张健 Dynamics of Nonlinear Schrodinger Equations with Potentials

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1、DynamicsofNonlinearSchrodingerEquations¨withPotentialsJianZhang(withGuanganChen,ZaihuiGan,XiaoguangLiandShihuiZhu)DepartmentofMathematics,SichuanNormalUniversityzhangjiancdv@sina.comx1Equationp1iut+4u+V(x;u)u+juju=0(1)Nu=u(t;x):[0;T)R!C;0

2、2BackgroundV(x;u)=~jxj2:Bose-EinsteinCondensateGross,E.P.(1963),Pitaevskii,L.P.(1961)2V(x;u)=E(juj2)=F1[1F[juj2]]:Davey-StewartsonSystem1jj2Davey,A.andStewartson,K.(1974)KnownStudiesforNLSwithV(x;u)=~jxj2:Wadati,M.andTsurumi,T.,1997;Oh,Y.G.1989;Cazenave,T

3、.,1989;Rabinowitz,P.H.,1992;Zhang,J.,1999;Fukuizumi,R.,2001;Ohta,M.andHirose,M.,2002;Carles,R.,2002;Chen,ShuandZhang,2002;Li,LiuandZhang,2006;ZhuandZhang,2008.KnownStudiesforNLSwithV(x;u)=E(juj2):1Ghidaglia,J.-M.andSaut,J.C.(1990):Localwell-posednessforN=2Guo

4、,B.L.andWang,B.X.(1999):Localwell-posednessforN=3Cipolatti,R.(1992,1993),Ohta,M.(1994,1995):ExistenceandstabilitiesofgroundstatePapanicolaou,G.C.,Sulem,C.andSulem,P-L.(1994),Ozawa,T.(1992):Existenceofblow-upsolutionsRichards,G.(2009):ConcentrationpropertiesforN=2,p=

5、3Yang,H.(2005,2012),ShenC.X.andGuoB.L.(2006):Globalwell-posednessinlowerregularspaceZhang,Gan,ShuandZhu(2006):Sharpthreshold,dynamicofblow-upsolutionsImposetheinitialdataNu(0;x)=u0;x2R:(2)Problems:Sharpcriteriafortheblow-upandglobalexistenceGlobaldynamic:stabilit

6、yofsolitons,asymptoticbehaviorBlow-updynamic:blow-uprate,concentration,limitingprofileArguments:Variationalmethods:Weinstein,1986;Tsutsumi,1990;Merle,1990;Zhang,1999Profiledecomposition:Gerard,1998,HmidiandKeraani,2005x2DynamicsofNLSwithV(x;u)=~jxj2LocalExis

7、tenceZ1N22:=fv2H(R)jjxjjvjdx<1gZ222222jjvjj:=jrvj+jvj+~jxjjvjdxZZZ212~2212+4E(v):=jrvjdx+jxjjvjdxjvjNdx2242+NLemma1(Oh,1989,JDE;Cazenave,1989,AMS)Letu02.)9!asolutionu(t;x)oftheCauchyproblem(1)–(2)inC([0;T);),eitherT=1(globalexistence)orelseT<1andlimjjujj=1(blo

8、w-up).Furthermore,forallt2[0;T),u(t;x)satisfiest!Tku(t)k2=ku0k2;E(u(t))=E(u0):4SharpCriteriaforp=1+NTheorem1(Zhang,2000,JSP)LetN1andu02.