Decay on expanding spheres as \(t\to +\infty\) of the solutions of semilinear wave equations
type: Item
instance of: scholarly article
title: Decay on expanding spheres as \(t\to +\infty\) of the solutions of semilinear wave equations
zbMATH Open document ID: 0685.35018
DOI: 10.1007/BF01766159
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 1988-01-01T00:00:00Z
review text: The paper is devoted to asymptotic properties of finite energy solutions of semilinear wave equations \(u_{tt}-\Delta u+q(x,t)| u|^{p- 1}u=0,\) when \(x\in R^ n\), \(n\geq 3\), \(1\leq p<1+4/(n-2)\), and \(q\geq 0\). The solutions are studied in a modified forward lightcone defined by \(| x| \leq (1-\epsilon)(t-M),\) \(t\geq M\) for \(0<\epsilon <1\) and \(M>0.\) Extending results and methods by C. Morawetz and W. A. Strauss it is shown that, with suitable assumptions upon q, the total energy in the region \(| x| \leq (1-\epsilon)(t-M)\) is \(O(t^{-2})\) as \(t\to \infty\). Conditions on q are given which ensure the boundedness of the total energy. The results are combined to prove exponential decay in the modified lightcone under suitable assumptions upon q and for \(n\geq 3\) and odd.
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35L70
zbMATH DE Number: 4122432
zbMATH Keywords: exponential decay
zbMATH Keywords: bounded total energy
zbMATH Keywords: finite energy solutions
zbMATH Keywords: semilinear wave equations
author: G. Perla Menzala
reviewed by: S. Leif Svensson
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4122432