Continuum of finite point blowup rates for the critical generalized Korteweg-de Vries equation

TL;DR

This paper constructs solutions to the mass critical gKdV equation that blow up at a finite point with a continuum of rates, revealing new blowup behaviors beyond previously known solutions.

Contribution

It demonstrates the existence of finite-time blowup solutions with a range of rates characterized by a parameter, expanding understanding of blowup phenomena in critical gKdV.

Findings

Existence of solutions blowing up with rate t^{- u} for u in (3/7, 1/2)

Blowup residue functions belong to H^1 for the full range of u

Contrast with previously known blowup solutions, including a special case at u=2/5

Abstract

For any $\nu\in(\frac 37,\frac12)$, we prove the existence of an $H^1$ solution $u$ of the mass critical generalized Korteweg-de Vries equation on the time interval $(0,T_0]$, for some $T_0>0$, which blows up at the time $t=0$ and at the point $x=0$ with the rate $\|\partial_x u (t,x)\|_{L^2} \approx t^{-\nu}$. Such a blowup rate is associated to a blowup residue of the form $r_\alpha(x)= x^{\alpha -\frac 12}$ for $x>0$ close to the blowup point, where $\alpha=\frac{3\nu-1}{2-4\nu}$. The condition $\nu\in(\frac37,\frac12)$ is equivalent to $\alpha>1$, which corresponds to the full range for which the residue $r_\alpha$ belongs to $H^1$. Such blowup at a finite point is in contrast with all the blowup solutions constructed for this equation, except the one constructed previously by the authors corresponding to the special value $\nu=\frac 25$. Finally, we present some open problems regarding the blowup phenomenon for the mass critical gKdV equation.