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, 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.