Asymptotic Decomposition for Nonlinear Damped Klein-Gordon Equations

Ze Liand Lifeng Zhao

1School of Mathematics and Statistics,Ningbo University,Ningbo 315211,China;

2School Mathematical Sciences,University of Science and Technology of China,Hefei 230026,China.

Abstract.In this paper,we prove that if the solution to the damped focusing Klein-Gordon equations is global forward in time with bounded trajectory,then it will decouple into the superposition of divergent equilibriums.The core ingredient of our proof is the existence of the “concentration-compact attractor” introduced by Tao which yields a finite number of asymptotic profiles.Using the damping effect,we can prove all the profiles are equilibrium points.

Key words:Nonlinear Klein-Gordon equations,damping,soliton resolution,global attractor.

1 Introduction

In this paper,we consider the following damped focusing Klein-Gordon equation:

Dynamical systems of the type(1.1)appear in a number of physical settings,for example it describes the behavior of waves propagating in a nonlinear medium with damping effect,see[25,26].

We focus on the dynamic behavior of solutions to(1.1)in this paper.Whenα=0,(1.1)is called nonlinear Klein-Gordon equation(NLKG).Whenα＞ 0,we call(1.1)the nonlinear damped Klein-Gordon equation(dNLKG).For NLKG,Cazenave[3]obtained the following dichotomy:solutions either blow up at finite time or are global forward in time and bounded in H,provided1＜p＜∞,whend=1,1＜p≤5,whend=2 andwhend≥3.For dNLKG,Feireisl[17]gave an independent proof of the boundedness of the trajectory to global solutions,forwhend≥3.For dNLKG,1＜p＜,Burq,Raugel,Schlag[2]proved that radial global solutions will converge to equilibrium points as time goes to infinity.A natural problem is what happens for non-radial solutions?It is widely conjectured that the solutions will decouple into the superposition of equilibrium points.In the positive direction,Feireisl[17]constructed a global solution to dNLKG which decouples into the superposition of finite number of divergent shifted ground states.Indeed,this problem is closely related to the soliton resolution conjecture in dispersive equations.The(imprecise sense)soliton resolution conjecture states that for“generic”large global solutions,the evolution asymptotically decouples into the superposition of divergent solitons,a free radiation term,and an error term tending to zero ast→∞.For more expression and history on the soliton resolution,see Soffer[32].

There are a lot of works devoted to the verification of the soliton resolution conjecture.Duyckaerts,Kenig,and Merle[10]first made a breakthrough on this topic.For radial data to three dimensional focusing energy-critical wave equations,they proved the solution with bounded trajectory asymptotically decouples into the superposition of a finite number of rescaled ground states plus a radiation term.One of the key ingredient of their arguments is the novel tool,called “channels of energy”introduced by[10,11].The method developed by them has been applied to many other situations,such as[7,8,22–24]for wave maps,[6,13,20,30]for semilinear wave equations.By a weak version of channel energy,the soliton resolution along a sequence of times for radial even dimensional critical wave equations and energy critical equivariant geometric wave equations such as Yang-Mills,wave maps was proved by[5,6,21].Recently Duyckaerts,Jia,Kenig,Merle[14,15]solved the soliton resolution along a sequence of times for the focusing energy critical wave equation.

It is known that(1.1)admits a ground state which is the radial positive stationary solution with the minimized energy among all the non-zero stationary solutions.Besides the ground state,(1.1)also has an infinite number of nodal solutions which own zero points.(see Berestycki,Lions[1]).The dynamics below and slightly above the ground state is relatively clear in the literature of dispersive PDEs.For NLKG and initial data with energy below the ground state,the dichotomy characterization of blow up v.s.global existence was given by Payne,Sattinger[29]and scattering v.s.blowup below the ground state was obtained by Ibrahim,Masmoudi,Nakanishi[19].Nakanishi,Schlag[27,28]obtained the nine set dynamics of the solutions to NLKG with energy slightly above the ground state.In fact they proved the trichotomy forward in time:the solution(i)either blows up at finite time(ii)or globally exists and scatters to zero(iii)or globally exists and scatters to the ground states.The main technical ingredient of their papers is the“one pass” theorem which excludes the existence of(almost)homoclinic orbits between the ground state and(almost)heteroclinic orbits connecting ground stateQwith-Q.

The dynamics for data far away from the ground state are very less understood except the critical wave equations.For dNLKG with the radial assumption,Burq,Raugel,Schlag[2]showed the dichotomy in forward time(i)the solution either blows up at finite time,(ii)or converges to some equilibrium point.The key of their proof is to prove theω-limit set of the trajectory is just one single point by using the theory of invariant manifolds and its foliation.For high dimensional nonlinear Schr¨odinger equations(NLS),Tao[33]proved any global solution with a bounded trajectory is attracted by a“concentration compact”attractor excluding a free wave part.And this result was refined in Tao[34]for NLS with a potential under the radial assumption.

