Blowup,Global Fast and Slow Solutions for a Semilinear Combustible System

YUAN Junli

Faculty of Science,Nantong University,Nantong 226007,China.

Blowup,Global Fast and Slow Solutions for a Semilinear Combustible System

YUAN Junli?

Faculty of Science,Nantong University,Nantong 226007,China.

.In this paper,we investigate a semilinear combustible systemut?duxx=vp,vt?dvxx=uqwith double fronts free boundary,wherep≥1,q≥1.For such a problem,we use the contraction mapping theorem to prove the local existence and uniqueness of the solution.Also we study the blowup and global existence property of the solution.Our results show that whenpq＞1 blowup occurs if the initial datum is large enough and the solution is global and slow,whose decay rate is at most polynomial if the initial value is suitably large,while whenp＞1,q＞1 there is a global and fast solution,which decays uniformly at an exponential rate if the initial datum is small.

Free boundary;blowup;global fast solution;global slow solution.

1 Introduction

In this paper,we consider the following system

wherep≥1,q≥1.This problem is usually used as a model to describe heat propagation in a two-component combustible mixture.Here bothx=g(t)andx=h(t)are moving boundaries to be determined,h0,d,μandρa(bǔ)re positive constants,and the initial functionu0,v0satisfies

The equation governing the free boundary,h′(t)=?μ(ux(t,h(t))+ρvx(t,h(t))),is a special case of the well-known Stefan condition,which was given by Josef Stefan in his papers appeared in 1889.The original Stefan problem treats the formation of ice in the polar seas.Until now,the Stefan condition has been used in the modeling of a number of applied problems.For example,it was used to describe the melting of ice in contact with water[1],in the modeling of oxygen in the muscle[2],and in wound healing[3]and tumor growth[4–6].There is a vast literature on the Stefan problem,and some important recent theoretical advances can be found in[7].

Similar free boundary model can also describe ecological dynamics.We can refer to several earlier papers,for example,[8–14]and[15].In[8],Du and Lin may be the first attempt to use the Stefan condition in the study of the spreading of populations.They proposed the following free boundary model

They gave a spreading-vanishing dichotomy for the solution of(1.3).Furthermore,they showed that when spreading occurs,for large time,the expanding front moves at a constant speed.

A corresponding work in a fixed domain with Dirichlet boundary condition can be found in[16]and[17],which considered the following nonlinear reaction-diffusion system

where ? is a bounded domain in Rnwith smooth boundary??,m1,n2≥0 andm2,n1＞0 andu0(x),v0(x)are nonnegative,continuous and bounded functions.In[18],the authors consider the case of ?=Rn.

For single equation,much previous mathematical investigation on the corresponding problem with the reaction termuphas been based on the fixed domain problem

Ifthe domain is unbounded,the corresponding problem is the following Cauchy problem

As a sortofintermediate between the cases ofbounded and unbounded intervals,the corresponding moving boundary problem was explored in[19–22]and[23].In[21],Zhang and Lin considered the following system

The result showed that whenp＞1 blowup occurs if the initial datum is large enough and that the solution is global and fast,which decays uniformly at an exponential rate if the initial datum is small,while there is a global and slow solution provided that the initial value is suitably large(the definition of slow or fast solution can be found in Section 4). In[24],Ling et al.studied the global existence and blow-up for a parabolic equation with a nonlocal source and absorption

with the same initial and boundary conditions as in(1.4).As far as the coupled system is concerned,Kim et al.in[25]considered the following mutualistic modelThey got the local existence and uniqueness of a classical solution and the asymptotic behavior of the solution and showed that the free boundary problem admits a global slow solution if the inter-specific competitions are strong,while if the inter-specific competitions are weak there exist the blowup solution and global fast solution.

In this paper,our main interests in studying the long time behaviors of the solution for the system(1.1)are motivated from the above discussions.The rest of this paper is organized as follows.In Section 2,we first prove the local existence and uniqueness of the solution to problem(1.1)by using the contraction mapping theorem.Then,we make use of the Hopf lemma and the maximum principle to show a significant property of the double free boundary fronts,see Theorem 2.2.Finally,a fundamental and useful comparison principle is displayed.Section 3 is devoted to the investigation ofthe blowup property.We take advantage of suitable estimates on the solution(see Lemma 3.1)to verify the main result that all solutions that exist for finite time would blow up in theL∞sense(see Theorem 3.1).Then,we prove that the solution blows up for large initial data. In Section 4,we mainly deal with the existence of global fast solution and global slow solution.The existence of global fast solution can be established easily by constructing a suitable upper solution(see Theorem 4.1),while for the global slow solution,we need to give some priori estimates for global solutions first(see Lemmas 4.1 and 4.2).

