1 Introduction

In this paper we investigate a blow-up result for the following quasilinear viscoelastic wave equation:

$$\begin{aligned}& \vert u_{t} \vert ^{\rho }u_{tt} - \Delta u + \int ^{t}_{0} g(t-s) \Delta u(s) \,ds -\Delta u_{t}= 0, \quad \text{in }\varOmega \times (0,\infty ), \end{aligned}$$
(1.1)
$$\begin{aligned}& u(x,t)= 0, \quad \text{on }\varGamma _{0} \times [0, \infty ), \end{aligned}$$
(1.2)
$$\begin{aligned}& \frac{\partial u}{\partial \nu } +\frac{\partial u_{t}}{\partial \nu }- \int _{0}^{t} g(t-s) \frac{\partial u}{\partial \nu }(s ) \,ds+ f(u _{t})= \vert u \vert ^{p-2}u, \quad \text{on } \varGamma _{1} \times [0, \infty ), \end{aligned}$$
(1.3)
$$\begin{aligned}& u(x, 0) = u_{0} (x) , \quad\quad u_{t}(x, 0) = u_{1} (x), \quad \text{in }\varOmega , \end{aligned}$$
(1.4)

where \(\varOmega \subset {\mathbb{R}}^{n}\) is a bounded domain with sufficiently smooth boundary \(\partial \varOmega =\varGamma _{0} \cup \varGamma _{1}\), \(\varGamma _{0} \cap \varGamma _{1} =\emptyset \), where \(\varGamma _{0} \) and \(\varGamma _{1} \) are measurable over ∂Ω, ν is the unit outward normal to ∂Ω, and g is a positive function.

For the case of \(\rho =0\), problem (1.1)–(1.4) arises in the theory of viscoelasticity and describes the spread of strain waves in a viscoelastic configuration [1,2,3]. Messaoudi [4] studied the following initial-boundary value problem:

$$\begin{aligned} \textstyle\begin{cases} u_{tt} -\Delta u +\int _{0}^{t} g(t-s) \Delta u(s) \,ds+a \vert u_{t} \vert ^{m-2} u_{t} =b \vert u \vert ^{p-2} u, \quad \text{in }\varOmega \times (0, \infty ), \\ u(x,t)=0, \quad \text{on } \partial \varOmega \times [0, \infty ), \end{cases}\displaystyle \end{aligned}$$

where \(m\geq 1\), \(p>2\), \(a,b >0\) are constants and \(g: \mathbb{R}^{+} \to \mathbb{R}^{+} \) is a nonincreasing function. Under suitable conditions on g, he proved that any weak solution with negative initial energy blows up in finite time if \(p>m\). He [5] also extended the blow-up result to certain solutions with positive initial energy. Song and Zhong [6] considered the viscoelastic wave equation with strong damping:

$$\begin{aligned} \textstyle\begin{cases} u_{tt} -\Delta u +\int _{0}^{t} g(t-s) \Delta u(s) \,d s -\Delta u_{t} = \vert u \vert ^{p-2} u , \quad \text{in }\varOmega \times [0,T], \\ u(x,t)=0, \quad x\in \partial \varOmega . \end{cases}\displaystyle \end{aligned}$$

Recently, Park et al. [7] showed the blow-up result of solution for the following viscoelastic wave equation with nonlinear damping:

$$\begin{aligned} \textstyle\begin{cases} u_{tt} -\Delta u +\int _{0}^{t} g(t-s)\Delta u(s) \,ds +h(u_{t}) = \vert u \vert ^{p-2}u, \quad \text{in }\varOmega \times (0, \infty ), \\ u(x,t)=0, \quad \text{on }\partial \varOmega \times (0, \infty ). \end{cases}\displaystyle \end{aligned}$$

They obtained the blow-up of solution with positive initial energy as well as nonpositive initial energy under a weaker assumption on the damping term. Liu and Yu [8] studied the following viscoelastic wave equation with boundary damping and source terms:

$$\begin{aligned} \textstyle\begin{cases} u_{tt} -\Delta u +\int _{0}^{t} g(t-s) \Delta u(s) \,ds =0, \quad \text{in }\varOmega \times (0, \infty ), \\ u(x,t)=0, \quad \text{on }\varGamma _{0} \times [0, \infty ), \\ \frac{\partial u}{\partial \nu } -\int _{0}^{t} g(t-s) \frac{\partial u}{\partial \nu } (s) \,ds + \vert u_{t} \vert ^{m-2} u_{t} = \vert u \vert ^{p-2} u , \quad \text{on }\varGamma _{1} \times [0, \infty ). \end{cases}\displaystyle \end{aligned}$$

They proved the blow-up result of solutions with nonpositive initial energy as well as positive initial energy for both the linear and nonlinear damping cases. In the absence of the viscoelastic term (\(g=0\)), the related problem has been extensively investigated, and results concerning the global existence of solution and nonexistence have been studied (see [9,10,11,12,13,14]).

