Full range of infinite point blow-up exponents for the critical generalized KdV equation

TL;DR

This paper constructs solutions to the critical generalized KdV equation that blow up at a range of rates, extending previous results to include all rates greater than 1/2, and introduces a modified energy functional for analysis.

Contribution

It proves the existence of finite-time blow-up solutions with a continuous range of rates for the critical gKdV, extending prior work restricted to higher blow-up exponents.

Findings

Solutions blow up with rates (T-t)^{- u} for (rac{1}{2}, 1)

Constructs solutions close to solitons concentrating at + in space

Introduces a modified virial-energy functional for broader (rac{1}{2}, 1)

Abstract

For the quintic, mass critical generalized Korteweg-de Vries equation, for any $\nu \in (\frac{1}{2}, 1)$, we prove the existence of solutions in the energy space that blow up in finite time $T>0$ with the blow-up rate $\|\partial_x u(t)\|_{L^2} \sim (T-t)^{-\nu}$ (infinite point blow-up). These solutions are constructed arbitrarily close to the family of solitons and correspond to the concentration of a soliton traveling at $+\infty$ in space as $t\uparrow T$. This complements the previous results obtained in the work of Martel, Merle, Rapha\"el in 2015 on infinite point exotic blow-up, which were valid under the technical restriction $\nu>\frac {11}{13}$. The value $\nu=\frac 12$ corresponds to a critical case to be treated elsewhere. At the technical level, we implement a modification of the virial-energy functional, to allow all $\nu > \frac 12$ and simplify the proof of energy estimates.