Small time periodic solutions of fully nonlinear telegraph equations in more spatial dimensions
type: Item
instance of: scholarly article
title: Small time periodic solutions of fully nonlinear telegraph equations in more spatial dimensions
zbMATH Open document ID: 0679.34038
DOI: 10.1016/S0294-1449(16)30323-7
published in: Annales de l'Institut Henri Poincaré. Analyse Non Linéaire
publication date: 1989-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/78175
review text: This paper is devoted to the study of the existence of a classical solution to the problem \((1)\quad Au+\partial_ tu+\kappa \partial^ 2_ tu+\sum^{n}_{r=1}\kappa_ r\partial_{x_ r}\partial_ tu+G(Ju)+f=0\) in \(\Omega\) \(\times {\mathbb{R}}\), (2) \(u(x,t)=0\) for \(x\in \partial \Omega\), \(t\in {\mathbb{R}}\), \((3)\quad u(x,t)=u(x,t+2\pi)\) for \(x\in \Omega\), \(t\in {\mathbb{R}}\), where \(\kappa\) and \(\kappa_ r\) are constants, A is an elliptic operator, J is the jet of all derivatives up to order 2 and f is a smooth and small function. The solution u to (1)-(3) is written in the form \(u(x,t)=\int^{t}_{0}v(x,\tau)d\tau +b(x),\) where, obviously, \(v=\partial_ tu\) and, by virtue of \(2\pi\)-periodicity of u in t, v satisfies \(\int^{2\pi}_{0}v(x,t)dt=0\). This device makes avoiding the occurence of the ``loss of derivatives'' possible. Hence, by using Schauder's theorem, the author is able to extend earlier results of Rabinowitz to more spatial variables and to suppress the explicit appearance of the small parameter, with, in addition, an improvement of the assumptions on the regularity of f and G.
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 35B10
Mathematics Subject Classification ID: 35L10
zbMATH DE Number: 4113055
zbMATH Keywords: telegraph equations
zbMATH Keywords: Schauder's theorem
author: Milan Stedry
reviewed by: Jean Mawhin
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4113055