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

We consider the nonlinear integrodifferential Benjamin-Bona-Mahony equation $$u_t-u_{txx}+u_x-{\int }_0^{\mathrm{\infty }}g(s)u_{xx}(t-s)\mathrm{d}s+u{u}_x=f$$ 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.