Asymptotic stability for monotone traveling kinks of the dissipative Boussinesq problem

We study the dissipative Boussinesq problem, which extends the "good" Boussinesq equation by incorporating viscosity effects.

It is well-known that this model supports monotone decreasing traveling kink solutions.

We show that these kinks are orbitally stable in $H^1(\rone)$.

Moreover, they are asymptotically stable in $\dot{H}^1(\rone)$ and $L^p(\rone), 2<p\leq \infty$.