For the case of \(\rho >0\), Cavalcanti et al. [15] considered the following problem:

$$\begin{aligned} \textstyle\begin{cases} \vert u_{t} \vert ^{\rho }u_{tt} -\Delta u -\Delta u_{tt}+\int _{0}^{t} g(t-s) \Delta u(s) \,ds -\gamma \Delta u_{t} =0, \quad \text{in }\varOmega \times (0, \infty ), \\ u(x,t) =0, \quad \text{on }\partial \varOmega \times (0, \infty ). \end{cases}\displaystyle \end{aligned}$$

They showed a global existence result for \(\gamma \geq 0\) and an exponential decay result for \(\gamma >0\). In the case of \(\gamma =0\), Liu [16] proved the nonlinear viscoelastic problem:

$$\begin{aligned} \textstyle\begin{cases} \vert u_{t} \vert ^{\rho }u_{tt} -\Delta u -\Delta u_{tt}+\int _{0}^{t} g(t-s) \Delta u(s) \,ds = b \vert u \vert ^{p-2} u, \quad \text{in }\varOmega \times (0, \infty ), \\ u(x,t) =0, \quad \text{on }\partial \varOmega \times (0, \infty ). \end{cases}\displaystyle \end{aligned}$$

He discussed the general decay result for the global solution and the finite time blow-up of solution. In the absence of the dispersion term \(-\Delta u_{tt}\), Song [17] investigated the nonexistence of global solutions with positive initial energy for the viscoelastic wave equation with nonlinear damping:

$$\begin{aligned} \textstyle\begin{cases} \vert u_{t} \vert ^{\rho } u_{tt} -\Delta u +\int _{0}^{t} g(t-s) \Delta u(s) \,ds + \vert u_{t} \vert ^{m-2} u_{t} = \vert u \vert ^{p-2} u, \quad \text{in }\varOmega \times [0,T], \\ u(x,t) =0, \quad x \in \partial \varOmega . \end{cases}\displaystyle \end{aligned}$$

Recently, Hao and Wei [18] established the blow-up result of solution with negative initial energy and some positive initial energy for the quasilinear viscoelastic wave equation with strong damping:

$$\begin{aligned} \textstyle\begin{cases} \vert u_{t} \vert ^{\rho }u_{tt} -\Delta u +\int _{0}^{t} g(t-s) \Delta u(s) \,ds -\Delta u_{t} = \vert u \vert ^{p-2} u , \quad \text{in }\varOmega \times (0, \infty ), \\ u(x,t)=0, \quad \text{on }\partial \varOmega \times (0, \infty ) . \end{cases}\displaystyle \end{aligned}$$

To our knowledge, there are few blow-up results of solution for quasilinear viscoelastic wave equation with boundary nonlinear damping and source terms. Motivated by the previous work, we study the blow-up of solution with nonpositive initial energy as well as positive initial energy for the quasilinear viscoelastic wave equation with strong damping and boundary weak damping.

This paper is organized as follows. In Sect. 2, we recall the notation, hypotheses, and some necessary preliminaries. In Sect. 3, we prove the blow-up of solution for (1.1)–(1.4).

2 Preliminaries

In this section we give notations, hypotheses, and some lemmas needed in our main result.

We impose the assumptions for problem (1.1)–(1.4):

\((H_{1})\) :

Hypotheses on g

The kernel \(g : [0, \infty ) \to [0, \infty )\) is a nonincreasing and differentiable function satisfying

$$ 1- \int ^{\infty }_{0} g(s)\,ds : = l >0 . $$
(2.1)
\((H_{2})\) :

Hypotheses on f

Let \(f : {\mathbb{R}} \to {\mathbb{R}} \) be a nondecreasing \(C^{1}\) function with \(f(0)=0 \). Assume that there exists a strictly increasing and odd function \(\xi : [-1, 1] \to {\mathbb{R}}\) such that

$$\begin{aligned}& \bigl\vert \xi (s) \bigr\vert \leq \bigl\vert f(s) \bigr\vert \leq \bigl\vert \xi ^{-1} (s) \bigr\vert \quad \text{for } \vert s \vert \leq 1 , \end{aligned}$$
(2.2)
$$\begin{aligned}& a_{1} \vert s \vert ^{m-1} \leq \bigl\vert f(s) \bigr\vert \leq a_{2} \vert s \vert ^{m-1} \quad \text{for } \vert s \vert >1 , \end{aligned}$$
(2.3)

where \(a_{1}\), \(a_{2}\) are positive constants and \(\xi ^{-1}\) represents the inverse function of ξ.

\((H_{3})\) :

Hypotheses on p, m, and ρ

$$\begin{aligned}& 2< m , 2< p \quad \text{if } n=1,2, \quad\quad 2 < m, p \leq \frac{2(n-1)}{n-2} \quad \text{if } n \geq 3, \end{aligned}$$
(2.4)
$$\begin{aligned}& 0< \rho \quad \text{if } n=1,2, \quad\quad 0< \rho \leq \frac{2}{n-2} \quad \text{if } n\geq 3 . \end{aligned}$$
(2.5)

