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Open Access 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:

(11)

(12)

(13)

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

(14)

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

(15)

satisfies

(16)

(17)

Let be the best constant of Sobolev embedding inequality

(18)

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

(19)

If , , then

(110)

2. Blow-Up Results

We set

(21)

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

(22)

and one can find that easily from

(23)

whence

(24)

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

(25)

and by Sobolev inequality (1.8), , , where

(26)

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

(27)

for all .

Theorem 2.2.

For , suppose that and satisfy

(28)

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

(29)

Proof.

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

(210)

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

(211)

Hence,

(212)

Let us compute the derivatives and . Thus one has

(213)

and

(214)

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 ,

(215)

Hence for all and by assumption (2.8) we have

(216)

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

(217)

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

(218)

which implies

(219)

Then using Lemma 1.1, one obtain that as

(220)

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

(221)

To choose , we may fix as

(222)

Thus, for the lifespan is estimated by

(223)

which completes the proof.

Theorem 2.3.

Assume that and the following conditions are valid:

(224)

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

Proof.

Let

(225)

then

(226)

(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

(228)

By using Poincare-Friedrich's inequality

(229)

and Holder's inequality

(230)

(231)

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

(232)

Since as so, there must be a such that

(233)

By inequality

(234)

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

(235)

where

(236)

Therefore, there exits a positive constant

(237)

such that

(238)

This completes the proof.

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