Global existence and blow-up phenomena for the periodic Hunter-Saxton equation parametrized by the speed \(\kappa \) of the Galilean frame
type: Item
instance of: scholarly article
title: Global existence and blow-up phenomena for the periodic Hunter-Saxton equation parametrized by the speed \(\kappa \) of the Galilean frame
zbMATH Open document ID: 1232.35134
author: Xuemei Wei
author: Zhaoyang Yin
published in: Nonlinear Analysis. Real World Applications
publication date: 2011-08-19T00:00:00Z
review text: \textit{J. K. Hunter} and \textit{R. Saxton} [SIAM J. Appl. Math. 51, No. 6, 1498--1521 (1991; Zbl 0761.35063)] proposed a nonlinear wave equation showing that weakly nonlinear waves are described by the equation \[ (u_t+ uu_x)_x= {1\over 2} u^2_x, \] where \(u(t,x)\) describes the director field of a nematic liquid crystal, \(x\) is a space variable in a reference frame moving with the linearized wave velocity, and \(t\) is a slow time variable. In this paper, the periodic Hunter-Saxton equation parametrized by the speed \(\kappa\) of the Galilean frame is under consideration. Firstly, the local existence and the blow-up scenario of strong solutions to the equation are obtained. After that, two blow-up results are established and the blow-up rates of strong solutions to the equation are presented. Finally, it is proved that the equation also has global solutions.
reviewed by: Elena Gavrilova
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 76A15
Mathematics Subject Classification ID: 35B44
Mathematics Subject Classification ID: 35D35
zbMATH DE Number: 5940420
zbMATH Keywords: global existence blow-up phenomena
zbMATH Keywords: periodic Hunter-Saxton
zbMATH Keywords: Galilean frame
MaRDI profile type: Publication
full work available at URL: j.nonrwa.2010.10.022
OpenAlex ID: W1964544385
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DOI: 10.1016/J.NONRWA.2010.10.022
description: scientific article; zbMATH DE number 5940420