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

Boundary Value Problems20092009:691496

DOI: 10.1155/2009/691496

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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq1_HTML.gif be a bounded domain in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq2_HTML.gif with smooth boundary http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq3_HTML.gif . We consider the initial boundary value problem for a nonlinear hyperbolic equation with Lewis function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq4_HTML.gif which depends on spacial variable:

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ1_HTML.gif
(11)
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ2_HTML.gif
(12)
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ3_HTML.gif
(13)

where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq5_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq6_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq7_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq8_HTML.gif 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

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ4_HTML.gif
(14)

in a Hilbert space where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq9_HTML.gif are http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq10_HTML.gif are positive linear operators defined on some dense subspace of the Hilbert space and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq11_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq12_HTML.gif . 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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq13_HTML.gif denotes the usual norm of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq14_HTML.gif .

The source term http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq15_HTML.gif in (1.1) with the primitive

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ5_HTML.gif
(15)

satisfies

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ6_HTML.gif
(16)
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ7_HTML.gif
(17)

Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq16_HTML.gif be the best constant of Sobolev embedding inequality

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ8_HTML.gif
(18)

from http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq17_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq18_HTML.gif .

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

Lemma 1.1.

Suppose that a positive, twice differentiable function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq19_HTML.gif satisfies for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq20_HTML.gif the inequality
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ9_HTML.gif
(19)
If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq21_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq22_HTML.gif , then
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ10_HTML.gif
(110)

2. Blow-Up Results

We set

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ11_HTML.gif
(21)

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

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ12_HTML.gif
(22)

and one can find that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq23_HTML.gif easily from

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ13_HTML.gif
(23)

whence

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ14_HTML.gif
(24)

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

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ15_HTML.gif
(25)

and by Sobolev inequality (1.8), http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq24_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq25_HTML.gif , where

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ16_HTML.gif
(26)

Note that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq26_HTML.gif has the maximum value http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq27_HTML.gif at http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq28_HTML.gif which are given in (2.1).

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

Lemma 2.1.

Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq29_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq30_HTML.gif . Then
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ17_HTML.gif
(27)

for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq31_HTML.gif .

Theorem 2.2.

For http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq32_HTML.gif , suppose that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq33_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq34_HTML.gif satisfy
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ18_HTML.gif
(28)
If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq35_HTML.gif , then the global solution of the problem (1.1)–(1.3) blows up in finite time and the lifespan
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ19_HTML.gif
(29)

Proof.

To prove the theorem, it suffices to show that the function
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ20_HTML.gif
(210)
satisfies the hypotheses of the Lemma 1.1, where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq36_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq37_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq38_HTML.gif to be determined later. To achieve this goal let us observe
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ21_HTML.gif
(211)
Hence,
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ22_HTML.gif
(212)
Let us compute the derivatives http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq39_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq40_HTML.gif . Thus one has
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ23_HTML.gif
(213)
and
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ24_HTML.gif
(214)
for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq41_HTML.gif . 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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq42_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ25_HTML.gif
(215)
Hence http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq43_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq44_HTML.gif and by assumption (2.8) we have
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ26_HTML.gif
(216)
Therefore http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq45_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq46_HTML.gif and by the construction of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq47_HTML.gif , it is clearly that
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ27_HTML.gif
(217)
whence, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq48_HTML.gif . Thus for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq49_HTML.gif , from (2.13), (2.15), and (2.17) we obtain
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ28_HTML.gif
(218)
which implies
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ29_HTML.gif
(219)
Then using Lemma 1.1, one obtain that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq50_HTML.gif as
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ30_HTML.gif
(220)
Now, we are in a position to choose suitable http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq51_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq52_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq53_HTML.gif be a number that depends on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq54_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq55_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq56_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq57_HTML.gif as
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ31_HTML.gif
(221)
To choose http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq58_HTML.gif , we may fix http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq59_HTML.gif as
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ32_HTML.gif
(222)
Thus, for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq60_HTML.gif the lifespan http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq61_HTML.gif is estimated by
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ33_HTML.gif
(223)

which completes the proof.

Theorem 2.3.

Assume that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq62_HTML.gif and the following conditions are valid:
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ34_HTML.gif
(224)

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

Proof.

Let
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ35_HTML.gif
(225)
then
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ36_HTML.gif
(226)
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ37_HTML.gif
(227)
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
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ38_HTML.gif
(228)
By using Poincare-Friedrich's inequality
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ39_HTML.gif
(229)
and Holder's inequality
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ40_HTML.gif
(230)
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ41_HTML.gif
(231)
where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq63_HTML.gif . Using (2.30) and (2.31), we find from (2.28) that
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ42_HTML.gif
(232)
Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq64_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq65_HTML.gif so, there must be a http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq66_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ43_HTML.gif
(233)
By inequality
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ44_HTML.gif
(234)
and by virtue of (2.33) and using (2.32), we get
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ45_HTML.gif
(235)
where
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ46_HTML.gif
(236)
Therefore, there exits a positive constant
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ47_HTML.gif
(237)
such that
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ48_HTML.gif
(238)

This completes the proof.

Authors’ Affiliations

(1)
Department of Mathematics, Shiraz University

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© Faramarz Tahamtani 2009

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.