On the Unboundedness of Higher Regularity Sobolev Norms of Solutions for the Critical Schr\"odinger-Debye System with Vanishing Relaxation Delay

Abstract

We consider the Schr\"odinger-Debye system in $\mathbb{R}^n$, for $n=3,4$. Developing on previously known local well-posedness results, we start by establishing global well-posedness in $H^1(\mathbb{R}^3)\times L^2(\mathbb{R}^3)$ 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 Schr\"odinger equation. We start by proving local well-posedness in $H^1(\mathbb{R}^4)\times H^1(\mathbb{R}^4)$. 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 $H^1(\mathbb{R}^4)\times H^1(\mathbb{R}^4)$ 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 $H^1(\mathbb{R}^4)\times H^1(\mathbb{R}^4)$ norm.