Continuum of finite point blowup rates for the critical generalized Korteweg-de Vries equation
Didier Pilod, Professor @ Department of Mathematics, 黑料吃瓜资源
Abstract
The generalized Korteweg-de Vries equation (gKdV) is obtained by changing the nonlinearity of the famous (quadratic) Korteweg-de Vries equation. We consider the quintic power nonlinearity,
for which the gKdV equation is critical for the L^2-norm.
It was established in the early 2000s that for initial data with sufficiently large mass, solutions can blow up in finite time through the concentration (or 鈥渂ubbling鈥) of a solitary wave. In all known constructions, the spatial location of this solitary wave goes to infinity as time approaches the blow-up time.
In the first part of this talk, I will review these classical blow-up results. In the second part, I will present new results obtained in collaboration with Yvan Martel, in which we construct solutions that blow up in finite time at a finite spatial point. These new constructions also yield blow-up rates that are closer to the self-similar regime. They are obtained through the interaction between a solitary wave and a cusp-shaped function.