Skip to main content

Lower bounds for blow-up time of a nonlinear viscoelastic wave equation


This paper deals with the blow-up for a class of nonlinear viscoelastic wave equation. Under certain conditions on the data, we construct a lower bound for the blow-up time when blow-up occurs.

1 Introduction

In this paper, we study the blow-up solution for the following nonlinear viscoelastic wave equation:

$$\begin{aligned} \textstyle\begin{cases} u_{tt}-\Delta u+ \int^{t}_{0}g(t-s)\triangle u(s)\,ds+|u_{t}|^{q-2}u_{t}=|u|^{p-2}u, &(x,t)\in\Omega\times (0,T),\\ u(x,t)=0, &(x,t)\in\partial\Omega\times(0,T),\\ u(x,0)=u_{0}(x), \qquad u_{t}(x,0)=u_{1}(x), &x\in\Omega, \end{cases}\displaystyle \end{aligned}$$

where Ω is a bounded domain in \(\mathbb{R}^{n}\) with a smooth boundary Ω, g is a positive function satisfying some conditions to be specified later, and

$$\begin{aligned} 2< p, q\leq \left \{ \textstyle\begin{array}{@{}l@{\quad}l} \infty, & \mbox{if } n=1,2, \\ \frac{2n-2}{n-2}, & \mbox{if } n\geq3. \end{array}\displaystyle \right . \end{aligned}$$

The blow-up properties of the solution to (1.1) has been studied by many authors (see [15]). For instance, Messaoudi [2] studied (1.1) and proved a blow-up result for solutions with negative initial energy if \(p>q\geq2\) and a global result for \(2\leq p\leq q\). This result has been later improved by the same author in [3] to accommodate certain solutions with positive initial energy. In [4], Song and Zhong considered (1.1) for strong damping \(-\Delta u_{t}\) and proved a blow-up result for solutions with positive initial energy by using the ideas of the ‘potential well’ theory introduced by Payne and Sattinger [6]. Wang [5] has investigated a sufficient conditions of the initial data with arbitrarily positive initial energy such that the corresponding solution of (1.1) with \(q=2\) blows up in finite time. For related results, we refer the reader to [710].

When blow-up occurs, the blow-up time \(T^{*}\) cannot usually be computed exactly. It is therefore of great importance in practice to determine lower and upper bounds for \(T^{*}\). The aim of this note is to derive a lower bound for \(T^{*}\) when blow-up occurs. We point out that it is, in general, very hard to obtain a lower bound estimate for viscoelastic wave equation problems, for the method to estimate the derivative of the control functional in parabolic cases is no longer effective and the memory part makes it difficult to estimate the energy. Our method is based on a first-order differential inequality technique for a suitably defined auxiliary function and makes use of some Sobolev-type inequality.

Before stating our main result, let us recall some results on the local existence, uniqueness, and blow-up of the solution

Theorem 1.1

(see [3])

Let \((u_{0}(x),u_{1}(x) )\in H_{0}^{1}(\Omega)\times L^{2}(\Omega)\) and p, q satisfy condition (1.2). Let \(g\in C^{1}[0,\infty)\) be a non-negative and non-increasing function satisfying

$$ 1-\int_{0}^{\infty}g(s)\,ds=l>0. $$

Then problem (1.1) has a unique local solution

$$u\in C \bigl([0, T_{m}); H_{0}^{1}(\Omega) \bigr), \qquad u_{t} \in C \bigl([0,T_{m}); L^{2}( \Omega) \bigr)\cap L^{q} \bigl(\Omega\times (0,T_{m}) \bigr) $$

for some \(T_{m}>0\).

Remark 1.1

Condition (1.3) is necessary to guarantee the hyperbolicity and well-posedness of system (1.1).

