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

*Boundary Value Problems*
**volume 2009**, Article number: 691496 (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.

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

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.

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

Proof.

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

Hence,

Let us compute the derivatives and . Thus one has

and

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.

Proof.

Let

then

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

where

Therefore, there exits a positive constant

such that

This completes the proof.

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Tahamtani, F. Blow-Up Results for a Nonlinear Hyperbolic Equation with Lewis Function.
*Bound Value Probl* **2009, **691496 (2009). https://doi.org/10.1155/2009/691496

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DOI: https://doi.org/10.1155/2009/691496

### Keywords

- Finite Time
- Initial Energy
- Initial Boundary
- Nonlinear Evolution Equation
- Nonlinear Wave Equation