A Survey on the Sharp Blow-up Criteria for a Class of Nonlinear Schrödinger Equations

ZHANG Jian, ZHU Shihui

(College of Mathematics and Software Science,Sichuan Normal University,Chengdu 610066,Sichuan)

1 Introduction

In this paper,we consider the following Cauchy problem of nonlinear Schrödinger equations

In fact,equation(1)is proposed by Davey and Stewartson in 1974 and is also called Davey-Stewartson system(see[1]).In fluid mechanics,Davey-Stewartson system models the evolution of weakly nonlinear water wave having one predominant direction of travel,and the wave amplitude is modulated slowly in two horizontal directions.

When the singular integral operatorEis replaced by the harmonic potential,i.e.E=-|x|2,equation(1)is called the nonlinear Schrödinger equation with a harmonic potential.This equation models the remarkable Bose-Einstein condensate with attractive inter-particle interactions under a magnetic trap(see[2-4]).Physicists and mathematicians are very interested in studying dynamics of this equation(see[2,4-5]).Oh[6]established the local well-posedness in the corresponding energy fieldΣ={u∈H1(RN)||x|u∈L2(RN)}.Cazenave[5],Zhang[3,7],Shu and Zhang[8],Chen and Zhang[9],Carles[10]studied the existence of blow-up solutions and sharp thresholds of blow-up and global existence.Carles[10]also gave the transformation which reveals the relationship between the nonlinear Schrödinger equation with and without a harmonic potential.Recently,Merle and Raphaël[11-13]obtained a large body of breakthrough work for the super-critical mass blow-up solutions with the help of the Spectral Properties[12],such as sharp blow-up rates,profiles of blow-up solutions,etc.Then,using this transformation proposed by Carles[10],Zhang,Li and Wu[14-16]obtained some dynamical properties of blowup solutions such as sharp blow-up rates,L2-concentration and rate ofL2-concentration etc.Zhu,Zhang and Li[17]obtained the limiting profile of blow-up solutions in the natural energy fieldΣ={u∈H1(RN)||x|u∈L2(RN)}.

For the Cauchy problem(1)～(2),Ghidaglia and Saut[18],Guo and Wang[19]established the local well-posedness in the energy spaceH1(RN)forN=2 andN=3 respectively(see[5,20]for a review).Cipolatti[21],Zhang and Zhu[22]studied the existence of the standing waves.Cipolatti[23],Ohta[24-25],Gan and Zhang[26]investigated the stability and instability of standing waves.Ghidaglia and Saut[18],Guo and Wang[19]studied the existence of blow-up solutions,and Wang and Guo[27]further discussed the scattering of global solutions.Ozawa[28]constructed some exact blow-up solutions.Richards[29], Papanicolaou et al[30],Gan and Zhang[26,31], Shu and Zhang[32],Zhang and Zhu[22]studied the sharp conditions of blow-up and global existence for the Cauchy problem(1)～(2).Li et al[33],Richards[29]obtained the mass-concentration properties of the blow-up solutions inL2-critical case whenN=2.

In the present paper,we are focusing on the sharp criteria of blow-up and global existence for the Cauchy problem(1)～(2).First,whenN=2,3,by constructing a type of cross-constrained variational problem and establishing so-called cross-constraint manifolds of the evolution flow,a sharp threshold for blow-up and global existence of the solutions to the Cauchy problemis given.Secondly,forN=2,by using the profile decomposition of bounded sequences inH1,a precisely sharp criterion of the blow-up solutions for the Cauchy problem(1)～(2)with 3≤p＜+∞ is given.Thirdly,forN=3,by using the profile decomposition of bounded sequences inH1,a precisely sharp threshold of the blow-up solutions for the Cauchy problem(1)～(2)withis given.We should point out that most of the above results has been published.There are two main aims of this survey.One is to give a collection about the sharp criteria of blow-up and global existence for equation(1),and it is convenient for readers to refer.The other is to give main sketch to obtain the above sharp criteria,and it may be quite useful for students to handle this method.

2 Preliminaries

The functionalH(u(t))is well-defined according to the Sobolev embedding theorem and the properties of the singular operatorE.Ghidaglia and Saut[18],Guo and Wang[19]established the local well-posedness of the Cauchy problem(1)～(2)in energy spaceH1.

Proposition 2.1LetN∈{2,3}andu0∈H1.There exists a unique solutionu(t,x)of the Cauchy problem(1)～(2)on the maximal time[0,T)such thatu(t,x)∈C([0,T)；H1)and eitherT=+∞(global existence),orT＜+∞and=+∞ (blow-up).Furthermore,for allt∈[0,T),u(t,x)satisfies the following conservation laws.

(i)Conservation of mass‖u(t)‖2=‖u0‖2.

(ii)Conservation of energyH(u(t))=H(u0).

By some basic calculations,we have the following proposition(see Ohta[25]).

3 Cross-constrained Variational Methods and Sharp Criteria

4 Profile Decomposition and Sharp Criteria for N=2

5 Profile Decomposition and Sharp Criteria for N=3

First,using the profile decomposition of bounded sequence inH1,we compute the best constant of a generalized Gagliardo-Nirenberg inequality in dimension three.More precisely,we have the following theorems.

(i)If‖∇u0‖2＜y0,then the solutionu(t,x)of the Cauchy problem(1)～(2)exists globally.Moreover,for all timet,u(t,x)satisfies

(ii)If‖∇u0‖2＞y0and|x|u0∈L2,then the solutionu(t,x)of the Cauchy problem(1)～(2)blows up in finite timeT＜+∞,wherey0is the unique positive solution of the equationg(y)=0 andg(y)is defined in(27).

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