# Study of solutions to an initial and boundary value problem for certain systems with variable exponents

- Yunzhu Gao
^{1}Email author and - Wenjie Gao
^{2}Email author

**2013**:76

**DOI: **10.1186/1687-2770-2013-76

© Gao and Gao; licensee Springer. 2013

**Received: **15 February 2013

**Accepted: **18 March 2013

**Published: **5 April 2013

## 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.

### Keywords

existence blow-up parabolic system hyperbolic system variable exponent## 1 Introduction

*∂*Ω and $0<T<\mathrm{\infty}$, ${Q}_{T}=\mathrm{\Omega}\times [0,T)$, ${S}_{T}$ denotes the lateral boundary of the cylinder ${Q}_{T}$, and the source terms ${f}_{1}$, ${f}_{2}$ are in the form

respectively, where ${p}_{1}$, ${p}_{2}$, ${a}_{1}$, ${a}_{2}$ are functions satisfying conditions (2.1) below.

In the case when ${p}_{1}$, ${p}_{2}$ 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 $x\in {\mathbb{R}}^{N}$ ($N\ge 1$), $t>0$, 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 [3]).

where $\mathrm{\Omega}\in {\mathbb{R}}^{n}$ is a bounded domain with smooth boundary *∂* Ω, and the source term is of the form $f(x,u)=a(x){u}^{p(x)}$ or $f(x,u)=a(x){\int}_{\mathrm{\Omega}}{u}^{q(y)}(y,t)\phantom{\rule{0.2em}{0ex}}dy$. 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).

## 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.

**Definition 2.1**We say that the solution $(u(x,t),v(x,t))$ for problem (1.1) blows up in finite time if there exists an instant ${T}^{\ast}<\mathrm{\infty}$ such that

Our first result here is the following.

**Theorem 2.1** *Let* $\mathrm{\Omega}\subset {\mathbb{R}}^{N}$ *be a bounded smooth domain*, ${p}_{1}(x)$, ${p}_{2}(x)$, ${a}_{1}(x)$, ${a}_{2}(x)$ *satisfy the conditions in* (2.1), *and assume that* ${u}_{0}(x)$ *and* ${v}_{0}(x)$ *are nonnegative*, *continuous and bounded*. *Then there exists a* ${T}^{0}$, $0<{T}^{0}\le \mathrm{\infty}$, *such that problem* (1.1) *has a nonnegative and bounded solution* $(u,v)$ *in* ${Q}_{{T}^{0}}$.

*Proof* We only prove the case when ${f}_{1}(u,v)={a}_{1}(x){v}^{{p}_{1}(x)}$ and ${f}_{2}(u,v)={a}_{2}(x){u}^{{p}_{2}(x)}$, and the proofs to the cases ${f}_{1}(u,v)={a}_{1}(x){\int}_{\mathrm{\Omega}}{v}^{{p}_{1}(y)}(y,t)\phantom{\rule{0.2em}{0ex}}dy$ and ${f}_{2}(u,v)={a}_{2}(x){\int}_{\mathrm{\Omega}}{u}^{{p}_{2}(y)}(y,t)\phantom{\rule{0.2em}{0ex}}dy$ are similar.

where $g(x,y,t)$ is the corresponding Green function. Then the existence and uniqueness of solutions for a given $({u}_{0}(x),{v}_{0}(x))$ could be obtained by a fixed point argument.

is a contraction in the set ${E}_{T}$ to be defined below.

where ${\mathrm{\Omega}}_{T}=\mathrm{\Omega}\times [0,T]$, $M>\u2980({u}_{0}(x),{v}_{0}(x))\u2980$ is a fixed positive constant.

It is obvious that $\mu (t)\to 0$ when $t\to {0}^{+}$.

*t*, we have

where $\theta <1$ 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 3.1**

*Let*$y(t)$

*be a solution of*

*where*$\lambda >0$, $a>0$, $r>1$

*and*$C>0$

*are given constants*.

*Then*,

*there exists a constant*$K>0$

*such that if*$y(0)\ge K$,

*then*$y(t)$

*cannot be globally defined*;

*in fact*,

*Proof*It is sufficient to take $K>0$ such that

By a direct integration to (3.2), then we get immediately (3.1), which gives an upper bound for the blow-up time ${t}^{\ast}=\frac{2{y}^{1-r}(0)}{a(r-1)}$. □

The next theorem gives the main result of this section.

