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