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Blow-up of solutions for a quasilinear system with degenerate damping terms

Abstract

In this work, we consider a quasilinear system of viscoelastic equations with degenerate damping and general source terms. According to some suitable hypothesis, we study the blow-up of solutions. This is the general case of the recent results of Boulaaras’ works (Bull. Malays. Math. Sci. Soc. 43:725–755, 2020) and (Appl. Anal. 99:1724–1748, 2020).

Introduction

In this paper, we consider the following problem:

$$ \textstyle\begin{cases} \vert u_{t} \vert ^{\eta }u_{tt}-M( \Vert \nabla u \Vert _{2}^{2}) \Delta u+\int _{0}^{t}h_{1}(t-s)\Delta u(s)\,ds-\Delta u_{tt}+ ( \vert u \vert ^{k}+ \vert v \vert ^{l} ) \vert u_{t} \vert ^{j-1}u_{t} \\ \quad =f_{1} ( u,v ) ,\quad ( x,t ) \in \Omega \times ( 0,T ) , \\ \vert v_{t} \vert ^{\eta }v_{tt}-M( \Vert \nabla v \Vert _{2}^{2}) \Delta v+\int _{0}^{t}h_{2}(t-s)\Delta v(s)\,ds-\Delta v_{tt}+ ( \vert v \vert ^{\theta }+ \vert u \vert ^{ \varrho } ) \vert v_{t} \vert ^{s-1}v_{t} \\ \quad =f_{2} ( u,v ) ,\quad ( x,t ) \in \Omega \times ( 0,T ) , \\ u ( x,t ) =v ( x,t ) =0, \quad ( x,t ) \in \partial \Omega \times ( 0,T ) , \\ u ( x,0 ) =u_{0} ( x ) ,\qquad u_{t} ( x,0 ) =u_{1} ( x ) , \quad x\in \Omega , \\ v ( x,0 ) =v_{0} ( x ) ,\qquad v_{t} ( x,0 ) =v_{1} ( x ) , \quad x\in \Omega , \end{cases} $$
(1.1)

where \(k,l,\theta ,\varrho \geq 0\); \(j,s\geq 1\) for \(N=1,2\), and \(0\leq j\), \(s\leq \frac{N+2}{N-2}\) for \(N\geq 3\); and \(\eta \geq 0\) for \(N=1,2\) and \(0<\eta \leq \frac{2}{N-2}\) for \(N\geq 3\), \(h_{i}(\cdot):R^{+}\rightarrow R^{+}\) \((i=1,2)\) are positive relaxation functions which will be specified later. \(( \vert ( \cdot ) \vert ^{a}+ \vert ( \cdot ) \vert ^{b} ) \vert ( \cdot ) _{t} \vert ^{\tau -1} ( \cdot ) _{t}\) and \(-\Delta ( \cdot ) _{tt}\) are the degenerate damping term and the dispersion term, respectively, and \(M(\sigma )\) is a nonnegative locally Lipschitz function for \(\gamma ,\sigma \geq 0\) like \(M(\sigma )=\alpha _{1}+\alpha _{2}\sigma ^{\gamma }\). Especially, we select \(\alpha _{1}=\alpha _{2}=1\), and

$$ \textstyle\begin{cases} f_{1}(u,v)=a_{1} \vert u+v \vert ^{2(p+1)}(u+v)+b_{1} \vert u \vert ^{p}.u. \vert v \vert ^{p+2}, \\ f_{2}(u,v)=a_{1} \vert u+v \vert ^{2(p+1)}(u+v)+b_{1} \vert v \vert ^{p}.v. \vert u \vert ^{p+2}. \end{cases} $$
(1.2)

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 (for further details, see the references [3, 59, 12, 17, 20]). If \(\eta \geq 0 \), this type of problem has been studied by many authors. For more depth, here are some papers that focused on the study of this damping (for example, see ([10, 11, 13, 16, 19, 25, 27, 28]). The effect of the degenerate damping terms often appears in many applications and practical problems and turns a lot of systems into different problems worth studying. Recently, the stability, the asymptotic behavior, and blowing up of evolution systems with time degenerate damping have been studied by many authors, see [1, 2, 18]. The most important is the source term with nonlinear functions \(f_{1}\) and \(f_{2}\) satisfying appropriate conditions. In physics they appear in several issues and theories. Many researchers also touched on this type of problem in several different issues, where the global existence of solutions, stability, and blow-up of solutions were studied. For more information, the reader is referred to [1, 11, 15, 18, 2224, 26].

Most recently, if \(\gamma =0\), \(\alpha _{1}=1\) our problem (1.1) was studied in [14]. Under some restrictions on the initial datum, standard conditions on relaxation functions, the authors established the global existence and proved the general decay of solutions. Based on all of the above results, we believe that the combination of these terms of damping (memory term, degenerate damping, dispersion, and the source terms) 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. In the last section we prove our main result.

Preliminaries

We prove the blow-up result under the following suitable assumptions.

(A1) \(h_{i}: \mathbb{R}_{+}\rightarrow \mathbb{R}_{+}\) are differentiable and decreasing functions such that

$$\begin{aligned} &h_{i}(t)\geq 0 , \quad 1- \int _{0}^{\infty }h_{i} ( s ) \,ds=l_{i}>0, \quad i=1,2. \end{aligned}$$
(2.1)

(A2) There exist constants \(\xi _{1},\xi _{2}>0\) such that

$$\begin{aligned} &h_{i}^{\prime } ( t ) \leq -\xi _{i} h_{i} ( t ) , \quad t\geq 0, i=1,2. \end{aligned}$$
(2.2)

Lemma 2.1

There exists a function \(F(u, v)\) such that

$$\begin{aligned} F(u, v) =&\frac{1}{2(\rho +2)} \bigl[u f_{1}(u, v)+v f_{2}(u, v) \bigr] \\ =&\frac{1}{2(\rho +2)} \bigl[a_{1} \vert u+v \vert ^{2(p+2)}+2 b_{1} \vert u v \vert ^{p+2} \bigr] \geq 0, \end{aligned}$$

where

$$\begin{aligned} \frac{\partial F}{\partial u}=f_{1}(u, v), \qquad \frac{\partial F}{\partial v}=f_{2}(u, v), \end{aligned}$$

we take \(a_{1}=b_{1} = 1 \) for convenience.

Lemma 2.2

[22] There exist two positive constants \(c_{0}\) and \(c_{1}\) such that

$$ \frac{c_{0}}{2(\rho +2)} \bigl( \vert u \vert ^{2(p+2)}+ \vert v \vert ^{2(p+2)} \bigr) \leq F(u, v) \leq \frac{c_{1}}{2(\rho +2)} \bigl( \vert u \vert ^{2(\rho +2)}+ \vert v \vert ^{2(p+2)} \bigr). $$
(2.3)

Theorem 2.3

Assume that (2.1) and (2.2) hold. Let

$$ \textstyle\begin{cases} -1< p< \frac{4-n}{n-2}, & n\geq 3; \\ p\geq -1, & n=1,2. \end{cases} $$
(2.4)

Then, for any initial data

$$ (u_{0},u_{1},v_{0},v_{1})\in \mathcal{H}, $$

problem (1.1) has a unique solution for some \(T>0\)

$$\begin{aligned}& u,v\in C\bigl([0,T]; H^{2}(\Omega )\cap H^{1}_{0}( \Omega )\bigr), \\& u_{t}\in C\bigl([0,T]; H^{1}_{0}(\Omega ) \bigr)\cap L^{j+1}(\Omega ), \\& v_{t}\in C\bigl([0,T]; H^{1}_{0}(\Omega ) \bigr)\cap L^{s+1}(\Omega ), \end{aligned}$$

where

$$\begin{aligned} \mathcal{H} =& H^{1}_{0}(\Omega )\times L^{2}(\Omega )\times H^{1}_{0}( \Omega ) \times L^{2}(\Omega ). \end{aligned}$$

Now, we define the energy functional.

