Remarks on blowing-up of solutions for some nonlinear Schrödinger equations
type: Item
instance of: scholarly article
title: Remarks on blowing-up of solutions for some nonlinear Schrödinger equations
zbMATH Open document ID: 0731.35015
author: Kazuhiro Kurata
author: Takayoshi Ogawa
published in: Tokyo Journal of Mathematics
publication date: 1990-01-01T00:00:00Z
review text: The paper is concerned with blowing up properties of nonlinear Schrödinger equations. Two types of nonlinearities are studied: the local interaction \(f(| u|^ 2)u\) and the convolution \((V*| u|^ 2)u\). For the special cases \(f(t^ 2)=t^{p-1}+t^{q-1}\) and \(V(x)=| x|^{-\lambda}+| x|^{-\mu}\), sufficient conditions are given which ensure blow-up of the solution of the Cauchy problem for suitable initial data. For certain exterior-domain problems with Dirichlet or Neumann boundary conditions, similar results are proved for \(f(t^ 2)=t^{p-1}\). Finally, it is shown by variational arguments that the Cauchy problem with \(V(x)=| x|^{-2}\) has a global solution for small initial data.
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35Q55
Mathematics Subject Classification ID: 35G10
Mathematics Subject Classification ID: 35K25
zbMATH DE Number: 4208622
zbMATH Keywords: convolution nonlinearity
zbMATH Keywords: blowing up
zbMATH Keywords: global solution
zbMATH Keywords: small initial data
reviewed by: Michael Plum
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1270132270
OpenAlex ID: W2076282385
DOI: 10.3836/TJM/1270132270
description: scientific article; zbMATH DE number 4208622