Nonlinear superposition and absorption of delta waves in one space dimension
type: Item
instance of: scholarly article
title: Nonlinear superposition and absorption of delta waves in one space dimension
zbMATH Open document ID: 0661.35058
DOI: 10.1016/0022-1236(87)90063-2
author: Jeffrey Rauch
author: M. C. Reed
published in: Journal of Functional Analysis
publication date: 1987-01-01T00:00:00Z
review text: The author deals with problems of the following types: let \(u^{\epsilon}\) be the solution of the semilinear strictly hyperbolic system \[ (*)\quad (\partial_ t+A(x,t)\partial_ x+B(x,t))u=f(x,t,u) \] with the initial data of the form \(g+h^{\epsilon}\), where g is ``classical'' and smooth \(h^{\epsilon}\) converge to a distribution \(\mu\). Then \(u^{\epsilon}\) converge to \(\bar u+\sigma\) in specified sense, where \(\bar u\) is a solution of (*), \(\bar u(t=0)=g\), and \(\sigma\) is a solution of (*) with \(f=0\), \(\sigma (t=0)=\mu\). This fact expresses a superposition principle: the singular part of the solution propagates linearly, the classical part propagates by the nonlinear equation. The distribution \(\mu\) is a singular measure for f sublinear or can be more singular for f bounded. If f satisfies condition of superlinear dissipation \(\lim_{| u_ j| \to \infty} f_ j(x,t,u)/u_ j=- \infty\) and \(sgn(u_ j)f_ j(x,t,u)\leq c(l+\sum | u_ i|)\) (the solution do not blow up in finite time), then \(u^{\epsilon}\to \bar u,\) i.e. the singular part is absorbed.
Mathematics Subject Classification ID: 35L60
Mathematics Subject Classification ID: 35L45
Mathematics Subject Classification ID: 35B30
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 4080036
zbMATH Keywords: distribution initial data
zbMATH Keywords: semilinear strictly hyperbolic
zbMATH Keywords: classical
zbMATH Keywords: smooth
zbMATH Keywords: superposition principle
zbMATH Keywords: singular part
zbMATH Keywords: singular measure
zbMATH Keywords: superlinear dissipation
reviewed by: Alexandr Doktor
MaRDI profile type: Publication
full work available at URL: 0022-1236(87)90063-2
OpenAlex ID: W2061740908
cites work: Weak limits of solutions to semilinear hyperbolic systems
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cites work: Nonlinear microlocal analysis of semilinear hyperbolic systems in one space dimension
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cites work: Q3900552
description: scientific article; zbMATH DE number 4080036