On the Cauchy problem of fourth-order nonlinear Schrödinger equations
type: Item
instance of: scholarly article
title: On the Cauchy problem of fourth-order nonlinear Schrödinger equations
zbMATH Open document ID: 1115.35121
author: Ai Guo
author: Shangbin Cui
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2007-04-19T00:00:00Z
review text: The paper deals with the Cauchy problem for the fourth-order semilinear Schrödinger equation \[ \begin{cases} i\partial_tu+a\Delta u +b\Delta^2u=\pm u^p,& (x,t)\in \mathbb R^{n+1},\\ u(0,x)=u_0(x),& x\in \mathbb R^{n}, \end{cases} \] where \(p\geq 1\) is a positive integer, \(1\leq n\leq 4,\) \(a\) and \(b\) are real constants with \(b\neq0\) and the initial data \(u_0(x)\) belong to a suitable Besov space. It is proved that for any \(q\in [{{4(p^2-1)}\over{(4-n)p+4+n}},\infty],\) the above Cauchy problem is locally well-posed in \(\dot B^{s_p}_{2,q}\) and \(\dot B^s_{2,q}\) where \(s_p=n/2-4/(p-1)\) and \(s>s_p;\) and almost globally well-posed in these spaces for any \(q\in[1,\infty]\) and small initial data. Global well-posedness is obtained when \(a=0\) for small initial data.
reviewed by: Dian K. Palagachev
Mathematics Subject Classification ID: 35Q55
Mathematics Subject Classification ID: 35R25
Mathematics Subject Classification ID: 35B35
zbMATH DE Number: 5144913
zbMATH Keywords: fourth order semilinear Schrödinger equation
zbMATH Keywords: Cauchy problem
zbMATH Keywords: well-posedness
zbMATH Keywords: Besov space
MaRDI profile type: Publication
full work available at URL: j.na.2006.04.020
OpenAlex ID: W1979766020
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DOI: 10.1016/J.NA.2006.04.020
description: scientific article; zbMATH DE number 5144913