Lemma 2.4

Assume that (2.1),(2.2), and (2.4) hold, let \((u,v)\) be a solution of (1.1), then \(E(t)\) is nonincreasing, that is,

$$\begin{aligned} E(t) =&\frac{1}{\eta +2} \bigl[ \Vert u_{t} \Vert _{\eta +2}^{\eta +2}+ \Vert v_{t} \Vert _{\eta +2}^{\eta +2} \bigr]+\frac{1}{2} \bigl[ \Vert \nabla u_{t} \Vert _{2}^{2}+ \Vert \nabla v_{t} \Vert _{2}^{2} \bigr] \\ &{}+\frac{1}{2(\gamma +1)} \bigl[ \Vert \nabla u \Vert _{2}^{2(\gamma +1)}+ \Vert \nabla v \Vert _{2}^{2(\gamma +1)} \bigr] \\ &{}+\frac{1}{2} \biggl[ \biggl(1- \int _{0}^{t}h_{1}(s)\,ds \biggr) \Vert \nabla u \Vert _{2}^{2}+ \biggl(1- \int _{0}^{t}h_{2}(s)\,ds \biggr) \Vert \nabla v \Vert _{2}^{2} \biggr] \\ &{}+\frac{1}{2} \bigl[(h_{1}o\nabla u) (t)+(h_{2}o\nabla v) (t) \bigr]- \int _{\Omega }F(u,v)\,dx \end{aligned}$$
(2.5)

satisfies

$$\begin{aligned} E'(t) \leq & \frac{1}{2} \bigl[\bigl(h'_{1}o \nabla u\bigr) (t)+\bigl(h'_{2}o\nabla v\bigr) \bigr](t)-\frac{1}{2} \bigl[h_{1}(t) \Vert \nabla u \Vert _{2}^{2}+h_{2}(t) \Vert \nabla v \Vert _{2}^{2} \bigr] \\ &{}- \int _{\Omega }\bigl( \vert u \vert ^{k}+ \vert v \vert ^{l}\bigr) \vert u_{t} \vert ^{j+1}\,dx- \int _{\Omega }\bigl( \vert v \vert ^{\theta }+ \vert u \vert ^{ \varrho }\bigr) \vert v_{t} \vert ^{s+1}\,dx \\ \leq & 0. \end{aligned}$$
(2.6)

Proof

By multiplying (1.1)1, (1.1)2 by \(u_{t}\), \(v_{t}\) and integrating over Ω, we get

$$\begin{aligned}& \frac{d}{dt} \biggl\{ \frac{1}{\eta +2} \Vert u_{t} \Vert _{\eta +2}^{ \eta +2}+\frac{1}{\eta +2} \Vert v_{t} \Vert _{\eta +2}^{\eta +2}+ \frac{1}{2} \Vert \nabla u_{t} \Vert _{2}^{2}+\frac{1}{2} \Vert \nabla v_{t} \Vert _{2}^{2} \\& \qquad {}+\frac{1}{2(\gamma +1)} \bigl[ \Vert \nabla u \Vert _{2}^{2(\gamma +1)}+ \Vert \nabla v \Vert _{2}^{2(\gamma +1)} \bigr] \\& \qquad {}+\frac{1}{2} \biggl(-1 \int _{0}^{t}h_{1}(s)\,ds \biggr) \Vert \nabla u \Vert _{2}^{2}+\frac{1}{2} \biggl(-1 \int _{0}^{t}h_{2}(s)\,ds \biggr) \Vert \nabla v \Vert _{2}^{2} \\& \qquad {}+\frac{1}{2}(h_{1}o\nabla u) (t)+ \frac{1}{2}(h_{2}o\nabla v) (t)- \int _{\Omega }F(u,v)\,dx \biggr\} \\& \quad =- \int _{\Omega }\bigl( \vert u \vert ^{k}+ \vert v \vert ^{l}\bigr) \vert u_{t} \vert ^{j+1}\,dx- \int _{\Omega }\bigl( \vert v \vert ^{\theta }+ \vert u \vert ^{ \varrho }\bigr) \vert v_{t} \vert ^{s+1}\,dx \\& \qquad {}+\frac{1}{2}\bigl(h_{1}'o\nabla u \bigr)-\frac{1}{2}h_{1}(t) \Vert \nabla u \Vert _{2}^{2}+\frac{1}{2}\bigl(h_{2}'o \nabla v\bigr)-\frac{1}{2}h_{2}(t) \Vert \nabla v \Vert _{2}^{2}, \end{aligned}$$
(2.7)

we obtain (2.5) and (2.6). □

Blow-up

In this section, we prove the blow-up result of solution of problem (1.1).

First, we define the functional

$$\begin{aligned} \mathbb{H}(t)=-E(t) =&-\frac{1}{\eta +2} \bigl[ \Vert u_{t} \Vert _{ \eta +2}^{\eta +2}+ \Vert v_{t} \Vert _{\eta +2}^{\eta +2} \bigr]- \frac{1}{2} \bigl[ \Vert \nabla u_{t} \Vert _{2}^{2}+ \Vert \nabla v_{t} \Vert _{2}^{2} \bigr] \\ &{}-\frac{1}{2(\gamma +1)} \bigl[ \Vert \nabla u \Vert _{2}^{2(\gamma +1)}+ \Vert \nabla v \Vert _{2}^{2(\gamma +1)} \bigr] \\ &{}-\frac{1}{2} \biggl[ \biggl(1- \int _{0}^{t}h_{1}(s)\,ds \biggr) \Vert \nabla u \Vert _{2}^{2}+ \biggl(1- \int _{0}^{t}h_{2}(s)\,ds \biggr) \Vert \nabla v \Vert _{2}^{2} \biggr] \\ &{}-\frac{1}{2} \bigl[(h_{1}o\nabla u) (t)+(h_{2}o\nabla v) (t) \bigr] \\ &{}+\frac{1}{2(p+2)}\bigl[ \Vert u+v \Vert _{2(p+2)}^{2(p+2)}+2 \Vert uv \Vert _{p+2}^{p+2}\bigr]. \end{aligned}$$
(3.1)

Theorem 3.1

Assume that (2.1)(2.2) and (2.4) hold, and suppose that \(E(0)<0\) and

$$ 2(p+2)>\max \bigl\{ k+j+1;l+j+1;\theta +s+1;\varrho +s+1;2(\gamma +1) \bigr\} . $$
(3.2)

Then the solution of problem (1.1) blows up in finite time.

