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Blow-up of arbitrarily positive initial energy solutions for a viscoelastic wave system with nonlinear damping and source terms

Boundary Value Problems20182018:35

https://doi.org/10.1186/s13661-018-0951-9

  • Received: 14 September 2017
  • Accepted: 5 March 2018
  • Published:

Abstract

This work is concerned with the Dirichlet initial boundary problem for a semilinear viscoelastic wave system with nonlinear weak damping and source terms. For nonincreasing positive functions g and h, we show the finite time blow-up of some solutions whose initial data have arbitrarily high initial energy.

Keywords

  • Viscoelastic wave system
  • Nonlinear damping
  • Blow-up
  • Arbitrarily positive initial energy

MSC

  • 35L53
  • 35B44

1 Introduction and main result

We consider a semilinear viscoelastic wave system with nonlinear damping and source terms,
$$\begin{aligned}& u_{tt}-\Delta u+ \int_{0}^{t} g(t-\tau)\Delta u(\tau)\,d\tau + \vert u_{t}\vert ^{m-1} u_{t}=f_{1} (u,v),\quad x\in \Omega,t>0, \end{aligned}$$
(1)
$$\begin{aligned}& v_{tt}-\Delta v+ \int_{0}^{t} h(t-\tau)\Delta v(\tau)\,d\tau + \vert v_{t}\vert ^{r-1} v_{t}=f_{2} (u,v),\quad x\in \Omega,t>0, \end{aligned}$$
(2)
subject to null Dirichlet boundary and initial conditions
$$\begin{aligned}& u(x,t)=v(x,t)=0, \quad x\in \partial \Omega,t>0, \end{aligned}$$
(3)
$$\begin{aligned}& \begin{aligned}&u(x,0)=u_{0}(x),\qquad u_{t}(x,0)=u_{1}(x), \\ &v(x,0)=v_{0}(x),\qquad v_{t}(x,0)=v_{1}(x),\quad x\in \Omega, \end{aligned} \end{aligned}$$
(4)
where \(\Omega \subset \mathbb{R}^{N}\) (\(N\geq 1\)) is a bounded domain with smooth boundary Ω, \(m>1\), \(r>1\), and the relaxation functions \(g:\mathbb{R}^{+}\rightarrow \mathbb{R}^{+}\) and \(h: \mathbb{R}^{+}\rightarrow \mathbb{R}^{+}\) are positive nonincreasing. Problems of this type arise in viscoelasticity and systems governing the longitudinal motion of a viscoelastic configuration obeying the nonlinear Boltzmann model.

During the past decades, there has been much work dealing with the well-posedness and qualitative properties of solutions for damped viscoelastic wave equation. In this paper, we would like to investigate the blow-up phenomena with high initial energy for a semilinear damped viscoelastic wave system. To motivate our work, let us recall some results regarding viscoelastic wave models. For the single viscoelastic wave equation, we refer the reader to [1, 2] (the case \(g=0\)) and [37] (the case \(g\neq 0\)), where blow-up solutions with initial negative energy, positive energy and arbitrarily positive energy are [17], respectively. Moreover, for general energy decay estimates on global solutions of a nonlinear abstract viscoelastic equation with variable density and the oscillation criteria and numerical solution of damped wave models, we refer the reader to [810].

Concerning wave systems without viscoelastic term (\(g=0\)), Agre and Rammaha [11] investigated the following coupled semilinear wave system with nonlinear damping terms:
$$\begin{aligned}& u_{tt}-\Delta u+\vert u_{t}\vert ^{m-1} u_{t}=(p+1) \bigl[ a\vert u+v\vert ^{p-1}(u+v)+b\vert u \vert ^{ \frac{p-3}{2}}u\vert v\vert ^{\frac{p+1}{2}} \bigr] , \\& v_{tt}-\Delta v+\vert v_{t}\vert ^{r-1} v_{t}=(p+1) \bigl[ a\vert u+v\vert ^{p-1}(u+v)+b\vert v \vert ^{ \frac{p-3}{2}}v\vert u\vert ^{\frac{p+1}{2}} \bigr] , \end{aligned}$$
in \(\Omega \times (0,\infty)\), where \(\Omega \subset \mathbb{R}^{N}\) (\(N=1,2,3\)), \(m\geq 1\), \(r\geq 1\), \(a>1\), \(b>0\), \(p\geq 3\). Using the Galerkin method and the method in [2] different from the concavity method we already know, that is, differential inequality techniques, they determined local and global existence of weak solutions and showed that any weak solution with negative initial energy blows up in finite time. Thereafter, Said-Houari [12] considered the blow-up result for a larger class of initial data with positive initial energy combining potential well method and differential inequality techniques ([2]). Pişkin [13] studied a coupled semilinear Klein–Gordon system with nonlinear damping terms,
$$\begin{aligned}& u_{tt}-\Delta u+m_{1}^{2}u+\vert u_{t}\vert ^{m-1} u_{t}=(p+1) \bigl[ a\vert u+v \vert ^{p-1}(u+v)+b\vert u\vert ^{ \frac{p-3}{2}}u\vert v\vert ^{\frac{p+1}{2}} \bigr] , \\& v_{tt}-\Delta v+m_{2}^{2}v+\vert v_{t} \vert ^{r-1} v_{t}=(p+1) \bigl[ a\vert u+v\vert ^{p-1}(u+v)+b\vert v\vert ^{ \frac{p-3}{2}}v\vert u\vert ^{\frac{p+1}{2}} \bigr] , \end{aligned}$$
in \(\Omega \times (0,\infty)\), where \(\Omega \subset \mathbb{R}^{N}\) (\(N=1,2,3\)), \(m\geq 1\), \(r\geq 1\), \(m_{1},m_{2}>0\), \(a,b>0\), \(p>1\). The decay estimates of the solution are established by using Nakao’s inequality. Meanwhile, similar to [2], he also proved the blow-up of the solution in finite time with negative initial energy, using the technique of appropriate modification for energy functional.
In the presence of the viscoelastic term (\(g\neq 0\)), Han and Wang [14] discussed semilinear coupled viscoelastic wave system with nonlinear damping terms,
$$\begin{aligned}& u_{tt}-\Delta u+ \int_{0}^{t} g(t-\tau)\Delta u(\tau)\,d\tau + \vert u_{t}\vert ^{m-1} u_{t} \\& \quad =(p+1) \bigl[ a \vert u+v\vert ^{p-1}(u+v)+b\vert u\vert ^{ \frac{p-3}{2}}u\vert v \vert ^{\frac{p+1}{2}} \bigr] , \\& v_{tt}-\Delta v+ \int_{0}^{t} h(t-\tau)\Delta v(\tau)\,d\tau + \vert v_{t}\vert ^{r-1} v_{t} \\& \quad =(p+1) \bigl[ a \vert u+v\vert ^{p-1}(u+v)+b\vert v\vert ^{ \frac{p-3}{2}}v\vert u \vert ^{\frac{p+1}{2}} \bigr] , \end{aligned}$$
in \(\Omega \times (0,\infty)\), where \(\Omega \subset \mathbb{R}^{N}\) (\(N=1,2,3\)), \(m\geq 1\), \(r\geq 1\), \(a>1\), \(b>0\), \(p\geq 3\). They established several results concerning the global existence, uniqueness and finite time blow-up of weak solutions with negative initial energy by utilizing the Galerkin and the concavity method. Recently, Messaoudi and Said-Hauari [15] dealt with our problem (1)–(4) and improved the result in [12] to a larger class of initial data for which the initial energy can take positive values. Besides, for the work on quasilinear wave equations, we refer the reader to [1618] and the references therein.

