Blowup of the Solutions for a Reaction-Advection-Diffusion Equation with Free Boundaries

YANG Jian

School of Mathematical Sciences,University of Jinan,Jinan 250022,China.

Abstract. We investigate a blowup problem of a reaction-advection-diffusion equation with double free boundaries and aim to use the dynamics of such a problem to describe the heat transfer and temperature change of a chemical reaction in advective environment with the free boundary representing the spreading front of the heat.We study the influence of the advection on the blowup properties of the solutions and conclude that large advection is not favorable for blowup.Moreover,we give the decay estimates of solutions and the two free boundaries converge to a finite limit for small initial data.

Key Words: Nonlinear reaction-advection-diffusion equation;one-phase Stefan problem;decay;blowup.

1 Introduction

We study the blowup solution for the following reaction-advection-diffusion equation

wherex=g(t),h(t)are free boundaries to be determined.p＞1 andh0,µ,βare some given positive constants.The initial functionu0satisfies

Problem(1.1)may be viewed as describing the heat transfer and temperature change in chemical reaction,u(t,x)may represent the temperature over a one-dimensional region and the initial functionu0(x)stands for the temperature distribution in each point at the initial time which occupy an initial region(-h0,h0).

Whenβ=0(i.e.there is no advection in the environment),the qualitative properties of the problem(1.1)were studied by Souplet and his collaborators[1-4],they study the blowup phenomenon of reaction-diffusion equation with a free boundary and prove the solution blows up in finite time inL∞norm and all global solutions are bounded and decay uniformly to 0.

Our concern is to discuss the blowup properties of the solutions of (1.1) withβ ＞0,which means the heat transfer is affected by advection.In the process of chemical reactions,we may use the free boundary to represent the spreading front of the heat.We know that chemical reactions with a high initial temperature produce a lot of heat,which will increase the temperature of chemical reactions,thus accelerating the rate of chemical reactions,and as a result making the reaction temperature higher and higher and even converge to infinity in finite time.Mathematically,it means the solutions will blow up.In the field of physics [5],the advective heat transfer is an important physical process.Advective heat transfer is caused by relative displacement of particles during fluid movement.In some cases,the fluid motion that occurs under the influence of external forces is called forced motion of fluid.Heat transfer and fluid motion are inextricably linked because of advective heat transfer.Therefore,advection due to external forces is a very important factor affecting heat transfer.

The main purpose of this paper is to investigate the effect of the advection on the properties of blowup solution of(1.1)provided the initial datum has compact supports.Our results indicate that larger advection makes blowup more difficult.We denoteT*as the blowup time throughout this paper:

IfT*＜∞,one has

The plan of the paper is as follows: some preliminary results,including the existence and the comparison principle of (1.1) are gathered in Section 2.Blowup and vanishing results are proved in Section 3.

2 Preliminaries

We begin by recalling the local existence and uniqueness result which can be proved by the similar method as in[6-8].

Theorem 2.1.For any u0satisfying(1.2)and any α∈(0,1),there exists a positive number T＞0such that problem(1.1)admits a unique solution

furthermore,

where GT:={(t,x)∈R2:x∈[g(t),h(t)],t∈(0,T]},C and T only depend on h0,α,‖u0‖C2([-h0,h0]).

and(u,g,h)is a solution to(1.1),then

Proof.The proof of Lemma 2.1 is identical to that of Lemma 5.7 in [7],so we omit the details here.

Remark 2.1.The triplein Lemma 2.1 is often called an upper solution of(1.1).A lower solution can be defined analogously by reversing all the inequalities.

3 Main Results

In this section,we mainly determine the asymptotic behavior of the solution of(1.1).Similar to the references[2,3,8],we will give some sufficient conditions that imply blowup(∞)or vanishing(0).We first construct the energy functional as follows:

The following lemma gives the basic energy identity.

Lemma 3.1.Let u be the solution of the problem(1.1),then we have the relations

Integrating by parts,we get

Integrating(3.7),we obtain(3.2).Now the proof is completed.

Proof.In order to get the estimates ofL,we assumevis the solution of the following auxiliary problem:

It is well-known thatvexists for allt＞0 and we can deduce from Lemma 2.1 thatu≥v≥0 andg(t)≤λ(t)≤-h0,h(t)≥σ(t)≥h0on(0,T*).By the same argument as in Lemma 3.1,denoting|v(t)|1=v(t,x)dx,we easily get

On the other hand,by the comparison principle,we havev ≤,wheresatisfies the following Cauchy problem

By theL1-L∞estimate for the equation,we have

Therefore,by(3.8),we have

Next we give the lower bound of the constantL.By using H¨older’s inequality,we deduce

Noticing thatg(t)≤λ(t)≤-h0,h(t)≥σ(t)≥h0on(0,T*)and(3.8),we have

By(3.9)and plugging the valuet=t0into(3.10),we have

The proof now is completed.

The following result indicates the conditions for blowup.

Lemma 3.3.Let u be the solution of the problem(1.1),then we have T*＜∞whenever

Proof.Arguing by contradiction,we assumeT*=∞.The assumption(3.11)and Lemma 3.2 imply that

for allt≥t0sufficiently large.

Define the function

Then we have

and

Using identity(3.1),we have

Noticing that(3.12)holds fort≥t0sufficiently large,we have

By applying the Cauchy-Schwarz inequality,we obtain

On the other hand,(3.13)implies that

so we have limt∞I(t)=∞.Then we obtain

for some larget1≥t0+1(sincep＞1).DefiningJ(t):=(t)fort≥t1,then we have

This implies thatJis concave,decreasing and positive fort≥t1,which is impossible.This contradiction shows thatT*＜∞.The proof is completed.

As a consequence of Lemmas 3.1-3.3,we immediately have the following blowup result.

Theorem 3.1.Let u be the solution of the problem(1.1),if

then the solution will blow up in finite time.

Remark 3.1.In Theorem 3.1,we denote

it is easy to obtain

andH(β)0 when∞.Thus we conclude that large advection is not favorable for blowup.

In the following part,we give the long-time behavior of the global solutions of the problem(1.1)and we obtain the decay rate of the global solution.

Theorem 3.2.Let u be the solution of the problem(1.1).If u0is so small such that

then T*=∞.Moreover,h∞＜∞,g∞＞-∞and there exist real numbers C and γ＞0depending on u0such that

Proof.We construct some suitable global upper solution of the problem(1.1).Motivated by[3,9],we define

whereα,γandϵto be fixed later.A direct calculation shows that

for allt≥0 andη(t)≤x≤ρ(t).Moreover,we haveρ′(t)=2h0αe-αt＞0 and-µx(t,ρ(t))=2µϵρ-1(t)eBy the definition ofρ(t),we can derive that 2h0≤ρ(t)≤4h0for allt≥0.If we choose

we have

Now we assume(3.14)holds and takeϵ=2e‖u0‖L∞≤ϵ0,then we haveu0(x)≤(0,x)for-h0≤x ≤h0.By the comparison principle,we obtainη(t)≤g(t),h(t)≤ρ(t) andu(t,x)≤(t,x)forη(t)≤x≤ρ(t),t＞0 as long asuexists.The proof is now complete.

Remark 3.2.Theorem 3.2 indicates that the double free boundaries converge to finite limits and the solutionudecays at a exponential rate astgoes to infinity provided the initial value is small.Moreover,we see that it is more difficult to obtain the solution in Theorem 3.2 provided the advection is larger.

Acknowledgement

This work is supported by Natural Science Foundation of China(No.11901238)and Natural Science Foundation of Shandong Province(No.ZR2019MA063).