Proof

From (2.5), we have

$$ E(t)\leq E(0)\leq 0. $$
(3.3)

Therefore

$$\begin{aligned} \mathbb{H}'(t)=-E'(t) \geq & \int _{\Omega }\bigl( \vert u \vert ^{k}+ \vert v \vert ^{l}\bigr) \vert u_{t} \vert ^{j+1}\,dx+ \int _{\Omega }\bigl( \vert v \vert ^{ \theta }+ \vert u \vert ^{\varrho }\bigr) \vert v_{t} \vert ^{s+1}\,dx. \end{aligned}$$
(3.4)

Hence

$$\begin{aligned}& \begin{gathered} \mathbb{H}'(t)\geq \int _{\Omega }\bigl( \vert u \vert ^{k}+ \vert v \vert ^{l}\bigr) \vert u_{t} \vert ^{j+1}\,dx\geq 0, \\ \mathbb{H}'(t)\geq \int _{\Omega }\bigl( \vert v \vert ^{\theta }+ \vert u \vert ^{\varrho }\bigr) \vert v_{t} \vert ^{s+1}\,dx\geq 0 . \end{gathered} \end{aligned}$$
(3.5)

By (3.1) and (2.3), we have

$$\begin{aligned} 0 \leq& \mathbb{H}(0)\leq \mathbb{H}(t)\leq \frac{1}{2(p+2)} \bigl( \Vert u+v \Vert _{2(p+2)}^{2(p+2)}+2 \Vert uv \Vert _{p+2}^{p+2} \bigr) \\ \leq & \frac{c_{1}}{2(p+2)} \bigl( \Vert u \Vert _{2(p+2)}^{2(p+2)}+ \Vert v \Vert _{2(p+2)}^{2(p+2)} \bigr) . \end{aligned}$$
(3.6)

We set

$$\begin{aligned} \mathcal{K}(t) =&\mathbb{H}^{1-\alpha }+\frac{\varepsilon }{\eta +1} \int _{\Omega } \bigl[u \vert u_{t} \vert ^{\eta }u_{t}+v \vert v_{t} \vert ^{ \eta }v_{t} \bigr]\,dx \\ &{}+\varepsilon \int _{\Omega } [\nabla u_{t}\nabla u+\nabla v_{t} \nabla v ]\,dx , \end{aligned}$$
(3.7)

where \(\varepsilon >0\) is to be assigned later, and we assume that

$$\begin{aligned} 0< \alpha < &\min \biggl\{ \biggl(1-\frac{1}{2(p+2)}- \frac{1}{\eta +2} \biggr),\frac{1+2\gamma }{4(\gamma +1)}, \frac{2p+3-(k+j)}{2j(p+2)}, \\ & \frac{2p+3-(l+j)}{2j(p+2)},\frac{2p+3-(\theta +s)}{2s(p+2)}, \frac{2p+3-(\varrho +s)}{2s(p+2)} \biggr\} < 1. \end{aligned}$$
(3.8)

By multiplying (1.1)1, (1.1)2 by u, v and with a derivative of (3.7), we get

$$\begin{aligned} \mathcal{K}'(t) = &(1-\alpha )\mathbb{H}^{-\alpha } \mathbb{H}'(t)+ \frac{\varepsilon }{\eta +1}\bigl( \Vert u_{t} \Vert _{\eta +2}^{\eta +2}+ \Vert v_{t} \Vert _{\eta +2}^{\eta +2}\bigr)+\varepsilon \bigl( \Vert \nabla u_{t} \Vert _{2}^{2}+ \Vert \nabla v_{t} \Vert _{2}^{2}\bigr) \\ &{}+ \underbrace{\varepsilon \int _{\Omega }\nabla u \int ^{t}_{0}g(t-s) \nabla u(s) \,dsdx}_{J_{1}} + \underbrace{\varepsilon \int _{\Omega }\nabla v \int ^{t}_{0}h(t-s) \nabla v(s) \,dsdx}_{J_{2}} \\ &{}- \underbrace{\varepsilon \int _{\Omega }\bigl( \vert u \vert ^{k}+ \vert v \vert ^{l}\bigr) \vert u_{t} \vert ^{j-1}u_{t}.udx}_{J_{3}}- \underbrace{\varepsilon \int _{\Omega }\bigl( \vert v \vert ^{\theta }+ \vert u \vert ^{\varrho }\bigr) \vert v_{t} \vert ^{s-1}v_{t}.vdx}_{J_{4}} \\ &{}-\varepsilon \bigl( \Vert \nabla u \Vert _{2}^{2}+ \Vert \nabla v \Vert _{2}^{2}\bigr)- \varepsilon \bigl( \Vert \nabla u \Vert ^{2(\gamma +1)}_{2}+ \Vert \nabla v \Vert _{2}^{2(\gamma +1)}\bigr) \\ &{}+ \underbrace{\varepsilon \bigl[ \Vert u+v \Vert _{2(p+2)}^{2(p+2)}+2 \Vert uv \Vert _{p+2}^{p+2} \bigr]}_{J_{5}}. \end{aligned}$$
(3.9)

We have

$$\begin{aligned} J_{1} =&\varepsilon \int _{0}^{t}h_{1}(t-s)\,ds \int _{\Omega }\nabla u.\bigl( \nabla u(s)-\nabla u(t)\bigr) \,dx\,ds+\varepsilon \int _{0}^{t}h_{1}(s)\,ds \Vert \nabla u \Vert _{2}^{2} \\ \geq & \frac{\varepsilon }{2} \int _{0}^{t}h_{1}(s)\,ds \Vert \nabla u \Vert _{2}^{2}-\frac{\varepsilon }{2}(h_{1}o \nabla u). \end{aligned}$$
(3.10)
$$\begin{aligned} J_{2} =&\varepsilon \int _{0}^{t}h_{2}(t-s)\,ds \int _{\Omega }\nabla v.\bigl( \nabla v(s)-\nabla v(t)\bigr) \,dx\,ds+\varepsilon \int _{0}^{t}h_{2}(s)\,ds \Vert \nabla v \Vert _{2}^{2} \\ \geq & \frac{\varepsilon }{2} \int _{0}^{t}h_{2}(s)\,ds \Vert \nabla v \Vert _{2}^{2}-\frac{\varepsilon }{2}(h_{2}o \nabla v). \end{aligned}$$
(3.11)

From (3.9), we find

$$\begin{aligned} \mathcal{K}'(t) \geq &(1-\alpha )\mathbb{H}^{-\alpha } \mathbb{H}'(t)+ \frac{\varepsilon }{\eta +1}\bigl( \Vert u_{t} \Vert _{\eta +2}^{\eta +2}+ \Vert v_{t} \Vert _{\eta +2}^{\eta +2}\bigr)+\varepsilon \bigl( \Vert \nabla u_{t} \Vert _{2}^{2}+ \Vert \nabla v_{t} \Vert _{2}^{2}\bigr) \\ &{}-\varepsilon \biggl[ \biggl(1-\frac{1}{2} \int _{0}^{t}h_{1}(s)\,ds \biggr) \Vert \nabla u \Vert _{2}^{2}+ \biggl(1- \frac{1}{2} \int _{0}^{t}h_{2}(s)\,ds \biggr) \Vert \nabla v \Vert _{2}^{2} \biggr] \\ &{}-\frac{\varepsilon }{2}(h_{1}o\nabla u)-\frac{\varepsilon }{2}(h_{2}o \nabla v)-\varepsilon \bigl( \Vert \nabla u \Vert ^{2(\gamma +1)}_{2}+ \Vert \nabla v \Vert _{2}^{2(\gamma +1)}\bigr) \\ &{}-J_{3}-J_{4}+J_{5} . \end{aligned}$$
(3.12)

