Regular attractor for damped KdV‐Burgers equations on R

We study the well‐posedness and dynamic behavior for the KdV‐Burgers equation with a force f ( x ) ∈ L p ( R ) ⋂ L q ( R ) ( 1 < p ⩽ 2 , 2 < q < + ∞ ) on R . We establish L p − L q estimates of the evolution e − t ( ∂ x 2 − ∂ x 3 ) , as an application we obtain the local well‐posedness. Then the global well‐posedness follows from a uniform estimate for solutions as t goes to infinity. Next, we prove the asymptotical regularity of solutions in space H 9 2 − 1 p − ( R ) and H 7 2 + 1 2 q − , q ( R ) by the smoothing effect of e − t ( ∂ x 2 − ∂ x 3 ) . The regularity and the asymptotical compactness in L 2 yields the asymptotical compactness in H 7 2 − 1 p⋂ H 3 , q ( R ) by an interpolation arguement. Finally, we conclude the existence of an ( L 2 ( R ) , H 7 2 − 1 p( R ) ⋂ H 3 , q ( R ) ) globalattractor.