4阶非线性薛定谔方程解的渐近行为（英文）
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摘要
本文综述了具有幂次非线性或耗散形式的非线性4阶薛定谔方程小振幅解的全局存在性和渐近行为的一些最新研究结果.考虑了散射问题的超临界非线性和临界非线性情况.
We survey our recent results on global existence and asymptotic behavior in time of small amplitude solutions to fourth-order Schr?dinger equations with power nonlinearities or nonlinearities of divergence form. Not only the super critical nonlinearities but also the critical ones from the view point of scattering problem are considered.
We survey our recent results on global existence and asymptotic behavior in time of small amplitude solutions to fourth-order Schr?dinger equations with power nonlinearities or nonlinearities of divergence form. Not only the super critical nonlinearities but also the critical ones from the view point of scattering problem are considered.
引文
[1]Aoki,K.,Hayashi,N.and Naumkin,P.I.,Global existence of small solutions for the fourth-order nonlinear Schr?dinger equation,NoDEA Nonlinear Differential Equations Appl.,2016,23(6):Article 65.
[2]Ben-Artzi,M.,Koch,H.and Saut,J.C.,Dispersion estimates for fourth order Schr?dinger equations,C.R.Acad.Sci.Paris Sér.I Math.,2000,330(2):87-92.
[3]Doi,S.,On the Cauchy problem for Schr?dinger type equations and the regularity of the solutions,J.Math.Kyoto Univ.,1994,34(2):319-328.
[4]Dysthe,K.B.,Note on a modification to the nonlinear Schr?dinger equation for application to deep water waves,Proc.Roy.Soc.A,1979,369(1736):105-114.
[5]Fukumoto,Y.and Moffatt,H.K.,Motion and expansion of a viscous vortex ring.Part 1.A higher-order asymptotic formula for the velocity,J.Fluid Mech.,2000,417:1-45.
[6]Hao,C.C.,Hsiao,L.and Wang,B.X.,Well-posedness of Cauchy problem for the fourth order nonlinear Schr?dinger equations in multi-dimensional spaces,J.Math.Anal.Appl.,2007,328(1):58-83.
[7]Hayashi,N.,Mendez-Navarro,J.A.and Naumkin,P.I.,Scattering of solutions to the fourth-order nonlinear Schr?dinger equation,Commun.Contemp.Math.,2016,18:Article 1550035,24 pages.
[8]Hayashi,N.,Mendez-Navarro,J.A.and Naumkin,P.I.,Asymptotics for the fourth-order nonlinear Schr?dinger equation in the critical case,J.Differential Equations,2016,261(9):5144-5179.
[9]Hayashi,N.and Naumkin,P.I.,Asymptotic properties of solutions to dispersive equation of Schr?dinger type,J.Math.Soc.Japan,2008,60(3):631-652.
[10]Hayashi,N.and Naumkin,P.I.,Large time asymptotics for the fourth-order nonlinear Schr?dinger equation,J.Differential Equations,2015,258(3):880-905.
[11]Hayashi,N.and Naumkin,P.I.,Global existence and asymptotic behavior of solutions to the fourth-order nonlinear Schr?dinger equation in the critical case,Nonlinear Anal.,2015,116:112-131.
[12]Hayashi,N.and Naumkin,P.I.,Factorization technique for the fourth-order nonlinear Schr?dinger equation,Z.Angew.Math.Phys.,2015,66(5):2343-2377.
[13]Hayashi,N.and Naumkin,P.I.,On the inhomogeneous fourth-order nonlinear Schr?dinger equation,J.Math.Phys.,2015,56(9):093502,25 pages.
[14]Huo,Z.H.and Jia,Y.L.,Well-posedness for the fourth-order nonlinear derivative Schr?dinger equation in higher dimension,J.Math.Pures Appl.(9),2011,96(2):190-206.
[15]Karpman,V.I.,Stabilization of soliton instabilities by higher-order dispersion:fourth order nonlinear Schr?dinger-type equations,Phys.Rev.E,1996,53(2):R1336-R1339.
[16]Karpman,V.I.and Shagalov,A.G.,Stability of soliton described by nonlinear Schr?dinger-type equations with higher-order dispersion,Phys.D,2000,144(1/2):194-210.
[17]Miao,C.X.,Xu,G.X.and Zhao,L.F.,Global well-posedness and scattering for the defocusing energy-critical nonlinear Schr?dinger equations of fourth order in dimensions d≥9,J.Differential Equations,2011,251(12):3381-3402.
[18]Pausader,B.,The focusing energy-critical fourth-order Schr?dinger equation with radial data,Discrete Contin.Dyn.Syst.,2009,24(4):1275-1292.
[19]Pausader,B.and Shao,S.L.,The mass-critical fourth-order Schr?dinger equation in high dimensions,J.Hyperbolic Differential Equations,2010,7(4):651-705.
[20]Pausader,B.and Xia,S.X.,Scattering theory for the fourth-order Schr?dinger equation in low dimensions,Nonlinearity,2013,26(8):2175-2191.
