Restriction theorems and semilinear Klein-Gordon equations in \((1+3)\)-dimensions
type: Item
instance of: scholarly article
title: Restriction theorems and semilinear Klein-Gordon equations in \((1+3)\)-dimensions
zbMATH Open document ID: 0865.35077
DOI: 10.1215/S0012-7094-96-08510-5
author: Hans Lindblad
author: Christopher D. Sogge
published in: Duke Mathematical Journal
publication date: 1996-12-19T00:00:00Z
review text: The authors consider the semilinear Klein-Gordon equation with right-hand side \(F_p(u)\), \[ |\partial^jF_p(u)|\leq C|u|^{p-j},\quad 0\leq j\leq [p], \qquad |F_p'(u)-F_p'(v)|\leq C|u-v|^{p-j}, \] if \(1<p<2\), and prove global and semiglobal existece results for the Cauchy problem. More precisely, in the case \(1\leq n\leq 3\), \(p>1+{2\over n}\) it is shown that for each compactly supported Cauchy data \((f,g)\in C^{1+[n/2]}\times C^{[n/2]}\cap H^{(n+2)/2}\times H^{n/2}\) one can find a global \(C^1\) solution if \(|f|_{H^{(n+2)/2}}+|g|_{H^{n/2}}\) is sufficiently small. A semiglobal existence result, i.e. \(u\in C^1([0,T_\varepsilon]\times\mathbb{R}^n)\), \(T_\varepsilon=\exp(c/\varepsilon^{p-1})\), \(c=\text{const}>0\), \(\varepsilon>0\) sufficiently small, is proved under the weaker condition \(p=1+{2/n}\).
Mathematics Subject Classification ID: 35L15
Mathematics Subject Classification ID: 35L70
zbMATH DE Number: 949892
zbMATH Keywords: semiglobal existence result
reviewed by: Peter R. Popivanov
MaRDI profile type: Publication
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full work available at URL: s0012-7094-96-08510-5
OpenAlex ID: W2045523850
description: scientific article; zbMATH DE number 949892