Global existence for quasi-linear dissipative wave equations with large data and small parameter
type: Item
instance of: scholarly article
title: Global existence for quasi-linear dissipative wave equations with large data and small parameter
zbMATH Open document ID: 0648.35051
DOI: 10.1007/BF01163297
author: Albert Milani
published in: Mathematische Zeitschrift
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173927
review text: We consider the hyperbolic quasi-linear initial value problem for \[ u=u(x,t),\quad v=v(x,t)\in {\mathbb{R}}^ 1,\quad x\in {\mathbb{R}},\quad t\in {\mathbb{R}}^+,\quad \epsilon >0, \] \[ \epsilon u_ t+u-\delta (v)_ x=0;\quad v_ t-u_ x=0, \] \[ u(x,0)=u_ 0(x);\quad v(x,0)=v_ 0(x), \] where \(\delta\) : \({\mathbb{R}}\to {\mathbb{R}}\) is smooth and satisfies \(\delta '(0)>0\), \(\delta\) ''(0)\(\neq 0\). We show that the smooth solution of such problem exists globally in time, provided that the \(L^{\infty}\) norm of \(u_ 0\), \(v_ 0\) and the parameter \(\epsilon\) be sufficiently small. In particular, the \(L^{\infty}\) norm of \(u_{0x}\) and \(v_{0x}\) is not required to be small, but the smallness of \(\epsilon\) will depend on the size of such norm.
Mathematics Subject Classification ID: 35L60
Mathematics Subject Classification ID: 35L45
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 35B30
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 4058065
zbMATH Keywords: global existence
zbMATH Keywords: quasi-linear initial value problem
zbMATH Keywords: smooth solution
zbMATH Keywords: parameter
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4058065