Yi JIANG（蔣毅）Chuan LUO（羅川）

VC/VR Lab and Department of Mathematics，Sichuan Normal University，Chengdu 610066，ChinaE-mail:yijiang103@163.com;luochuancd@126.com

OPTIMAL CONDITIONS OF GLOBAL EXISTENCE AND BLOW-UP FOR A NONLINEAR PARABOLIC EQUATION?

Yi JIANG（蔣毅）Chuan LUO（羅川）

VC/VR Lab and Department of Mathematics，Sichuan Normal University，Chengdu 610066，China
E-mail:yijiang103@163.com;luochuancd@126.com

According to the variational analysis and the potential well argument，we get the optimal conditions of global existence and blow-up for a type of nonlinear parabolic equations. Furthermore，we give its application in the instability of the steady states.

nonlinear parabolic equation;variational analysis;potential well argument; global existence;blow up

2010 MR Subject Classification35A15;35K15;35K55

1　Introduction

In this paper，we study the Cauchy problem for a type of nonlinear parabolic equation in Sobolev spaces H1（RN）with N≥2，

On the view-point of physics，we focus to find the optimal conditions of blow-up and global existence for the parabolic equation（1.1）.By the comparing principle，this result was obtained in Wang and Ding［1］as follows:

It shows two differences in comparison to the result of Wang and Ding［1］.The first point is the integral form result inis easy to verify than a local result with point to point inas an example to illustrate，one can make sure that u0satisfiesHowever， one could not verify whetherSecond，there are no constrained conditions such as

To get the optimal conditions of global existence and blow-up for the Cauchy problem（1.1），first，we establish a constrained optimal problem.Then we characterize the optimal solution by the positive symmetric solution of the nonlinear elliptic equation.By the potential well argument and variatial analysis，we get the optimal conditions of global existence and blow-up and its applications in the instability of the steady states.It is worth to pointing out the the method used in this paper comes from（see［2-5］）.

2　Preliminaries

Now，we recall the local well-posedness of the Cauchy problem for the parabolic equation（1.1）from［6-9］.

is the energy functional.

Moreover，we define the functional for

and the constrained optimal problem as follows:

3　Optimal Conditions

In this section，the first target is to characterize the optimal value by the positive symmetric solution of the nonlinear elliptic equation（1.2）.

Lemma 3.1The optimal value d＞0 provided 1＜p＜N+2/（N-2）+.

One gets that there exists some λ1≥0 such that

where λuniquely depends on u and satisfies

and p＞1，equality（3.2）implies thatThen

In terms of the Sobolev embedding inequality（see［10］）

where C＞0 is the Sobolev constant，it implies that

Therefore d＞0.

Theorem 3.2There exists a positive symmetric solution D of the elliptic equation（1.2）such that

ProofFrom Lemma 3.1，we may letbe a minimizing sequence for（2.5），that is

In terms of Definition 2.4 andis bounded for allHence，there exists a subsequencesuch that

For simplicity，we still denoteThe compactness lemma（see［11］）makes sure that the imbeddingis compact.Therefore，one has

According to（3.4）and K（un）=0，it implies that

By equation（3.4），we have

Now，we establish the invariant sets as follows:

Lemma 3.3Let

Then R1and R2are invariant sets of the Cauchy problem（1.1）.That is，whenor，respectively，for all，whereis the solution of the Cauchy problem（1.1）and T is the maximal existence time.

In terms of the definition（2.3），one has

Then，for all λ＞0，we obtain thatTherefore，are nonempty.

From（3.15）and（3.16），we get

It contradicts definitions（2.5）and（3.6）.Hence R1is invariant under the flow generated by the Cauchy problem（1.1）.By the same argument as the above，we can show that R2is invariant under the flow generated by the Cauchy problem（1.1）.

In the following，we give the optimal conditions of blowup and global solution for the Cauchy problem of the parabolic equation（1.1）.

Then we have

First，we prove result（I）.The equal condition holds only forthenis a global solution of the parabolic equation（1.1）.

Now，we prove result（II）.According to Lemma 3.3，（3.17）and（3.20），one has

By definition（2.3）equality（2.2），we obtain that

According to inequality（3.23），it impliesWhich tegother with the H¨older inequality，one has

Hence，

Thus，from Proposition 2.1，one hasand

4　Application in Instability

In this section，we use the main result Theorem 3.2 to prove instability of steady states. We call D（x）is the steady states of equation（1.1）:if D（x）is the positive solution of the elliptic equation（1.2），then u（t，x）=D（x）for all t∈R+is a global solution of the equation（1.2）with the initial datum u（0，x）=D（x）.The instability of steady states means the solution u（t，x）of the parabolic equation（1.1）goes away from the orbit associated with D（x）to infinity in a finite time.

Theorem 4.1Let D（x）be the unique positive symmetric solution of the elliptic equation（1.2）.Then for any ε＞0，there existssuch that

and with the property:the solution u（t，x）of the Cauchy problem for the parabolic equation（1.1）with the initial datum u0（x）is defined forsatisfying

Then，there exists λ＞1 such that

From definition（2.3），we can get

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?February 6，2015;revised November 22，2015.The

was supported by National Natural Science Foundation of China（11126336 and 11201324），New Teachers’Fund for Doctor Stations，Ministry of Education（20115134120001），F(xiàn)ok Ying Tuny Education Foundation（141114），Youth Fund of Sichuan Province（2013JQ0027）