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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 byestablishing global well-posedness in $H^1(\mathbb{R}^3)\timesL^2(\mathbb{R}^3)$ for a broad class of initial data. We then concentrate onthe initial value problem in $n=4$, which is the energy-critical dimension forthe corresponding cubic nonlinear Schr\"odinger equation. We start by provinglocal well-posedness in $H^1(\mathbb{R}^4)\times H^1(\mathbb{R}^4)$. Then, forthe focusing case of the system, we derive a virial type identity and use it toprove that for radially symmetric smooth initial data with negative energy,there is a positive time $T_0$, depending only on the data, for which, eitherthe $H^1(\mathbb{R}^4)\times H^1(\mathbb{R}^4)$ 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 globalwell-posedness result for regular initial data which is small in the$H^1(\mathbb{R}^4)\times H^1(\mathbb{R}^4)$ norm.

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