At this point, we use Young’s inequality for \(\delta >0\)

$$ XY\leq \frac{\delta ^{\alpha }X^{\alpha }}{\alpha }+ \frac{\delta ^{-\beta }X^{\beta }}{\beta }, \quad \alpha ,\beta >0, \frac{1}{\alpha }+\frac{1}{\beta }=1, $$
(3.13)

we get, for \(\delta _{1},\delta _{2}>0\),

$$\begin{aligned} \bigl\vert u \vert u_{t} \vert ^{j-1}u_{t} \bigr\vert \leq &\frac{\delta _{1}^{j+1}}{j+1} \vert u \vert ^{j+1}+ \frac{j}{j+1}\delta _{1}^{-(\frac{j+1}{j})} \vert u_{t} \vert ^{j+1}, \\ \bigl\vert v \vert v_{t} \vert ^{s-1}v_{t} \bigr\vert \leq &\frac{\delta _{2}^{s+1}}{s+1} \vert v \vert ^{s+1}+ \frac{s}{s+1}\delta _{2}^{-(\frac{s+1}{s})} \vert v_{t} \vert ^{s+1}. \end{aligned}$$
(3.14)

Hence, we have

$$ \begin{gathered} J_{3}\leq \varepsilon \frac{\delta _{1}^{j+1}}{j+1} \int _{\Omega }\bigl( \vert u \vert ^{k}+ \vert v \vert ^{l}\bigr) \vert u \vert ^{j+1}\,dx+ \varepsilon \frac{j\delta _{1}^{-(\frac{j+1}{j})}}{j+1} \int _{\Omega }\bigl( \vert u \vert ^{k}+ \vert v \vert ^{l}\bigr) \vert u_{t} \vert ^{j+1}\,dx, \\ J_{4}\leq \varepsilon \frac{\delta _{2}^{s+1}}{s+1} \int _{\Omega }\bigl( \vert v \vert ^{\theta }+ \vert u \vert ^{\varrho }\bigr) \vert v \vert ^{s+1}\,dx+ \varepsilon \frac{s\delta _{2}^{-(\frac{s+1}{s})}}{s+1} \int _{\Omega }\bigl( \vert v \vert ^{\theta }+ \vert u \vert ^{\varrho }\bigr) \vert v_{t} \vert ^{s+1}\,dx. \end{gathered} $$
(3.15)

Therefore, using (3.5) and by setting \(\delta _{1}\), \(\delta _{1}\) so that

$$ \frac{j\delta _{1}^{-(\frac{j+1}{j})}}{j+1}= \frac{\kappa \mathbb{H}^{-\alpha }(t)}{2}, \qquad \frac{s\delta _{2}^{-(\frac{s+1}{s})}}{s+1}= \frac{\kappa \mathbb{H}^{-\alpha }(t)}{2}, $$

substituting in (3.12), we get

$$\begin{aligned} \mathcal{K}'(t) \geq &\bigl[(1-\alpha )-\varepsilon \kappa \bigr] \mathbb{H}^{- \alpha }\mathbb{H}'(t)+\frac{\varepsilon }{\eta +1} \bigl( \Vert u_{t} \Vert _{\eta +2}^{\eta +2}+ \Vert v_{t} \Vert _{\eta +2}^{\eta +2} \bigr) \\ &{}-\varepsilon \biggl[ \biggl(1-\frac{1}{2} \int _{0}^{t}h_{1}(s)\,ds \biggr) \Vert \nabla u \Vert _{2}^{2}+ \biggl(1- \frac{1}{2} \int _{0}^{t}h_{2}(s)\,ds \biggr) \Vert \nabla v \Vert _{2}^{2} \biggr] \\ &{}+\varepsilon \bigl( \Vert \nabla u_{t} \Vert _{2}^{2}+ \Vert \nabla v_{t} \Vert _{2}^{2} \bigr)-\frac{\varepsilon }{2}(h_{1}o \nabla u)- \frac{\varepsilon }{2}(h_{2}o\nabla v) \\ &{}-\varepsilon C_{1}(\kappa )\mathbb{H}^{\alpha j}(t) \int _{\Omega }\bigl( \vert u \vert ^{k}+ \vert v \vert ^{l}\bigr) \vert u \vert ^{j+1}\,dx \\ &{}-\varepsilon C_{2}(\kappa )\mathbb{H}^{\alpha s}(t) \int _{\Omega }\bigl( \vert v \vert ^{\theta }+ \vert u \vert ^{\varrho }\bigr) \vert v \vert ^{s+1}\,dx \\ &{}-\varepsilon \bigl( \Vert \nabla u \Vert ^{2(\gamma +1)}_{2}+ \Vert \nabla v \Vert _{2}^{2(\gamma +1)}\bigr)+J_{5} , \end{aligned}$$
(3.16)

where

$$ C_{1}(\kappa ):= \biggl(\frac{2j}{\kappa (j+1)} \biggr)^{j+1} \frac{1}{j+1}, \qquad C_{2}(\kappa ):= \biggl( \frac{2s}{\kappa (s+1)} \biggr)^{s+1} \frac{1}{s+1}, $$
(3.17)

we have

$$\begin{aligned} \int _{\Omega }\bigl( \vert u \vert ^{k}+ \vert v \vert ^{l}\bigr) \vert u \vert ^{j+1}\,dx =& \Vert u \Vert ^{k+j+1}_{k+j+1}+ \int _{\Omega } \vert v \vert ^{l} \vert u \vert ^{j+1}\,dx, \\ \int _{\Omega }\bigl( \vert v \vert ^{\theta }+ \vert u \vert ^{\varrho }\bigr) \vert v \vert ^{s+1}\,dx =& \Vert v \Vert ^{\theta +s+1}_{\theta +s+1}+ \int _{ \Omega } \vert u \vert ^{\varrho } \vert v \vert ^{s+1}\,dx. \end{aligned}$$
(3.18)

By Young’s inequality, we find for \(\delta _{3},\delta _{4}>0\)

$$ \begin{gathered} \int _{\Omega } \vert v \vert ^{l} \vert u \vert ^{j+1}\,dx\leq \frac{l}{l+j+1} \delta _{3}^{(\frac{l+j+1}{l})} \Vert v \Vert ^{l+j+1}_{l+j+1}+ \frac{j+1}{l+j+1}\delta _{3}^{-(\frac{l+j+1}{l})} \Vert u \Vert ^{l+j+1}_{l+j+1}, \\ \int _{\Omega } \vert u \vert ^{\varrho } \vert v \vert ^{s+1}\,dx\leq \frac{\varrho }{\varrho +s+1} \delta _{4}^{(\frac{\varrho +s+1}{\varrho })} \Vert u \Vert ^{\varrho +s+1}_{ \varrho +s+1}+ \frac{s+1}{\varrho +s+1}\delta _{4}^{-( \frac{\varrho +s+1}{\varrho })} \Vert v \Vert ^{\varrho +s+1}_{\varrho +s+1}. \end{gathered} $$
(3.19)

