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 $u_t + u_x - u_{xxt} + uu_x + \mathscr{L}_γ(u) = 0$. The proposed dampings $\mathscr{L}_γ$ generalize standards ones, as parabolic ($\mathscr{L}_γ(u)=-Δu$) or weak damping ($\mathscr{L}_γ(u)=γu$) and allows us to consider a greater range. After establish the local well-posedness in the energy space, we investigate some numerical properties.