Slowly travelling infinite point blow-up for the critical generalized KdV equation

Abstract

We study the finite time blow up phenomenon for the quintic, mass critical gKdV equation. We prove the existence of a class of solutions $U$ with infinite point, finite time blow up behavior, at the particular blow up rate $$\|\partial_x U(t)\|_{L^2} \sim (T-t)^{-ν}\quad \text{as} \quad t \uparrow T,$$ where $ν= \frac 12$, $T$ is the blow up time and where the travel speed of the blow up bubble is logarithmic. Therefore, we call this behaviour slowly travelling infinite point blow up. The special blow up rate $ν=\frac 12$ is a threshold which separates finite and infinite point bubbling. In a previous work arXiv:2511.13538, the author constructed other infinite point blow up solutions for the continuum of rates $ν\in(\frac 12,1)$, using polynomial tails in the space variable and extending the results in arXiv:1209.2510, restricted to $ν> \frac{11}{13}$. However, that work suggested a change of the tail for the threshold case. In the present paper, we consider an exponentially decaying tail on the right in space. As in arXiv:2511.13538, the initial data can be taken arbitrarily close to the ground state in $H^1$. From a technical perspective, in addition to the change of tail, we have to adapt the energy-virial functional to the presence of the exponential tail, by modifying a scaling term used to control the right-hand side of the blow up solution.