In view of the work mentioned above, one can find that research on the blow-up phenomena of the solutions with high initial energy for a semilinear damped viscoelastic wave system (1)–(4) has not been started yet. Since the viscoelastic terms, nonlinear damping and source terms are included in the system, the classical method employed in single equation cannot be directly used to prove the blow-up result. The main difficulty of the present paper is to find the technique to deal with nonlinear damping and source terms. In order to overcome the difficulty, combining an argument of contradiction, property of convex function ([7]) and important inequalities in [15] (cf. Lemma 2.1), we consider problem (1)–(4) and prove a blow-up result of certain solutions at a high energy level.

Firstly, let us present some notations and assumptions used throughout this article.

Taking
$$\begin{aligned}& f_{1} (u,v)= \bigl[ a\vert u+v\vert ^{2(p+1)}(u+v)+b\vert u\vert ^{p}u\vert v\vert ^{p+2} \bigr] , \\& f_{2} (u,v)= \bigl[ a\vert u+v\vert ^{2(p+1)}(u+v)+b\vert v\vert ^{p}v\vert u\vert ^{p+2} \bigr] ,\quad a,b>0, \end{aligned}$$
one can easily verify that
$$uf_{1} (u,v)+vf_{2} (u,v)=2(p+2)F(u,v),\quad \forall (u,v)\in \mathbb{R}^{2}, $$
where
$$F(u,v)=\frac{1}{2(p+2)} \bigl[ a\vert u+v\vert ^{2(p+2)}+2b\vert uv \vert ^{(p+2)} \bigr] . $$
For the relaxation functions \(g(s)\), \(h(s)\) and real number p, we give the following assumptions:
(\(\mathrm{H}_{1}\)): 
\(g\in C^{1}([0,\infty ])\), \(h\in C^{1}([0, \infty ])\) are nonnegative functions satisfying
$$\begin{aligned}& g'(s)\leq 0, \quad 1- \int_{0}^{\infty }g(s)\,ds=l>0, \\& h'(s)\leq 0,\quad 1- \int_{0}^{\infty }h(s)\,ds=k>0. \end{aligned}$$
(\(\mathrm{H}_{2}\)): 
$$\begin{aligned}& -1< p< \infty,\quad N=1,2, \\& -1< p\leq \frac{3-N}{N-2},\quad N\geq 3. \end{aligned}$$

Remark 1

Condition (\(\mathrm{H}_{1}\)) is necessary to guarantee the hyperbolicity and well-posedness of the system (1)–(4).

Note that we easily obtain the following local existence and uniqueness of weak solution for problem (1)–(4) by using the Faedo–Galerkin approximation methods and the Banach contraction mapping principle, which is similar to [2] with slight modification. The process of this proof is standard, so we omit it here.

Proposition

Under the assumptions (\(\mathrm{H}_{1}\)) and (\(\mathrm{H}_{2}\)), let the initial data \((u_{0},u_{1})\in H_{0}^{1}(\Omega)\times L^{2}(\Omega)\) and \((v_{0},v_{1})\in H_{0}^{1}(\Omega)\times L^{2}(\Omega)\) are given, then the problem (1)(4) has a unique local solution
$$\begin{aligned}& (u,v)\in C \bigl( [0,T];H_{0}^{1}(\Omega) \bigr) \times C \bigl( [0,T];H _{0}^{1}(\Omega) \bigr), \\& (u_{t},v_{t})\in C \bigl( [0,T];L^{2}(\Omega) \bigr) \cap L^{m+1} \bigl( \Omega \times (0,T) \bigr) \times C \bigl( [0,T];L^{2}(\Omega) \bigr) \cap L ^{m+1} \bigl( \Omega \times (0,T) \bigr), \end{aligned}$$
for the maximum existence time \(T>0\), where \(T\in (0,\infty ]\).
The energy related to problem (1)–(4) is
$$\begin{aligned} E(t) =&\frac{1}{2} \bigl( \Vert u_{t}\Vert _{2}^{2}+\Vert v_{t}\Vert _{2}^{2} \bigr) + \frac{1}{2} \biggl( 1- \int_{0}^{t} g(s)\,ds \biggr) \Vert \nabla u\Vert _{2}^{2} \\ &{}+ \frac{1}{2} \biggl( 1- \int_{0}^{t} h(s)\,ds \biggr) \Vert \nabla v\Vert _{2}^{2} \\ &{}+\frac{1}{2} \bigl[ (g\circ \nabla u)+(h\circ \nabla v) \bigr] - \int_{\Omega }F(u,v)\,dx, \end{aligned}$$
(5)
where
$$(g\circ \nabla v) (t)= \int_{0}^{t} g(t-\tau)\bigl\Vert v(t)-v(\tau) \bigr\Vert _{2}^{2}\,d\tau. $$

Now we are in a position to state our main result.

Theorem 1

Under the assumptions (\(\mathrm{H}_{1}\)) and (\(\mathrm{H}_{2}\)), assume that \(m>1\), \(r>1\), \(2(p+2)>\max \{m+1,r+1\}\), and
$$\begin{aligned}& \max \biggl\{ \int_{0}^{\infty }g(s)\,ds, \int_{0}^{\infty }h(s)\,ds \biggr\} < \frac{p+1}{p+1+ \frac{1}{4(p+2)}}. \end{aligned}$$
(6)
Let \((u,v)\) be a solution of Eqs. (1)(4), satisfying
$$\begin{aligned}& \int_{\Omega }u(0)u_{t}(0)\,dx+ \int_{\Omega }v(0)v_{t}(0)\,dx>ME(0)>0, \end{aligned}$$
(7)
then \((u,v)\) blows up in finite time, where
$$M=\frac{\sigma }{\sigma +1} \biggl( \frac{1-\xi }{2c_{0}\varepsilon_{0}(p+2)} \biggr) ^{\frac{1}{\sigma^{\ast }}}, $$
\(\varepsilon_{0}\in (0,1)\) is a root of the equation \(\frac{\sigma }{ \sigma +1} ( \frac{1-\xi }{2c_{0}\varepsilon_{0}(p+2)} ) ^{\frac{1}{ \sigma^{\ast }}}=\frac{2(p+2)(1-\varepsilon_{0})}{\alpha (\varepsilon _{0})}\), and
$$\begin{aligned}& \sigma =\max \{m,r\},\qquad \sigma^{\ast }=\min \{m,r\}, \\& \xi =\frac{2(p+2)-(\sigma +1)}{2(p+2)-2},\qquad \xi^{\ast }=\frac{2(p+2)-( \sigma^{\ast }+1)}{2(p+2)-2}, \\& \alpha (\varepsilon)=2\sqrt{ \bigl( (p+2) (1-\varepsilon)+1 \bigr) \biggl( k( \varepsilon)\lambda_{1}-\frac{c_{0}\varepsilon (p+2)\xi^{ \ast }}{1-\xi } \biggr) }, \\& k(\varepsilon)= \bigl( (p+2) (1-\varepsilon)-1 \bigr) \min \{k,l\}- \frac{1- \min \{k,l\}}{4(p+2)(1-\varepsilon)}, \end{aligned}$$
\(\lambda_{1}\) being the first eigenvalue of −Δ.