Hence

$$\begin{aligned} \mathbb{H}^{\alpha j}(t) \int _{\Omega }\bigl( \vert u \vert ^{k}+ \vert v \vert ^{l}\bigr) \vert u \vert ^{j+1}\,dx \leq & \mathbb{H}^{\alpha j}(t) \Vert u \Vert ^{k+j+1}_{k+j+1}+ \frac{l\mathbb{H}^{\alpha j}(t)}{l+j+1}\delta _{3}^{( \frac{l+j+1}{l})} \Vert v \Vert ^{l+j+1}_{l+j+1} \\ &{}+\frac{(j+1)\mathbb{H}^{\alpha j}(t)}{l+j+1}\delta _{3}^{-( \frac{l+j+1}{l})} \Vert u \Vert ^{l+j+1}_{l+j+1}, \\ \mathbb{H}^{\alpha s}(t) \int _{\Omega }\bigl( \vert v \vert ^{\theta }+ \vert u \vert ^{\varrho }\bigr) \vert v \vert ^{s+1}\,dx \leq & \mathbb{H}^{\alpha s}(t) \Vert v \Vert ^{\theta +s+1}_{\theta +s+1}+ \frac{\varrho \mathbb{H}^{\alpha s}(t)}{\varrho +s+1}\delta _{4}^{( \frac{\varrho +s+1}{\varrho })} \Vert u \Vert ^{\varrho +s+1}_{\varrho +s+1} \\ &{}+\frac{(s+1)\mathbb{H}^{\alpha s}(t)}{\varrho +s+1}\delta _{4}^{-( \frac{\varrho +s+1}{\varrho })} \Vert v \Vert ^{\varrho +s+1}_{\varrho +s+1}. \end{aligned}$$
(3.20)

Since (2.4) holds, we obtain by using (3.6) and (3.8)

$$\begin{aligned}& \mathbb{H}^{\alpha j}(t) \Vert u \Vert _{k+j+1}^{k+j+1} \leq c_{1}\bigl( \Vert u \Vert ^{2\alpha j(p+2)+k+j+1}_{2(p+2)}+ \Vert v \Vert ^{2\alpha j(p+2)}_{2(p+2)} \Vert u \Vert _{k+j+1}^{k+j+1}\bigr), \\& \mathbb{H}^{\alpha j}(t) \Vert v \Vert _{k+j+1}^{k+j+1} \leq c_{2}\bigl( \Vert v \Vert ^{2\alpha j(p+2)+k+j+1}_{2(p+2)}+ \Vert u \Vert ^{2\alpha j(p+2)}_{2(p+2)} \Vert v \Vert _{k+j+1}^{k+j+1}\bigr), \\& \mathbb{H}^{\alpha s}(t) \Vert v \Vert _{\theta +s+1}^{\theta +s+1} \leq c_{3}\bigl( \Vert v \Vert ^{2\alpha s(p+2)+\theta +s+1}_{2(p+2)}+ \Vert u \Vert ^{2\alpha s(p+2)}_{2(p+2)} \Vert v \Vert _{\theta +s+1}^{ \theta +s+1}\bigr), \\& \mathbb{H}^{\alpha s}(t) \Vert u \Vert _{\theta +s+1}^{\theta +s+1} \leq c_{4}\bigl( \Vert u \Vert ^{2\alpha s(p+2)+\theta +s+1}_{2(p+2)}+ \Vert v \Vert ^{2\alpha s(p+2)}_{2(p+2)} \Vert u \Vert _{\theta +s+1}^{ \theta +s+1}\bigr) \end{aligned}$$
(3.21)

for some positive constants \(c_{i},i=1,\ldots,4\). By using (3.8) and the algebraic inequality

$$ B^{\varsigma }\leq (B+1)\leq \biggl(1+\frac{1}{b}\biggr) (B+b), \quad \forall B>0, 0< \varsigma < 1, b>0, $$
(3.22)

we have \(\forall t>0\)

$$\begin{aligned}& \begin{gathered} \Vert u \Vert ^{2\alpha j(p+2)+k+j+1}_{2(p+2)}\leq d\bigl( \Vert u \Vert ^{2(p+2)}_{2(p+2)}+ \mathbb{H}(0)\bigr)\leq d\bigl( \Vert u \Vert ^{2(p+2)}_{2(p+2)}+\mathbb{H}(t)\bigr) , \\ \Vert v \Vert ^{2\alpha j(p+2)+k+j+1}_{2(p+2)}\leq d\bigl( \Vert v \Vert ^{2(p+2)}_{2(p+2)}+ \mathbb{H}(0)\bigr)\leq d\bigl( \Vert v \Vert ^{2(p+2)}_{2(p+2)}+\mathbb{H}(t)\bigr), \\ \Vert v \Vert ^{2\alpha s(p+2)+\theta +s+1}_{2(p+2)}\leq d\bigl( \Vert v \Vert ^{2(p+2)}_{2(p+2)}+\mathbb{H}(0)\bigr)\leq d\bigl( \Vert v \Vert ^{2(p+2)}_{2(p+2)}+ \mathbb{H}(t)\bigr), \\ \Vert u \Vert ^{2\alpha s(p+2)+\theta +s+1}_{2(p+2)}\leq d\bigl( \Vert u \Vert ^{2(p+2)}_{2(p+2)}+\mathbb{H}(0)\bigr)\leq d\bigl( \Vert u \Vert ^{2(p+2)}_{2(p+2)}+ \mathbb{H}(t)\bigr), \end{gathered} \end{aligned}$$
(3.23)

where \(d=1+\frac{1}{\mathbb{H}(0)}\). Also, since

$$ (X+Y)^{\gamma } \leq C\bigl(X^{\gamma }+Y^{\gamma }\bigr), \quad X,Y>0, \gamma >0, $$
(3.24)

we conclude

$$\begin{aligned} \begin{gathered} \Vert v \Vert ^{2\alpha j(p+2)}_{2(p+2)} \Vert u \Vert _{k+j+1}^{k+j+1} \leq c_{5}\bigl( \Vert v \Vert ^{2(p+2)}_{2(p+2)}+ \Vert u \Vert ^{2(p+2)}_{2(p+2)} \bigr), \\ \Vert u \Vert ^{2\alpha j(p+2)}_{2(p+2)} \Vert v \Vert _{k+j+1}^{k+j+1} \leq c_{6}\bigl( \Vert u \Vert ^{2(p+2)}_{2(p+2)}+ \Vert v \Vert ^{2(p+2)}_{2(p+2)} \bigr), \\ \Vert u \Vert ^{2\alpha s(p+2)}_{2(p+2)} \Vert v \Vert _{\theta +s+1}^{ \theta +s+1}\leq c_{7}\bigl( \Vert v \Vert ^{2(p+2)}_{2(p+2)}+ \Vert u \Vert ^{2(p+2)}_{2(p+2)} \bigr), \\ \Vert v \Vert ^{2\alpha s(p+2)}_{2(p+2)} \Vert u \Vert _{\theta +s+1}^{ \theta +s+1}\leq c_{8}\bigl( \Vert v \Vert ^{2(p+2)}_{2(p+2)}+ \Vert u \Vert ^{2(p+2)}_{2(p+2)} \bigr). \end{gathered} \end{aligned}$$
(3.25)

