This issuePrevious ArticleOn the existence of quasi periodic and almost periodic solutions of neutral functional differential equationsNext ArticleThe global solution of an initial boundary value problem for the damped Boussinesq equation
Existence of the global attractor for weakly damped, forced KdV equation on Sobolev spaces of negative index
In this paper, we treat the weakly damped, forced KdV equation on $\dot{H}^s$.
We are interested in the lower bound of $s$ to assure the existence
of the global attractor.
The KdV equation has infinite conservation laws, each of which is defined
in $H^j(j\in\mathbb Z, j\ge 0)$.
The existence of the global attractor is usually proved by using those
conservation laws.
Because the KdV equation on $\dot{H}^s$ has no conservation law for $s<0$,
it seems a natural question whether we can
show the existence of the global attractor for $s<0$.
Moreover, because the conservation laws restrict the behavior
of solutions,
the time global behavior of solutions for $s<0$
may be different from that for $s\ge 0$.
By using a modified energy,
we prove the existence of the global attractor for $s > -3/8$,
which is identical to the global attractor for $s \ge 0$.