The outline of the paper is as follows. In Sect. 2, we introduce three lemmas related to the study of problem (1)–(4). Section 3 devoted to the proof of our main result.

2 Preliminary results

In the section, we give some lemmas which are useful for the proof of our blow-up result.

Lemma 1

Assume (\(\mathrm{H}_{1}\)) and (\(\mathrm{H}_{2}\)) hold. Let \((u,v)\) be a solution of (1)(4), then \(E(t)\) is nonincreasing, that is, \(E'(t)\leq 0\).

Proof

By the multiplier method, multiplying (1), (2) by \(u_{t}\), \(v_{t}\), respectively, and then using (5), we get
$$\begin{aligned} E'(t) =&- \bigl( \Vert u_{t}\Vert _{m+1}^{m+1}+ \Vert v_{t}\Vert _{r+1}^{r+1} \bigr) - \frac{1}{2} g(t)\Vert \nabla u\Vert _{2}^{2}- \frac{1}{2} h(t)\Vert \nabla v\Vert _{2} ^{2} \\ &{}+\frac{1}{2} \bigl[ \bigl(g'\circ \nabla u\bigr) (t)+ \bigl(h'\circ \nabla v\bigr) (t) \bigr] , \end{aligned}$$
for \(t\geq 0\), \(E'(t)\leq 0\). Moreover, the following energy inequality holds:
$$E'(t)\leq - \bigl( \Vert u_{t}\Vert _{m+1}^{m+1}+\Vert v_{t}\Vert _{r+1}^{r+1} \bigr),\quad \forall t\geq 0. $$
 □

Lemma 2

([15], Lemma 2.1)

There exist two positive constants \(c_{0}\) and \(c_{1}\) such that
$$\begin{aligned}& \frac{c_{0}}{2(p+2)} \bigl( \vert u\vert ^{2(p+2)}+\vert v\vert ^{2(p+2)} \bigr) \leq F(u,v) \leq \frac{c_{1}}{2(p+2)} \bigl( \vert u \vert ^{2(p+2)}+\vert v\vert ^{2(p+2)} \bigr). \end{aligned}$$
(8)

Next, we present the following crucial lemma which repeats the same one of Han and Wang [14], Theorem 2.4, with slight modification, so we will omit its proof.

Lemma 3

([14])

Under the assumptions (\(\mathrm{H}_{1}\)) and (\(\mathrm{H}_{2}\)), assume that \(m>1\), \(r>1\), \(2(p+2)>\max \{m+1,r+1\}\) and satisfying (6). If \(\exists t_{0}\geq 0\) such that \(E(t_{0})<0\), then the solution of the problem (1)(4) blows up in finite time.

3 Proof of Theorem 1

In the section, using an argument of contradiction and the property of a convex function, we prove our main result.