Let λ be the best constant of the Sobolev embedding \(H_{0}^{1}(\Omega)\hookrightarrow L^{p}(\Omega)\) and \(\beta=\lambda/l^{\frac{1}{2}}\). We set

$$\alpha=\beta^{-\frac{p}{p-2}}, \quad E_{1}= \biggl(\frac{1}{2}- \frac{1}{p} \biggr)\alpha^{2}. $$

Define the energy functional \(E(t)\) associated to our system (1.1),

$$E(t)=\frac{1}{2}\bigl\| u_{t}(t)\bigr\| ^{2}_{2}+ \frac{1}{2}\biggl(1-\int_{0}^{t} g(s)\,ds \biggr)\bigl\| \nabla u(t)\bigr\| ^{2}_{2} +\frac{1}{2}(g\circ\nabla u) (t)-\frac{1}{p}\|u\|_{p}^{p}, $$


$$g\circ w(t)=\int^{t}_{0}g(t-s)\bigl\| w(s)- w(t) \bigr\| _{2}^{2}\,ds. $$

Moreover, we assume that g satisfies

$$ \int_{0}^{\infty}g(s)\,ds< \frac{p^{2}-2p}{p^{2}-2p+1}. $$

Then we have the following blow-up result.

Theorem 1.2

(see [3])

Assume that p, q satisfy condition (1.2) and g satisfies (1.3) and (1.4). If \(p>q\) and the initial data \((u_{0},u_{1})\) satisfies

$$E(0)>E_{1}, \qquad \|\nabla u_{0}\|_{2}>\alpha, $$

then any solution of (1.1) blows up in finite time.

2 The main result

In this section, we switch to discuss the lower bound of the blow-up time for the blow-up solution of (1.1). Before we state and prove our main result, we need the following lemma.

Lemma 2.1

Suppose that (1.2), (1.3), and (1.4) hold. Let u be a solution of (1.1). Then energy functional \(E(t)\) is non-increasing, that is, \(E'(t)\leq0\).


By multiplying (1.1) by \(u_{t}\) and integrating over Ω, we obtain

$$\begin{aligned} &\frac{d}{dt} \biggl( \frac{1}{2}\|u_{t} \|_{2}^{2}+ \frac{1}{2}\|\nabla u\|_{2}^{2}- \frac{1}{p}\|u\|_{p}^{p} \biggr)-\int _{0}^{t} g(t-s)\int_{\Omega}\nabla u(s)\cdot\nabla u_{t}(t)\,dx\,ds \\ &\quad=-\int_{0}^{t} \bigl\| u_{t}(s)\bigr\| _{q}^{q} \end{aligned}$$

for any regular solution. This result remains valid for weak solutions by a simple density argument. For the last term on the left side of (2.1), we have

$$\begin{aligned} &\int_{0}^{t}g(t-s)\int _{\Omega}\nabla u_{t}(t)\cdot\nabla u(s)\,dx \,ds \\ &\quad=\int_{0}^{t}g(t-s)\int _{\Omega}\nabla u_{t}(t) \bigl(\nabla u(s)-\nabla u(t) \bigr)\,dx\,ds+\int_{0}^{t}g(t-s)\int _{\Omega}\nabla u_{t}(t)\cdot\nabla u(t)\,dx\,ds \\ &\quad=-\frac{1}{2}\int_{0}^{t}g(t-s) \frac{d}{dt}\int_{\Omega}\bigl|\nabla u(s)-\nabla u(t)\bigr|^{2}\,dx\,ds+\frac{1}{2}\int_{0}^{t}g(s) \frac{d}{dt}\int_{\Omega}\bigl|\nabla u(t)\bigr|^{2} \,dx\,ds \\ &\quad=\frac{1}{2}\frac{d}{dt} \biggl(\int_{0}^{t}g(s)\,ds \bigl\| \nabla u(t)\bigr\| ^{2}_{2}-(g\circ\nabla u) (t) \biggr)+ \frac{1}{2}\bigl(g'\circ\nabla u\bigr) (t)- \frac{1}{2}g(t)\bigl\| \nabla u(t)\bigr\| ^{2}_{2}. \end{aligned}$$

