Pohozaev Identities for the Semilinear Parabolic Equations and Applications

FENG Tingfu(冯廷福),ZHU Yan(朱艳),MU Gui(母贵)

(School of Mathematics,Kunming University,Kunming 650214,China)

Abstract: Inspired by Pohozaev(1965),we establish the Pohozaev identity for the semilinear parabolic equation by the divergence theorem and Green’s formula in this paper.Using this identity,we prove that there has no nontrivial solutions for some semilinear parabolic equations and systems under suitable conditions,respectively.Our results extend some earlier nonexistence results from semilinear elliptic equations to semilinear parabolic equations.

Key words: Semilinear parabolic equation;Pohozaev identity;Nonexistence

1.Introduction and Main Results

In this paper,we consider the following semilinear parabolic equation

whereΩis a smooth domain.

In 1965,Pohozaev[1]studied the existence of solutions for a class of semilinear elliptic partial differential equation

an important identity

was established by using the divergence theorem,whereν=ν(x)denotes the outward normal to∂Ωatx,in addition nonexistence of nontrivial solutions was obtained by using this identity whenin a strictly star-shaped bounded smooth domain.Later people called this kind of identity as the Pohozaev identity,a lot of literature shows that the Pohozaev identity plays a very important role in proving the existence and nonexistence results for the nonlinear elliptic partial differential equations,there are some applications of Pohozaev identity.For the elliptic equation without the condition of Ambrosetti-Rabinowitz,LI and YE[2]established a new manifold by combining Pohozaev identity with Nehari manifold,in which the existence of solutions of the limit equation was proved.More recently,Devillanova and Solimini[3],Struwe[4]noted that the strong convergence of approximate solutions can be obtained by an exact estimate of each term of nonlinear elliptic equation with critical exponents by using the global compact property and the local Pohozaev identity,thus the multiple solutions of the equation was obtained.

Brezis and Nirenberg[5]considered the following nonlinear elliptic equation involving critical Sobolev exponent

and they obtained a necessary condition for the existence of positive solutions of the above nonlinear elliptic equation by the Pohozaev identity.Pucci and Serrin[6]studied a kind of variational problem

and established corresponding to the Pohozaev identity and some nonexistence results of special equations are obtained by selecting proper test vector functionhin the Pohozaev identity,which extended to vector-valued extremals and higher-order equations.In particular,in [6]there has new results was obtained for the system

and for the semilinear pluriharmonic equation

ZHENG,MA and ZHANG[7]established the Pohozaev identity for the biharmonic equation.Using this identity,they proved that nonexistence of positive solutions for a class of fourth order elliptic systems in positive domainsΩ ⊂Rnwith smooth boundary∂Ω,where has a continuous positive vector field

andV(x) satisfies divV(x)=nand such thatV(x)·ν >0 on∂Ω,which is a new class of domains more general than star-shaped domain and was introduced by AN in [8-11].

PENG,WANG and YAN[12]dealed with the following nonlinear elliptic equation

By combining a finite reduction argument and the local Pohozaev identity,they proved that the above nonlinear elliptic equation has infinite many solutions.This method overcomes the difficulty of appearing in using the standard reduction method to locate the concentrated points of the solutions.

Since the fractional Laplacian is nonlocal,the Pohozaev identity of fractional Laplacian equation is also much more complex.Ros-oton and Serra[13]established the corresponding Pohozaev identity and obtained nonexistence results of nontrivial solutions.These nonexistence results,when the region is unbounded,often correspond to an important class of Liouville type theorem,which is often very important in the study of the theory of elliptic problems as a prior estimate and the technique of blow-up .

The above extensive literature shows that Pohozaev identity plays a very important role in proving the existence and nonexistence results for the nonlinear elliptic partial differential equations in bounded domains or unbounded domains.However,as far as we know,there has no results on the Pohozaev identity for the semilinear parabolic equation,which is the main research direction of this paper.Inspired by Pohozaev[1],we establish the Pohozaev identity for the semilinear parabolic equation by the divergence theorem and Green’s formula in this paper.Using this identity,we prove that there has no nontrivial solutions for some semilinear parabolic equations and systems under suitable conditions,respectively.Our results extend some earlier nonexistence results from semilinear elliptic equations to semilinear parabolic equations.

Theorem 1.1Letu(x,t) be a solution to (1.1).Then it satisfies

Corollary 1.1In the special caseΩ=Rnin Theorem 1.1,we have by (1.2) that

In the special caseΩ=Rnandf(x,u) is replaced byf(u) in Theorem 1.1,we also have by(1.2) that

which is a parabolic analogue of the Derrick-Pohozaev identity in [14](see also [15]) for the solutions to semilinear parabolic equationut −∆u=f(u) in Rn×(0,+∞),whereuand its derivatives go to zero rapidly as|x|→∞,satisfiesF(0)=0.

Corollary 1.2For eachT >0,integrating (1.2) from 0 toT,we have the following weighted energy estimate

This paper is organized as follows.The proof of Theorem 1.1 is given in Section 2;Section 3 is devoted to give some applications of Theorem 1.1.

2.Proof of Theorem 1.1

Proof of Theorem 1.1First,taking the inner product of (1.1) withand integrating overΩ,we have

and by Green’s formula we obtain

Sinceut=0 on∂Ω,the boundary integral in(2.3)vanishes.Now,from(2.1),(2.2)and(2.3),we have

In virtue ofF(x,u)=F(x,0)=0 forx ∈∂Ωand the divergence theorem,it gives

Now,from (2.5),(2.6) and (2.7),we have

Final,combining (2.4) and (2.8),we derive (1.2).

3.Some Applications of Theorem 1.1

Example 3.1Letu(x,t) solve the following semilinear parabolic equation

whereΩis a strictly star-shaped bounded smooth domain with respect to the origin.If

then (3.1) has no any nontrivial solutions.

ProofBy Theorem 1.1,we derive that

SinceΩis a strictly star-shaped with respect to the origin,it shows thatx·ν >0 forx ∈∂Ω.If (3.2) holds,we will derive a contradiction with (3.3).

Example 3.2Letu(x,t) solve the following semilinear parabolic equation

then (3.4) has no any nontrivial solutions.

ProofBy Corollary 1.1,one has

Example 3.3Let (u(x,t),v(x,t)) solve the following semilinear parabolic systems

then (3.7) has no any nontrivial solutions.

ProofBy Corollary 1.1,we have

Combining (3.11) with (3.12),we have (3.8).If (3.8) holds,then we derive is a contradiction with (3.9) and (3.10).