**Theorem 3.1**

*Let*$\mathrm{\Omega}\subset {\mathbb{R}}^{N}$

*be a bounded smooth domain*,

*and let*$(u,v)$

*be a positive solution of problem*(1.1),

*with*${p}_{1}(x)$, ${p}_{2}(x)$, ${a}_{1}(x)$, ${a}_{2}(x)$

*satisfying conditions in*(2.1).

*Then any solutions of problem*(1.1)

*will blow up at finite time*${T}^{\ast}$

*if the initial datum*$({u}_{0}(x),{v}_{0}(x))$

*satisfies*

*where* $\phi >0$ *is the first eigenfunction of the homogeneous Dirichlet Laplacian on* Ω *and* $C>0$ *is a constant depending only on the domain* Ω *and the bounds* ${C}_{1}$, ${C}_{2}$ *given in condition *(2.1).

*Proof*Let ${\lambda}_{1}$ be the first eigenvalue of

*φ*be a positive function satisfying

where $p=min\{{p}_{1}^{-},{p}_{2}^{-}\}$ and $\gamma =min\{{c}_{1},{c}_{2}\}$, $\mathrm{\Gamma}=max\{{C}_{1},{C}_{2}\}$.

Hence, for $\eta (0)$ big enough, the result follows from Lemma 3.1.

*φ*, we get

*γ*,

*p*and ${\parallel \phi \parallel}_{\mathrm{\infty}}$, $|\mathrm{\Omega}|$ denotes the measure of Ω. Hence,

By Lemma 3.1, the proof is complete. □

## 4 Blow-up of solutions for a hyperbolic system

**Lemma 4.1** [15]

*Let*$y(t)\in {C}^{2}$

*satisfying*

*and*$h(s)\ge 0$

*for all*$s\ge a$.

*Then*${y}^{\prime}(t)>0$

*whenever*

*y*

*exists*;

*and*

where ${u}_{0}(x),{v}_{0}(x),{u}_{1}(x),{v}_{1}(x)\ge 0$ and they are not identically zero, and ${f}_{1}(u,v)$, ${f}_{2}(u,v)$ as above respectively.

**Theorem 4.1** *Let* $(u,v)\in {C}^{2}\times {C}^{2}$ *be a solution of problem* (4.2), *and let the conditions in* (2.1) *hold*. *Then there exist sufficiently large initial data* ${u}_{0}$, ${v}_{0}$, ${u}_{1}$, ${v}_{1}$ *such that any solutions of problem* (4.1) *blew up at finite time* ${T}^{\ast}$.

*Proof*Let $({\lambda}_{1},\phi )$ be the first eigenvalue and eigenfunction of Laplacian in Ω with homogeneous Dirichlet boundary conditions as before. We assume that ${f}_{1}(u,v)={a}_{1}(x){v}^{{p}_{1}(x)}$, ${f}_{2}(u,v)={a}_{2}(x){u}^{{p}_{2}(x)}$, the other is similar. We also define the function $\eta (t)={\int}_{\mathrm{\Omega}}(u+v)\phi $, so we have

Hence, $(u,v)$ blows up before the maximal time of existence defined in inequality (4.1) is reached. □

