On the unboundedness of higher regularity Sobolev norms of solutions for the critical Schrodinger-Debye system with vanishing relaxation delay

摘要

We consider the Schrodinger-Debye system in R-n, for n = 3,4. Developing on previously known local well-posedness results, we start by establishing global well-posedness in H-1(R-3) x L-2(R-3) for a broad class of initial data. We then concentrate on the initial value problem in n = 4, which is the energy-critical dimension for the corresponding cubic nonlinear Schrodinger equation. We start by proving local well-posedness in H-1(R-4) x H-1(R-4). Then, for the focusing case of the system, we derive a virial type identity and use it to prove that for radially symmetric smooth initial data with negative energy, there is a positive time T-0, depending only on the data, for which, either the H-1(R-4) x H-1(R-4) solutions blow-up in [0, T-0], or the higher regularity Sobolev norms are unbounded on the intervals [0, T], for T > T-0, as the delay parameter vanishes. We finish by presenting a global well-posedness result for regular initial data which is small in the H-1(R-4) x H-1(R-4) norm.