Inserting (2.2) into (2.1), we get

$$E'(t)=-\int_{0}^{t} \bigl\| u_{t}(s)\bigr\| _{q}^{q}+\frac{1}{2} \bigl(g'\circ\nabla u\bigr) (t)-\frac{1}{2}g(t)\bigl\| \nabla u(t) \bigr\| ^{2}_{2}\leq0, $$

where we also use g being non-negative and non-increasing function. □

Theorem 2.2

Assume that the conditions in Theorem 1.2 hold. Let \(u(x, t)\) be the solution of problem (1.1), which blows up at a finite time \(T^{*}\). Then

$$T^{*}\geq\int^{\infty}_{F(0)} \frac{1}{C_{3}y^{k}+y+C_{4}}\,dy, $$

where the constants \(C_{3}\), \(C_{4}\), and the exponent k will be defined in (2.9), and \(F(0)=\int_{\Omega}|u_{0}|^{p}\,dx\).


Define \(F(t)=\int_{\Omega}|u(t)|^{p}\,dx\). Then

$$ F'(t)=p\int_{\Omega}|u|^{p-2}uu_{t}\,dx \leq \frac{p}{2} \biggl(\int_{\Omega}|u|^{2p-2}\,dx+ \int_{\Omega}|u_{t}|^{2}\,dx \biggr). $$

To estimate the first term on the right side of inequality (2.3), we consider the following two cases.

Case 1. \(2< p\leq\frac{2n}{n-1}\). Let \(\gamma=2p-2\), \(\mu=n(p-2)\), \(2^{*}=\frac{2n}{n-2}\). Applying Hölder’s inequality and the embedding inequality, we have

$$ \int_{\Omega}|u|^{\gamma}\,dx=\int_{\Omega}|u|^{\gamma\theta}|u|^{\gamma (1-\theta)}\,dx \leq \biggl(\int_{\Omega}|u|^{\mu}\,dx \biggr)^{\frac{\gamma\theta}{\mu}} \biggl(\int_{\Omega}|u|^{2^{*}}\,dx \biggr)^{\frac{\gamma(1-\theta)}{2^{*}}}, $$

where θ satisfies

$$ \frac{\gamma\theta}{\mu}+\frac{\gamma(1-\theta)}{2^{*}}=1. $$

A straightforward computation shows

$$\begin{aligned}& \theta=\frac{1-\frac{\gamma}{2^{*}}}{\frac{\gamma}{\mu}-\frac{\gamma }{2^{*}}}=\frac{\mu(2^{*}-\gamma)}{\gamma(2^{*}-\mu)}, \\& \frac{\gamma\theta}{\mu}=\frac{2^{*}-\gamma}{2^{*}-\mu}=\frac {2}{n},\qquad \frac{\gamma-\theta\gamma}{2^{*}}=1- \frac{2}{n}, \end{aligned}$$

and then we have

$$\begin{aligned} \|u\|_{\gamma}^{\gamma} \leq&\|u\|_{\mu}^{\gamma\theta} \|u\|_{2^{*}}^{\gamma(1-\theta)} =\|u\|_{\mu}^{\frac{2\mu}{n}}\|u \|_{2^{*}}^{2} \\ \leq& C_{\ast}^{2}\bigl(1+|\Omega|^{\frac{2(p-\mu)}{np}}\bigr)\|u\| _{p}^{\frac{2\mu}{n}}\|\nabla u\|_{2}^{2} \\ \leq& C_{\ast}^{2}\bigl(1+|\Omega|^{\frac{2(p-\mu)}{np}}\bigr) \bigl(\|u\| _{p}^{\frac{2\mu}{n}\cdot s}+\|\nabla u\|_{2}^{2t} \bigr) \\ \leq& C_{1}\bigl(\| u\|_{p}^{p}+\|\nabla u \|_{2}^{2}\bigr)^{k_{1}}, \end{aligned}$$

