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

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 $0<\epsilon \ll 1$ . 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.

38 pages

References in corpus (1)

• Self-similar solutions to super-critical gKdV

Cited by in corpus (5)

• Numerical study of the transverse stability of line solitons of the Zakharov-Kuznetsov equations
• Numerical study of blow-up and stability of line solitons for the Novikov-Veselov equation
• On asymptotic dynamics for $L^2$ critical generalized KdV equations with a saturated perturbation
• Nonlinear Profile Decomposition and the Concentration Phenomenon for Supercritical Generalized KdV Equations
• Blow-up solutions for $L^2$-supercritical gKdV equations with exactly $k$ blow-up points