Asymptotic stability of a Korteweg–de Vries equation with a two-dimensional center manifold

Local asymptotic stability analysis is conducted for an initial-boundary-value problem of a Korteweg–de Vries equation posed on a finite interval [0,2⁢π⁢7/3]{[0,2\pi\sqrt{7/3}]}. The equation comes with a Dirichlet boundary condition at the left end-point and both the Dirichlet and Neumann homogeneo...

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Bibliographic Details
Main Authors: Tang Shuxia, Chu Jixun, Shang Peipei, Coron Jean-Michel
Format: Article
Language:English
Published: De Gruyter 2018-11-01
Series:Advances in Nonlinear Analysis
Subjects:
Online Access:https://doi.org/10.1515/anona-2016-0097
Description
Summary:Local asymptotic stability analysis is conducted for an initial-boundary-value problem of a Korteweg–de Vries equation posed on a finite interval [0,2⁢π⁢7/3]{[0,2\pi\sqrt{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.
ISSN:2191-9496
2191-950X