2 Local existence and uniqueness

Firstly,we prove the local existence and uniqueness of the solution to(1.1)by applying the contraction mapping theorem.

Theorem 2.1.For any given(u0,v0)satisfying(1.2)and any α∈(0,1),there is a T＞0such that problem(1.1)admits a unique solution

where DT={(t,x)∈R2:t∈[0,T],x∈(g(t),h(t))},C and T only depend on h0,α,‖u0‖C2([?h0,h0])and‖v0‖C2([?h0,h0]).

Proof.As in[8],we first straighten the free boundaries.Letζ(y)be a function inC3(R) satisfying

and setξ(y)=ζ(?y).Consider the transformation

Notice that as long as

the above transformation is a diffeomorphism from R onto R.Moreover,it changes the free boundariesx=g(t),x=h(t)to the linesy=?h0andy=h0respectively.Now,direct calculations show that

whereA=A(g(t),h(t),y),B=B(g(t),h(t),y),C=C(g(t),g′(t),h(t),h′(t),y).Let

For 0＜T＜h0/16min{1/(1+|k1|),1/(1+|k2|)},we define?T=[0,T]×[?h0,h0],and

It is easily seen that ΓT=WT×GT×HTis a complete metric space with the metric

Next,we shall prove the existence and uniqueness result by using the contraction mapping theorem.First,we observe that due to our choice ofT,for any given(w,g,h)∈ΓT,we have

Therefore the transformationintroduced at the beginning of the proof is well de fined.Applying standardLptheory and then the Sobolev imbedding theorem[26],we find that for anythe following initial boundary value problem

whereA,BandCare the same as that in(2.1),C1is a constant depending onα,h0,‖v0‖C2([?h0,h0]).Also,the following initial boundary value problem

whereA,BandCare the same as that in(2.1),C2is a constant depending onα,h0,‖u0‖C2([?h0,h0]).Define

We now define F:ΓT→C(?T)×C1([0,T])×C1([0,T])by

Clearly(w,g,h)∈ΓTis a fixed point of F if and only if(w,z,g,h)solves(2.1).By(2.2)and (2.3),we have

Next we prove that F is a contraction mapping on ΓTforT＞0 sufficiently small.In fact,letThen it follows from (2.2),(2.3)that

whereC4depend onC2,C3and the functionsA,B,C.Taking the difference of equations forresults in

We may assume thatT≤1.Combining(2.4)and(2.5)and applying the mean value theorem,it yields

whereC5depends onC4,μandρ.Then for

The above arguments show that the operator F is contractive on ΓT.It now follows from the contraction mapping theorem that F has a unique fixed point(w,g,h)in ΓT. Moreover by theLpestimates,we have additional regularity for(w,z,g,h)as a solution of problem(2.1),namely,g(t),h(t)∈C1+α/2([0,T])andu,v∈C1+α/2,2+α((0,T]×(g(t),h(t))). Thus(u(t,x),v(t,x),g(t),h(t))is a classical solution of problem(1.1).

Next we present a description of the monotone behavior of the moving boundariesg(t)andh(t).

Theorem 2.2.The two free boundary fronts g(t)and h(t)to problem(1.1)are strictly monotone decreasing and increasing respectively,i.e.,for any solution in(0,T],we have

Proof.By applying the maximum principle and the Hopf lemma to the problem(1.1),we can deduce that

Then,combining the above two inequalities with the Stefan conditions gives the desired result.

Now we shall prove a comparison principles which can be used to estimate the solution(u(t,x),v(t,x))and the free boundariesx=g(t)andx=h(t).

Lemma 2.1.(The Comparison Principle).Let

then the solution(u,v,g,h)of(1.1)satisfies

where DTis defined as in Theorem2.1.

