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 - Funding Information:
Funding: The authors were supported by European Research Council advanced grant 266907 (CPDENL) of the 7th Research Framework Programme (FP7). In addition, J. Chu was supported by the National Natural Science Foundation of China (no. 11401021) and the Doctoral Program of Higher Education (no. 20130006120011). P. Shang was supported by the National Natural Science Foundation of China (no. 11301387) and the Doctoral Program of Higher Education (no. 20130072120008).
Publisher Copyright:
© 2018 Walter de Gruyter GmbH, Berlin/Boston.
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 -