Asymptotically self-similar global solutions of the nonlinear Schrödinger and heat equations
type: Item
instance of: scholarly article
title: Asymptotically self-similar global solutions of the nonlinear Schrödinger and heat equations
zbMATH Open document ID: 0916.35109
DOI: 10.1007/PL00004606
author: Thierry Cazenave
author: Fred B. Weissler
published in: Mathematische Zeitschrift
publication date: 1999-07-19T00:00:00Z
review text: The authors study global solution, including self-similar solutions of the initial value problem for nonlinear Schrödinger equations. The methods are also applied to the nonlinear heat equation. The proof of the existence of the global solutions is obtained by using norms analogous to those used by Cannone and Planchon for the Navier-Stokes systems with Besov norms. It is analyzed the action of the linear Schrödinger group on homogeneous functions. That allows to construct self-similar solutions and also a class of classical \(H^1\) solutions that are asymptotically self-similar as \(t\to\infty\). Also, the authors obtain solutions on \((-\infty,0)\) which have an asymptotically self-similar blow up behavior at \(t=0\).
Mathematics Subject Classification ID: 35Q55
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35A05
zbMATH DE Number: 1185894
zbMATH Keywords: global solution
zbMATH Keywords: self-similar solutions
zbMATH Keywords: nonlinear Schrödinger equations
zbMATH Keywords: nonlinear heat equation
reviewed by: L. Vázquez
MaRDI profile type: Publication
full work available at URL: pl00004606
OpenAlex ID: W2090662918
description: scientific article; zbMATH DE number 1185894