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arXiv · 1604.01128

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

Shuxia Tang·Jixun Chu·Peipei Shang·Jean-Michel Coron
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Abstract

Local asymptotic stability analysis is conducted for an initial-boundary-value problem of a Korteweg-de Vries equation posed on a finite interval $\left[0, 2\pi \sqrt{7/3}\right]$. The equation comes with a Dirichlet boundary condition at the left end-point and both of 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. By analyzing the Korteweg-de Vries equation restricted on the local center manifold, a polynomial decay rate of the solution is obtained.

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Shuxia Tang, Jixun Chu, Peipei Shang, Jean-Michel Coron. 2016-04-05. Asymptotic stability of a Korteweg-de Vries equation with a two-dimensional center manifold. https://arxiv.org/abs/1604.01128

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