Proof.For smallbe the unique solution of(1.1)withreplaced byrespectively,whereand

For any 0＜σ≤T,let

The strong maximum principle yieldsNote thatfollows thatThis contradicts(2.6).Therefore,for allWe may now apply the usual comparison principle overto conclude that

Since the unique solution of(1.1)depends continuously on the parameters in(1.1),asconverges tothe unique solution of(1.1).The desired result then follows by lettingε→0 in the inequalitiesThe proof is complete.

Remark 1.The pairin Lemma 2.1 is usually called an upper solution of the problem(1.1).We can define a lower solution of the problem(1.1)by reversing all the inequalities in the obvious places.Furthermore,one can easily prove an analogue of Lemma 2.1 for lower solutions.

3 Finite time blowup

In this section,we assumepq＞1.We primarily discuss the blowup behavior of the solution to problem(1.1).In what follows,we aim to verify that all solutions that exist for finite time would blow up in theL∞sense.Before giving this result,we need to make some estimates firstly.

Lemma 3.1.Assume(u,v,g,h)is a solution to problem(1.1)defined for t∈[0,T)for some T∈(0,∞),and there exists a positive number M1(＞1)such that u(t,x)≤M1,v(t,x)≤M1for t∈[0,T)and x∈(g(t),h(t)),then there exists a positive number M2independent of T,such that

Proof.We define

for some appropriateMover the region

Next,we prove by suitably choosing ofM,wis a upper solution.Direct calculations show that,for(t,x)∈?M,

It follows that

Withoutloss ofgenerality,we can assumesince we can letM1be sufficiently large.Therefore,setting

We can apply the maximum principle tow?uandw?vover?Mand deduce thatIt would then follow thathence

Similarly,it can be proved that

The proof is complete.

Definition 3.1.If Tmax＜∞and

then we say that(u,v)blows up in finite time.If(3.1)and(3.2)holds,then we say u and v blow up simultaneously.

For convenience,we set

Theorem 3.1.Let[0,Tmax)be the maximal time interval in which the solution(u,v,g,h)of(1.1)exists.If Tmax＜∞,then u and v blow up simultaneously.

Proof.Firstly,we show that if one ofuandvblows up,so does another.Without loss of generality,we assume thatuexists globally,then the solutionˉvof the following linear parabolic problem

exists globally.On the other hand,the solutionis the upper solution of the following problem

According to the comparison principle,we haveexists globally.Souandvblow up simultaneously.In what follows we use the contradiction argument.AssumeTmax＜∞and

then there existM1＞0 such thatfor allThen according to the above lemma,there exists a constantM2independent ofTmaxsuch that

Now we fixδ∈(0,Tmax)and according to standardLpestimates,the Sobolev embedding theorem and the H¨older estimates for parabolic equations,we can findM3＞0 depending only onδ0,M1andM2such thatM3fort∈[δ0,Tmax).It then follows from the proof of Theorem 2.1 that there exists aτ＞0 depending only onMi,i=1,2,3,such that the solution of problem(1.1)with the initial timeTmax?τ/2 can be extended uniquely to the timeTmax?τ/2+τ,which is a contradiction to the definition ofTmaxitself.Hence the desired result follows.The proof is complete.

In what follows,we will give the blowup result of the solution of(1.1).

Theorem 3.2.Let(u,v,g,h)be a solution of problem(1.1).Then for large initial data the solution u and v blow up simultaneously.

Proof.Consider the following auxiliary problem,

The next blowup result will be used to derive the global slow solution.

Theorem 3.3.Let ?(x)be the first eigenfunction of the eigenvalue problem

withThen the solution of problem(1.1)with theinitial function u0(x)and v0(x)in the form ofΛ?(x)ceases to exist in finite time provided that

where C is the same constants of Theorem2.3in[16].

Proof.From the proof of Theorem 3.2,we haveu≥w,v≥z,(w,z)satisfies(3.3).Ifp≤q, using the second estimate of Theorem 2.3 of[16],we have

The usual comparison principle implies

Hence,we have

For the casep≥q,the corresponding result is similarly obtained.The proof is complete.

4 Global fast solution and global slow solution

In this section,we mainly investigate the long time behavior of the global solutions of problem(1.1).Similar as the work in[19]and[20],we also aim to find global fast solution and global slow solution.Here,we first give the definitions of global fast solution and global slow solution.

