TY - JOUR
T1 - Asymptotic stability of a Korteweg-de Vries equation with a two-dimensional center manifold
AU - Tang, Shuxia
AU - Chu, Jixun
AU - Shang, Peipei
AU - Coron, Jean Michel
N1 - Publisher Copyright:
© 2018 Walter de Gruyter GmbH, Berlin/Boston.
Copyright:
Copyright 2018 Elsevier B.V., All rights reserved.
PY - 2018/11/1
Y1 - 2018/11/1
N2 - Local asymptotic stability analysis is conducted for an initial-boundary-value problem of a Korteweg-de Vries equation posed on a finite interval [0, 2π √7/3]. The equation comes with a Dirichlet boundary condition at the left end-point and both the Dirichlet and Neumann homogeneous boundary conditions at the right end-point. It is known that the associated linearized equation around the origin is not asymptotically stable. In this paper, the nonlinear Korteweg-de Vries equation is proved to be locally asymptotically stable around the origin through the center manifold method. In particular, the existence of a two-dimensional local center manifold is presented, which is locally exponentially attractive. Analyzing the Korteweg-de Vries equation restricted on the local center manifold, we obtain a polynomial decay rate of the solution.
AB - Local asymptotic stability analysis is conducted for an initial-boundary-value problem of a Korteweg-de Vries equation posed on a finite interval [0, 2π √7/3]. The equation comes with a Dirichlet boundary condition at the left end-point and both the Dirichlet and Neumann homogeneous boundary conditions at the right end-point. It is known that the associated linearized equation around the origin is not asymptotically stable. In this paper, the nonlinear Korteweg-de Vries equation is proved to be locally asymptotically stable around the origin through the center manifold method. In particular, the existence of a two-dimensional local center manifold is presented, which is locally exponentially attractive. Analyzing the Korteweg-de Vries equation restricted on the local center manifold, we obtain a polynomial decay rate of the solution.
KW - Korteweg-de Vries equation
KW - asymptotic stability
KW - center manifold
KW - nonlinearity
KW - polynomial decay rate
UR - http://www.scopus.com/inward/record.url?scp=85054391817&partnerID=8YFLogxK
U2 - 10.1515/anona-2016-0097
DO - 10.1515/anona-2016-0097
M3 - Article
AN - SCOPUS:85054391817
VL - 7
SP - 497
EP - 515
JO - Advances in Nonlinear Analysis
JF - Advances in Nonlinear Analysis
SN - 2191-9496
IS - 4
ER -