Proof of Theorem 1

Assume \((u,v)\) is a global solution of problem (1)–(4). Multiplying (1), (2) by u, v, respectively, and integrating over Ω, we derive that
$$\begin{aligned}& (u_{tt},u)+\Vert \nabla u\Vert _{2}^{2}- \int_{0}^{t} g(t-\tau) \int_{\Omega }\nabla u(t)\nabla u(\tau)\,dx\,d\tau + \int_{\Omega }\vert u_{t}\vert ^{m-1}u_{t}u \,dx \\& \quad = \int_{\Omega }uf_{1} (u,v)\,dx, \\& (v_{tt},v)+\Vert \nabla v\Vert _{2}^{2}- \int_{0}^{t} h(t-\tau) \int_{\Omega }\nabla v(t)\nabla v(\tau)\,dx\,d\tau + \int_{\Omega }\vert v_{t}\vert ^{r-1}v_{t}v \,dx = \int_{\Omega }vf_{2} (u,v)\,dx. \end{aligned}$$
Thus the following equalities are obtained:
$$\begin{aligned}& \frac{d}{dt}(u,u_{t})=\Vert u_{t}\Vert _{2}^{2}-\Vert \nabla u\Vert _{2}^{2}+ \int_{0} ^{t} g(t-\tau) \int_{\Omega }\nabla u(t)\nabla u(\tau)\,dx\,d\tau + \int_{\Omega }uf_{1} (u,v)\,dx \\& \hphantom{\frac{d}{dt}(u,u_{t})=}{}- \int_{\Omega }\vert u_{t}\vert ^{m-1}u_{t}u \,dx, \end{aligned}$$
(9)
$$\begin{aligned}& \frac{d}{dt}(v,v_{t})=\Vert v_{t}\Vert _{2}^{2}-\Vert \nabla v\Vert _{2}^{2}+ \int_{0} ^{t} h(t-\tau) \int_{\Omega }\nabla v(t)\nabla v(\tau)\,dx\,d\tau + \int_{\Omega }vf_{2} (u,v)\,dx \\& \hphantom{\frac{d}{dt}(v,v_{t})=}{}- \int_{\Omega }\vert v_{t}\vert ^{r-1}v_{t}v \,dx. \end{aligned}$$
(10)
Using the Cauchy inequality, we estimate the third terms on the right side of (9) and (10), for \(\forall \varepsilon \in (0,1)\),
$$\begin{aligned}& \int_{0}^{t} g(t-\tau) \int_{\Omega }\nabla u(t)\nabla u(\tau)\,dx\,d \tau \\& \quad = \int_{0}^{t} g(t-\tau) \int_{\Omega }\nabla u(t) \bigl[ \nabla u( \tau)-\nabla u(t) \bigr]\,dx\,d\tau + \int_{0}^{t} g(\tau)\,d\tau \Vert \nabla u\Vert _{2}^{2} \\& \quad \geq -\frac{2(p+2)(1-\varepsilon)}{2}(g\circ \nabla u) (t)-\frac{1}{4(p+2)(1- \varepsilon)} \int_{0}^{t} g(\tau)\,d\tau \Vert \nabla u\Vert _{2}^{2} \\& \qquad {}+ \int_{0}^{t} g(\tau)\,d\tau \Vert \nabla u\Vert _{2}^{2}, \end{aligned}$$
(11)
$$\begin{aligned}& \int_{0}^{t} h(t-\tau) \int_{\Omega }\nabla v(t)\nabla v(\tau)\,dx\,d \tau \\& \quad = \int_{0}^{t} h(t-\tau) \int_{\Omega }\nabla v(t) \bigl[ \nabla v( \tau)-\nabla v(t) \bigr]\,dx\,d\tau + \int_{0}^{t} h(\tau)\,d\tau \Vert \nabla v\Vert _{2}^{2} \\& \quad \geq -\frac{2(p+2)(1-\varepsilon)}{2}(h\circ \nabla v) (t)-\frac{1}{4(p+2)(1- \varepsilon)} \int_{0}^{t} h(\tau)\,d\tau \Vert \nabla v\Vert _{2}^{2} \\& \quad \quad {} + \int_{0}^{t} h(\tau)\,d\tau \Vert \nabla v\Vert _{2}^{2}. \end{aligned}$$
(12)
Combining (11) and (12), we derive that
$$\begin{aligned} \frac{d}{dt}(u,u_{t})+\frac{d}{dt}(v,v_{t}) \geq& - \biggl( 1- \int_{0}^{t} g(s)\,ds \biggr) \Vert \nabla u\Vert _{2}^{2}- \biggl( 1- \int_{0}^{t} h(s)\,ds \biggr) \Vert \nabla v\Vert _{2}^{2} \\ &{} +\Vert u_{t}\Vert _{2}^{2}+\Vert v_{t}\Vert _{2}^{2}-\frac{2(p+2)(1-\varepsilon)}{2} \bigl( (g \circ \nabla u) (t)+(h \circ \nabla v) (t) \bigr) \\ &{} - \int_{\Omega }\vert v_{t}\vert ^{r-1}v_{t}v \,dx-\frac{1}{4(p+2)(1-\varepsilon)} \int_{0}^{t} g(\tau)\,d\tau \Vert \nabla u\Vert _{2}^{2} \\ &{} -\frac{1}{4(p+2)(1-\varepsilon)} \int_{0}^{t} h(\tau)\,d\tau \Vert \nabla v\Vert _{2}^{2}- \int_{\Omega }\vert u_{t}\vert ^{m-1}u_{t}u \,dx \\ &{} +2(p+2) \int_{\Omega }F(u,v)\,dx. \end{aligned}$$
(13)
For the right side of (13) to add \(2(p+2)(1-\varepsilon)E(t)\), one can get
$$\begin{aligned}& \frac{d}{dt}(u,u_{t})+\frac{d}{dt}(v,v_{t}) \\& \quad \geq \bigl( (p+2) (1- \varepsilon)-1 \bigr) \biggl( 1- \int_{0}^{t} g(s)\,ds \biggr) \Vert \nabla u\Vert _{2}^{2} \\& \qquad {}+ \bigl( (p+2) (1-\varepsilon)+1 \bigr) \bigl( \Vert u_{t}\Vert _{2}^{2}+\Vert v_{t}\Vert _{2}^{2} \bigr) - \int_{\Omega }\vert u_{t}\vert ^{m-1}u_{t}u \,dx \\& \qquad {}+ \bigl( (p+2) (1-\varepsilon)-1 \bigr) \biggl( 1- \int_{0}^{t} h(s)\,ds \biggr) \Vert \nabla v\Vert _{2}^{2}- \int_{\Omega }\vert v_{t}\vert ^{r-1}v_{t}v \,dx \\& \qquad {}+2(p+2)\varepsilon \int_{\Omega }F(u,v)\,dx-2(p+2) (1-\varepsilon)E(t) \\& \qquad {}-\frac{1}{4(p+2)(1-\varepsilon)} \int_{0}^{t} g(\tau)\,d\tau \Vert \nabla u\Vert _{2}^{2} \\& \qquad {}-\frac{1}{4(p+2)(1-\varepsilon)} \int_{0}^{t} h(\tau)\,d\tau \Vert \nabla v\Vert _{2}^{2}. \end{aligned}$$
(14)
For the third and fifth terms on the right side of (14), Hölder’s and Young’s inequalities give us
$$\biggl\vert \int_{\Omega }\vert u_{t}\vert ^{m-1}u_{t}u \,dx\biggr\vert \leq \Vert u\Vert _{m+1} \Vert u_{t} \Vert _{m+1}^{m}\leq \varepsilon_{1}^{m+1} \frac{\Vert u\Vert _{m+1}^{m+1}}{m+1} +\varepsilon_{1}^{-\frac{m+1}{m}} \frac{m}{m+1} \Vert u_{t}\Vert _{m+1}^{m+1}. $$
By the convexity of the function \(\frac{u^{y}}{y}\) in y, for \(u\geq 0\) and \(y>0\), we have
