Scattering for semilinear wave equations with small data in three space dimensions
type: Item
instance of: scholarly article
title: Scattering for semilinear wave equations with small data in three space dimensions
zbMATH Open document ID: 0627.35064
DOI: 10.1007/BF01163296
author: Hartmut Pecher
published in: Mathematische Zeitschrift
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173926
review text: Solutions of a class of semilinear wave equations in three space dimensions with power-type nonlinearity are shown to have the same asymptotic behaviour as suitable solutions of the corresponding linear problem as \(t\to \pm \infty\). The main assumptions are that the Cauchy data are small and smooth enough and that the nonlinearity vanishes of order \(p>1+\sqrt{2}\) at the origin. Thus the scattering operator is shown to exist in a dense set in the neighbourhood of the origin in energy space. The result is sharp in the sense that blow-up occurs in the case of the nonlinearity \(| u| ^ p\) if \(p<1+\sqrt{2}\) as was shown by F. John.
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35L15
zbMATH DE Number: 4019610
zbMATH Keywords: semilinear wave equations
zbMATH Keywords: power-type nonlinearity
zbMATH Keywords: asymptotic behaviour
zbMATH Keywords: Cauchy data
zbMATH Keywords: scattering oprator
zbMATH Keywords: energy space
zbMATH Keywords: blow-up
MaRDI profile type: Publication
cites work: Q3482143
cites work: Blow-up of solutions of nonlinear wave equations in three space dimensions
cites work: Weighted L∞ and L1 estimates for solutions to the classical wave equation in three space dimensions
cites work: The scattering theory for the nonlinear wave equation with small data
cites work: The scattering theory for the nonlinear wave equation with small data. II
cites work: Nonlinear scattering theory at low energy
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description: scientific article; zbMATH DE number 4019610