Asymptotic behavior for dissipative Korteweg-de Vrie equations

Abstract

We study the large time behavior of solutions to the dissipative Korteweg-de Vrie equations $u_t+u_{xxx}+|D{|}^{\alpha }u+u{u}_x=0$ with $0<\alpha <2$. We find $v$ such that $u-v$ decays like $t^{-r(\alpha )}$ as $t\to \infty$ in various Sobolev norm.