Open Access

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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq1_HTML.gif be a bounded domain in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq2_HTML.gif with smooth boundary https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq4_HTML.gif which depends on spacial variable:

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

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

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

in a Hilbert space where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq9_HTML.gif are https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq13_HTML.gif denotes the usual norm of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq14_HTML.gif .

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

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

satisfies

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

Let https://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

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

from https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq17_HTML.gif to https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq19_HTML.gif satisfies for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq20_HTML.gif the inequality
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ9_HTML.gif
(19)
If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq21_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq22_HTML.gif , then
https://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

https://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

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

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

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

whence

https://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

https://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), https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq24_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq25_HTML.gif , where

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

Note that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq26_HTML.gif has the maximum value https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq27_HTML.gif at https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq29_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq30_HTML.gif . Then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ17_HTML.gif
(27)

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

Theorem 2.2.

For https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq32_HTML.gif , suppose that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq33_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq34_HTML.gif satisfy
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ18_HTML.gif
(28)
If https://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
https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq36_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq37_HTML.gif and https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ21_HTML.gif
(211)
Hence,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ22_HTML.gif
(212)
Let us compute the derivatives https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq39_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq40_HTML.gif . Thus one has
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ23_HTML.gif
(213)
and
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ24_HTML.gif
(214)
for all https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq42_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ25_HTML.gif
(215)
Hence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq43_HTML.gif for all https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ26_HTML.gif
(216)
Therefore https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq45_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq46_HTML.gif and by the construction of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq47_HTML.gif , it is clearly that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ27_HTML.gif
(217)
whence, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq48_HTML.gif . Thus for all https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ28_HTML.gif
(218)
which implies
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq50_HTML.gif as
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq51_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq52_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq53_HTML.gif be a number that depends on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq54_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq55_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq56_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq57_HTML.gif as
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ31_HTML.gif
(221)
To choose https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq58_HTML.gif , we may fix https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq59_HTML.gif as
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ32_HTML.gif
(222)
Thus, for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq60_HTML.gif the lifespan https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq61_HTML.gif is estimated by
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq62_HTML.gif and the following conditions are valid:
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ35_HTML.gif
(225)
then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ36_HTML.gif
(226)
https://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
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ39_HTML.gif
(229)
and Holder's inequality
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ40_HTML.gif
(230)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ41_HTML.gif
(231)
where https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ42_HTML.gif
(232)
Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq64_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq65_HTML.gif so, there must be a https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_IEq66_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ43_HTML.gif
(233)
By inequality
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ45_HTML.gif
(235)
where
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F691496/MediaObjects/13661_2009_Article_873_Equ47_HTML.gif
(237)
such that
https://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.