Stable self-similar blow-up dynamics for slightly $L^2$-supercritical generalized KdV equations

TL;DR

This paper proves the existence and stability of self-similar blow-up solutions for slightly supercritical generalized KdV equations, providing detailed descriptions of singularity formation in the energy space.

Contribution

It introduces a new stable self-similar blow-up dynamic for the slightly supercritical gKdV equations, expanding understanding of singularity formation.

Findings

Existence of stable self-similar blow-up solutions

Detailed description of singularity formation near blow-up time

Extension of blow-up analysis to slightly supercritical regime

Abstract

In this paper we consider the slightly $L^2$-supercritical gKdV equations $\partial_t u+(u_{xx}+u|u|^{p-1})_x=0$, with the nonlinearity $5<p<5+\varepsilon$ and $0<\varepsilon\ll 1$ . We will prove the existence and stability of a blow-up dynamic with self-similar blow-up rate in the energy space $H^1$ and give a specific description of the formation of the singularity near the blow-up time.