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Open Access Open Badges Research Article

Blow-Up Results for a Nonlinear Hyperbolic Equation with Lewis Function

Faramarz Tahamtani

Author Affiliations

Department of Mathematics, Shiraz University, Shiraz 71454, Iran

Boundary Value Problems 2009, 2009:691496  doi:10.1155/2009/691496

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2009/1/691496

Received:17 February 2009
Accepted:28 September 2009
Published:11 October 2009

© 2009 The Author(s)

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The initial boundary value problem for a nonlinear hyperbolic equation with Lewis function in a bounded domain is considered. In this work, the main result is that the solution blows up in finite time if the initial data possesses suitable positive energy. Moreover, the estimates of the lifespan of solutions are also given.

1. Introduction

Let be a bounded domain in with smooth boundary . We consider the initial boundary value problem for a nonlinear hyperbolic equation with Lewis function which depends on spacial variable:




where , , , and is a continuous function.

The large time behavior of solutions for nonlinear evolution equations has been considered by many authors (for the relevant references one may consult with [114].)

In the early 1970s, Levine [3] considered the nonlinear wave equation of the form


in a Hilbert space where are are positive linear operators defined on some dense subspace of the Hilbert space and is a gradient operator. He introduced the concavity method and showed that solutions with negative initial energy blow up in finite time. This method was later improved by Kalantarov and Ladyzheskaya [4] to accommodate more general cases.

Very recently, Zhou [10] considered the initial boundary value problem for a quasilinear parabolic equation with a generalized Lewis function which depends on both spacial variable and time. He obtained the blowup of solutions with positive initial energy. In the case with zero initial energy Zhou [11] obtained a blow-up result for a nonlinear wave equation in . A global nonexistence result for a semilinear Petrovsky equation was given in [14].

In this work, we consider blow-up results in finite time for solutions of problem (1.1)-(1.3) if the initial datas possesses suitable positive energy and obtain a precise estimate for the lifespan of solutions. The proof of our technique is similar to the one in [10]. Moreover, we also show the blowup of solution in finite time with nonpositive initial energy.

Throughout this paper denotes the usual norm of .

The source term in (1.1) with the primitive





Let be the best constant of Sobolev embedding inequality


from to .

We need the following lemma in [4, Lemma 2.1].

Lemma 1.1.

Suppose that a positive, twice differentiable function satisfies for the inequality


If , , then


2. Blow-Up Results

We set


The corresponding energy to the problem (1.1)-(1.3) is given by


and one can find that easily from




We note that from (1.6) and (1.7), we have


and by Sobolev inequality (1.8), , , where


Note that has the maximum value at which are given in (2.1).

Adapting the idea of Zhou [10], we have the following lemma.

Lemma 2.1.

Suppose that and . Then


for all .

Theorem 2.2.

For , suppose that and satisfy


If , then the global solution of the problem (1.1)–(1.3) blows up in finite time and the lifespan



To prove the theorem, it suffices to show that the function


satisfies the hypotheses of the Lemma 1.1, where , and to be determined later. To achieve this goal let us observe




Let us compute the derivatives and . Thus one has




for all . In the above assumption (1.7), the definition of energy functionals (2.2) and (2.4) has been used. Then, due to (2.1) and (2.7) and taking ,


Hence for all and by assumption (2.8) we have


Therefore for all and by the construction of , it is clearly that


whence, . Thus for all , from (2.13), (2.15), and (2.17) we obtain


which implies


Then using Lemma 1.1, one obtain that as


Now, we are in a position to choose suitable and . Let be a number that depends on , , , and as


To choose , we may fix as


Thus, for the lifespan is estimated by


which completes the proof.

Theorem 2.3.

Assume that and the following conditions are valid:


Then the corresponding solution to (1.1)–(1.3) blows up in finite time.







where the left-hand side of assumption (1.7) and the energy functional (2.2) have been used. Taking the inequality (2.27) and integrating this, we obtain


By using Poincare-Friedrich's inequality


and Holder's inequality



where . Using (2.30) and (2.31), we find from (2.28) that


Since as so, there must be a such that


By inequality


and by virtue of (2.33) and using (2.32), we get




Therefore, there exits a positive constant


such that


This completes the proof.


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