$$\frac{\Vert u\Vert _{m+1}^{m+1}}{m+1}\leq \theta \frac{\Vert u\Vert _{2}^{2}}{2}+(1- \theta)\frac{\Vert u\Vert _{2(p+2)}^{2(p+2)}}{2(p+2)}, $$
where \(\theta =\frac{2(p+2)-(m+1)}{2(p+2)-2}\), then one can get
$$\begin{aligned}& \biggl\vert \int_{\Omega }\vert u_{t}\vert ^{m-1}u_{t}u \,dx\biggr\vert \leq \varepsilon _{1}^{m+1} \biggl( \theta \frac{\Vert u\Vert _{2}^{2}}{2}+(1-\theta)\frac{\Vert u\Vert _{2(p+2)}^{2(p+2)}}{2(p+2)} \biggr) +\varepsilon_{1}^{-\frac{m+1}{m}} \frac{m}{m+1}\Vert u_{t}\Vert _{m+1}^{m+1}. \end{aligned}$$
(15)
Similarly,
$$\begin{aligned}& \biggl\vert \int_{\Omega }\vert v_{t}\vert ^{r-1}v_{t}v \,dx\biggr\vert \leq \varepsilon _{1}^{r+1} \biggl( \eta \frac{\Vert v\Vert _{2}^{2}}{2}+(1-\eta)\frac{\Vert v\Vert _{2(p+2)} ^{2(p+2)}}{2(p+2)} \biggr) +\varepsilon_{1}^{-\frac{r+1}{r}} \frac{r}{r+1}\Vert v_{t}\Vert _{r+1}^{r+1}, \end{aligned}$$
(16)
where \(\eta =\frac{2(p+2)-(r+1)}{2(p+2)-2}\).
Take
$$\begin{aligned}& \sigma =\max \{m,r\},\qquad \sigma^{\ast }=\min \{m,r\}, \\& \xi =\frac{2(p+2)-(\sigma +1)}{2(p+2)-2}, \\& \xi^{\ast }=\frac{2(p+2)-( \sigma^{\ast }+1)}{2(p+2)-2}. \end{aligned}$$
By (14)–(16) and Lemma 1, we have
$$\begin{aligned}& \frac{d}{dt} \biggl( (u,u_{t})+(v,v_{t})- \varepsilon_{1}^{-\frac{ \sigma^{\ast }+1}{\sigma^{\ast }}}\frac{\sigma }{\sigma +1}E(t) \biggr) \\& \quad \geq \frac{d}{dt} \bigl( (u,u_{t})+(v,v_{t}) \bigr) + \varepsilon_{1}^{-\frac{ \sigma^{\ast }+1}{\sigma^{\ast }}}\frac{\sigma }{\sigma +1} \bigl( \Vert u_{t}\Vert _{m+1}^{m+1} +\Vert v_{t} \Vert _{r+1}^{r+1} \bigr) \\& \quad \geq \frac{d}{dt} \bigl( (u,u_{t})+(v,v_{t}) \bigr) + \varepsilon_{1}^{- \frac{m+1}{m}}\frac{m}{m+1}\Vert u_{t} \Vert _{m+1}^{m+1} +\varepsilon_{1}^{- \frac{r+1}{r}} \frac{r}{r+1}\Vert v_{t}\Vert _{r+1}^{r+1} \\& \quad \geq \bigl( (p+2) (1-\varepsilon)+1 \bigr) \bigl( \Vert u_{t} \Vert _{2}^{2}+\Vert v_{t}\Vert _{2}^{2} \bigr) -2(p+2) (1-\varepsilon)E(t) \\& \qquad {} +2(p+2)\varepsilon \int_{\Omega }F(u,v)\,dx \\& \qquad {} -\varepsilon_{1}^{m+1} \biggl( \theta \frac{\Vert u\Vert _{2}^{2}}{2}+(1- \theta)\frac{\Vert u\Vert _{2(p+2)}^{2(p+2)}}{2(p+2)} \biggr) - \varepsilon_{1}^{r+1} \biggl( \eta \frac{\Vert v\Vert _{2}^{2}}{2}+(1-\eta) \frac{\Vert v\Vert _{2(p+2)}^{2(p+2)}}{2(p+2)} \biggr) \\& \qquad {}+ \biggl( \bigl( (p+2) (1-\varepsilon)-1 \bigr) \biggl( 1- \int_{0} ^{t} g(s)\,ds \biggr) -\frac{1}{4(p+2)(1-\varepsilon)} \int_{0}^{t} g( \tau)\,d\tau \biggr) \Vert \nabla u\Vert _{2}^{2} \\& \qquad {} + \biggl( \bigl( (p+2) (1-\varepsilon)-1 \bigr) \biggl( 1- \int_{0} ^{t} h(s)\,ds \biggr) -\frac{1}{4(p+2)(1-\varepsilon)} \int_{0}^{t} h( \tau)\,d\tau \biggr) \Vert \nabla v\Vert _{2}^{2}. \end{aligned}$$
For the formula above, using Lemma 2 and the Poincaré inequality, we get
$$\begin{aligned}& \frac{d}{dt} \biggl( (u,u_{t})+(v,v_{t}) - \varepsilon_{1}^{-\frac{ \sigma^{\ast }+1}{\sigma^{\ast }}}\frac{\sigma }{\sigma +1}E(t) \biggr) \\& \quad \geq \bigl( (p+2) (1-\varepsilon)+1 \bigr) \bigl( \Vert u_{t} \Vert _{2} ^{2}+\Vert v_{t}\Vert _{2}^{2} \bigr) -2(p+2) (1-\varepsilon)E(t) \\& \quad \quad {} +c_{0}\varepsilon \bigl( \Vert u\Vert _{2(p+2)}^{2(p+2)}+ \Vert v\Vert _{2(p+2)} ^{2(p+2)} \bigr) \\& \qquad {} + \biggl( \bigl( (p+2) (1- \varepsilon)-1 \bigr) l-\frac{1-l}{4(p+2)(1- \varepsilon)} \biggr) \Vert \nabla u\Vert _{2}^{2} \\& \qquad {} -\varepsilon_{1}^{m+1} \biggl( \theta \frac{\Vert u\Vert _{2}^{2}}{2}+(1- \theta)\frac{\Vert u\Vert _{2(p+2)}^{2(p+2)}}{2(p+2)} \biggr) \\& \qquad {} + \biggl( \bigl( (p+2) (1-\varepsilon)-1 \bigr) k-\frac{1-k}{4(p+2)(1- \varepsilon)} \biggr) \Vert \nabla v\Vert _{2}^{2} \\& \qquad {} -\varepsilon_{1}^{r+1} \biggl( \eta \frac{\Vert v\Vert _{2}^{2}}{2}+(1- \eta)\frac{\Vert v\Vert _{2(p+2)}^{2(p+2)}}{2(p+2)} \biggr) \\& \quad \geq \bigl( (p+2) (1-\varepsilon)+1 \bigr) \bigl( \Vert u_{t} \Vert _{2}^{2}+\Vert v_{t}\Vert _{2}^{2} \bigr) + \biggl( k_{1}(\varepsilon) \lambda_{1} -\frac{ \varepsilon_{1}^{m+1}\theta }{2} \biggr) \Vert u\Vert _{2}^{2} \\& \qquad {} + \biggl( k_{2}(\varepsilon)\lambda_{1} - \frac{\varepsilon _{1}^{r+1}\eta }{2} \biggr) \Vert v\Vert _{2}^{2}-2(p+2) (1-\varepsilon)E(t) \\& \qquad {} + \biggl( c_{0}\varepsilon -\frac{\varepsilon_{1}^{m+1}(1- \theta)}{2(p+2)} \biggr) \Vert u \Vert _{2(p+2)}^{2(p+2)}+ \biggl( c_{0}\varepsilon - \frac{\varepsilon_{1}^{r+1}(1-\eta)}{2(p+2)} \biggr) \Vert v\Vert _{2(p+2)} ^{2(p+2)}, \end{aligned}$$
(17)
where \(k_{1}(\varepsilon)= ( (p+2)(1-\varepsilon)-1) l-\frac{1-l}{4(p+2)(1- \varepsilon)}\), \(k_{2}(\varepsilon)= ( (p+2)(1-\varepsilon)-1) k-\frac{1-k}{4(p+2)(1- \varepsilon)}\), \(\lambda_{1}\) being the first eigenvalue of −Δ.
