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

Abstract

Abstract Local asymptotic stability analysis is conducted for an initial-boundary-value problem of a Korteweg–de Vries equation posed on a finite interval <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>[</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi>π</m:mi> <m:mo>⁢</m:mo> <m:msqrt> <m:mrow> <m:mn>7</m:mn> <m:mo>/</m:mo> <m:mn>3</m:mn> </m:mrow> </m:msqrt> </m:mrow> <m:mo>]</m:mo> </m:mrow> </m:math> {[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.