Blow-up solutions for $L^2$-supercritical gKdV equations with exactly $k$ blow-up points

TL;DR

This paper investigates the formation of multiple blow-up points in slightly $L^2$-supercritical gKdV equations, extending previous work on single blow-up solutions and describing the singularity development near blow-up time.

Contribution

It proves the existence of solutions with exactly $k$ blow-up points and characterizes the singularity formation in these multi-blow-up scenarios.

Findings

Existence of solutions with multiple blow-up points.

Description of singularity formation near blow-up time.

Extension of stable blow-up dynamics to multi-point cases.

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 $0<\varepsilon\ll 1$ . In the previous work of the author we know that there exists an stable self-similar blow-up dynamics for slightly $L^2$-supercritical gKdV equations. Such solution can be viewed as solutions with single blow-up point. In this paper we will prove the existence of solutions with multiple blow-up points, and give a description of the formation of the singularity near the blow-up time.