Stability of blow-up profile and lower bounds for blow-up rate for the critical generalized KdV equation

Details Stability of blow-up profile and lower bounds for blow-up rate for the critical generalized KdV equation Stability of blow-up profile and lower bounds for blow-up rate for the critical generalized KdV equation

2004-05-12

arxiv.org/abs/math/0405229v1

Ann. of Math. (2), Vol. 155 (2002), no. 1, 235--280

Mathematics

Analysis of PDEs

46 pages, published version

Scientific paper

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 {\it 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} \to +\infty as T\uparrow T, where T\le +\infty (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 {\it 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.

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