Global singularity structure of weak solutions to 3-D semilinear dispersive wave equations with discontinuous initial data.
type: Item
instance of: scholarly article
title: Global singularity structure of weak solutions to 3-D semilinear dispersive wave equations with discontinuous initial data.
zbMATH Open document ID: 1036.35136
DOI: 10.1016/S0022-0396(03)00106-2
author: Ingo Witt
author: Huicheng Yin
published in: Journal of Differential Equations
publication date: 2004-01-26T00:00:00Z
review text: The commutator technique introduced in \textit{J.-M. Bony's} paper [Taniguchi Symposium, Katata, 11--49 (1984; Zbl 0669.35073)] and Strichartz' inequalities are used to describe the singularity set for the solutions of the three dimensional Cauchy problem for the semilinear wave equation \(\partial_{t}^{2}{u}-\Delta_{x}u+g(u)=0\) with discontinuous Cauchy data \((0, u_{1}(x))\), \(u_{1}\in{C^{\infty}(\overline{B(0,1)})}\) and \(u_{1}(x)=0\) for \(|{x}|>1\). It is assumed that \(g(0)=0\), \(\int^{u}_{0}{g(s)\,ds}\geq0\), \(|{g^{(j)}(u)}|\leq{C_{j}(1+|{u}|)^{p-j}}\), \(1<p\leq5\). One proves that the weak solution is smooth outside \(\Sigma_{1}\cup\Sigma_{2}\), where \(\Sigma_{1,2}=\{(t,x)\); \(t\geq{0}\), \((t\pm1)^{2}=|{x}|^{2}\}\).
reviewed by: Mihai Pascu
Mathematics Subject Classification ID: 35L67
Mathematics Subject Classification ID: 35A21
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 35L15
zbMATH DE Number: 2033004
zbMATH Keywords: discontinuous Cauch data
zbMATH Keywords: Strichartz' inequality
zbMATH Keywords: commutator technique
MaRDI profile type: Publication
cites work: Decay estimates for the critical semilinear wave equation
cites work: Self-spreading and strength of singularities for solutions to semilinear wave equations
cites work: Q3963412
cites work: Q3822600
cites work: Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation
cites work: The global Cauchy problem for the nonlinear Klein-Gordon equation
cites work: Regularity and asymptotic behaviour of the wave equation with a critical nonlinearity
cites work: Das Anfangswertproblem im Großen für eine Klasse nichtlinearer Wellengleichungen
cites work: Interaction of nonlinear progressing waves for semilinear wave equations
cites work: Q4317280
full work available at URL: s0022-0396(03)00106-2
OpenAlex ID: W2000888432
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description: scientific article; zbMATH DE number 2033004