Stability of Blow-Up Profile and Lower Bounds for Blow-Up Rate for the Critical Generalized KdV Equation

The generalized Korteweg-de Vries equations are a class of Hamiltonian systems in infinite dimension derived from the KdV equation where the quadratic term is replaced by a higher order power term.These equations have two conservation laws in the energy space H 1 (L 2 norm and energy).We consider in this paper the critical generalized KdV equation, which corresponds to the smallest power of the nonlinearity such that the two conservation laws do not imply a bound in H 1 uniform in time for all H 1 solutions (and thus global existence).From [15], there do exist for this equation solutions u(t) such that |u(t)| H 1 → +∞ as t ↑ T , where T ≤ +∞ (we call them blow-up solutions).The question is to describe, in a qualitative way, how blow up occurs.For solutions with L 2 mass close to the minimal mass allowing blow up and with decay in L 2 at the right, we prove after rescaling and translation which leave invariant the L 2 norm that the solution converges to a universal profile locally in space at the blow-up time T .From the nature of this profile, we improve the standard lower bound on the blow-up rate for finite time blow-up solutions.