Department of Mathematics and Statistics, Beihua University, Jilin City, P.R. China

Institute of Mathematics, Jilin University, Changchun, 130012, P.R. China

Abstract

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.

1 Introduction

In this paper, we first consider the initial and boundary value problem to the following nonlinear parabolic system with variable exponents:

where

or

respectively, where

In the case when

The motivation of this work is due to

where

Besides, this work is also motivated by

where

We also study the following nonlinear hyperbolic system of equations:

The aim of this paper is to extend the results in

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

**Definition 2.1** We say that the solution

where

Our first result here is the following.

**Theorem 2.1**

Let us consider the equivalent systems of (1.1)

where

We introduce the following iteration scheme:

and the convergence of the sequence

is a contraction in the set

Now, we define

where

We denote

and for arbitrary

where

We claim that Ψ is a contraction on

and we always have

Now, we define

It is obvious that

Then, by using inequality (2.2), we get

Hence, for sufficiently small

where

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 3.1**

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 3.1**

with the homogeneous Dirichlet boundary condition, and let

We introduce the function

We now deal with the term

Then we have

where

From the convex property of the function

Then we get

Note that

Hence, for

Next, we state briefly the proof to the theorem in the case

In view of the property of

According to the convex property of the function

where

By Lemma 3.1, the proof is complete. □

4 Blow-up of solutions for a hyperbolic system

**Lemma 4.1**

Now, let us study the following problem:

where

**Theorem 4.1**

The term

By virtue of the convex property of the function

Then we have

Now, we can apply Lemma (4.1) for

Moreover,

Hence,

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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.

Acknowledgements

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.