Existence and stability of the log-log blow-up dynamics for the L2-critical nonlinear Schrodinger equation in a domain

Let iu(t) = -Delta u - |u| 4/N u be the L-2-critical nonlinear Schrodinger equation, in a domain Omega subset of R-N with initial data in H-0(1)(Omega) (Dirichlet boundary condition) and N <= 4. We prove existence and stability of finite time blow-up dynamics with the log-log blow-up speed vertical bar del u(t)vertical bar(L2) similar to root log vertical bar log(T- t)vertical bar/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.