The nonlinear Schrodinger equation with combined power-type nonlinearities

doi:10.1080/03605300701588805

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	The nonlinear Schrodinger equation with combined power-type nonlinearities
	Tao, Terence ; Visan, Monica ; Zhang, Xiaoyi
	2007
发表期刊	COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS
ISSN	0360-5302
卷号	32 期号:7-9 页码:1281-1343
摘要	We undertake a comprehensive study of the nonlinear Schrodinger equation iu(t) + Delta u = lambda(1)vertical bar u vertical bar(p1)u + lambda(2 vertical bar)u vertical bar(p2)u, where u(t, x) is a complex-valued function in spacetime IRt x IRx", lambda(1), and lambda(2) are nonzero real constants, and 0 < p(1) < p(2) <= 4/n-2 We address questions related to local and global well-posedness, finite time blowup, and asymptotic behaviour. Scattering is considered both in the energy space H-1(IRn) and in the pseudoconformal space Sigma : = {f is an element of H-1 (IRn); xf is an element of L-2 (IRn)}. Of particular interest is the case when both nonlinearities are defocusing and correspond to the L-x(2)-critical, respectively H-x(1)-critical NLS, that is, lambda(1),lambda(2) > 0 and p(1) = 4/n, p(2) = 4/n-2. The results at the endpoint p(1) = 4/n are conditional on a conjectured global existence and spacetime estimate for the L-x(2)-critical nonlinear Schrodinger equation, which has been verified in dimensions n >= 2 for radial data in Tao et al. (to appear a,b) and Killip et al. (preprint). As an off-shoot of our analysis, we also obtain a new, simpler proof of scattering in H-x(1) for solutions to the nonlinear Schrodinger equation iu(t) + Delta u = vertical bar u vertical bar(p)u, with 4/n < p < 4/n-2 which was first obtained by Ginibre and Velo (1985).
关键词	energy-critical mass-critical nonlinear Schrodinger equation well-posedness
DOI	10.1080/03605300701588805
语种	英语
WOS研究方向	Mathematics
WOS类目	Mathematics, Applied ; Mathematics
WOS记录号	WOS:000250012900013
出版者	TAYLOR & FRANCIS INC
引用统计
文献类型	期刊论文
条目标识符	http://ir.amss.ac.cn/handle/2S8OKBNM/5247
专题	中国科学院数学与系统科学研究院
通讯作者	Visan, Monica
作者单位	1.Inst Adv Study, Sch Math, Princeton, NJ 08540 USA 2.Univ Calif Los Angeles, Dept Math, Los Angeles, CA 90024 USA 3.Chinese Acad Sci, Acad Math & Syst Sci, Beijing, Peoples R China
推荐引用方式 GB/T 7714	Tao, Terence,Visan, Monica,Zhang, Xiaoyi. The nonlinear Schrodinger equation with combined power-type nonlinearities[J]. COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS,2007,32(7-9):1281-1343.
APA	Tao, Terence,Visan, Monica,&Zhang, Xiaoyi.(2007).The nonlinear Schrodinger equation with combined power-type nonlinearities.COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS,32(7-9),1281-1343.
MLA	Tao, Terence,et al."The nonlinear Schrodinger equation with combined power-type nonlinearities".COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS 32.7-9(2007):1281-1343.