Global attractors for the Benjamin-Bona-Mahony equation with memory

We consider the nonlinear integrodifferential Benjamin-Bona-Mahony equation ut − utxx + ux − Z0∞ g(s)uxx(t − s)ds + uux = f, where the dissipation is entirely contributed by the memory term. Under a suitable smallness assumption on the external force f, we show that the related solution semigroup possesses the global attractor in the natural weak energy space. The result is obtained by means of a nonstandard approach based on the construction of a suitable family of attractors on certain invariant sets of the phase space.