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)\(H1) --> + infinity as t up arrow T, where T less than or equal to + infinity (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.