Blow up for the critical gKdV equation II: minimal mass dynamics

TL;DR

This paper analyzes the near soliton dynamics for the mass critical gKdV equation, classifying blow-up behavior, proving stability of certain regimes, and describing the universal minimal mass blow-up solution and its attractor properties.

Contribution

It provides a complete classification of near soliton dynamics, including existence, uniqueness, and behavior of minimal mass blow-up solutions for the critical gKdV equation.

Findings

Classified three main regimes: blow-up, soliton, exit.

Proved stability of blow-up and exit regimes.

Described the universal minimal mass blow-up solution and its attractor property.

Abstract

We fully revisit the near soliton dynamics for the mass critical (gKdV) equation. In Part I, for a class of initial data close to the soliton, we prove that only three scenario can occur: (BLOW UP) the solution blows up in finite time $T$ in a universal regime with speed $1/(T-t)$; (SOLITON) the solution is global and converges to a soliton in large time; (EXIT) the solution leaves any small neighborhood of the modulated family of solitons in the scale invariant $L^2$ norm. Regimes (BLOW UP) and (EXIT) are proved to be stable. We also show in this class that any nonpositive energy initial data (except solitons) yields finite time blow up, thus obtaining the classification of the solitary wave at zero energy. In Part II, we classify minimal mass blow up by proving existence and uniqueness (up to invariances of the equation) of a minimal mass blow up solution $S(t)$. We also completely describe the blow up behavior of $S(t)$. Second, we prove that $S(t)$ is the universal attractor in the (EXIT) case, i.e. any solution as above in the (EXIT) case is close to $S$ (up to invariances) in $L^2$ at the exit time. In particular, assuming scattering for $S(t)$ (in large positive time), we obtain that any solution in the (EXIT) scenario also scatters, thus achieving the description of the near soliton dynamics.