Damping to prevent the blow-up of the Korteweg-de Vries equation

Abstract

We study the behavior of the solution of a generalized damped KdV equation $u_t+u_x+u_{xxx}+u^p{u}_x+{\mathcal{L}}_{\gamma }(u)=0$. We first state results on the local well-posedness. Then when $p\ge 4$, conditions on ${\mathcal{L}}_{\gamma }$ are given to prevent the blow-up of the solution. Finally, we numerically build such sequences of damping.