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:
where is a bounded domain with smooth boundary ∂Ω and , , denotes the lateral boundary of the cylinder , and the source terms , are in the form
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].
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.
Throughout the paper, we assume that the exponents and the continuous functions satisfy the following conditions:
Definition 2.1 We say that the solution for problem (1.1) blows up in finite time if there exists an instant such that
Our first result here is the following.
Theorem 2.1Let 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.
Let us consider the equivalent systems of (1.1)
where is the corresponding Green function. Then the existence and uniqueness of solutions for a given could be obtained by a fixed point argument.
We introduce the following iteration scheme:
and the convergence of the sequence follows by showing that
is a contraction in the set to be defined below.
Now, we define
and for arbitrary , define the set
where , is a fixed positive constant.
We claim that Ψ is a contraction on . In fact, for any fixed, we have
and we always have
Now, we define
It is obvious that when .
Then, by using inequality (2.2), we get
Hence, for sufficiently small t, we have
where is a constant. Then Ψ is a strict contraction. □
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.
Lemma 3.1Let be a solution of
where , , and are given constants. Then, there exists a constant such that if , then cannot be globally defined; in fact,
Proof It is sufficient to take such that
Hence, we have
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.
Theorem 3.1Let be a bounded smooth domain, and let be 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 time if the initial datum satisfies
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).
Proof Let be the first eigenvalue of
with the homogeneous Dirichlet boundary condition, and let φ be a positive function satisfying
We introduce the function . First of all, we consider the case , . Then
We now deal with the term . For each , we divide Ω into the following four sets:
Then we have
where and , .
From the convex property of the function , and Jensen’s inequality, we obtain
Then we get
Hence, for big enough, the result follows from Lemma 3.1.
Next, we state briefly the proof to the theorem in the case and . We repeat the previous argument under defining , and we obtain in much the same way
In view of the property of φ, we get
According to the convex property of the function , , and by using Jensen’s inequality, by considering again , , , as before, we obtain
where depends only on γ, p and , denotes the measure of Ω. Hence,
By Lemma 3.1, the proof is complete. □
4 Blow-up of solutions for a hyperbolic system
, , and for all . Then wheneveryexists; and
Now, let us study the following problem:
where and they are not identically zero, and , as above respectively.
Theorem 4.1Let 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 .
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
The term is dealt with as before, then we get
By virtue of the convex property of the function , , and Jensen’s inequality, we still obtain
Then we have
Now, we can apply Lemma (4.1) for , large enough such that , and note that
Hence, blows up before the maximal time of existence defined in inequality (4.1) is reached. □
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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