Global attractor for the damped BBM equation in the sharp low regularity space

The long-term behavior of low regularity solutions to the damped BBM equation with a distribution force on the torus is studied. Since the energy equation fails to hold for the low regularity solutions, the existence of a bounded absorbing set is not a trivial. This difficulty is overcome by splitting the solution into five parts, where some parts decay exponentially in gradually higher regularity spaces, the final remainder belongs the energy space and thus enjoys the dissipative effect. In this way, the existence of a global attractor is proved in the sharp low regularity space. Moreover, the attractor is shown to have a finite fractal dimension based on the quasi-stable estimate method.