Take \(\varepsilon_{1}= ( \frac{2c_{0}\varepsilon (p+2)}{1-\xi } ) ^{\frac{1}{\sigma^{\ast }+1}}\), we have
$$\begin{aligned}& \begin{aligned}&\frac{d}{dt} \biggl( (u,u_{t})+(v,v_{t}) - \varepsilon_{1}^{-\frac{ \sigma^{\ast }+1}{\sigma^{\ast }}}\frac{\sigma }{\sigma +1}E(t) \biggr) \\ &\quad \geq \bigl( (p+2) (1-\varepsilon)+1 \bigr) \bigl( \Vert u_{t} \Vert _{2}^{2}+\Vert v_{t}\Vert _{2}^{2} \bigr) -2(p+2) (1-\varepsilon)E(t) \\ &\qquad {} + \biggl( k_{1}(\varepsilon)\lambda_{1} - \frac{\varepsilon_{1}^{m+1} \theta }{2} \biggr) \Vert u\Vert _{2}^{2}+ \biggl( k_{2}(\varepsilon)\lambda_{1} -\frac{ \varepsilon_{1}^{r+1}\eta }{2} \biggr) \Vert v\Vert _{2}^{2}, \\ &\frac{d}{dt} \biggl( (u,u_{t})+(v,v_{t})- \biggl( \frac{2c_{0}\varepsilon (p+2)}{1- \xi } \biggr) ^{-\frac{1}{\sigma^{\ast }}}\frac{\sigma }{\sigma +1}E(t) \biggr) \\ &\quad \geq \bigl( (p+2) (1-\varepsilon)+1 \bigr) \bigl( \Vert u_{t} \Vert _{2}^{2}+\Vert v_{t}\Vert _{2}^{2} \bigr) -2(p+2) (1-\varepsilon)E(t) \\ &\qquad {} + \biggl( k_{1}(\varepsilon)\lambda_{1} - \frac{c_{0}\varepsilon (p+2)\theta }{1-\xi } \biggr) \Vert u\Vert _{2}^{2}+ \biggl( k_{2}(\varepsilon) \lambda_{1} -\frac{c_{0}\varepsilon (p+2)\eta }{1-\xi } \biggr) \Vert v\Vert _{2}^{2}. \end{aligned} \end{aligned}$$
(18)
Since
$$\begin{aligned}& \max \biggl\{ \int_{0}^{\infty }g(s)\,ds, \int_{0}^{\infty }h(s)\,ds \biggr\} < \frac{p+1}{p+1+ \frac{1}{4(p+2)}}, \\& \delta_{1}= \bigl( 2(p+2)-2 \bigr) l-\frac{1-l}{2(p+2)}>0, \\& \delta_{2}= \bigl( 2(p+2)-2 \bigr) k-\frac{1-k}{2(p+2)}>0. \end{aligned}$$
Then we can take ε small enough such that
$$\begin{aligned}& k_{1}(\varepsilon)\lambda_{1}-\frac{c_{0}\varepsilon (p+2) \theta }{1-\xi }>0, \\& k_{2}(\varepsilon)\lambda_{1}-\frac{c_{0}\varepsilon (p+2) \eta }{1-\xi }>0. \end{aligned}$$
The Cauchy inequality gives us
$$\begin{aligned}& \bigl( (p+2) (1-\varepsilon)+1 \bigr) \Vert u_{t}\Vert _{2}^{2}+ \biggl( k_{1}( \varepsilon) \lambda_{1} -\frac{c_{0}\varepsilon (p+2)\theta }{1- \xi } \biggr) \Vert u\Vert _{2}^{2} \\& \quad \geq 2\sqrt{ \bigl( (p+2) (1-\varepsilon)+1 \bigr) \biggl( k_{1}( \varepsilon)\lambda_{1} -\frac{c_{0}\varepsilon (p+2)\theta }{1-\xi } \biggr) } ( u,u_{t} ), \end{aligned}$$
(19)
$$\begin{aligned}& \bigl( (p+2) (1-\varepsilon)+1 \bigr) \Vert v_{t}\Vert _{2}^{2}+ \biggl( k_{2}( \varepsilon) \lambda_{1} -\frac{c_{0}\varepsilon (p+2)\eta }{1-\xi } \biggr) \Vert v\Vert _{2}^{2} \\& \quad \geq 2\sqrt{ \bigl( (p+2) (1-\varepsilon)+1 \bigr) \biggl( k_{2}( \varepsilon)\lambda_{1} -\frac{c_{0}\varepsilon (p+2)\eta }{1-\xi } \biggr) } ( v,v _{t} ). \end{aligned}$$
(20)
Combining (19) and (20), we get
$$\begin{aligned}& \frac{d}{dt} \biggl( (u,u_{t})+(v,v_{t})- \biggl( \frac{2c_{0}\varepsilon (p+2)}{1- \xi } \biggr) ^{-\frac{1}{\sigma^{\ast }}}\frac{\sigma }{\sigma +1}E(t) \biggr) \\& \quad \geq \alpha_{1}(\varepsilon) ( u,u_{t} ) + \alpha_{2}(\varepsilon) ( v,v_{t} ) -2(p+2) (1-\varepsilon)E(t) \\& \quad \geq \alpha (\varepsilon) \biggl( ( u,u_{t} ) + ( v,v_{t} ) -\frac{2(p+2)(1- \varepsilon)}{\alpha (\varepsilon)}E(t) \biggr), \end{aligned}$$
(21)
where
$$\begin{aligned}& \alpha_{1}(\varepsilon)=2\sqrt{ \bigl( (p+2) (1-\varepsilon)+1 \bigr) \biggl( k_{1}(\varepsilon)\lambda_{1}- \frac{c_{0}\varepsilon (p+2) \theta }{1-\xi } \biggr) }, \\& \alpha_{2}(\varepsilon)=2\sqrt{ \bigl( (p+2) (1-\varepsilon)+1 \bigr) \biggl( k_{2}(\varepsilon)\lambda_{1}- \frac{c_{0}\varepsilon (p+2) \eta }{1-\xi } \biggr) }, \\& \alpha (\varepsilon)=2\sqrt{ \bigl( (p+2) (1-\varepsilon)+1 \bigr) \biggl( k( \varepsilon)\lambda_{1}-\frac{c_{0}\varepsilon (p+2)\xi^{ \ast }}{1-\xi } \biggr) }, \\& k(\varepsilon)= \bigl( (p+2) (1-\varepsilon)-1 \bigr) \min \{k,l\}- \frac{1- \min \{k,l\}}{4(p+2)(1-\varepsilon)}. \end{aligned}$$
It is easy to see that
$$\begin{aligned}& \begin{aligned} &\alpha (\varepsilon)\rightarrow \sqrt{2(p+3)\min \{\delta_{1}, \delta_{2}\}\lambda_{1}},\qquad \biggl( \frac{2c_{0}\varepsilon (p+2)}{1- \xi } \biggr) ^{-\frac{1}{\sigma^{\ast }}}\frac{\sigma }{\sigma +1} \rightarrow +\infty, \\ &\frac{2(p+2)(1-\varepsilon)}{\alpha (\varepsilon)}\rightarrow \frac{2(p+2)}{\sqrt{2(p+3) \min \{\delta_{1},\delta_{2}\}\lambda_{1}}},\quad \mbox{as }\varepsilon \rightarrow 0^{+}. \end{aligned} \end{aligned}$$
(22)
On the other hand, by the definition of \(k(\varepsilon)\), we have
$$\begin{aligned}& k(\varepsilon)\rightarrow -\infty,\qquad k(\varepsilon)\lambda_{1}- \frac{c _{0}\varepsilon (p+2)\xi^{\ast }}{1-\xi }\rightarrow -\infty,\quad \mbox{as }\varepsilon \rightarrow 1^{-}, \\& k(\varepsilon)\lambda_{1}-\frac{c_{0}\varepsilon (p+2)\xi^{\ast }}{1- \xi }\rightarrow \frac{\min \{\delta_{1},\delta_{2}\}\lambda_{1}}{2}, \quad \mbox{as }\varepsilon \rightarrow 0^{+}. \end{aligned}$$
