Long-time behavior of solutions of a BBM equation with generalized damping

Abstract

We study the long-time behavior of the solution of a damped BBM equation $u_t+u_x-u_{xxt}+u{u}_x+{\mathcal{L}}_{\gamma }(u)=0$. The proposed dampings ${\mathcal{L}}_{\gamma }$ generalize standards ones, as parabolic (${\mathcal{L}}_{\gamma }(u)=-\Delta u$) or weak damping (${\mathcal{L}}_{\gamma }(u)=\gamma u$) and allows us to consider a greater range. After establish the local well-posedness in the energy space, we investigate some numerical properties.