Stable self-similar blow-up dynamics for slightly L 2 -supercritical generalized KDV equations

In this paper we consider the slightly L2-supercritical gKdV equations ∂tu + (uxx + u|u|p−1)x = 0, with the nonlinearity 5 < p < 5 + ε and 0 < ε 1 . We will prove the existence and stability of a blow-up dynamics with self-similar blow-up rate in the energy space H1 and give a specific description of the formation of the singularity near the blow-up time.