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Lower bounds for blow-up time of a nonlinear viscoelastic wave equation
Boundary Value Problems volume 2015, Article number: 219 (2015)
Abstract
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:
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
The blow-up properties of the solution to (1.1) has been studied by many authors (see [1–5]). 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 [7–10].
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
Then problem (1.1) has a unique local solution
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
Define the energy functional \(E(t)\) associated to our system (1.1),
where
Moreover, we assume that g satisfies
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
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, \(E'(t)\leq0\).
Proof
By multiplying (1.1) by \(u_{t}\) and integrating over Ω, we obtain
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
Inserting (2.2) into (2.1), we get
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
where the constants \(C_{3}\), \(C_{4}\), and the exponent k will be defined in (2.9), and \(F(0)=\int_{\Omega}|u_{0}|^{p}\,dx\).
Proof
Define \(F(t)=\int_{\Omega}|u(t)|^{p}\,dx\). Then
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
where θ satisfies
A straightforward computation shows
and then we have
where we have used the Hölder inequality,
and
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
with \(k_{2}=p-1\), \(C_{2}=C^{\gamma}_{\ast}(1+|\Omega|^{1-\frac{\gamma }{2^{*}}})\).
From Lemma 2.1, we have
Recalling the definition of \(E(t)\), (1.4), and (2.6), we have
where
Applying Theorem 1.2, we have
According to (2.8) and (2.10), we obtain
 □
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Acknowledgements
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.
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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.
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Lu, Y., Fei, L. & Zhenhua, G. Lower bounds for blow-up time of a nonlinear viscoelastic wave equation. Bound Value Probl 2015, 219 (2015). https://doi.org/10.1186/s13661-015-0479-1
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DOI: https://doi.org/10.1186/s13661-015-0479-1