As usual, \((\cdot , \cdot )\) and \(\Vert \cdot \Vert _{p}\) denote the inner product in the space \(L^{2}(\varOmega )\) and the norm of the space \(L^{p}(\varOmega )\), respectively. For brevity, we denote \(\Vert \cdot \Vert _{2}\) by \(\Vert \cdot \Vert \). We introduce the notations:

$$ (u, v)_{\varGamma _{1}}= \int _{\varGamma _{1}} u(x) v(x) \,d\varGamma , \quad\quad \Vert \cdot \Vert _{q, \varGamma _{1}} = \Vert \cdot \Vert _{L^{q} (\varGamma _{1})}, \quad 1\leq q\leq \infty , $$

the Hilbert space

$$ V =\bigl\{ u\in H^{1} (\varOmega ): u\vert _{\varGamma _{0}} =0\bigr\} $$

(\(u\vert _{\varGamma _{0}}\) is the trace sense), endowed the equivalent norm \(\Vert \nabla u\Vert \). We recall the trace Sobolev embedding

$$ V\hookrightarrow L^{q} (\varGamma _{1}) \quad \text{for } 2\leq q < r=\frac{2(n-1)}{(n-2)} $$

and the embedding inequality

$$\begin{aligned} \Vert v \Vert _{q,\varGamma _{1}} \leq B \Vert \nabla v \Vert \quad \text{for } v \in V, \end{aligned}$$
(2.6)

where \(B >0\) is the optimal constant. We define the energy associated with problem (1.1)–(1.4) by

$$ \begin{aligned}[b] E(t) ={}& \frac{1}{\rho +2} \bigl\Vert u_{t}(t) \bigr\Vert _{\rho +2}^{\rho +2} + \frac{1}{2} \biggl( 1 - \int ^{t}_{0} g(s) \,ds \biggr) \bigl\Vert \nabla u(t) \bigr\Vert ^{2} \\ &{}+ \frac{1}{2} (g \circ \nabla u) (t) - \frac{1}{p} \bigl\Vert u(t) \bigr\Vert ^{p}_{p, \varGamma _{1}} , \end{aligned} $$
(2.7)

where \((g \circ \nabla u)(t) = \int ^{t}_{0} g(t-s) \Vert \nabla u(t) - \nabla u(s) \Vert ^{2} \,ds \). It is easy to find that

$$ E ' (t) = - \bigl(f\bigl( u_{t} (t) \bigr), u_{t} (t)\bigr)_{\varGamma _{1}} + \frac{1}{2} \bigl(g' \circ \nabla u\bigr) (t) - \frac{g(t)}{2} \bigl\Vert \nabla u(t) \bigr\Vert ^{2} - \Vert \nabla u _{t} \Vert ^{2} \leq 0 . $$
(2.8)

Therefore, E is a nonincreasing function.

Next, we define the functionals:

$$\begin{aligned}& I(t) = \biggl( 1 - \int ^{t}_{0} g(s) \,ds \biggr) \bigl\Vert \nabla u(t) \bigr\Vert ^{2} + (g \circ \nabla u) (t) - \bigl\Vert u(t) \bigr\Vert ^{p}_{p, \varGamma _{1}} , \\& H(t) = \frac{1}{2} \biggl[ \biggl( 1 - \int ^{t}_{0} g(s) \,ds \biggr) \bigl\Vert \nabla u(t) \bigr\Vert ^{2} + (g \circ \nabla u) (t) \biggr]- \frac{1}{p} \bigl\Vert u(t) \bigr\Vert ^{p} _{p, \varGamma _{1}} . \end{aligned}$$

Similar as in [8], for \(t\geq 0\), we define

$$ h(t)={\inf_{u\in V, u\neq 0 }\sup_{\lambda \geq 0}H(\lambda u)} . $$

Then, we have

$$ 0< h_{0} \leq h(t) \leq \sup_{\lambda \geq 0}H(\lambda u) \quad \text{for } t\geq 0, $$

where

$$\begin{aligned}& h_{0} = \frac{p-2}{2p} \biggl( \frac{l}{B^{2}} \biggr)^{p/(p-2)}, \\& \sup_{\lambda \geq 0}H(\lambda u) = \frac{p-2}{2p} \biggl( \frac{( 1 - \int ^{t}_{0} g(s) \,ds ) \Vert \nabla u(t) \Vert ^{2} + (g \circ \nabla u)(t)}{ \Vert u(t) \Vert _{p, \varGamma _{1}}^{2} } \biggr)^{p/(p-2)} . \end{aligned}$$

Lemma 2.1

(Lemma 4.1 of [8])

Suppose that \((H_{1})\)–\((H_{3})\) hold. For any fixed number \(\delta <1\), assume that \((u_{0}, u_{1}) \in V\times L^{2}(\varOmega )\) and satisfy

$$\begin{aligned} I(0)< 0, \quad\quad E(0) < \delta h_{0} . \end{aligned}$$
(2.9)

Assume further that g satisfies

$$\begin{aligned} \int ^{\infty }_{0} g(s) \,ds < \frac{ p-2 }{ p-2+\frac{1}{[(1- \hat{\delta })^{2} (p-2) +2(1-\hat{\delta })] } }, \end{aligned}$$
(2.10)

where \(\hat{\delta }=\max \{ 0, \delta \}\). Then we have \(I(t)<0 \) for all \(t\in [0, T)\), and

$$\begin{aligned} \begin{aligned}[b] h_{0} &< \frac{p-2}{2p} \biggl[ \biggl( 1 - \int ^{t}_{0} g(s) \,ds \biggr) \bigl\Vert \nabla u(t) \bigr\Vert ^{2} + (g \circ \nabla u) (t) \biggr] \\&< \frac{p-2}{2p} \bigl\Vert u(t) \bigr\Vert ^{p} _{p, \varGamma _{1}} , \quad t \in [0, T).\end{aligned} \end{aligned}$$
(2.11)

Throughout this paper, we define

$$\begin{aligned} K(t) = \hat{ \delta } h_{0} - E(t) , \end{aligned}$$
(2.12)

which, from (2.8), is an increasing function. Using (2.7), (2.9), and (2.11), we see that

$$\begin{aligned}& 0 < K(0) \leq K(t) \leq \hat{ \delta } h_{0} +\frac{1}{p} \bigl\Vert u(t) \bigr\Vert ^{p} _{p, \varGamma _{1}} \leq p_{0} \bigl\Vert u(t) \bigr\Vert ^{p}_{p, \varGamma _{1}} , \quad t\in [0, T), \end{aligned}$$
(2.13)

where \(p_{0}=\frac{ (p-2)\hat{ \delta } }{2p} + \frac{1}{p}\).

