Existence and stability of the log-log blow-up dynamics for the L 2-critical nonlinear Schrödinger equation in a domain

Abstract

Let iut = -Δu - |u|4/N u be the L 2-critical nonlinear Schrödinger equation, in a domain Ω ⊂ ℝN with initial data in H10 (Ω) (Dirichlet boundary condition) and N ≤ 4. We prove existence and stability of finite time blow-up dynamics with the log-log blow-up speed |Δ u(t)|L2 ∼ √log|log(T-t)|/T-t. Moreover, for a suitable class of finite time blow-up solutions, we derive global rigidity properties which turn out to be modeled after the ℝN ones. © 2007 Birkhäuser Verlag Basel/Switzerland.