l+2} \bigl(h_{1}(0) \bigr)^{l+1} \biggl[\delta 2^{2(l+1)}h_{1}(0) \bigl\Vert \nabla u(t) \bigr\Vert ^{2(l+1)}- \frac{1}{4\delta } \bigl(h_{1}^{\prime }o \nabla u \bigr) (t) \biggr] \\ & \quad{}+\frac{(l+1)^{-1}}{l+2}c_{s}^{l+2} \bigl(h_{2}(0) \bigr)^{l+1} \biggl[\delta 2^{2(l+1)}h_{2}(0) \bigl\Vert \nabla v(t) \bigr\Vert ^{2(l+1)}-\frac{1}{4\delta } \bigl(h_{2}^{\prime }o\nabla v \bigr) (t) \biggr] \\ &\leq \frac{1}{l+2} \bigl( \Vert u_{t} \Vert _{l+2}^{l+2}+ \Vert u_{t} \Vert _{l+2}^{l+2} \bigr) \\ &\quad{} + \frac{(l+1)^{-1}}{l+2}c_{s}^{l+2} \bigl(h_{1}(0) \bigr)^{l+1} \biggl[\delta 2^{2(l+1)}h_{1}(0)M \bigl( \bigl\Vert \nabla u(t) \bigr\Vert ^{2} \bigr) \bigl\Vert \nabla u(t) \bigr\Vert ^{2} \\ & \quad{} -\frac{1}{4\delta } \bigl(h_{1}^{\prime }o\nabla u \bigr) (t) \biggr] \\ & \quad{}+\frac{(l+1)^{-1}}{l+2}c_{s}^{l+2} \bigl(h_{2}(0) \bigr)^{l+1} \biggl[\delta 2^{2(l+1)}h_{2}(0)M \bigl( \bigl\Vert \nabla v(t) \bigr\Vert ^{2} \bigr) \bigl\Vert \nabla v(t) \bigr\Vert ^{2} \\ &\quad{}- \frac{1}{4\delta } \bigl(h_{2}^{\prime }o\nabla v \bigr) (t) \biggr]. \end{aligned}$$
(3.20)

For \(I_{6}\), we have

$$\begin{aligned}& \begin{aligned}[b] I_{6}&=\alpha \int _{\varOmega }v(t) \int _{0}^{t}h_{1}(t-s) \bigl(u(t)-u(s) \bigr) \,ds\,dx+\alpha \int _{\varOmega }u(t) \int _{0}^{t}h_{2}(t-s) \bigl(v(t)-v(s) \bigr)\,ds\,dx \\ &\quad{} +b_{1} \int _{\varOmega } \vert v \vert ^{q+1} \vert u \vert ^{p-1}u \int _{0}^{t}h_{1}(t-s) \bigl(u(t)-u(s) \bigr)\,ds\,dx\\ &\quad{} +b_{2} \int _{\varOmega } \vert u \vert ^{p+1} \vert v \vert ^{q-1}v \int _{0}^{t}h_{2}(t-s) \bigl(v(t)-v(s) \bigr) \,ds\,dx\\ &=I_{6}^{0}+b_{1}I_{6}^{1}+b_{2}I_{6}^{2}, \end{aligned} \end{aligned}$$
(3.21)
$$\begin{aligned}& \bigl\vert I^{0}_{6} \bigr\vert \leq \frac{\alpha c^{2}_{s}}{2} \bigl( \bigl\Vert \nabla u(t) \bigr\Vert ^{2}+ \bigl\Vert \nabla v(t) \bigr\Vert ^{2}+(a-k) \bigl[(h_{1}o \nabla u) (t)+(h_{2}o\nabla v) (t) \bigr] \bigr) , \end{aligned}$$
(3.22)

and

$$ \bigl\vert I^{1}_{6} \bigr\vert \leq \frac{1}{2} \int _{\varOmega } \bigl( \vert v \vert ^{2(q+1)}+ \vert u \vert ^{2p} \bigr) \int _{0}^{t} \bigl\vert h_{1}(t-s) (u(t)- u(s) \bigr\vert \,ds\,dx=I^{11}_{6}+I^{12}_{6}. $$
(3.23)

