Scattering theory for semilinear wave equations with small data in two space dimensions
type: Item
instance of: scholarly article
title: Scattering theory for semilinear wave equations with small data in two space dimensions
zbMATH Open document ID: 0767.35055
author: Kimitoshi Tsutaya
published in: Proceedings of the Japan Academy. Series A
publication date: 1993-06-29T00:00:00Z
review text: We consider a scattering problem for the semilinear wave equation \[ u_{tt}-\Delta u=F(u),\qquad (x,t)\in\mathbb{R}^ n\times\mathbb{R},\tag{1} \] where \(F(u)=\lambda| u|^ p\) or \(\lambda| u|^{p-1} u\), \(\lambda\in\mathbb{R}\), \(p>1\) and \(n=2\). We compare the asymptotic behavior as \(t\to\pm\infty\) of the solution of (1) with those of suitable solutions of the Cauchy problem for the free wave equation \(u_{tt}-\Delta u=0\), \((x,t)\in\mathbb{R}^ n\times\mathbb{R}\), in the sense of energy norm. The aim of this paper is to prove that if \(p>(3+\sqrt{17})/2\), the scattering operator for (1) exists for smooth and small data in two space dimensions.
Mathematics Subject Classification ID: 35P25
Mathematics Subject Classification ID: 35L05
zbMATH DE Number: 221221
zbMATH Keywords: existence of scattering operator
zbMATH Keywords: asymptotic behavior
zbMATH Keywords: Cauchy problem
zbMATH Keywords: free wave equation
MaRDI profile type: Publication
full work available at URL: pjaa.68.227
OpenAlex ID: W2063707086
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DOI: 10.3792/PJAA.68.227
description: scientific article; zbMATH DE number 221221