The cubic fourth-order Schrodinger equation

被引：137
作者：
Pausader, Benoit [1 ]
机构：
[1] Univ Cergy Pontoise, Dept Math, CNRS, UMR 8088, F-95302 Cergy Pontoise, France
来源：
JOURNAL OF FUNCTIONAL ANALYSIS | 2009年 / 256卷 / 08期
关键词：
Fourth-order dispersive equation; Scattering; Energy-critical equation; GLOBAL WELL-POSEDNESS; CAUCHY-PROBLEM; SCATTERING; INSTABILITY; REGULARITY;
D O I：
10.1016/j.jfa.2008.11.009
中图分类号：
O1 [数学];
学科分类号：
0701 ; 070101 ;
摘要：
Fourth-order Schrodinger equations have been introduced by Karpman and Shagalov to take into account the role of small fourth-order dispersion terms in the propagation of intense laser beams in a bulk medium with Kerr nonlinearity. In this paper we investigate the cubic defocusing fourth-order Schrodinger equation i partial derivative(t)u + Delta(2)u + vertical bar u vertical bar(2)u = 0 in arbitrary space dimension R-n for arbitrary initial data. We prove that the equation is globally well-posed when n <= 8 and ill-posed when n >= 9, with the additional important information that scattering holds true when 5 n <= 8. (C) 2008 Elsevier Inc. All rights reserved.
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收藏
页码：2473 / 2517
页数：45
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