By using the Young and Hölder inequalities and Lemma 1.1, we find

$$ \begin{aligned}[b] I^{11}_{6}&=\frac{1}{2} \int _{\varOmega } \vert v \vert ^{2(q+1)} \int _{0}^{t} \bigl\vert h_{1}(t-s) \bigl( u(t)- u(s) \bigr) \bigr\vert \,ds\,dx\\ &\leq \frac{\delta }{2} \int _{\varOmega } \vert v \vert ^{4(q+1)}\,dx+ \frac{1}{8\delta } \int _{\varOmega } \biggl[ \int _{0}^{t}h_{1}(t-s) \bigl\vert u(t)- u(s) \bigr\vert \,ds\biggr]^{2}\,dx\\ &\leq \frac{c^{4(q+1)}_{s}\delta }{2} \Vert \nabla v \Vert ^{4(q+1)}+ \frac{1}{8\delta }(a-k) (h_{1}o\nabla u) (t) \\ &\leq \frac{c^{4(q+1)}_{s}\delta }{2}M \bigl( \Vert \nabla v \Vert ^{2} \bigr) \Vert \nabla v \Vert ^{2}+ \frac{1}{8\delta }(a-k) (h_{1}o\nabla u) (t). \end{aligned}$$
(3.24)

Also by following a similar technique to above, we get

$$ \begin{aligned} \bigl\vert I^{12}_{6} \bigr\vert &\leq \frac{c^{4p}_{s}\delta }{2}M \bigl( \Vert \nabla u \Vert ^{2} \bigr) \Vert \nabla u \Vert ^{2}+\frac{1}{8\delta }(a-k) (h_{1}o\nabla u) (t). \end{aligned}$$
(3.25)

Hence

$$ \begin{aligned}[b] \bigl\vert I^{1}_{6} \bigr\vert &\leq \frac{c^{4(q+1)}_{s}\delta }{2}M \bigl( \Vert \nabla v \Vert ^{2} \bigr) \Vert \nabla v \Vert ^{2}+ \frac{c^{4p}_{s}\delta }{2}M \bigl( \Vert \nabla u \Vert ^{2} \bigr) \Vert \nabla u \Vert ^{2} \\ &\quad{} +\frac{1}{4\delta }(a-k) (h_{1}o\nabla u) (t). \end{aligned}$$
(3.26)

Similarly

$$ \begin{aligned}[b] \bigl\vert I^{2}_{6} \bigr\vert &\leq \frac{c^{4(p+1)}_{s}\delta }{2}M \bigl( \Vert \nabla u \Vert ^{2} \bigr) \Vert \nabla u \Vert ^{2}+ \frac{c^{4q}_{s}\delta }{2}M \bigl( \Vert \nabla v \Vert ^{2} \bigr) \Vert \nabla v \Vert ^{2} \\ &\quad{} +\frac{1}{4\delta }(a-k) (h_{2}o\nabla v) (t). \end{aligned}$$
(3.27)

Summing (3.22), (3.26) and (3.27), we get

$$\begin{aligned} \begin{aligned}[b] I_{6}&\leq \biggl(b_{2}\frac{c^{4(p+1)}_{s}\delta }{2}+ \frac{c^{4p}_{s}\delta }{2}b_{1} \biggr)M \bigl( \Vert \nabla u \Vert ^{2} \bigr) \Vert \nabla u \Vert ^{2} \\ &\quad{} + \biggl(b_{2} \frac{c^{4q}_{s}\delta }{2}+b_{1} \frac{c^{4(q+1)}_{s}\delta }{2} \biggr)M \bigl( \Vert \nabla v \Vert ^{2} \bigr) \Vert \nabla v \Vert ^{2} \\ &\quad{}+ \biggl(\frac{\alpha c^{2}_{s}}{2}+\frac{1}{4\delta } \biggr) (a-k) \bigl((h_{1}o\nabla u) (t)+(h_{2}o \nabla v) (t) \bigr)+ \frac{\alpha c^{2}_{s}}{2} \bigl( \Vert \nabla u \Vert ^{2}+ \Vert \nabla v \Vert ^{2} \bigr). \end{aligned} \end{aligned}$$
(3.28)

Combining (3.16) and (3.17)–(3.28), we complete the proof. □

Proof of Theorem 3.1

Now, for \(M,\varepsilon _{1}>0\), we introduce the following functional:

$$ F(t)=ME(t)+I(t)+\psi (t)+\varepsilon _{1}\phi (t) .$$
(3.29)

Firstly we prove that \(F(t)\) is equivalent to \(E(t)\); for this we show that \(F(t)\) verified the following boundedness:

$$ \kappa _{1} E(t)\leq F(t)\leq \kappa _{2} E(t) $$
(3.30)

for some positive constants \(\kappa _{2}\), \(\kappa _{2}\).

