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.
Keywords:existence; blow-up; parabolic system; hyperbolic system; variable exponent
In this paper, we first consider the initial and boundary value problem to the following nonlinear parabolic system with variable exponents:
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].
The motivation of this work is due to , where the following system of equations is studied.
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 ).
Besides, this work is also motivated by  in which the following problem is considered:
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.
We also study the following nonlinear hyperbolic system of equations:
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).
2 Existence of solutions
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 2.1Letbe a bounded smooth domain, , , , satisfy the conditions in (2.1), and assume thatandare nonnegative, continuous and bounded. Then there exists a, , such that problem (1.1) has a nonnegative and bounded solutionin.
Let us consider the equivalent systems of (1.1)
We introduce the following iteration scheme:
Now, we define
and we always have
Now, we define
Then, by using inequality (2.2), we get
Hence, for sufficiently small t, we have
3 Blow-up of solutions
In this section, we study the blow-up property of the solutions to problem (1.1). We need the following lemma.
Hence, we have
The next theorem gives the main result of this section.
Theorem 3.1Letbe a bounded smooth domain, and letbe a positive solution of problem (1.1), with, , , satisfying conditions in (2.1). Then any solutions of problem (1.1) will blow up at finite timeif the initial datumsatisfies
with the homogeneous Dirichlet boundary condition, and let φ be a positive function satisfying
Then we have
Then we get
In view of the property of φ, we get
By Lemma 3.1, the proof is complete. □
4 Blow-up of solutions for a hyperbolic system
Now, let us study the following problem:
Theorem 4.1Letbe 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.
Proof Let be the first eigenvalue and eigenfunction of Laplacian in Ω with homogeneous Dirichlet boundary conditions as before. We assume that , , the other is similar. We also define the function , so we have
Then we have
The authors declare that they have no competing interests.
YG performed the calculations and drafted the manuscript. WG supervised and participated in the design of the study and modified the draft versions. All authors read and approved the final manuscript.
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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