Global attractor for weakly damped gKdV equations in higher sobolev spaces

Abstract

Long time behavior of solutions for weakly damped gKdV equations on the real line is studied. With some weak regularity assumptions on the force $f$, we prove the existence of global attractor in $H^s$ for any $s\geq 1$. The asymptotic compactness of solution semigroup is shown by Ball's energy method and Goubet's high-low frequency decomposition if $s$ is an integer and not an integer, respectively.