Moreover, similar as in [5], we can show the following lemma which is needed later.

Lemma 2.2

Let the conditions of Lemma 2.1 hold. Then the solution u of problem (1.1)–(1.4) satisfies

$$\begin{aligned}& \bigl\Vert u(t) \bigr\Vert ^{s}_{p, \varGamma _{1}} \leq C_{0} \bigl\Vert u(t) \bigr\Vert ^{p}_{p,\varGamma _{1}} , \quad t\in [0,T), \textit{ for any } 2\leq s \leq p, \end{aligned}$$
(2.14)

where \(C_{0}\) is a positive constant.

Proof

If \(\Vert u(t)\Vert _{p,\varGamma _{1}} \geq 1\), then \(\Vert u(t)\Vert _{p,\varGamma _{1}} ^{s} \leq \Vert u(t)\Vert _{p, \varGamma _{1}}^{p}\).

If \(\Vert u(t)\Vert _{p,\varGamma _{1}} \leq 1\), then

$$ \bigl\Vert u(t) \bigr\Vert _{p,\varGamma _{1}}^{s} \leq \bigl\Vert u(t) \bigr\Vert _{p, \varGamma _{1}}^{2} \leq B ^{2} \bigl\Vert \nabla u(t) \bigr\Vert ^{2} , $$

where we used (2.6). Then there exists a positive constant \(C_{1}=\max \{1, {B^{2}}\} \) such that

$$\begin{aligned} \bigl\Vert u(t) \bigr\Vert ^{s}_{p, \varGamma _{1}} \leq C_{1} \bigl( \bigl\Vert u(t) \bigr\Vert ^{p}_{p, \varGamma _{1}} + \bigl\Vert \nabla u(t) \bigr\Vert ^{2} \bigr) \quad \text{for any } 2 \leq s \leq p. \end{aligned}$$
(2.15)

By (2.1), (2.7), (2.12), and (2.13), we get

$$\begin{aligned} \frac{l}{2} \bigl\Vert \nabla u(t) \bigr\Vert ^{2} &\leq \frac{1}{2} \biggl( 1- \int _{0} ^{t} g(s) \,ds \biggr) \bigl\Vert \nabla u(t) \bigr\Vert ^{2} \\ &\leq \hat{\delta } h_{0} -K(t)- \frac{1}{\rho +2} \bigl\Vert u_{t}(t) \bigr\Vert _{ \rho +2}^{\rho +2} - \frac{1}{2} (g \circ \nabla u) (t) + \frac{1}{p} \bigl\Vert u(t) \bigr\Vert ^{p} _{p, \varGamma _{1}} \\ &\leq \hat{\delta } h_{0} +\frac{1}{p} \bigl\Vert u(t) \bigr\Vert _{p,\varGamma _{1}}^{p}. \end{aligned}$$
(2.16)

Using (2.13), (2.15), and (2.16), we have the desired result (2.14). □

We state the well-posedness which can be established by the arguments of [5, 19,20,21].

Theorem 2.1

Assume \((H_{1})\)–\((H_{3})\) hold. Then, for every \((u_{0} , u_{1}) \in V\times L^{2}(\varOmega )\), there exists a unique local solution in the class

$$ u \in C\bigl([0, T);V \bigr), \quad\quad u_{t} \in C\bigl([0, T); L^{2} (\varOmega ) \bigr) \cap L^{m} \bigl( \varGamma _{1}\times [0, T) \bigr) \quad \textit{for some } T >0. $$

3 A blow-up result

To show the blow-up result for solutions with nonpositive initial energy as well as positive initial energy, we use the similar method of [8]. Our main result reads as follows.

Theorem 3.1

Let \((H_{1})\)–\((H_{3})\) hold and \(p> {m, }\) and the conditions of Lemma 2.1 hold. Moreover, we assume

$$\begin{aligned} \xi ^{-1}(1) < \biggl( \frac{ \theta \hat{\delta } h_{0} p \eta }{(p-1) \vert \varGamma _{1} \vert } \biggr)^{ \frac{p-1}{p} } , \end{aligned}$$
(3.1)

where \(0 < \theta < \min \{2 c_{1} , 2c_{2} \}\), \(0<\eta < \theta ^{\frac{1}{p-1}} \), and \(c_{1}\) and \(c_{2}\) will be specified later. Then the solution of problem (1.1)–(1.4) blows up in finite time.

Proof

To show this theorem, we use the idea given in [4, 5]. We assume that there exists some positive constant M such that, for \(t>0\), the solution \(u(t)\) of (1.1)–(1.4) satisfies

$$ \bigl\Vert u_{t}(t) \bigr\Vert _{\rho +2}^{\rho +2} + \bigl\Vert \nabla u(t) \bigr\Vert ^{2} + \bigl\Vert u(t) \bigr\Vert _{p, \varGamma _{1}}^{p} \leq M. $$
(3.2)

Let us consider the following function:

$$\begin{aligned} L(t) = K^{1-\sigma } (t) + \frac{\varepsilon }{\rho +1} \int _{\varOmega } \bigl\vert u_{t}(t) \bigr\vert ^{\rho }u_{t}(t) u(t) \,dx +\frac{\varepsilon }{2} \bigl\Vert \nabla u(t) \bigr\Vert ^{2} , \quad \varepsilon >0, \end{aligned}$$
(3.3)

where

$$\begin{aligned} 0 < \sigma < \min \biggl\{ 1,\frac{1}{\rho +2}, \frac{p-m}{p(m-1)} \biggr\} . \end{aligned}$$
(3.4)

From (1.1)–(1.3), we obtain

$$\begin{aligned} L'(t) &= (1- \sigma ) K^{-\sigma } (t) K' (t) + \frac{\varepsilon }{ \rho +1} \bigl\Vert u_{t} (t) \bigr\Vert _{\rho +2}^{\rho +2} - \varepsilon \bigl\Vert \nabla u(t) \bigr\Vert ^{2} \\ &\quad{} + \varepsilon \int ^{t}_{0} g(t-s) \bigl(\nabla u(s) , \nabla u(t) \bigr) \,ds - \varepsilon \bigl(f\bigl(u_{t} (t)\bigr), u(t) \bigr)_{\varGamma _{1}} + \varepsilon \bigl\Vert u(t) \bigr\Vert ^{p} _{p, \varGamma _{1}} . \end{aligned}$$
(3.5)

Using Young’s inequality, we get

$$\begin{aligned} &\int ^{t}_{0} g(t-s) \bigl(\nabla u(s) , \nabla u(t) \bigr) \,ds \\ &\quad = \int ^{t}_{0} g(t-s) \bigl\Vert \nabla u(t) \bigr\Vert ^{2} \,ds + \int ^{t}_{0} g(t-s) \bigl(\nabla u(s) - \nabla u(t) , \nabla u(t)\bigr) \,ds \\ &\quad \geq \biggl(1- \frac{1}{4\tau } \biggr) \int ^{t}_{0} g(s) \,ds \bigl\Vert \nabla u(t) \bigr\Vert ^{2} - \tau (g\circ \nabla u) (t) \end{aligned}$$
(3.6)

for some number \(\tau >0\). By (3.5) and (3.6), we have

$$\begin{aligned} L'(t) &\geq (1- \sigma ) K^{-\sigma } (t) K' (t) + \frac{\varepsilon }{\rho +1} \bigl\Vert u_{t} (t) \bigr\Vert _{\rho +2}^{\rho +2} - \varepsilon \bigl\Vert \nabla u(t) \bigr\Vert ^{2} \\ &\quad{} + \varepsilon \biggl( 1- \frac{1}{4 \tau } \biggr) \int ^{t}_{0} g(s) \,ds \bigl\Vert \nabla u(t) \bigr\Vert ^{2} \\ &\quad {}- \varepsilon \tau (g\circ \nabla u) (t) - \varepsilon \bigl(f \bigl(u_{t} (t)\bigr), u(t)\bigr)_{ \varGamma _{1}} + \varepsilon \bigl\Vert u(t) \bigr\Vert ^{p}_{p,\varGamma _{1}} . \end{aligned}$$
(3.7)

Applying (2.7) and (2.12) to the last term \(\Vert u(t)\Vert ^{p} _{p,\varGamma _{1}}\) in the right-hand side of (3.7), we find that

$$\begin{aligned} L'(t) \geq &(1- \sigma ) K^{-\sigma } (t) K' (t) + \frac{\varepsilon }{\rho +1} \bigl\Vert u_{t} (t) \bigr\Vert _{\rho +2}^{\rho +2} - \varepsilon \bigl\Vert \nabla u(t) \bigr\Vert ^{2} \\ & {} + \varepsilon \biggl( 1- \frac{1}{4\tau } \biggr) \int ^{t}_{0} g(s) \,ds \bigl\Vert \nabla u(t) \bigr\Vert ^{2} - \varepsilon \tau (g\circ \nabla u) (t) - \varepsilon \bigl(f\bigl(u_{t} (t)\bigr), u(t)\bigr)_{\varGamma _{1}} \\ & {} + \varepsilon \biggl(p K(t) + \frac{p}{\rho +2} \bigl\Vert u_{t}(t) \bigr\Vert _{\rho +2} ^{\rho +2} + \frac{p}{2} \biggl( 1 - \int ^{t}_{0} g(s) \,ds \biggr) \bigl\Vert \nabla u(t) \bigr\Vert ^{2} \\ & {} + \frac{p}{2} (g \circ \nabla u) (t) -p \hat{\delta } h_{0} \biggr) \\ =& (1- \sigma ) K^{-\sigma } (t) K' (t) + \varepsilon \biggl( \frac{1}{ \rho +1} + \frac{p}{\rho +2} \biggr) \bigl\Vert u_{t} (t) \bigr\Vert _{\rho +2}^{\rho +2} \\ & {}+ \varepsilon \biggl( \frac{p}{2} - \tau \biggr) (g \circ \nabla u) (t) + \varepsilon p K(t) - \varepsilon \bigl(f\bigl(u_{t} (t)\bigr), u(t) \bigr)_{\varGamma _{1}} \\ & {}+ \varepsilon \biggl\{ \biggl( \frac{p}{2} -1 \biggr) - \biggl( \frac{p}{2} -1 + \frac{1}{4\tau } \biggr) \int ^{t}_{0} g(s) \,ds \biggr\} \bigl\Vert \nabla u(t) \bigr\Vert ^{2} -\varepsilon p \hat{\delta }h_{0} . \end{aligned}$$

From (2.11), we see that

$$\begin{aligned} L'(t) \geq& (1- \sigma ) K^{-\sigma } (t) K' (t) + \varepsilon \biggl( \frac{1}{\rho +1} + \frac{p}{\rho +2} \biggr) \bigl\Vert u_{t} (t) \bigr\Vert _{ \rho +2}^{\rho +2} \\ & {} + \varepsilon \biggl[ (1- \tau )+(1-\hat{\delta }) \biggl( \frac{p}{2}-1 \biggr) \biggr] (g \circ \nabla u) (t) + \varepsilon p K(t) - \varepsilon \bigl(f\bigl(u_{t} (t)\bigr), u(t)\bigr)_{\varGamma _{1}} \\ & {} + \varepsilon \biggl[ (1-\hat{\delta }) \biggl( \frac{p}{2} -1 \biggr) - \biggl\{ (1-\hat{\delta }) \biggl( \frac{p}{2} -1 \biggr) + \frac{1}{4\tau } \biggr\} \int ^{t}_{0} g(s) \,ds \biggr] \bigl\Vert \nabla u(t) \bigr\Vert ^{2} \end{aligned}$$
(3.8)

for some τ with \(0<\tau <1+ (1-\hat{\delta }) (\frac{p}{2}-1 ) \). Using (2.10), (3.8) can be rewritten by

$$\begin{aligned} \begin{aligned}[b] L'(t) &\geq (1- \sigma ) K^{-\sigma } (t) K' (t) + \varepsilon \biggl(\frac{ 1}{\rho +1} + \frac{p}{\rho +2} \biggr) \bigl\Vert u_{t} (t) \bigr\Vert _{ \rho +2}^{\rho +2} \\ &\quad {} + \varepsilon c_{1} (g \circ \nabla u) (t) + \varepsilon c_{2} \bigl\Vert \nabla u(t) \bigr\Vert ^{2} + \varepsilon p K(t) - \varepsilon \bigl(f\bigl(u_{t} (t)\bigr), u(t) \bigr)_{ \varGamma _{1}} , \end{aligned} \end{aligned}$$
(3.9)

where

$$ c_{1} =(1-\tau )+ (1-\hat{\delta }) \biggl(\frac{p}{2}-1\biggr) >0, $$

and

$$ c_{2} = (1-\hat{\delta }) \biggl( \frac{p}{2} -1 \biggr) - \biggl\{ (1- \hat{\delta }) \biggl( \frac{p}{2} -1 \biggr) + \frac{1}{4\tau } \biggr\} \int ^{t}_{0} g(s) \,ds >0 . $$

Let us put \(\varGamma _{11} = \{ x\in \varGamma _{1} : \vert u_{t} (x,t)\vert \leq 1 \}\) and \(\varGamma _{12} = \{ x\in \varGamma _{1} : \vert u_{t} (x,t)\vert > 1 \} \). Then we obtain the inequalities which are given in [21]:

$$\begin{aligned} \int _{\varGamma _{11}} f\bigl(u_{t} (x, t)\bigr) u(x,t) \,dx \leq \frac{\eta ^{p-1} }{p} \bigl\Vert u(t) \bigr\Vert ^{p}_{p, \varGamma _{1}} + \frac{(p-1) \vert \varGamma _{1} \vert }{p \eta } \bigl(\xi ^{-1}(1) \bigr)^{\frac{p}{p-1}} , \quad \eta >0 , \end{aligned}$$

and

$$\begin{aligned} \int _{\varGamma _{12}} f\bigl(u_{t} (x, t)\bigr) u(x,t) \,dx \leq \frac{\mu ^{m}}{m} \bigl\Vert u(t) \bigr\Vert ^{m} _{p, \varGamma _{1}} +\frac{(m-1) }{m \mu ^{\frac{m}{m-1}}} K'(t) , \quad \mu >0 . \end{aligned}$$

Inserting these into (3.9), we obtain

$$\begin{aligned} L'(t) \geq & \biggl\{ (1- \sigma ) K^{-\sigma } (t) - \frac{ \varepsilon (m-1) }{m \mu ^{\frac{m}{m-1}}} \biggr\} K' (t) \\ & {}+ \varepsilon \biggl( \frac{1}{ \rho +1} + \frac{p}{\rho +2} \biggr) \bigl\Vert u_{t} (t) \bigr\Vert _{\rho +2}^{\rho +2} + \varepsilon c_{1} (g \circ \nabla u) (t) \\ & {}+ \varepsilon c_{2} \bigl\Vert \nabla u(t) \bigr\Vert ^{2} +\varepsilon p K(t) - \frac{ \varepsilon \eta ^{p-1}}{p} \bigl\Vert u(t) \bigr\Vert ^{p}_{p, \varGamma _{1}} \\ & {}-\frac{\varepsilon (p-1) \vert \varGamma _{1} \vert }{p \eta } \bigl(\xi ^{-1}(1) \bigr)^{\frac{p}{p-1}} - \frac{\varepsilon \mu ^{m}}{m} \bigl\Vert u(t) \bigr\Vert ^{m}_{p, \varGamma _{1}} . \end{aligned}$$
(3.10)