Substituting (3.23) and (3.25) in (3.21), we get

$$\begin{aligned} \mathbb{H}^{\alpha j}(t) \Vert u \Vert _{k+j+1}^{k+j+1} \leq &c_{9}\bigl( \Vert v \Vert ^{2(p+2)}_{2(p+2)}+ \Vert u \Vert ^{2(p+2)}_{2(p+2)}+ \mathbb{H}(t)\bigr), \\ \mathbb{H}^{\alpha j}(t) \Vert v \Vert _{k+j+1}^{k+j+1} \leq &c_{10}\bigl( \Vert v \Vert ^{2(p+2)}_{2(p+2)}+ \Vert u \Vert ^{2(p+2)}_{2(p+2)}+ \mathbb{H}(t)\bigr), \\ \mathbb{H}^{\alpha s}(t) \Vert v \Vert _{\theta +s+1}^{\theta +s+1} \leq &c_{11}\bigl( \Vert v \Vert ^{2(p+2)}_{2(p+2)}+ \Vert u \Vert ^{2(p+2)}_{2(p+2)}+ \mathbb{H}(t)\bigr), \\ \mathbb{H}^{\alpha s}(t) \Vert u \Vert _{\theta +s+1}^{\theta +s+1} \leq &c_{12}\bigl( \Vert v \Vert ^{2(p+2)}_{2(p+2)}+ \Vert u \Vert ^{2(p+2)}_{2(p+2)}+ \mathbb{H}(t)\bigr). \end{aligned}$$
(3.26)

Hence, by fixed \(\delta _{3},\delta _{4}>0\), we get

$$\begin{aligned}& \begin{gathered} \mathbb{H}^{\alpha j}(t) \int _{\Omega }\bigl( \vert u \vert ^{k}+ \vert v \vert ^{l}\bigr) \vert u \vert ^{j+1}\,dx \\ \quad \leq M_{1} \biggl(1+\frac{l\delta _{3}^{(\frac{l+j+1}{l})}}{l+j+1}+ \frac{(j+1)\delta _{3}^{-(\frac{l+j+1}{l})}}{l+j+1} \biggr) \bigl( \Vert v \Vert ^{2(p+2)}_{2(p+2)}+ \Vert u \Vert ^{2(p+2)}_{2(p+2)}+\mathbb{H}(t) \bigr), \\ \mathbb{H}^{\alpha s}(t) \int _{\Omega }\bigl( \vert v \vert ^{\theta }+ \vert u \vert ^{\varrho }\bigr) \vert v \vert ^{s+1}\,dx \\ \quad \leq M_{2} \biggl(1+ \frac{\varrho \delta _{4}^{(\frac{\varrho +s+1}{\varrho })}}{\varrho +s+1}+ \frac{(s+1)\delta _{4}^{-(\frac{\varrho +s+1}{\varrho })}}{\varrho +s+1} \biggr) \bigl( \Vert v \Vert ^{2(p+2)}_{2(p+2)}+ \Vert u \Vert ^{2(p+2)}_{2(p+2)}+ \mathbb{H}(t) \bigr) \end{gathered} \end{aligned}$$
(3.27)

for some constants \(M_{1},M_{2}>0\).

Now, for \(0< a<1\), from (3.1)

$$\begin{aligned} J_{5} =&\varepsilon \bigl[ \Vert u+v \Vert _{2(p+2)}^{2(p+2)}+2 \Vert uv \Vert _{p+2}^{p+2}\bigr] \\ =&\varepsilon a \bigl[ \Vert u+v \Vert _{2(p+2)}^{2(p+2)}+2 \Vert uv \Vert _{p+2}^{p+2} \bigr] \\ &{}+\frac{2\varepsilon (p+2)(1-a)}{\eta +2}\bigl( \Vert u_{t} \Vert _{\eta +2}^{ \eta +2}+ \Vert v_{t} \Vert _{\eta +2}^{\eta +2}\bigr) \\ &{}+\varepsilon (p+2) (1-a) \bigl( \Vert \nabla u_{t} \Vert _{2}^{2}+ \Vert \nabla v_{t} \Vert _{2}^{2}\bigr) \\ &{}+\varepsilon (p+2) (1-a) \biggl(1- \int _{0}^{t}g(s)\,ds\biggr) \Vert \nabla u \Vert _{2}^{2} \\ &{}+\varepsilon (p+2) (1-a) \biggl(1- \int _{0}^{t}h(s)\,ds\biggr) \Vert \nabla v \Vert _{2}^{2} \\ &{}+\varepsilon (p+2) (1-a) \bigl((h_{1}o\nabla u)+(h_{2}o \nabla v)\bigr) \\ &{}+\frac{\varepsilon (p+2)(1-a)}{\gamma +1}\bigl( \Vert \nabla u \Vert ^{2( \gamma +1)}_{2}+ \Vert \nabla v \Vert _{2}^{2(\gamma +1)}\bigr) \\ &{}+\varepsilon 2(p+2) (1-a)\mathbb{H}(t). \end{aligned}$$
(3.28)

Substituting in (3.16), and by using (2.3), we get

$$\begin{aligned} \mathcal{K}'(t) \geq & \bigl\{ (1-\alpha )-\varepsilon \kappa \bigr\} \mathbb{H}^{-\alpha }\mathbb{H}'(t)+\varepsilon \bigl\{ (p+2) (1-a)+1 \bigr\} \bigl( \Vert \nabla u_{t} \Vert _{2}^{2}+ \Vert \nabla v_{t} \Vert _{2}^{2}\bigr) \\ &{}+\varepsilon \biggl\{ \frac{2\varepsilon (p+2)(1-a)}{\eta +2}+ \frac{1}{\eta +1} \biggr\} \bigl( \Vert u_{t} \Vert _{\eta +2}^{\eta +2}+ \Vert v_{t} \Vert _{\eta +2}^{\eta +2}\bigr) \\ &{}+\varepsilon \biggl\{ (p+2) (1-a) \biggl(1- \int _{0}^{t}h_{1}(s)\,ds \biggr)- \biggl(1-\frac{1}{2} \int _{0}^{t}h_{1}(s)\,ds \biggr) \biggr\} \Vert \nabla u \Vert _{2}^{2} \\ &{}+\varepsilon \biggl\{ (p+2) (1-a) \biggl(1- \int _{0}^{t}h_{2}(s)\,ds \biggr)- \biggl(1-\frac{1}{2} \int _{0}^{t}h_{2}(s)\,ds \biggr) \biggr\} \Vert \nabla v \Vert _{2}^{2} \\ &{}+\varepsilon \biggl\{ (p+2) (1-a)-\frac{1}{2} \biggr\} (h_{1}o\nabla u+h_{2}o \nabla v) \\ &{}+\varepsilon \biggl\{ \frac{(p+2)(1-a)}{\gamma +1}-1 \biggr\} \bigl( \Vert \nabla u \Vert ^{2(\gamma +1)}_{2}+ \Vert \nabla v \Vert _{2}^{2(\gamma +1)}\bigr) \\ &{}+\varepsilon \bigl\{ c_{0}a- \bigl(M_{3}C_{1}( \kappa )+M_{4}C_{2}( \kappa ) \bigr) \bigr\} \bigl( \Vert u \Vert _{2(p+2)}^{2(p+2)}+ \Vert v \Vert _{2(p+2)}^{2(p+2)} \bigr) \\ &{}+\varepsilon \bigl\{ ( 2(p+2) (1-a)- \bigl(M_{3}C_{1}( \kappa )+M_{4}C_{2}( \kappa ) \bigr) \bigr\} \mathbb{H}(t), \end{aligned}$$
(3.29)

