Scattering of solutions of nonlinear Klein-Gordon equations in three space dimensions
type: Item
instance of: scholarly article
title: Scattering of solutions of nonlinear Klein-Gordon equations in three space dimensions
zbMATH Open document ID: 0554.35095
author: Masayoshi Tsutsumi
published in: Journal of the Mathematical Society of Japan
publication date: 1983-01-01T00:00:00Z
review text: This paper is devoted to the theory of scattering for the non-linear Klein-Gordon equation \[ (*)\quad u_{tt}-\Delta u+m^ 2u+h(u)=0 \] in space dimension \(n=3\), with \(h\in {\mathcal C}^ 1\) and h satisfying a power estimate \(| h'(u)| \leq C| u|^{p-1}\) with \(3<p<5\). The only regularity assumption on the solutions (actually on the initial data) is that they have finite energy. The main results are: (1) the existence of dispersive solutions at -\(\infty\), in the sense that for any finite energy solution \(u_-\) of the free equation \(u_{tt}-\Delta u+m^ 2u=0\), there exists a global solution u of (*) such that \(u(t)- u_-(t)\to 0\) in the energy norm as \(t\to -\infty\). This implies the existence of the wave operator \(\Omega_-\) defined in the whole energy space. (Similar results hold of course at \(+\infty)\); (2) the fact that for small initial data, all solutions thereby obtained are dispersive both at \(+\infty\) and -\(\infty\), namely asymptotic completeness for small initial data. This paper complements earlier work of \textit{W. Strauss} [J. Funct. Anal. 41, 110-133 (1981; Zbl 0466.47006) and ibid. 43, 281-293 (1981; Zbl 0494.35068)] where the same results are proved (for any dimension \(n\geq 2)\) in the range 4/n\(\leq p-1\leq 4/(n-1)\) (i.e. 4/3\(\leq p- 1\leq 2\) for \(n=3)\). The method consists in solving the integral equation \[ u_ s(t)=u_-(t)+\sum^{t}_{s}d\tau E(t-\tau)h(u_ s(\tau)) \] where \(E(t)=\omega^{-1}\sin \omega t\), \(\omega =(-\Delta +m^ 2)^{1/2}\), and taking the limit of \(u_ s\) as \(s\to -\infty\) by a compactness method. The basic tools are (1) space time integrability properties of solutions of the free equation proved by Strichartz, and (2) Besov space estimates of E(t) due to Brenner and Pecher. Similar results, actually for \(2\leq n\leq 5\), have been obtained independently in [\textit{H. Pecher}, Math. Z. 185, 261-270 (1984; Zbl 0538.35063)]. They have been extended to arbitrary dimension in [\textit{H. Pecher}, Low energy…
Mathematics Subject Classification ID: 35P25
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35L15
Mathematics Subject Classification ID: 47A40
Mathematics Subject Classification ID: 35Q99
zbMATH DE Number: 3882887
zbMATH Keywords: scattering
zbMATH Keywords: non-linear Klein-Gordon equation
zbMATH Keywords: finite energy
zbMATH Keywords: existence of dispersive solutions
zbMATH Keywords: wave operator
zbMATH Keywords: space time integrability properties
zbMATH Keywords: Besov space estimates
reviewed by: Jean Ginibre
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/03530521
OpenAlex ID: W1997548832
DOI: 10.2969/JMSJ/03530521
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description: scientific article; zbMATH DE number 3882887