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

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 $5<p<5+\epsilon$ and $0<\epsilon \ll 1$ . 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.

35 Pages. Minor revision