Continuum of finite point blowup rates for the critical generalized Korteweg-de Vries equation
Yvan Martel, Didier Pilod
TL;DR
This work proves the existence of finite-point blowup solutions for the mass-critical gKdV with a continuum of blowup rates in the range ν ∈ (3/7, 1/2). The authors construct a two-component blowup: a concentrating soliton bubble encoding the rate Γ and a fixed background residue r_α that induces a controlled forcing, linked by a refined blowup profile W_b and a modulation analysis. Central to the approach are the two-scale interactions between the shrinking bubble and the residue, a detailed construction of the background residue Θ(t,x) with μ=t^ν and ρ=t^β, and a backward-in-time bootstrap around an advanced approximate solution, culminating in the capture of a blowup residue r_α ∈ H^1 when α>1. The results extend the landscape of exotic finite-time blowups beyond previously known isolated rates (e.g., ν=2/5) and pose open problems on stability, full rate spectra, and broader residue constructions, with potential implications for other critical dispersive equations.
Abstract
For any $ν\in(\frac 37,\frac12)$, 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 $\|\partial_x u (t,x)\|_{L^2} \approx t^{-ν}$. Such a blowup rate is associated to a blowup residue of the form $r_α(x)= x^{α-\frac 12}$ for $x>0$ close to the blowup point, where $α=\frac{3ν-1}{2-4ν}$. The condition $ν\in(\frac37,\frac12)$ is equivalent to $α>1$, which corresponds to the full range for which the residue $r_α$ 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 $ν=\frac 25$. Finally, we present some open problems regarding the blowup phenomenon for the mass critical gKdV equation.