We recall (3.3), (3.5), and (3.9) and, using the fact that \(l\leq \gamma \), we get

$$ \begin{aligned} &\bigl\vert I(t)+\psi (t)+\varepsilon _{1} \phi (t) \bigr\vert \\ &\quad \leq \frac{\varepsilon _{1} +1}{l+2} \bigl( \Vert u_{t} \Vert _{l+2}^{l+2}+ \Vert v_{t} \Vert _{l+2}^{l+2} \bigr)+ \frac{\varepsilon _{1}+1}{2} \bigl( \Vert \nabla u_{t} \Vert ^{2} \bigr)+ \Vert \nabla v_{t} \Vert ^{2})) \\ &\quad \quad{}+ \biggl(\frac{\varepsilon _{1}c}{2}+\frac{(l+1)^{-1}}{(l+2)}c^{l+2}_{s} \bigl(\varepsilon _{1}+2^{2l+1}(a-k)^{l+2} \bigr) \biggr) \bigl( \Vert \nabla u \Vert ^{2( \gamma +1)}+ \Vert \nabla v \Vert ^{2(\gamma +1)} \bigr) \\ &\quad \quad{}+\frac{a-k}{2} \biggl(1+\frac{(l+1)^{-1}}{(l+2)}(a-k)^{l}c^{l+2}_{s} \biggr) \bigl((h_{1}o\nabla u) (t)+(h_{2}o\nabla v) (t) \bigr)+ \frac{1}{\xi }E(t) \\ &\quad \leq \kappa E(t) , \end{aligned}$$

where \(\kappa >0\) depending the \(\varepsilon _{1}\), a, b, l, c, \(c_{s}\), kξ. For the choice of \(M=\kappa +\epsilon \) with \(\epsilon >0\), we get \(F(t)\sim E(t) \).

By recalling (1.9), (3.4), (3.6), (3.10) and (A2), we deduce that

$$\begin{aligned} F'(t)&\leq \biggl(\mu ^{2}\delta +\varepsilon _{1}\frac{\mu ^{2}}{4\eta }-(1-d)e^{-2 \tau _{1}}-M\beta \biggr) \bigl( \bigl\Vert \nabla z_{1}(x,1,t)\bigr) \bigr\Vert ^{2}+ \bigl\Vert \nabla z_{2}(x,1,t)) \bigr\Vert ^{2} ) \\ &\quad{} -2\tau (t)e^{-2\tau _{1}} \int _{0}^{1} \bigl( \Vert \nabla z_{1} \Vert ^{2}+ \Vert \nabla z_{2} \Vert ^{2} \bigr)\,d\rho \\ &\quad{}- (\varepsilon _{1} \biggl[k-\eta (a-k+1)- \biggl( \frac{b_{1}+b_{2}}{2}+\alpha \biggr)c_{s}^{2} \biggr] \\ &\quad{}- \biggl(2\delta (a-k)^{2}-\delta {} \biggl[ (a-k)+ \frac{(l+1)^{-1}}{(l+2)}(h_{2}c_{s})^{l+2}2^{2(l+1)}+ \omega \biggr]M_{0} \biggr) \bigl( \Vert \nabla u \Vert ^{2}+ \Vert \nabla v \Vert ^{2} \bigr) \\ &\quad{}-\frac{1}{l+1}(h_{0}-1-\varepsilon _{1}) \bigl( \Vert u_{t} \Vert _{l+2}^{l+2}+ \Vert v_{t} \Vert _{l+2}^{l+2} \bigr) \\ &\quad{} -(h_{0}- \delta -M\lambda -1- \varepsilon _{1}) \bigl( \Vert \nabla u_{t} \Vert ^{2}+ \Vert \nabla v_{t} \Vert ^{2} \bigr) \\ &\quad{}+ \biggl[(\frac{M}{2}-\frac{h_{1}}{4\delta } \biggl(1+ \frac{(l+1)^{-1}}{(l+2)}h_{1}^{l}c_{s}^{l+2} \biggr)-\frac{1}{\zeta } \biggl( \frac{\varepsilon _{1}}{4\eta }+\frac{M_{0}}{4\delta } \\ &\quad{}+ \biggl(2\delta +\frac{1}{3\delta }+\frac{\alpha c^{2}_{s}}{2} \biggr) (a-k) \biggr) \biggr] \bigl( \bigl(h_{1}'o\nabla u \bigr) (t)+ \bigl(h_{2}'o\nabla v \bigr) (t) \bigr),\quad \forall t\geq t_{0}>0, \end{aligned}$$

where \(M_{0}=\max \{M \Vert \nabla u \Vert ^{2},M \Vert \nabla v \Vert ^{2}\}\), \(h_{0}=\min \{\int _{0}^{t_{0}}h_{1}(s)\,ds,\int _{0}^{t_{0}}h_{2}(s)\,ds\}\), \(h_{1}=\min \{h_{1}(0), h_{2}(0)\}\), \(h_{2}=\max \{h_{1}(0),h_{2}(0)\}\), \(\omega =\max \{b_{1}\frac{c_{s}^{4(q+1)}}{2}+\frac{c_{s}^{4q}}{2}b_{2},b_{2}\frac{c_{s}^{4(p+1)}}{2}+\frac{c_{s}^{4p}}{2}b_{1}\}\) and \(\zeta =\max \{\zeta _{1}, \zeta _{1}\}\).