where

$$\begin{aligned}& M_{3}:=M_{1} \biggl(1+\frac{l\delta _{3}^{(\frac{l+j+1}{l})}}{l+j+1}+ \frac{(j+1)\delta _{3}^{-(\frac{l+j+1}{l})}}{l+j+1} \biggr)>0 . \\& M_{4}:=M_{2} \biggl(1+ \frac{\varrho \delta _{4}^{(\frac{\varrho +s+1}{\varrho })}}{\varrho +s+1}+ \frac{(s+1)\delta _{4}^{-(\frac{\varrho +s+1}{\varrho })}}{\varrho +s+1} \biggr)>0. \end{aligned}$$

At this stage, we take \(a>0\) small enough so that

$$ (p+2) (1-a)>1+\gamma , $$

we have

$$\begin{aligned}& \lambda _{1}:=(p+2) (1-a)-1>0 \\& \lambda _{2}:=(p+2) (1-a)-\frac{1}{2}>0 \\& \lambda _{3}:=\frac{(p+2)(1-a)}{\gamma +1}-1>0 \end{aligned}$$

and we assume that

$$ \max \biggl\{ \int _{0}^{\infty }h_{1}(s)\,ds, \int _{0}^{\infty }h_{2}(s)\,ds \biggr\} < \frac{(p+2)(1-a)-1}{((p+2)(1-a)-\frac{1}{2})}= \frac{2\lambda _{1}}{2\lambda _{1}+1} $$
(3.30)

gives

$$\begin{aligned} \lambda _{4} =& \biggl\{ \bigl((p+2) (1-a)-1 \bigr)- \int _{0}^{t}h_{1}(s)\,ds \biggl((p+2) (1-a)-\frac{1}{2} \biggr) \biggr\} >0, \\ \lambda _{5} =& \biggl\{ \bigl((p+2) (1-a)-1 \bigr)- \int _{0}^{t}h_{2}(s)\,ds \biggl((p+2) (1-a)-\frac{1}{2} \biggr) \biggr\} >0, \end{aligned}$$

then we choose κ so large that

$$\begin{aligned} \lambda _{6} =&ac_{0}- \bigl(M_{3}C_{1}( \kappa )+M_{4}C_{2}(\kappa ) \bigr)>0, \\ \lambda _{7} =&2(p+2) (1-a)- \bigl(M_{3}C_{1}( \kappa )+M_{4}C_{2}( \kappa ) \bigr)>0. \end{aligned}$$

Finally, we fix κ, a, and we appoint ε small enough so that

$$ \lambda _{8}=(1-\alpha )-\varepsilon \kappa >0. $$

Thus, for some \(\beta >0\), estimate (3.29) becomes

$$\begin{aligned} \mathcal{K}'(t) \geq &\beta \bigl\{ \mathbb{H}(t)+ \Vert u_{t} \Vert _{ \eta +2}^{\eta +2}+ \Vert v_{t} \Vert _{\eta +2}^{\eta +2}+ \Vert \nabla u_{t} \Vert _{2}^{2}+ \Vert \nabla v_{t} \Vert _{2}^{2} + \Vert \nabla u \Vert _{2}^{2}+ \Vert \nabla v \Vert _{2}^{2} \\ &{} + \Vert \nabla u \Vert ^{2(\gamma +1)}_{2}+ \Vert \nabla v \Vert _{2}^{2( \gamma +1)} +(h_{1}o \nabla u)+(h_{2}o\nabla v)+ \Vert u \Vert _{2(p+2)}^{2(p+2)}+ \Vert u \Vert _{2(p+2)}^{2(p+2)} \bigr\} . \end{aligned}$$
(3.31)

By (2.3), for some \(\beta _{1}>0\), we obtain

$$\begin{aligned} \mathcal{K}'(t) \geq &\beta _{1} \bigl\{ \mathbb{H}(t)+ \Vert u_{t} \Vert _{\eta +2}^{\eta +2}+ \Vert v_{t} \Vert _{\eta +2}^{\eta +2}+ \Vert \nabla u_{t} \Vert _{2}^{2}+ \Vert \nabla v_{t} \Vert _{2}^{2} + \Vert \nabla u \Vert _{2}^{2} \\ &{}+ \Vert \nabla v \Vert _{2}^{2}+ \Vert \nabla u \Vert ^{2(\gamma +1)}_{2}+ \Vert \nabla v \Vert _{2}^{2(\gamma +1)}+(h_{1}o\nabla u)+(h_{2}o \nabla v) \\ &{}+ \Vert u+v \Vert _{2(p+2)}^{2(p+2)}+2 \Vert uv \Vert _{p+2}^{p+2} \bigr\} \end{aligned}$$
(3.32)

and

$$ \mathcal{K}(t)\geq \mathcal{K}(0)>0, \quad t>0. $$
(3.33)

Next, using Holder’s and Young’s inequalities, we have

$$\begin{aligned} \biggl\vert \int _{\Omega }\bigl(u \vert u_{t} \vert ^{\eta } u_{t}+v \vert v_{t} \vert ^{\eta }v_{t}\bigr)\,dx \biggr\vert ^{\frac{1}{1-\alpha }} \leq &C \bigl[ \Vert u \Vert _{2(p+2)}^{\frac{\theta }{1-\alpha }}+ \Vert u_{t} \Vert _{ \eta +2}^{\frac{\mu }{1-\alpha }} \\ & + \Vert v \Vert _{2(p+2)}^{\frac{\theta }{1-\alpha }}+ \Vert v_{t} \Vert _{ \eta +2}^{\frac{\mu }{1-\alpha }} \bigr] , \end{aligned}$$
(3.34)

where \(\frac{1}{\mu }+\frac{1}{\theta }=1\).

