Global attractor for Camassa-Holm type equations with dissipative term
type: Item
instance of: scholarly article
title: Global attractor for Camassa-Holm type equations with dissipative term
zbMATH Open document ID: 1092.35017
DOI: 10.1016/S0252-9602(17)30201-1
author: Ling'e Yang
author: Boling Guo
published in: Acta Mathematica Scientia. Series B. (English Edition)
publication date: 2006-02-21T00:00:00Z
review text: The global existence of classical solution and global attractor for Camassa-Holm type equations with dissipative term are established by using fixed-point theorem and a priori estimates. The authors consider the Camassa-Holm type equations with dissipative term \[ u_t-u_{xxt}+\delta u_{xxxx}+f(u)_x= \varepsilon ({2u_x u_{xx}+uu_{xxx} })+g(x),\quad x \in [0,1],\;t > 0.\tag{1} \] Under nonlinear boundary condition \[ u({0,t})= u_x ({1,t})= u_{xx} ({0,t})= 0,\quad t > 0,\tag{2} \] \[ \delta u_{xxx} ({1,t})-\varepsilon u({1,t})u_{xx} ({1,t})= \varphi ({u({1,t})}),\quad t > 0, \tag{3} \] and the initial value \[ u({x,0})= u_0 (x),\quad x \in [0,1], \tag{4} \] where \(\delta > 0,\) and \(\varepsilon \) are real numbers. When \(\delta > 0, \varepsilon= 1,\quad f(u)= \frac{3}{2}u^2,\) and \(g(x)= 0,\) equation (1) becomes the Camassa-Holm equation which was proposed as a model for one directional long waves of small-amplitude in shallow water. In this paper, the existence of global classical solution and global attractor for problem (1)--(4) is studied. In Section 2, the local existence of classical solution is proved. In Section 3, the authors establish the global existence of smooth solution and in Section 4, they prove the existence of global attractor.
reviewed by: Leonid B. Chubarov
Mathematics Subject Classification ID: 35B41
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 35A05
Mathematics Subject Classification ID: 35A07
Mathematics Subject Classification ID: 35Q35
zbMATH DE Number: 5008209
zbMATH Keywords: Camassa-Holm type equations
zbMATH Keywords: classical solution
zbMATH Keywords: global attractor
zbMATH Keywords: global existence
zbMATH Keywords: almost conservation law
zbMATH Keywords: fixed-point theorem
zbMATH Keywords: nonlinear boundary condition
zbMATH Keywords: long waves of small-amplitude
MaRDI profile type: Publication
full work available at URL: s0252-9602(17)30201-1
OpenAlex ID: W2783098589
description: scientific article; zbMATH DE number 5008209