## Declarations

### 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.

## Authors’ Affiliations

## References

- Chen Y, Levine S, Rao M: Variable exponent, linear growth functions in image restoration.
*SIAM J. Appl. Math.*2006, 66: 1383-1406. 10.1137/050624522MATHMathSciNetView Article - Escobedo M, Herrero MA: Boundedness and blow up for a semilinear reaction-diffusion system.
*J. Differ. Equ.*1991, 89: 176-202. 10.1016/0022-0396(91)90118-SMATHMathSciNetView Article - Escobedo M, Herrero MA: A semilinear parabolic system in a bounded domain.
*Ann. Mat. Pura Appl.*1998, CLXV: 315-336.MathSciNetMATH - Friedman A, Giga Y: A single point blow up for solutions of nonlinear parabolic systems.
*J. Fac. Sci. Univ. Tokyo Sect. I*1987, 34(1):65-79.MATHMathSciNet - Galaktionov VA, Kurdyumov SP, Samarskii AA: A parabolic system of quasiliner equations I.
*Differ. Equ.*1983, 19(12):2133-2143.MathSciNet - Galaktionov VA, Vázquez JL Progress in Nonlinear Differential Equations and Their Applications 56. In
*A Stability Technique for Evolution Partial Differential Equations*. Birkhäuser, Boston; 2004.View Article - Kufner A, Oldrich J, Fucik S:
*Function Space*. Kluwer Academic, Dordrecht; 1977.MATH - Pinasco JP: Blow-up for parabolic and hyperbolic problems with variable exponents.
*Nonlinear Anal.*2009, 71: 1094-1099. 10.1016/j.na.2008.11.030MATHMathSciNetView Article - Andreu-Vaillo F, Caselles V, Mazón JM Progress in Mathematics 223. In
*Parabolic Quasilinear Equations Minimizing Linear Growth Functions*. Birkhäuser, Basel; 2004.View Article - Antontsev, SN, Shmarev, SI: Anisotropic parabolic equations with variable nonlinearity. CMAF, University of Lisbon, Portugal 013, 1-34 (2007)MATH
- Antontsev, SN, Shmarev, SI: Blow-up of solutions to parabolic equations with nonstandard growth conditions. CMAF, University of Lisbon, Portugal 02, 1-16 (2009)MATH
- Antontsev SN, Shmarev SI: Parabolic equations with anisotropic nonstandard growth conditions.
*Int. Ser. Numer. Math.*2007, 154: 33-44. 10.1007/978-3-7643-7719-9_4MathSciNetView ArticleMATH - Antontsev SN, Shmarev S: Blow-up of solutions to parabolic equations with nonstandard growth conditions.
*J. Comput. Appl. Math.*2010, 234: 2633-2645. 10.1016/j.cam.2010.01.026MATHMathSciNetView Article - Erdem D: Blow-up of solutions to quasilinear parabolic equations.
*Appl. Math. Lett.*1999, 12: 65-69.MATHMathSciNetView Article - Glassey RT: Blow-up theorems for nonlinear wave equations.
*Math. Z.*1973, 132: 183-203. 10.1007/BF01213863MATHMathSciNetView Article - Kalashnikov AS: Some problems of the qualitative theory of nonlinear degenerate second-order parabolic equations.
*Russ. Math. Surv.*1987, 42(2):169-222. 10.1070/RM1987v042n02ABEH001309MATHMathSciNetView Article - Levine HA:Some nonexistence and instability theorems for solutions of formally parabolic equations of the form $P{u}_{t}=-Au+F(u)$.
*Arch. Ration. Mech. Anal.*1973, 51: 371-386.MATHView ArticleMathSciNet - Levine HA, Payne LE: Nonexistence of global weak solutions for classes of nonlinear wave and parabolic equations.
*J. Math. Anal. Appl.*1976, 55: 329-334. 10.1016/0022-247X(76)90163-3MATHMathSciNetView Article - Lian SZ, Gao WJ, Cao CL, Yuan HJ: Study of the solutions to a model porous medium equation with variable exponents of nonlinearity.
*J. Math. Anal. Appl.*2008, 342: 27-38. 10.1016/j.jmaa.2007.11.046MATHMathSciNetView Article - Ruzicka M Lecture Notes in Math. 1748. In
*Electrorheological Fluids: Modelling and Mathematical Theory*. Springer, Berlin; 2000.View Article - Simon J:Compact sets in the space ${L}^{p}(0,T;B)$.
*Ann. Mat. Pura Appl.*1987, 4(146):65-96.MathSciNetMATH - Tsutsumi M: Existence and nonexistence of global solutions for nonlinear parabolic equations.
*Publ. Res. Inst. Math. Sci.*1972, 8: 211-229. 10.2977/prims/1195193108MATHMathSciNetView Article - Zhao JN:Existence and nonexistence of solutions for ${u}_{t}=div({|\mathrm{\nabla}u|}^{p-2}\mathrm{\nabla}u)+f(\mathrm{\nabla}u,u,x,t)$.
*J. Math. Anal. Appl.*1993, 172: 130-146. 10.1006/jmaa.1993.1012MATHMathSciNetView Article

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