We take \(\mu =(\eta +2)(1-\alpha )\) to get

$$ \frac{\theta }{1-\alpha }=\frac{\eta +2}{(1-\alpha )(\eta +2)-1}\leq 2(p+2). $$

Subsequently, by using (3.8), (3.6), and (3.22), we obtain

$$\begin{aligned} \Vert u \Vert _{2(p+2)}^{\frac{\eta +2}{(1-\alpha )(\eta +2)-1}} \leq &d\bigl( \Vert u \Vert _{2(p+2)}^{2(p+2)}+\mathbb{H}(t)\bigr) , \\ \Vert v \Vert _{2(p+2)}^{\frac{\eta +2}{(1-\alpha )(\eta +2)-1}} \leq &d\bigl( \Vert v \Vert _{2(p+2)}^{2(p+2)}+\mathbb{H}(t)\bigr), \quad \forall t \geq 0. \end{aligned}$$

Therefore,

$$\begin{aligned}& \biggl\vert \int _{\Omega }\bigl(u \vert u_{t} \vert ^{\eta } u_{t}+v \vert v_{t} \vert ^{\eta }v_{t}\bigr)\,dx \biggr\vert ^{\frac{1}{1-\alpha }} \\& \quad \leq c_{13} \bigl\{ \Vert u \Vert _{2(p+2)}^{2(p+2)}+ \Vert v \Vert _{2(p+2)}^{2(p+2)}+ \Vert u_{t} \Vert _{\eta +2}^{\eta +2}+ \Vert v_{t} \Vert _{\eta +2}^{ \eta +2}+\mathbb{H}(t) \bigr\} . \end{aligned}$$
(3.35)

Similarly, we have

$$\begin{aligned} \biggl\vert \int _{\Omega }(\nabla u\nabla u_{t}+\nabla v \nabla v_{t})\,dx \biggr\vert ^{\frac{1}{1-\alpha }} \leq &C \bigl[ \Vert \nabla u \Vert _{2}^{ \frac{\theta }{1-\alpha }}+ \Vert \nabla u_{t} \Vert _{2}^{ \frac{\mu }{1-\alpha }} + \Vert \nabla v \Vert _{2}^{\frac{\theta }{1-\alpha }}+ \Vert \nabla v_{t} \Vert _{2}^{\frac{\mu }{1-\alpha }} \bigr] , \end{aligned}$$

where \(\frac{1}{\mu }+\frac{1}{\theta }=1\).

We take \(\theta =2(\gamma +1)(1-\alpha )\) to get

$$\begin{aligned}& \frac{\mu }{1-\alpha }=\frac{2(\gamma +1)}{2(1-\alpha )(\gamma +1)-1} \leq 2, \\& \begin{aligned}[b] \biggl\vert \int _{\Omega }(\nabla u\nabla u_{t}+\nabla v \nabla v_{t})\,dx \biggr\vert ^{\frac{1}{1-\alpha }}\leq{} &c_{14} \bigl\{ \Vert \nabla u \Vert ^{2(\gamma +1)}_{2}+ \Vert \nabla v \Vert ^{2(\gamma +1)}_{2} \\ & {} + \Vert \nabla u_{t} \Vert _{2}^{2}+ \Vert \nabla v_{t} \Vert _{2}^{2} \bigr\} . \end{aligned} \end{aligned}$$
(3.36)

Hence, by (3.35) and (3.36),

$$\begin{aligned} \mathcal{K}^{\frac{1}{1-\alpha }}(t) =& \biggl(\mathbb{H}^{1-\alpha }+ \frac{\varepsilon }{\eta +1} \int _{\Omega }\bigl(u \vert u_{t} \vert ^{\eta }u_{t}+v \vert v_{t} \vert ^{\eta }v_{t}\bigr)\,dx \\ &{}+\varepsilon \int _{\Omega }(\nabla u_{t}\nabla u+\nabla v_{t} \nabla v)\,dx \biggr)^{\frac{1}{1-\alpha }} \\ \leq &c \biggl(\mathbb{H}(t)+ \biggl\vert \int _{\Omega }\bigl(u \vert u_{t} \vert ^{ \eta } u_{t}+v \vert v_{t} \vert ^{\eta }v_{t}\bigr) \,dx \biggr\vert ^{ \frac{1}{1-\alpha }}+ \Vert \nabla u \Vert _{2}^{\frac{2}{1-\alpha }}+ \Vert \nabla v \Vert _{2}^{\frac{2}{1-\alpha }} \\ & + \Vert \nabla u_{t} \Vert _{2}^{\frac{2}{1-\alpha }}+ \Vert \nabla v_{t} \Vert _{2}^{\frac{2}{1-\alpha }} \biggr) \\ \leq &c \bigl(\mathbb{H}(t)+ \Vert u_{t} \Vert _{\eta +2}^{\eta +2}+ \Vert v_{t} \Vert _{\eta +2}^{\eta +2}+ \Vert \nabla u \Vert ^{\gamma +1}_{2}+ \Vert \nabla v \Vert _{2}^{\gamma +1}+ \Vert \nabla u_{t} \Vert _{2}^{2} \\ &{}+ \Vert \nabla v_{t} \Vert _{2}^{2}+(h_{1}o \nabla u)+(h_{2}o\nabla v)+ \Vert u \Vert _{2(p+2)}^{2(p+2)}+ \Vert v \Vert _{2(p+2)}^{2(p+2)} \bigr) \\ \leq &c \bigl(\mathbb{H}(t)+ \Vert u_{t} \Vert _{\eta +2}^{\eta +2}+ \Vert v_{t} \Vert _{\eta +2}^{\eta +2}+ \Vert \nabla u \Vert ^{\gamma +1}_{2}+ \Vert \nabla v \Vert _{2}^{\gamma +1}+ \Vert \nabla u_{t} \Vert _{2}^{2} \\ &{}+ \Vert \nabla v_{t} \Vert _{2}^{2}+ \Vert \nabla u \Vert _{2}^{2}+ \Vert \nabla v \Vert _{2}^{2}+(h_{1}o\nabla u)+(h_{2}o\nabla v) \\ &{}+ \Vert u \Vert _{2(p+2)}^{2(p+2)}+ \Vert v \Vert _{2(p+2)}^{2(p+2)} \bigr) . \end{aligned}$$
(3.37)

From (3.31) and (3.37), it gives

$$ \mathcal{K}'(t)\geq \lambda \mathcal{K}^{\frac{1}{1-\alpha }}(t), $$
(3.38)

where \(\lambda > 0 \), this depends only on β and c.

By integration of (3.38), we obtain

$$ \mathcal{K}^{\frac{\alpha }{1-\alpha }}(t)\geq \frac{1}{\mathcal{K}^{\frac{-\alpha }{1-\alpha }}(0)-\lambda \frac{\alpha }{(1-\alpha )} t}. $$

Hence, \(\mathcal{K}(t)\) blows up in time

$$ T\leq T^{*}= \frac{1-\alpha }{\lambda \alpha \mathcal{K}^{\alpha /(1-\alpha )}(0)}. $$

Then the proof is completed. □

Conclusion

In this paper, we are interested in the blow-up for a quasilinear system of viscoelastic equations with degenerate damping and general source terms according to some suitable hypothesis. This work is a general case of the recent results of Boulaaras’ works in [11, 21] using the energy method. Next we will prove the result of local existence of this studied problem based on the recent result in [4].

Availability of data and materials

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Acknowledgements

The authors would like to thank the anonymous referees and the handling editor for their careful reading and for relevant remarks/suggestions which helped to improve the paper. The fifth author extends their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through Research Group Project under Grant no. (R.G.P-2/53/42).

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Correspondence to Salah Boulaaras.

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Boulaaras, S., Choucha, A., Agarwal, P. et al. Blow-up of solutions for a quasilinear system with degenerate damping terms. Adv Differ Equ 2021, 446 (2021). https://doi.org/10.1186/s13662-021-03609-0

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MSC

  • 35B40
  • 35L45
  • 93D15
  • 93D20

Keywords

  • Viscoelastic equation
  • Blow-up
  • Strong damping
  • Distributed delay