Modified scattering for the critical nonlinear Schrodinger equation

被引:37
作者
Cazenave, Thierry [1 ,2 ]
Naumkin, Ivan [3 ]
机构
[1] Univ Paris 06, F-75252 Paris 05, France
[2] CNRS, Lab Jacques Louis Lions, BC 187,4 Pl Jussieu, F-75252 Paris 05, France
[3] Univ Nice Sophia Antipolis, CNRS, Lab JA Dieudonne, UMR 7351, Parc Valrose, F-06108 Nice 02, France
关键词
Nonlinear Schrodinger equation; Pseudo-conformal transformation; Modified scattering; LONG-RANGE SCATTERING; LARGE TIME; DIMENSIONS; BEHAVIOR;
D O I
10.1016/j.jfa.2017.10.022
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
We consider the nonlinear Schrodinger equation iu(t) + Delta u = lambda broken vertical bar u broken vertical bar(2/N) u in all dimensions N >= 1, where lambda is an element of C and (sic)lambda <= 0. We construct a class of initial values for which the corresponding solution is global and decays as t -> infinity, like t(-N/2) if (sic)lambda = 0 and like (t log t)(-N/2) if (sic)lambda < 0. Moreover, we give an asymptotic expansion of those solutions as t -> infinity. We construct solutions that do not vanish, so as to avoid any issue related to the lack of regularity of the nonlinearity at u = 0. To study the asymptotic behavior, we apply the pseudo-conformal transformation and estimate the solutions by allowing a certain growth of the Sobolev norms which depends on the order of regularity through a cascade of exponents. (c) 2017 Elsevier Inc. All rights reserved.
引用
收藏
页码:402 / 432
页数:31
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