The scattering theory for the nonlinear wave equation with small data. II
type: Item
instance of: scholarly article
title: The scattering theory for the nonlinear wave equation with small data. II
zbMATH Open document ID: 0662.35078
author: K. Mochizuki
author: Takahiro Motai
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1987-01-01T00:00:00Z
review text: [For part I see J. Math. Kyoto Univ. 25, 703-715 (1985; Zbl 0605.35069).] In Part I we studied a scattering problem for the nonlinear wave equation \[ (0.1)\quad \partial^ 2_ tw(t)=\Delta w(t)+f(w(t))=0\quad in\quad (x,t)\in {\mathbb{R}}^ n\times {\mathbb{R}}. \] Here \(n\geq 2\) and \(\Delta =\sum^{n}_{j=1}\partial^ 2_{x_ j}\). For purposes of illustration, let \(f(w)=\lambda | w|^{\rho -1}w\) (\(\lambda\in {\mathbb{R}}\), \(\rho >1)\). It was then proved that the scattering operator exists on a dense set of a neighborhood of 0 in the energy space if \(\rho\) is restricted to satisfy \((0.2)\quad \rho \geq 2\) and \[ (0.3)\quad \gamma (n)\equiv (n^ 2+3n-2+\sqrt{(n^ 2+3n-2)^ 2-8n(n-1)})/(2n(n- 1))<\rho \leq (n+3)/(n-1). \] The lower bound \(\gamma\) (n) of \(\rho\) is smaller than those obtained in \textit{W. A. Strauss} [J. Funct. Anal. 41, 110-133 (1981; Zbl 0466.47006)] and \textit{S. Klainerman} [Arch. Ration. Mech. Anal. 78, 73-98 (1982; Zbl 0502.35015)], and it is expected to give a critical power. On the other hand, (0.2) is too restrictive. For it contradicts (0.3) in case of high space dimension (n\(\geq 5)\). The main purpose of the present paper is to eliminate this restriction.
Mathematics Subject Classification ID: 35P25
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 35B35
Mathematics Subject Classification ID: 35B30
zbMATH DE Number: 4082125
zbMATH Keywords: small data
zbMATH Keywords: scattering
zbMATH Keywords: nonlinear wave equation
zbMATH Keywords: scattering operator
zbMATH Keywords: energy space
zbMATH Keywords: lower bound
zbMATH Keywords: critical power
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195176032
OpenAlex ID: W2029213432
cites work: On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case
cites work: Finite-time blow-up for solutions of nonlinear wave equations
cites work: Blow-up of solutions of nonlinear wave equations in three space dimensions
cites work: Long-time behavior of solutions to nonlinear evolution equations
cites work: The scattering theory for the nonlinear wave equation with small data
cites work: \(L^p\)-Abschätzungen und klassische Lösungen für nichtlineare Wellengleichungen. I
cites work: \(L^p\) estimates for the waves equation
cites work: Nonlinear scattering theory at low energy
cites work: Scattering for semilinear wave equations with small data in three space dimensions
DOI: 10.2977/PRIMS/1195176032
Recommended article: The scattering theory for the nonlinear wave equation with small data
Recommended article: Q3760980
Recommended article: Q3971448
Recommended article: Scattering theory for semilinear wave equations with small data in two space dimensions
Recommended article: Scattering Theory for Semilinear Wave Equations with Small Data in Two Space Dimensions
Recommended article: Scattering states for the nonlinear wave equation with small data
Recommended article: Nonlinear small data scattering for the wave and Klein-Gordon equation
Recommended article: On small data scattering for 2-dimensional semilinear wave equations
Recommended article: Q5699171
Recommended article: Q4518843
description: scientific article; zbMATH DE number 4082125