A decay estimate for a class of hyperbolic pseudo-differential equations
type: Item
instance of: scholarly article
title: A decay estimate for a class of hyperbolic pseudo-differential equations
zbMATH Open document ID: 1009.35092
author: Sandra Lucente
author: Guido Ziliotti
published in: Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Serie IX. Rendiconti Lincei. Matematica e Applicazioni
publication date: 2002-10-31T00:00:00Z
review text: Summary: We consider the equation \(u_t- i\Lambda u= 0\), where \(\Lambda= \lambda(D_x)\) is a first-order pseudodifferential operator with real symbol \(\lambda(\xi)\). Under a suitable convexity assumption on \(\lambda\) we find the decay properties for \(u(t,x)\). These can be applied to the linear Maxwell system in anisotropic media and to the nonlinear Cauchy problem \(u_t- i\Lambda u= f(u)\), \(u(0,x)= g(x)\). If \(f(u)\) is a smooth function which satisfies \(f(u)\simeq|u|^p\) near \(u= 0\), and \(g\) is small in suitably Sobolev norm, we prove global existence theorems provided \(p\) is greater than a critical exponent.
Mathematics Subject Classification ID: 35S10
Mathematics Subject Classification ID: 35L60
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 1822110
zbMATH Keywords: decay estimate
zbMATH Keywords: nonlinear hyperbolic equation
zbMATH Keywords: small data
zbMATH Keywords: Maxwell system in anisotropic media
zbMATH Keywords: nonlinear Cauchy problem
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1822110