Exponential stabilization of periodic solutions of a system of KdV equations

Abstract

We consider a coupled nonlinear dispersive system of Korteweg–de Vries type in the presence of a dissipative mechanism. First we prove that the Cauchy problem is globally well posed in a suitable periodic Sobolev space and our main result says that the $L^2$ and $L^{\infty }$ norms of the solutions decay exponentially fast as $t\to +\infty$.