[21]Segata,J.,Well-posedness and existence of standing waves for the fourth order nonlinear Schr?dinger type equation,Discrete Contin.Dyn.Syst.,2010,27(3):1093-1105.
[22]Segata,J.and Shimomura,A.,Asymptotics of solutions to the fourth order Schr?dinger type equation with a dissipative nonlinearity,J.Math.Kyoto Univ.,2006,46(2):439-456.
[23]Shimomura,A.,Asymptotic behavior of solutions for Schr?dinger equations with dissipative nonlinearities,Comm.Partial Differential Equations,2006,31(9):1407-1423.
[2]Ben-Artzi,M.,Koch,H.and Saut,J.C.,Dispersion estimates for fourth order Schr?dinger equations,C.R.Acad.Sci.Paris Sér.I Math.,2000,330(2):87-92.
[3]Doi,S.,On the Cauchy problem for Schr?dinger type equations and the regularity of the solutions,J.Math.Kyoto Univ.,1994,34(2):319-328.
[4]Dysthe,K.B.,Note on a modification to the nonlinear Schr?dinger equation for application to deep water waves,Proc.Roy.Soc.A,1979,369(1736):105-114.
[5]Fukumoto,Y.and Moffatt,H.K.,Motion and expansion of a viscous vortex ring.Part 1.A higher-order asymptotic formula for the velocity,J.Fluid Mech.,2000,417:1-45.
[6]Hao,C.C.,Hsiao,L.and Wang,B.X.,Well-posedness of Cauchy problem for the fourth order nonlinear Schr?dinger equations in multi-dimensional spaces,J.Math.Anal.Appl.,2007,328(1):58-83.
[7]Hayashi,N.,Mendez-Navarro,J.A.and Naumkin,P.I.,Scattering of solutions to the fourth-order nonlinear Schr?dinger equation,Commun.Contemp.Math.,2016,18:Article 1550035,24 pages.
[8]Hayashi,N.,Mendez-Navarro,J.A.and Naumkin,P.I.,Asymptotics for the fourth-order nonlinear Schr?dinger equation in the critical case,J.Differential Equations,2016,261(9):5144-5179.
[9]Hayashi,N.and Naumkin,P.I.,Asymptotic properties of solutions to dispersive equation of Schr?dinger type,J.Math.Soc.Japan,2008,60(3):631-652.
[10]Hayashi,N.and Naumkin,P.I.,Large time asymptotics for the fourth-order nonlinear Schr?dinger equation,J.Differential Equations,2015,258(3):880-905.
[11]Hayashi,N.and Naumkin,P.I.,Global existence and asymptotic behavior of solutions to the fourth-order nonlinear Schr?dinger equation in the critical case,Nonlinear Anal.,2015,116:112-131.
[12]Hayashi,N.and Naumkin,P.I.,Factorization technique for the fourth-order nonlinear Schr?dinger equation,Z.Angew.Math.Phys.,2015,66(5):2343-2377.
[13]Hayashi,N.and Naumkin,P.I.,On the inhomogeneous fourth-order nonlinear Schr?dinger equation,J.Math.Phys.,2015,56(9):093502,25 pages.
[14]Huo,Z.H.and Jia,Y.L.,Well-posedness for the fourth-order nonlinear derivative Schr?dinger equation in higher dimension,J.Math.Pures Appl.(9),2011,96(2):190-206.
[15]Karpman,V.I.,Stabilization of soliton instabilities by higher-order dispersion:fourth order nonlinear Schr?dinger-type equations,Phys.Rev.E,1996,53(2):R1336-R1339.
[16]Karpman,V.I.and Shagalov,A.G.,Stability of soliton described by nonlinear Schr?dinger-type equations with higher-order dispersion,Phys.D,2000,144(1/2):194-210.
[17]Miao,C.X.,Xu,G.X.and Zhao,L.F.,Global well-posedness and scattering for the defocusing energy-critical nonlinear Schr?dinger equations of fourth order in dimensions d≥9,J.Differential Equations,2011,251(12):3381-3402.
[18]Pausader,B.,The focusing energy-critical fourth-order Schr?dinger equation with radial data,Discrete Contin.Dyn.Syst.,2009,24(4):1275-1292.
[19]Pausader,B.and Shao,S.L.,The mass-critical fourth-order Schr?dinger equation in high dimensions,J.Hyperbolic Differential Equations,2010,7(4):651-705.
[20]Pausader,B.and Xia,S.X.,Scattering theory for the fourth-order Schr?dinger equation in low dimensions,Nonlinearity,2013,26(8):2175-2191.
[21]Segata,J.,Well-posedness and existence of standing waves for the fourth order nonlinear Schr?dinger type equation,Discrete Contin.Dyn.Syst.,2010,27(3):1093-1105.
[22]Segata,J.and Shimomura,A.,Asymptotics of solutions to the fourth order Schr?dinger type equation with a dissipative nonlinearity,J.Math.Kyoto Univ.,2006,46(2):439-456.
[23]Shimomura,A.,Asymptotic behavior of solutions for Schr?dinger equations with dissipative nonlinearities,Comm.Partial Differential Equations,2006,31(9):1407-1423.