where we have used the Hölder inequality,

$$ \|u\|_{\mu}^{\frac{2\mu}{n}}\leq|\Omega|^{\frac{2(p-\mu)}{np}}\|u\| _{p}^{\frac{2\mu}{n}}\leq\bigl(1+|\Omega|^{\frac{2(p-\mu)}{np}}\bigr)\|u\| _{p}^{\frac{2\mu}{n}}, $$


$$ \|u\|_{2^{*}}\leq C_{\ast}\|\nabla u\|_{2}, $$

here \(C_{\ast}\) is the best constant of the Sobolev embedding \(H^{1}_{0}(\Omega)\hookrightarrow L^{2^{\ast}}(\Omega)\); \(\frac {1}{s}+\frac{1}{t}=1\), letting \(t=\frac{2\mu}{pn}s\), from which we can deduce \(k_{1}=\frac{3p-4}{p}\), \(C_{1}=C_{\ast}^{2}(1+|\Omega|^{\frac{2(p-\mu)}{np}})\).

Case 2. \(\frac{2n}{n-1}< p\leq\frac{2(n-1)}{n-2}\). Following the lines of the proof of inequality (2.4), we have

$$\begin{aligned} \|u\|_{\gamma}^{\gamma}\leq|\Omega|^{1-\frac{\gamma}{2^{*}}}\|u \| ^{\gamma}_{2^{*}} \leq& C^{\gamma}_{\ast}\bigl(1+| \Omega|^{1-\frac{\gamma }{2^{*}}}\bigr)\|\nabla u\|_{2}^{\gamma} \\ \leq&C^{\gamma}_{\ast}\bigl(1+|\Omega|^{1-\frac{\gamma}{2^{*}}}\bigr) \bigl(\|\nabla u\|_{2}^{\gamma}+\|u\|^{p(p-1)}_{p} \bigr) \\ \leq& C_{2}\bigl(\| u\|_{p}^{p}+\|\nabla u \|_{2}^{2}\bigr)^{k_{2}}, \end{aligned}$$

with \(k_{2}=p-1\), \(C_{2}=C^{\gamma}_{\ast}(1+|\Omega|^{1-\frac{\gamma }{2^{*}}})\).

From Lemma 2.1, we have

$$ E(t)\leq E(0)=\frac{1}{2}\| u_{1} \|^{2}_{2}+\frac{1}{2}\|\nabla u_{0} \|^{2}_{2}-\frac{1}{p}\| u_{0} \|^{p}_{p} = E_{0},\quad t\in\bigl[0,T^{*}\bigr). $$

Recalling the definition of \(E(t)\), (1.4), and (2.6), we have

$$\begin{aligned} &\bigl\| u_{t}(t)\bigr\| ^{2}_{2}+ \frac{1}{(p-1)^{2}}\bigl\| \nabla u(t)\bigr\| ^{2}_{2}+g\circ u(t) \\ &\quad\leq\bigl\| u_{t}(t)\bigr\| ^{2}_{2}+ \biggl(1-\int ^{t}_{0}g(s)\,ds \biggr)\bigl\| \nabla u(t) \bigr\| ^{2}_{2}+g\circ u(t) \\ &\quad= \frac{2}{p}\bigl\| u(t)\bigr\| ^{p}_{p}+2E(t)\leq \frac {2}{p}F(t)+2E_{0}. \end{aligned}$$

Combining (2.3)-(2.7), we get

$$\begin{aligned} F'(t) \leq&\frac{p}{2} \biggl(C_{i} \bigl(\| u\|_{p}^{p}+\|\nabla u\| _{2}^{2} \bigr)^{k_{i}}+\frac{2}{p}F(t)+2E_{0} \biggr) \\ \leq&\frac{p}{2} \biggl(C_{i} \biggl(F(t)+C_{0} \biggl(\frac{2}{p}F(t)+2E_{0}\biggr) \biggr)^{k_{i}}+ \frac{2}{p}F(t)+2E_{0} \biggr) \\ \leq&\frac{p}{2} \biggl(C_{i} \biggl(\biggl(1+ \frac{2C_{0}}{p}\biggr)F(t)+2C_{0}E_{0} \biggr)^{k_{i}}+\frac{2}{p}F(t)+2E_{0} \biggr) \\ \leq&\frac{pC_{i}}{2}2^{k_{i}-1} \biggl(\biggl(1+\frac {2C_{0}}{p} \biggr)^{k_{i}}F(t)^{k_{i}}+(2C_{0}E_{0})^{k_{i}} \biggr) +F(t)+pE_{0} \\ =&C_{3}F(t)^{k_{i}}+F(t)+C_{4}, \end{aligned}$$