Adding and subtracting \(\varepsilon \theta K(t) \) in the right-hand side of (3.10) and applying (2.7) and (2.12), we get

$$\begin{aligned} L'(t) \geq & \biggl\{ (1- \sigma ) K^{-\sigma } (t) - \frac{ \varepsilon (m-1) }{m \mu ^{\frac{m}{m-1}}} \biggr\} K' (t) + \varepsilon \biggl( \frac{1}{ \rho +1}+ \frac{p}{\rho +2} - \frac{\theta }{\rho +2} \biggr) \bigl\Vert u_{t} (t) \bigr\Vert _{ \rho +2}^{\rho +2} \\ & {}+ \varepsilon \biggl( c_{1} - \frac{\theta }{2} \biggr) (g \circ \nabla u) (t) + \varepsilon \biggl\{ c_{2} - \frac{ \theta }{2} \biggl( 1 - \int ^{t}_{0} g(s) \,ds \biggr) \biggr\} \bigl\Vert \nabla u(t) \bigr\Vert ^{2} +\varepsilon ( p - \theta ) K(t) \\ & {}+ \varepsilon \biggl( \frac{ \theta }{p} - \frac{\eta ^{p-1}}{p} \biggr) \bigl\Vert u(t) \bigr\Vert ^{p}_{p, \varGamma _{1}} - \frac{\varepsilon \mu ^{m}}{m} \bigl\Vert u(t) \bigr\Vert ^{m} _{p, \varGamma _{1}} +\varepsilon \theta \hat{\delta } h_{0} \\ & {}- \frac{ \varepsilon (p-1) \vert \varGamma _{1} \vert }{p \eta } \bigl(\xi ^{-1}(1) \bigr)^{ \frac{p}{p-1}} . \end{aligned}$$
(3.11)

Now, we choose \(\mu = ( k K^{-\sigma }(t) )^{- \frac{m-1}{m}}\), \(k>0\) will be specified later. Using (2.13), (2.14), and (3.4), the seventh term in the right-hand side of (3.11) is estimated as

$$\begin{aligned} - \frac{\varepsilon \mu ^{m}}{m} \bigl\Vert u(t) \bigr\Vert ^{m}_{p, \varGamma _{1}} =& - \frac{ \varepsilon k^{1-m}}{m} K^{\sigma (m-1)}(t) \bigl\Vert u(t) \bigr\Vert ^{m}_{p, \varGamma _{1}} \\ \geq &- \frac{\varepsilon k^{1-m}}{m} p_{0}^{\sigma (m-1)} \bigl\Vert u(t) \bigr\Vert _{p, \varGamma _{1}}^{\sigma p(m-1) +m} \geq - {\varepsilon C_{2} k^{1-m}} \bigl\Vert u(t) \bigr\Vert _{p, \varGamma _{1}}^{p} , \end{aligned}$$
(3.12)

where \(C_{2}= \frac{C_{0}p_{0}^{\sigma (m-1)}}{m} \). From (3.11) and (3.12), we obtain

$$\begin{aligned} L'(t) \geq & \biggl\{ (1- \sigma ) - \frac{ \varepsilon k (m-1) }{m} \biggr\} K^{-\sigma } (t) K' (t) + \varepsilon \biggl( \frac{1}{\rho +1}+ \frac{p}{\rho +2} - \frac{\theta }{\rho +2} \biggr) \bigl\Vert u_{t} (t) \bigr\Vert _{ \rho +2}^{\rho +2} \\ &{} + \varepsilon \biggl( c_{1} - \frac{\theta }{2} \biggr) (g \circ \nabla u) (t)+ \varepsilon \biggl( c_{2} - \frac{ \theta }{2} \biggr) \bigl\Vert \nabla u(t) \bigr\Vert ^{2} \\ & {}+ \varepsilon \biggl( \frac{ \theta }{p} - \frac{ \eta ^{p-1}}{p} -C_{2} k^{1-m} \biggr) \bigl\Vert u(t) \bigr\Vert ^{p}_{p,\varGamma _{1}} \\ &{} + \varepsilon ( p - \theta ) K(t) +\varepsilon \theta \hat{\delta } h_{0} - \frac{ \varepsilon (p-1) \vert \varGamma _{1} \vert }{p \eta } \bigl(\xi ^{-1}(1) \bigr)^{\frac{p}{p-1}}. \end{aligned}$$
(3.13)

We take θ such that

$$ 0 < \theta < \min \{ 2c_{1}, 2c_{2} \}, $$
(3.14)

then we can choose \(\eta >0\) sufficiently small so that \({ \theta } - {\eta ^{p-1}} >0 \). And then, we select \(k>0\) large enough such that

$$ \frac{ \theta }{p} - \frac{\eta ^{p-1}}{p} - C_{2} k^{1-m} >0 , $$

and then take \(\varepsilon >0\) with

$$ (1- \sigma ) - \frac{ \varepsilon k (m-1) }{m} >0 , \quad\quad K^{1-\sigma }(0)+ \frac{ \varepsilon }{\rho +1} \int _{\varOmega } \vert u_{1} \vert ^{\rho }u _{1} u_{0} \,dx+\frac{ \varepsilon }{2} \Vert \nabla u_{0} \Vert ^{2} >0. $$

Condition (3.1) gives that

$$\begin{aligned} \theta \hat{\delta } h_{0} - \frac{ (p-1) \vert \varGamma _{1} \vert }{p \eta } \bigl( \xi ^{-1}(1) \bigr)^{\frac{p}{p-1}} >0 . \end{aligned}$$
(3.15)

