ABSTRACT This work considers the well‐posedness issues of some inhomogeneous nonlinear equations of Schrödinger type. The goal is to develop a local theory in the critical regime and a global theory for small data. Indeed, in both cases: A critical inhomogeneous local source term and a critical inhomogeneous non‐local source term, we obtain the global existence of a unique solution in , for small data. Here, is the index of the invariant Sobolev norm under standard dilatation. To overcome the difficulty coming from the singular inhomogeneous term, we use Lorentz Sobolev spaces which satisfy coupled with fractional chain rules. The main novelty here is to avoid the mass conservation law.
Tarek Saanouni (Fri,) studied this question.