GAO YUN-ZHUAND GAO WEN-JIE

(1.School of Mathematics,Jilin University,Changchun,130012)

(2.Department of Mathematics,Beihua University,Jilin City,Jilin,132013)

Existence and Blow-up of Solutions for a Nonlinear Parabolic System with Variable Exponents*

GAO YUN-ZHU1,2AND GAO WEN-JIE1

(1.School of Mathematics,Jilin University,Changchun,130012)

(2.Department of Mathematics,Beihua University,Jilin City,Jilin,132013)

In this paper,we study a nonlinear parabolic system with variable exponents.The existence of classical solutions to an initial and boundary value problem is obtained by a f i xed point theorem of the contraction mapping,and the blow-up property of solutions in f i nite time is obtained with the help of the eigenfunction of the Laplace equation and a delicated estimate.

blow-up,existence,nonlinear parabolic system,variable exponent

1 Introduction

In this paper,we study the existence and blow-up of solutions to the nonlinear parabolic system

where α＞0,β＞0 are constants,Ω?RNis a bounded domain with smooth boundary?Ω and QT=Ω×[0,T)with 0＜T＜∞,STdenotes the lateral boundary of the cylinder QT,and

are source terms.We also assume that the exponents

satisfy the following conditions:

When p1,p2are constants,Escobedo and Herrero[1]investigated the boundedness and blow up of solutions to the problem(1.1).Furthermore,the authors also studied the uniqueness and global existence for some solutions(see[2]),and there have been also many results about the existence,boundedness and blow up property of the solutions(see[3–6]).

The motivation of our work is mainly due to[7],where the authors studied the following parabolic problem involving a variable exponent:

where Ω∈Rnis a bounded domain with smooth boundary?Ω,and the source term is of the form

or

The parabolic problems with sources as the one in(1.4)can be used to model chemical reactions,heat transfer or population dynamic,etc.The readers can refer to[8–14]and the references therein.

However,to the best of our knowledge,there are no results about the existence,blow-up properties of solutions to the systems of parabolic equations with variable exponents.Our main aim in this paper is to study the problem(1.1)and to obtain the existence and blow up results of the solutions.

Our main results are stated in the next section,including some preliminary results and local existence of solutions to the problem(1.1).The blow-up of solutions is obtained and proved in Section 3.

2 Existence of Solutions

This section is devoted to the proof of existence of solutions to the problem(1.1).We give the following def i nition.

Def i nition 2.1 We say that the solution(u(x,t),v(x,t))for the problem(1.1)blows up in f i nite time if there exists an instant T*＜∞ such that

where

and

Our f i rst result is the following theorem.

Theorem 2.1 Let Ω?RNbe a bounded smooth domain.Assume that p1(x)and p2(x) satisfy the conditions(1.2)-(1.3),and u0(x)and v0(x)are nonnegative,continuous and bounded.Then there exists a T0with 0＜ T0≤ ∞ such that the problem(1.1)has a nonnegative and bounded solution(u,v)in QT0.

Proof. Let us consider the equivalence systems of(1.1)

where gi(x,y,t)(i=1,2)are the corresponding Green functions.Then the existence and uniqueness of solutions for a given(u0(x),v0(x))could be obtained by a f i xed point theorem.

We introduce the following iteration scheme:

The convergence of the sequence{(un,vn)}follows by showing that

is a contraction mapping in the set ETwhich will be def i ned later.Now,we def i ne

where

We also denote

For an arbitrary T＞0,def i ne

Then we claim that Ψ is a strict contraction on ET.In fact,for any f i xed x∈Ω,we have

and

where

and

And we always have

Now,we def i ne

It is obvious that

Then by(2.1),we get

where Γ=max{α,β}.Sinceμi(t)(i=1,2)are sufficiently small as t→0,we get

Hence,Ψ is a strict contraction mapping.This completes the proof.

3 Blow up of Solutions

This section is devoted to the blow up of solutions to the problem(1.1).We f i rst give the following lemma.

Lemma 3.1 Let y(t)be a solution of

where r＞1 and a＞0.Then y(t)cannot be globally def i ned,and

This lemma follows by a direct integration,and gives an upper bound for the blow up time.The following theorem gives the main result of this section.

Theorem 3.1 Let Ω?RNbe a bounded smooth domain and(u,v)a positive solution of the problem(1.1)with p1(x)and p2(x)satisfying the conditions(1.2)-(1.3).Then for a sufficiently large initial datum(u0(x),v0(x)),there exists a f i nite time T*＞0 such that

Proof. Let λ1be the f i rst eigenvalue of

with homogeneous Dirichlet boundary condition and φ a positive eigenfunction satisfying

Set

We get

We now deal with the term

For each t＞0,we divide Ω into the following four sets:

Then,we have

where

In view of the convex property of

and by using Jensen's inequality again,we obtain

Then,we get

Since

we know that the result follows from Lemma 3.1 for η(0)large enough.The proof is completed.

Remark 3.1 Assume that the source terms in(1.1)are of the form

Then Theorem 3.1 also holds.

In fact,def i ning

and repeating the previous argument,we can obtain the same result.

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tion:35K15,35K20,35K55

A

1674-5647(2013)01-061-07

*Received date:Aug.18,2010.

The NSF(10771085)of China.