Let \(\epsilon >0\) be sufficiently small so M is fixed, we take \(h_{0}-M\lambda -1>\varepsilon _{1}\) and δ small enough such that

$$\begin{aligned} a_{3}=h_{0}-1-\varepsilon _{1}>0\quad \text{and}\quad a_{4}=h_{0}-\delta -M\lambda -1-\varepsilon _{1}. \end{aligned}$$

Further, we choose η small enough such that

$$\begin{aligned}& a_{1}=\mu _{1}^{2}\delta +\varepsilon _{1}\frac{\mu ^{2}}{4\eta }-(1-d)e^{-2\tau _{1}}-M\beta >0, \\& \begin{aligned} a_{2}={}&\varepsilon _{1} \biggl[k-\eta (a-k+1)- \biggl( \frac{b_{1}+b_{2}}{2}+\alpha \biggr)c_{s}^{2} \biggr]-2 \delta (a-k)^{2} \\ &{}-\delta \biggl((a-k)+\frac{(l+1)^{-1}}{(l+2)}(h_{2}c_{s})^{l+2}2^{2(l+1)}+ \omega \biggr)M_{0}>0, \end{aligned} \end{aligned}$$

and

$$ a_{5}=\frac{M}{2}-\frac{h_{1}}{4\delta } \biggl(1+ \frac{(l+1)^{-1}}{(l+2)}h_{1}^{l}c_{s}^{l+2} \biggr)-\frac{1}{\zeta } \biggl( \frac{\varepsilon _{1}}{4\eta }+\frac{M_{0}}{4\delta }+ \biggl(2\delta +\frac{1}{3\delta }+ \frac{\alpha c_{s}^{2}}{2} \biggr) (a-k) \biggr)< 0. $$

Thus

$$ \begin{aligned}[b] F^{\prime }(t)&\leq -a_{3} \frac{1}{l+2} \bigl( \Vert u_{t} \Vert _{l+2}^{l+2}+ \Vert v_{t} \Vert _{l+2}^{l+2} \bigr)-a_{2} \bigl( \Vert \nabla u \Vert ^{2}+ \Vert \nabla v \Vert ^{2} \bigr) \\ &\quad{} -2\tau (t)e^{-2\tau _{1}} \int _{0}^{1} \bigl( \Vert \nabla z_{1} \Vert ^{2}+ \Vert \nabla z_{2} \Vert ^{2} \bigr)\,d\rho \\ & \quad{}+a_{1} \bigl( \bigl\Vert \nabla z_{1}(x,1,t) \bigr\Vert ^{2}+ \bigl\Vert \nabla z_{2}(x,1,t) \bigr\Vert ^{2} \bigr)-a_{4} \bigl( \Vert \nabla u_{t} \Vert ^{2}+ \Vert \nabla v_{t} \Vert ^{2} \bigr) \\ &\quad{} +a_{5} \bigl[ \bigl(h_{1}^{\prime }o \nabla u \bigr) (t)+ \bigl(h_{2}^{\prime }o \nabla v \bigr) (t) \bigr] \\ &\leq -mE(t)-cE^{\prime }(t), \end{aligned} $$
(3.31)

where \(m=\min \{\frac{2e^{-2\tau _{1}}}{\xi },2\frac{a_{2}}{a},a_{3}\}\) and \(c=\min \{\frac{a_{1}}{\beta },\frac{a_{4}}{\lambda },-2a_{5}\}\).

Let \(L(t)=F(t)+cE(t)\sim E(t)\). From (3.31), we get

$$ L^{\prime }(t)\leq -c^{\prime }L(t),\quad \forall t\geq t_{0}, $$
(3.32)

for some \(c^{\prime }>0\). A simple integration over \((t_{0},t)\) yields

$$ L(t)\leq L(t_{0})e^{-c^{\prime }(t-t_{0})},\quad \forall t\geq t_{0}. $$
(3.33)

Thanks to the equivalence between L and E, we obtain (3.1). □

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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, 90 (2020). https://doi.org/10.1186/s13661-020-01390-9

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MSC

  • 35L90
  • 74G25
  • 35B40
  • 26A51

Keywords

  • Viscoelastic term
  • Global existence
  • Lyapunov functional
  • Delay function
  • Faedo–Galerkin method
  • Exponential decay