Hence, there exists \(\varepsilon_{\ast }\in (0,1)\) such that
$$\alpha (\varepsilon_{\ast })=0 \quad \mbox{and}\quad \alpha (\varepsilon)>0,\quad \forall \varepsilon \in (0,\varepsilon_{\ast }). $$
This implies that
$$\begin{aligned}& \begin{aligned}&\alpha (\varepsilon)\rightarrow 0,\qquad \frac{2(p+2)(1-\varepsilon)}{ \alpha (\varepsilon)}\rightarrow +\infty, \\ &\biggl( \frac{2c_{0}\varepsilon (p+2)}{1-\xi } \biggr) ^{-\frac{1}{ \sigma^{\ast }}}\frac{\sigma }{\sigma +1}\rightarrow \biggl( \frac{2c _{0}\varepsilon_{\ast }(p+2)}{1-\xi } \biggr) ^{-\frac{1}{\sigma^{ \ast }}}\frac{\sigma }{\sigma +1},\quad \mbox{as } \varepsilon \rightarrow \varepsilon_{\ast }^{-}. \end{aligned} \end{aligned}$$
(23)
Using (22), (23) and the continuity in ε of \(\frac{2(p+2)(1- \varepsilon)}{\alpha (\varepsilon)}\) and \(( \frac{2c_{0}\varepsilon (p+2)}{1-\xi } ) ^{-\frac{1}{\sigma^{\ast }}}\frac{\sigma }{ \sigma +1}\), there exists \(\varepsilon_{0}\in (0,\varepsilon_{ \ast })\subset (0,1)\) such that
$$\biggl( \frac{2c_{0}\varepsilon_{0}(p+2)}{1-\xi } \biggr) ^{-\frac{1}{ \sigma^{\ast }}}\frac{\sigma }{\sigma +1}= \frac{2(p+2)(1-\varepsilon _{0})}{\alpha (\varepsilon_{0})}. $$
Then (21) can be rewritten as
$$\begin{aligned}& \frac{d}{dt} \biggl( (u,u_{t})+(v,v_{t})- \biggl( \frac{2c_{0}\varepsilon _{0}(p+2)}{1-\xi } \biggr) ^{-\frac{1}{\sigma^{\ast }}}\frac{\sigma }{ \sigma +1}E(t) \biggr) \\& \quad \geq \alpha (\varepsilon_{0}) \biggl( ( u,u_{t} ) + ( v,v_{t} ) - \biggl( \frac{2c_{0}\varepsilon_{0}(p+2)}{1-\xi } \biggr) ^{-\frac{1}{ \sigma^{\ast }}} \frac{\sigma }{\sigma +1}E(t) \biggr). \end{aligned}$$
(24)
Now, setting \(H(t)=(u,u_{t})+(v,v_{t})- ( \frac{2c_{0}\varepsilon _{0}(p+2)}{1-\xi } ) ^{-\frac{1}{\sigma^{\ast }}}\frac{\sigma }{ \sigma +1}E(t)\). Then we exploit (7), this tells us that
$$H(0)= \int_{\Omega }u(0)u_{t}(0)\,dx+ \int_{\Omega }v(0)v_{t}(0)\,dx- \biggl( \frac{2c_{0}\varepsilon_{0}(p+2)}{1-\xi } \biggr) ^{-\frac{1}{ \sigma^{\ast }}}\frac{\sigma }{\sigma +1}E(0)>0 $$
and
$$\begin{aligned}& \frac{d}{dt}H(t)\geq \alpha (\varepsilon_{0})H(t). \end{aligned}$$
(25)
A simple integration of (25) over \((0,t)\) then yields
$$H(t)\geq e^{\alpha (\varepsilon_{0})t}H(0),\quad \forall t\geq 0. $$
Since \((u,v)\) is global, by Lemma 2 and Lemma 3, for \(t\geq 0\), we have \(0\leq E(t)\leq E(0)\). Hence, we obtain
$$(u,u_{t})+(v,v_{t})\geq e^{\alpha (\varepsilon_{0})t}H(0). $$
So, we get the estimate
$$\begin{aligned} \Vert u_{t}\Vert _{2}^{2}+\Vert v_{t}\Vert _{2}^{2} =&\bigl\Vert u(0)\bigr\Vert _{2}^{2}+\bigl\Vert v(0)\bigr\Vert _{2}^{2}+2 \int_{0}^{t} \bigl[ (u,u_{t})+(v,v_{t}) \bigr]\,d\tau \\ \geq& \bigl\Vert u(0)\bigr\Vert _{2}^{2}+\bigl\Vert v(0)\bigr\Vert _{2}^{2}+2 \int_{0}^{t} e^{\alpha ( \varepsilon_{0})t}H(0)\,d\tau \\ \geq& \bigl\Vert u(0)\bigr\Vert _{2}^{2}+\bigl\Vert v(0)\bigr\Vert _{2}^{2}+\frac{2}{\alpha ( \varepsilon_{0})} \bigl( e^{\alpha (\varepsilon_{0})t}-1 \bigr) H(0). \end{aligned}$$
(26)
On the other hand, by Lemma 1, Lemma 3 and the Hölder inequality, we derive
$$\begin{aligned} \Vert u_{t}\Vert _{2}+\Vert v_{t}\Vert _{2} \leq& \bigl\Vert u(0)\bigr\Vert _{2}+\bigl\Vert v(0) \bigr\Vert _{2}+ \int_{0} ^{t} \bigl\Vert u_{t}(\tau) \bigr\Vert _{2}\,d\tau + \int_{0}^{t} \bigl\Vert v_{t}(\tau) \bigr\Vert _{2}\,d\tau \\ \leq& \bigl\Vert u(0)\bigr\Vert _{2}+\bigl\Vert v(0)\bigr\Vert _{2}+C_{0} \biggl( \int _{0}^{t} \bigl\Vert u_{t}(\tau) \bigr\Vert _{m+1}\,d\tau + \int_{0}^{t} \bigl\Vert v_{t}(\tau) \bigr\Vert _{r+1}\,d\tau \biggr) \\ \leq& \bigl\Vert u(0)\bigr\Vert _{2}+\bigl\Vert v(0)\bigr\Vert _{2}+C_{0}t^{\frac{m}{m+1}} \int_{0}^{t} \bigl\Vert u_{t}(\tau) \bigr\Vert _{m+1}^{m+1}\,d\tau^{\frac{1}{m+1}} \\ &{}+C_{0}t^{\frac{r}{r+1}} \int_{0}^{t} \bigl\Vert v_{t}(\tau) \bigr\Vert _{r+1}^{r+1}\,d\tau^{\frac{1}{r+1}} \\ \leq& \bigl\Vert u(0)\bigr\Vert _{2}+\bigl\Vert v(0)\bigr\Vert _{2}+Ct^{\frac{\sigma }{\sigma +1}} \int_{0}^{t} \bigl( \bigl\Vert u_{t}( \tau)\bigr\Vert _{m+1}^{m+1}+\bigl\Vert v_{t}( \tau)\bigr\Vert _{r+1}^{r+1} \bigr)\,d\tau^{\frac{1}{\sigma^{\ast }+1}} \\ \leq& \bigl\Vert u(0)\bigr\Vert _{2}+\bigl\Vert v(0)\bigr\Vert _{2}+Ct^{\frac{\sigma }{\sigma +1}} \bigl( E(0)-E(t) \bigr) ^{\frac{1}{\sigma^{\ast }+1}} \\ \leq& \bigl\Vert u(0)\bigr\Vert _{2}+\bigl\Vert v(0)\bigr\Vert _{2}+Ct^{\frac{\sigma }{\sigma +1}}E(0) ^{\frac{1}{\sigma^{\ast }+1}}. \end{aligned}$$
This contradicts (26) and we get the finite time blow-up result. □

