We consider the model of Time Petri Nets where time is associated with transitions. Two semantics for time elapsing can be considered: the strong one, for which all transitions are urgent, and the weak one, for which time can elapse arbitrarily. It is well known that many verification problems such as the marking reachability are undecidable with the strong semantics. In this paper, we focus on Time Petri Nets with weak semantics equipped with three different memory policies for the firing of transitions. We prove that the reachability problem is decidable for the most common memory policy (intermediate) and becomes undecidable otherwise. Moreover, we study the relative expressiveness of these memory policies and obtain partial and surprising results.