In this paper,we aim to study the long time behaviors of damped Klein-Gordon equations without the radial assumption.Let

Remark 1.1.There are three types of soliton resolution result in the literature:(a)resolution along all time;(b)resolution along a time sequence;(c)resolution along any time sequence up to selecting subsequence.We emphasize that our result is exactly the(c)case which is stronger than(b).Moreover,the(c)case is indeed the compactness of trajectory modular invariant group actions,which can be applied to prove resolution along all time with Lojasiewicz-Simon inequality when the group action is ruled out(such as the radial case).

Recently,C?te,Martel[9]constructed multi-travelling waves consisting of ground state and excited states for the nonlinear Klein-Gordon equations.

Remark 1.4.In the radial case,by the abstract framework of Chill[4]and the spectrum analysis of the linearized operator around the equilibrium done in[2],Theorem 1.1 can further show that any radial global solution with bounded trajectory converges to some equilibrium ast→∞,i.e.,

This recovers part of the results in[2]except the fact that global solutions have bounded trajectory which was verified by[2]in the radial case.And we remark that although the classical Lojasiewicz-Simon inequality(LS inequality)obtained by Simon[31]was usually applied in the analytic setting,see for instance the abstract framework of Haraux,Jendoubi[18],Chill[4]had showed that under some spectrum assumption Lojasiewicz-Simon inequality can also be applied to non-analytical nonlinearities.

Remark 1.5.After submitting this paper into the arxiv,we submitted a paper entitled“Long time behaviors for 3D cubic damped Klein-Gordon equations in inhomogeneous media”to the arxiv,and it is marked by arXiv:1512.02755.We have withdrawn arX-iv:1512.02755 in the arxiv and will never submit it to any journal for publication.

In order to describe our proof,the following notions introduced by[33]are needed:

We sayE?His G-precompact withJcomponents ifE?J(GK)for some compactK?H andJ≥1.

Our proof is divided into three parts.In the first step,we prove the trajectory ofu(t)is attracted by a G-precompact set withJcomponents,namely the existence of concentration-compact attractor.The key ingredient in this step is the frequency localization and the spatial localisation.The idea of“concentration compact”attractor was introduced by Tao[33]in the study of dynamics of NLS.In the second step,for any time sequencetn→∞,we prove up to a subsequence there exist a finite number of asymptotic profiles whose sum can be viewed as a linear decomposition ofu(tn).Then by applying the perturbation theorem,we obtain a nonlinear profile decomposition for(u(t+tn),?tu(t+tn)).Using the damping effect of(1.1),we can show all the asymptotic profiles are exactly equilibrium points.

Our paper is organized as follows:In Section 2,we recall some preliminaries,such as the Strichartz estimates,the local well-posedness theory and the perturbation theorem.In Section 3,we prove the frequency localization and the spatial localization.In Section 4,we prove the existence of concentration-compact attractor.In Section 5,we extract the profiles and finish our proof by using the damping effect.

NotationsWe will use the notationX?Ywhenever there exists some positive constantCsuch thatX≤CY.Similarly,we will useX～YifX?Y?X.

We define the Fourier transform on Rdto be

For dyadic numberN,PNis the usual Littlewood-Paley decomposition operator with frequency truncated inN.Similarly,we useP≤NandP≥N.Sometimes,we denoteP＜μubyu＜μ.All the constants are denoted byCand they can change from line to line.

The constantp*is defined in Lemma 2.1.And ford≥3 we define 2*by

2 Preliminaries

As explained in Remark 1.4,we only need to consider

In this section,we recall the Strichartz estimates,local well-posedness and perturbation theorem.And we closely follow notations in[2]for reader’s convenience.Consider the linear equation,

3 Frequency localization and Spatial localization

4 Concentration compact attractor

In this section,we first prove the existence of the concentrate-compact attractor,then we prove Theorem 1.1.

4.1 Concentration-compactness attractor

5 Proof of theorem 1.1

Step 1.Combining Corollary 4.1 with Lemma B.7 in Tao[33],we have for anytn→∞,up to a subsequence there existJ1,J2,...,JMandwm∈Jm(GK)such that

wherexm,n∈Rdand they satisfies the separation property:

Step 2.By the linear energy decoupling property,we have

Then by the local well-posedness theory,there existsT＞0 such that the solutionWjto(1.1)with initial data(wj,vj)is well-posed on[0,T].By the perturbation theorem and the separation ofxm,n,we obtain

Therefore,Wjis an equilibrium and the same holds forwj.Thus we have proved there exist a finite number of equilibrium points{Qm}such that for any sequencetn→∞,there exists{xm,n}for which

By contradiction,we obtain our theorem.

6 Acknowledgement

We thank Prof.Yvan Martel for recommending our work to announce in Centre de Math′ematiques Laurent Schwartz.We appreciate the referee’s helpful comments which have deeply and intensively improved the present work.