4 Conclusion

We prove the finite time blow-up of some solutions for a semilinear viscoelastic wave system with nonlinear weak damping and source terms whose initial data have arbitrarily high initial energy. We point out that the methods for a single equation in [6, 7] are not necessarily applicable to our system. We also notice that the result in Theorem 1 extends the results for the system in [14, 15].

Declarations

Acknowledgements

The authors would like to deeply thank all the reviewers for their insightful and constructive comments.

Authors’ contributions

All authors contributed equally to the manuscript and read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
School of Mathematical Sciences, Ocean University of China, Qingdao, P.R. China

References

  1. Levine, H.A.: Instability and nonexistence of global solutions to nonlinear wave equations of the form \(Pu_{tt}=-Au+F(u)\). Trans. Am. Math. Soc. 192, 1–21 (1974) MATHGoogle Scholar
  2. Georgiev, V., Todorova, G.: Existence of a solution of the wave equation with nonlinear damping and source terms. J. Differ. Equ. 109, 295–308 (1994) MathSciNetView ArticleMATHGoogle Scholar
  3. Messaoudi, S.A.: Blow up and global existence in a nonlinear viscoelastic wave equation. Math. Nachr. 260, 58–66 (2003) MathSciNetView ArticleMATHGoogle Scholar
  4. Messaoudi, S.A.: Blow-up of positive-initial-energy solutions of a nonlinear viscoelastic hyperbolic equation. J. Math. Anal. Appl. 320, 902–915 (2006) MathSciNetView ArticleMATHGoogle Scholar
  5. Kafinia, M., Messaoudi, S.A.: A blow-up result in a Cauchy viscoelastic problem. Appl. Math. Lett. 21, 549–553 (2008) MathSciNetView ArticleMATHGoogle Scholar
  6. Wang, Y.J.: A global nonexistence theorem for viscoelastic equations with arbitrary positive initial energy. Appl. Math. Lett. 22, 1394–1400 (2009) MathSciNetView ArticleMATHGoogle Scholar
  7. Song, H.T.: Blow up of arbitrarily positive initial energy solutions for a viscoelastic wave equation. Nonlinear Anal., Real World Appl. 26, 306–314 (2015) MathSciNetView ArticleMATHGoogle Scholar
  8. Cavalcanti, M.M., Cavalcanti, V.N.D., Lasiecka, I., Webler, C.M.: Intrinsic decay rates for the energy of a nonlinear viscoelastic equation modeling the vibrations of thin rods with variable density. Adv. Nonlinear Anal. 6, 121–145 (2017) MathSciNetMATHGoogle Scholar
  9. Grace, S.R.: Oscillation criteria for third order nonlinear delay differential equations with damping. Opusc. Math. 35(4), 485–497 (2015) MathSciNetView ArticleMATHGoogle Scholar
  10. Kumar, S., Kumar, D., Singh, J.: Fractional modelling arising in unidirectional propagation of long waves in dispersive media. Adv. Nonlinear Anal. 5(4), 383–394 (2016) MathSciNetMATHGoogle Scholar
  11. Agre, K., Rammaha, M.A.: Systems of nonlinear wave equations with damping and source terms. Differ. Integral Equ. 19(11), 1235–1270 (2006) MathSciNetMATHGoogle Scholar
  12. Said-Houari, B.: Global nonexistence of positive initial-energy solutions of a system of nonlinear wave equations with damping and source terms. Differ. Integral Equ. 23(1–2), 79–92 (2010) MathSciNetMATHGoogle Scholar
  13. Pişkin, E.: Uniform decay and blow-up of solutions for coupled nonlinear Klein–Gordon equations with nonlinear damping terms. Math. Methods Appl. Sci. 37, 3036–3047 (2014) MathSciNetView ArticleMATHGoogle Scholar
  14. Han, X.S., Wang, M.X.: Global existence and blow-up of solutions for a system of nonlinear viscoelastic wave equations with damping and source. Nonlinear Anal. 71(11), 5427–5450 (2009) MathSciNetView ArticleMATHGoogle Scholar
  15. Messaoudi, S.A., Said-Hauari, B.: Global nonexistence of positive initial-energy solutions of a system of nonlinear viscoelastic wave equations with damping and source terms. J. Math. Anal. Appl. 365(1), 277–287 (2010) MathSciNetView ArticleMATHGoogle Scholar
  16. Liang, F., Gao, H.J.: Global nonexistence of positive initial-energy solutions for coupled nonlinear wave equations with damping and source terms. Abstr. Appl. Anal. 2011(4), 430 (2011) MathSciNetMATHGoogle Scholar
  17. Hao, J.H., Niu, S.S., Meng, H.H.: Global nonexistence of solutions for nonlinear coupled viscoelastic wave equations with damping and source terms. Bound. Value Probl. 2014(1), 1 (2014) MathSciNetView ArticleMATHGoogle Scholar
  18. Li, G., Hong, L.H., Liu, W.J.: Global nonexistence of solutions for the viscoelastic wave equation of Kirchhoff type with high energy. J. Funct. Spaces Appl. 89(8), 1–15 (2011) Google Scholar

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