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

- Faramarz Tahamtani
^{1}Email author

**2009**:691496

**DOI: **10.1155/2009/691496

© Faramarz Tahamtani 2009

**Received: **17 February 2009

**Accepted: **28 September 2009

**Published: **11 October 2009

## Abstract

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 [1–14].)

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

satisfies

Let be the best constant of Sobolev embedding inequality

from to .

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

Lemma 1.1.

## 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

whence

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.

for all .

Theorem 2.2.

Proof.

which completes the proof.

Theorem 2.3.

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

Proof.

This completes the proof.

## Authors’ Affiliations

## References

- Áng DD, Dinh APN: On the strongly damped wave equation: .
*SIAM Journal on Mathematical Analysis*1988, 19(6):1409–1418. 10.1137/0519103MATHMathSciNetView ArticleGoogle Scholar - Nishihara K: Asymptotic behavior of solutions of quasilinear hyperbolic equations with linear damping.
*Journal of Differential Equations*1997, 137(2):384–395. 10.1006/jdeq.1997.3268MATHMathSciNetView ArticleGoogle Scholar - Levine HA: Instability and nonexistence of global solutions to nonlinear wave equations of the form .
*Transactions of the American Mathematical Society*1974, 192: 1–21.MATHMathSciNetGoogle Scholar - Kalantarov VK, Ladyzhenskaya OA: The occurrence of collapase for quasi-linear equations of parabolic and hyperbolic type.
*Journal of Soviet Mathematics*1978, 10: 53–70. 10.1007/BF01109723View ArticleGoogle Scholar - Can M, Park SR, Aliyev F: Nonexistence of global solutions of some quasilinear hyperbolic equations.
*Journal of Mathematical Analysis and Applications*1997, 213(2):540–553. 10.1006/jmaa.1997.5557MATHMathSciNetView ArticleGoogle Scholar - Ono K: Global existence, asymptotic behaviour, and global non-existence of solutions for damped non-linear wave equations of Kirchhoff type in the whole space.
*Mathematical Methods in the Applied Sciences*2000, 23(6):535–560. 10.1002/(SICI)1099-1476(200004)23:6<535::AID-MMA125>3.0.CO;2-HMATHMathSciNetView ArticleGoogle Scholar - Tan Z: The reaction-diffusion equation with Lewis function and critical Sobolev exponent.
*Journal of Mathematical Analysis and Applications*2002, 272(2):480–495. 10.1016/S0022-247X(02)00166-XMATHMathSciNetView ArticleGoogle Scholar - Zhijian Y: Global existence, asymptotic behavior and blowup of solutions for a class of nonlinear wave equations with dissipative term.
*Journal of Differential Equations*2003, 187(2):520–540. 10.1016/S0022-0396(02)00042-6MATHMathSciNetView ArticleGoogle Scholar - Polat N, Kaya D, Tutalar HI: Blow-up of solutions for a class of nonlinear wave equations.
*Proceedings of the International Conference on Dynamic Systems and Applications, July 2004*572–576. - Zhou Y: Global nonexistence for a quasilinear evolution equation with a generalized Lewis function.
*Journal for Analysis and Its Applications*2005, 24(1):179–187.MATHMathSciNetGoogle Scholar - Zhou Y: A blow-up result for a nonlinear wave equation with damping and vanishing initial energy in .
*Applied Mathematics Letters*2005, 18(3):281–286. 10.1016/j.aml.2003.07.018MATHMathSciNetView ArticleGoogle Scholar - Wu S-T, Tsai L-Y: Blow-up of solutions for evolution equations with nonlinear damping.
*Applied Mathematics E-Notes*2006, 6: 58–65.MATHMathSciNetGoogle Scholar - Messaoudi SA, Houari BS: A blow-up result for a higher-order nonlinear Kirchhoff-type hyperbolic equation.
*Applied Mathematics Letters*2007, 20(8):866–871. 10.1016/j.aml.2006.08.018MATHMathSciNetView ArticleGoogle Scholar - Chen W, Zhou Y: Global nonexistence for a semilinear Petrovsky equation.
*Nonlinear Analysis: Theory, Methods & Applications*2009, 70(9):3203–3208. 10.1016/j.na.2008.04.024MATHMathSciNetView ArticleGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. 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.