- Open Access
Study of solutions to an initial and boundary value problem for certain systems with variable exponents
© Gao and Gao; licensee Springer. 2013
- Received: 15 February 2013
- Accepted: 18 March 2013
- Published: 5 April 2013
In this paper, the existence and blow-up property of solutions to an initial and boundary value problem for a nonlinear parabolic system with variable exponents is studied. Meanwhile, the blow-up property of solutions for a nonlinear hyperbolic system is also obtained.
- parabolic system
- hyperbolic system
- variable exponent
respectively, where , , , are functions satisfying conditions (2.1) below.
In the case when , are constants, system (1.1) provides a simple example of a reaction-diffusion system. It can be used as a model to describe heat propagation in a two-component combustible mixture. There have been many results about the existence, boundedness and blow-up properties of the solutions; we refer the readers to the bibliography given in [1–7].
where (), , and p, q are positive numbers. The authors investigated the boundedness and blow-up of solutions to problem (1.2). Furthermore, the authors also studied the uniqueness and global existence of solutions (see ).
where is a bounded domain with smooth boundary ∂ Ω, and the source term is of the form or . The author studied the blow-up property of solutions for parabolic and hyperbolic problems. Parabolic problems with sources like the ones in (1.3) appear in several branches of applied mathematics, which can be used to model chemical reactions, heat transfer or population dynamics etc. We also refer the interested reader to [9–23] and the references therein.
The aim of this paper is to extend the results in [2, 8] to the case of parabolic system (1.1) and hyperbolic system (1.4). As far as we know, this seems to be the first paper, where the blow-up phenomenon is studied with variable exponents for the initial and boundary value problem to some parabolic and hyperbolic systems. The main method of the proof is similar to that in [3, 8].
We conclude this introduction by describing the outline of this paper. Some preliminary results, including existence of solutions to problem (1.1), are gathered in Section 2. The blow-up property of solutions are stated and proved in Section 3. Finally, in Section 4, we prove the blow-up property of solutions for hyperbolic problem (1.4).
In this section, we first state some assumptions and definitions needed in the proof of our main result and then prove the existence of solutions.
Our first result here is the following.
Theorem 2.1 Let be a bounded smooth domain, , , , satisfy the conditions in (2.1), and assume that and are nonnegative, continuous and bounded. Then there exists a , , such that problem (1.1) has a nonnegative and bounded solution in .
Proof We only prove the case when and , and the proofs to the cases and are similar.
where is the corresponding Green function. Then the existence and uniqueness of solutions for a given could be obtained by a fixed point argument.
is a contraction in the set to be defined below.
where , is a fixed positive constant.
It is obvious that when .
where is a constant. Then Ψ is a strict contraction. □
In this section, we study the blow-up property of the solutions to problem (1.1). We need the following lemma.
By a direct integration to (3.2), then we get immediately (3.1), which gives an upper bound for the blow-up time . □
The next theorem gives the main result of this section.
where is the first eigenfunction of the homogeneous Dirichlet Laplacian on Ω and is a constant depending only on the domain Ω and the bounds , given in condition (2.1).
where and , .
Hence, for big enough, the result follows from Lemma 3.1.
By Lemma 3.1, the proof is complete. □
Lemma 4.1 
where and they are not identically zero, and , as above respectively.
Theorem 4.1 Let be a solution of problem (4.2), and let the conditions in (2.1) hold. Then there exist sufficiently large initial data , , , such that any solutions of problem (4.1) blew up at finite time .
Hence, blows up before the maximal time of existence defined in inequality (4.1) is reached. □
Supported by NSFC (11271154) and by Key Lab of Symbolic Computation and Knowledge Engineering of Ministry of Education and by the 985 program of Jilin University.
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