Asymptotic smoothing and attractors for the generalized Benjamin-Bona-Mahony equation on R3

We study the asymptotic behavior of the solutions of the Benjamin-Bona-Mahony equation defined on R3. We first provide a sufficient condition to verify the asymptotic compactness of an evolution equation defined in an unbounded domain, which involves the Littlewood-Paley projection operators. We then prove the existence of an attractor for the Benjamin-Bona-Mahony equation in the phase space H1(R3) by showing the solutions are point dissipative and asymptotic compact. Finally, we establish the regularity of the attractor and show that the attractor is bounded in H2(R3).