Damping to prevent the blow-up of the Korteweg-de Vries equation (Q523856)

A generalized damped KdV equation \(u_t +u_x +u_{xxx} + u^pu_x + \mathcal{L}_\gamma(u) = 0\) is considered. Using the semigroup method and an energy method, the local well-posedness and the global well-posedness of the damped KdV equation are studied. It is shown that if \(p < 4\), the solution does not blow-up whereas if \(p \geq 4\), it can. Moreover, an abstract result is illustrated numerically.