De finition 4.1.Let(u,v,g,h)be the solution of(1.1).Ifand the left and rightfree boundaries converge to a finite limit,respectively;namely,then we say that(u,v,g,h)is a global fast solution of(1.1).

De finition 4.2.Let(u,v,g,h)be the solution of(1.1).Ifand the left and right freeboundaries both grow up to infinity,namely,then we say that(u,v,g,h)is a globalslow solution of(1.1).

In the following theorems we give the existence results of the global fast solution and the global slow solution,respectively.

Theorem 4.1.Assume p＞1,q＞1.Let(u,v,g,h)be a solution of problem(1.1).If u0,v0is so small such that

thenand there exist numbers C,β＞0depending on u0andv0such that

i.e.(u,v,g,h)is a global fast solution of(1.1),which decays uniformly at exponential rate.

Proof.Itsuffices to constructa suitable globalsupersolution.Motivated by[27],we define

whereγ,βandε＞0 to be determined later.A direct computation yields:

Similarly,we have

On the otherhand,we can easilyit follows that

In the following,we give the existence of global slow solution.Firstly,we apply a theorem of[16]to prove a priori estimate for the global solution.

Lemma 4.1.Assume pq＞1.Let(u,v,g,h)be a solution to the problem(1.1)withandThen there exists a constantsuch that

where C remains bounded forbounded.

Proof.Ifp≥q,using the first estimate of Theorem 2.3 of[16],we have

whereC1is a positive constant independent ofbe a solution of

The usual comparison principle implies thatis an upper solution of problem(1.1), hence,it suffices to show thatC.In fact,the first equation in(4.1)is independent ofand the exponent ofis equal to

According to the proposition 1 of[21],we havewhich implyRecalling the first estimate of theorem 2.3 of[16]again, we can obtain

For the casep＜q,the corresponding bound is similarly obtained.The proof is complete.

Similar as the proof of the Lemma 4.1,we can obtain a result much stronger than Lemma 4.1,so,the proof is omitted here.

Lemma 4.2.Under the same condition as in Lemma4.1,the solution satisfies

Theorem 4.2.Assume pq＞1.Let ?(x)satisfy the same condition as in Theorem(3.3).Then there exists λ＞0such that the solution of(1.1)with initial data u0=λ?,v0=λ? is a global slow solution,which satis fies that h∞=?g∞=∞.

Proof.We denote the solution to(1.1)byu(u0,v0;·),v(u0,v0;·)to emphasize the dependence ofu,von the initial data when necessary.So dog(t),h(t),g∞,h∞and the maximal existence timeT.

Recalling the assumption that the initial functionu0in the form ofλ?is symmetric on[?h0,h0],we claim thath(t)=?g(t),which implies thatg(t)andh(t)are both finite or in finite at the same time.

Motivated by[19],we de fine

According to Theorem 4.1,we knowλ∈Σ ifλis small,so Σ is not empty.Whenλis large enough,it follows from Theorem 3.3 that the corresponding solution will blow up,i.e.T(λ?)＜∞,hence Σ is bounded.Let

First of all we sayτ=∞.In fact,by continuous dependence(see[29]and[28]),for any fixedconverges toHere we extendit follows from Lemma 4.1 thatThus we haveτ=∞since nonglobal solutions should satisfy

Next we claimσ∞=∞.In what follows we use the contradiction argument.We assumeσ∞＜∞.Sinceby Lemma 4.2,we can chooset0suf ficiently large such thatBy continuous dependence,we can deduce that

forλ＞λ?sufficiently close toλ?.But this implies thatg∞(λ?)＜∞andh∞(λ?)＜∞by Theorem 4.1,which is a contradiction to the definition ofλ?.This completes the proof.

Acknowledgement

This research is supported by National Science Foundation of China(No.11271209,1137-1370).

We would like to thank you for following the instructions above very closely in advance.It will definitely save us lot of time and expedite the process of your papers publication.

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Received 30 January 2015;Accepted 19 April 2015

?Corresponding author.Email address:yuanjunli@ntu.edu.cn(J.L.Yuan)

AMS Subject Classifications:35K20,35R35,92B05

Chinese Library Classifications:O175