Description
Blow-up phenomenon for the focusing nonlinear Schrödinger equation have been intensively studied since the 90s, and several blow-up profiles are now well characterized. In the critical mass regime in R^d, there are blow-up profiles in the form of solitary waves concentrating at one or more points. In bounded domains, with homogeneous Dirichlet boundary conditions, the existence of these explosion profiles have been proved in dimensions 2 and 3.
In this talk, we will prove that the explosion of the latter profiles, in bounded domain, can be prevented with the help of localized open and closed-loop controls around the explosion points. Furthermore, the time of existence of these controlled solutions can be then extended to infinity. Finally, assuming that the control regions satisfy GCC, we show that these explosion profiles can be controlled to zero in small time.