$$\begin{aligned} &C_{0}=\frac{1}{1-\int^{\infty}_{0}g(s)\,ds}, \\ &C_{3}=\frac{pC_{i}2^{k_{i}}}{4}\biggl(1+\frac{2C_{0}}{p} \biggr)^{k_{i}}, \\ &C_{4}=pE_{0}+\frac{pC_{i}}{4}(4C_{0}E_{0})^{k_{i}}, \quad i=1,2. \end{aligned}$$

Applying Theorem 1.2, we have

$$ \lim_{t\rightarrow T^{*}}\int_{\Omega}|u|^{p}\,dx=+ \infty. $$

According to (2.8) and (2.10), we obtain

$$ \int^{\infty}_{F(0)}\frac{1}{C_{3}y^{k}+y+C_{4}}\,dy\leq T^{*}. $$



  1. Kafini, M, Messaoudi, SA: A blow-up result in a Cauchy viscoelastic problem. Appl. Math. Lett. 21, 549-553 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  2. Messaoudi, SA: Blow up and global existence in a nonlinear viscoelastic wave equation. Math. Nachr. 260, 58-66 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  3. Messaoudi, SA: Blow up of positive-initial-energy solutions of a nonlinear viscoelastic hyperbolic equation. J. Math. Anal. Appl. 320, 902-915 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  4. Song, HT, Zhong, CK: Blow-up of solutions of a nonlinear viscoelastic wave equation. Nonlinear Anal. 11, 3877-3883 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  5. Wang, YJ: A global nonexistence theorem for viscoelastic equations with arbitrarily positive initial energy. Appl. Math. Lett. 22, 1394-1400 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  6. Payne, L, Sattinger, D: Saddle points and instability on nonlinear hyperbolic equations. Isr. J. Math. 22, 273-303 (1975)

    Article  MathSciNet  Google Scholar 

  7. Levine, HA, Ro Park, S: Global existence and global nonexistence of solutions of the Cauchy problem for a nonlinearly damped wave equation. J. Math. Anal. Appl. 228, 181-205 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  8. Vitillaro, E: Global nonexistence theorems for a class of evolution equations with dissipation. Arch. Ration. Mech. Anal. 149, 155-182 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  9. Yang, ZJ: Existence and asymptotic behavior of solutions for a class of quasi-linear evolution equations with non-linear damping and source terms. Math. Methods Appl. Sci. 25, 795-814 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  10. Messaoudi, SA, Said-Houari, B: Blow up of solutions of a class of wave equations with nonlinear damping and source terms. Math. Methods Appl. Sci. 27, 1687-1696 (2004)

    Article  MATH  MathSciNet  Google Scholar 

Download references


The authors are indebted to the referee for giving some important suggestions which improved the presentations of this paper. This work is supported in part by China NSF Grant No. 11501442, the China Postdoctoral Science Foundation Grant No. 2013M540767, the Shanxi Provincial Postdoctoral Science Foundation, the scientific research program funded by Shanxi Provincial education department No. 14JK1474, and the doctor scientific research start fund project of Xi An University of Science and Technology Grant No. 2014QDJ042.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Liang Fei.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The article is a joint work of the three authors, who contributed equally to the final version of the paper. All authors read and approved the final manuscript.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, 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.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lu, Y., Fei, L. & Zhenhua, G. Lower bounds for blow-up time of a nonlinear viscoelastic wave equation. Bound Value Probl 2015, 219 (2015).

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: