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

Abstract

We consider the Schrödinger–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(R3) × L^2(R3) 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 Schrödinger equation. We start by proving local well-posedness in H1^(R4) × H1^(R4). 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 T0, depending only on the data, for which, either the H1^(R4) × H1^(R4) solutions blow-up in [0, T0], or the higher regularity Sobolev norms are unbounded on the intervals [0, T ], for T > T0, 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(R4) × H^1(R4) norm.