Using (3.13)–(3.15) and \(2c_{1} < p\), we have

$$\begin{aligned} L'(t) \geq C \bigl( \bigl\Vert u_{t} (t) \bigr\Vert _{\rho +2}^{\rho +2} + \bigl\Vert \nabla u(t) \bigr\Vert ^{2}+ \bigl\Vert u(t) \bigr\Vert ^{p}_{p,\varGamma _{1}} + K(t) \bigr), \end{aligned}$$
(3.16)

here and in the sequel C denotes a generic positive constant. By arguments similar to those in [18], we get

$$\begin{aligned} \bigl(L(t)\bigr)^{\frac{1}{1-\sigma }} \leq C \bigl( K(t) + \bigl\Vert u_{t} (t) \bigr\Vert _{ \rho +2}^{\rho +2} + \bigl\Vert \nabla u(t) \bigr\Vert ^{2}+ \bigl\Vert u(t) \bigr\Vert ^{p}_{p,\varGamma _{1}} \bigr) . \end{aligned}$$
(3.17)

Indeed, using Young’s inequality and

$$\begin{aligned} \biggl\vert \int _{\varOmega } \bigl\vert u_{t}(t) \bigr\vert ^{\rho }u_{t} (t)u(t) \,dx \biggr\vert \leq \bigl\Vert u _{t}(t) \bigr\Vert _{\rho +2}^{\rho +1} \bigl\Vert u(t) \bigr\Vert _{\rho +2} , \end{aligned}$$

we obtain

$$\begin{aligned} \biggl( \biggl\vert \int _{\varOmega } \bigl\vert u_{t}(t) \bigr\vert ^{\rho }u_{t} (t)u(t) \,dx \biggr\vert \biggr)^{\frac{1}{1-\sigma }} \leq C \bigl( \bigl\Vert u_{t}(t) \bigr\Vert _{\rho +2}^{\frac{ \rho +1}{1-\sigma }\gamma }+ \bigl\Vert u(t) \bigr\Vert _{\rho +2}^{\frac{1}{1-\sigma } \beta } \bigr), \end{aligned}$$
(3.18)

where \(\frac{1}{\gamma }+\frac{1}{\beta }=1\). By using (3.4) and taking \(\gamma =\frac{(1-\sigma )(\rho +2)}{\rho +1}>1 \), we get \(\frac{ \beta }{1-\sigma } = \frac{\rho +2}{(1-\sigma )(\rho +2)-(\rho +1)} \). Since K is an increasing function, from (2.13) and (3.2), we have

$$\begin{aligned} \bigl\Vert u(t) \bigr\Vert _{\rho +2}^{\frac{\beta }{1-\sigma }} \leq B_{1}^{\frac{ \beta }{1-\sigma }} \bigl\Vert \nabla u(t) \bigr\Vert ^{\frac{\beta }{1-\sigma }} \leq \frac{(B _{1}^{2}M)^{\frac{\beta }{2(1-\sigma )}}}{K(0)}K(t) \leq \frac{p_{0}(B _{1}^{2}M)^{\frac{\beta }{2(1-\sigma )}}}{K(0)} \bigl\Vert u(t) \bigr\Vert ^{p}_{p,\varGamma _{1}}, \end{aligned}$$
(3.19)

where \(B_{1}\) is the embedding constant. Similarly, from (2.13) and (3.2), we obtain

$$\begin{aligned} \bigl\Vert \nabla u(t) \bigr\Vert ^{\frac{2}{1-\sigma }} \leq M^{\frac{1}{1-\sigma }} \leq \frac{p_{0} M^{\frac{1}{1-\sigma }}}{K(0)} \bigl\Vert u(t) \bigr\Vert ^{p}_{p,\varGamma _{1}}. \end{aligned}$$
(3.20)

From (3.3), (3.18)–(3.20), we find that (3.17) holds. Combining (3.16) and (3.17), we conclude that

$$\begin{aligned} L'(t) \geq C \bigl(L(t)\bigr)^{\frac{1}{1-\sigma }} \quad \text{for } t \geq 0 . \end{aligned}$$

Consequently, \(L(t)\) blows up in time \(T^{*} \leq \frac{1-\sigma }{c \sigma (L(0))^{\frac{\sigma }{1-\sigma }}} \). Furthermore, we have

$$ \lim_{t\to T^{*-}} \bigl( \bigl\Vert u_{t}(t) \bigr\Vert _{\rho +2}^{\rho +2} + \bigl\Vert \nabla u(t) \bigr\Vert ^{2} + \bigl\Vert u(t) \bigr\Vert _{p,\varGamma _{1}}^{p} \bigr)=\infty . $$

This leads to a contradiction with (3.2). Thus the solution of (1.1)–(1.4) blows up in finite time. □

4 Conclusions

In this paper, we study the blow-up of solutions for the quasilinear viscoelastic wave equation with strong damping and boundary nonlinear damping. In recent years, there has been published much work concerning the wave equation with nonlinear boundary damping. But as far as we know, there was no blow-up result of solutions to the viscoelastic wave equation with nonlinear boundary damping and source terms. Therefore, we prove a finite time blow-up result of solution with positive initial energy as well as nonpositive initial energy. Moreover, we generalize the earlier result under a weaker assumption on the damping term.