On the semilinear Schrödinger equation with time dependent coefficients

We consider nonlinear Schrödinger equation with time dependent coefficients. Fanelli 5] found a transformation between solutions of the original equation and of the usual Schrödinger equation with power nonlinearity involving time dependent coefficients in some Lorentz spaces. In this paper we extend the results in 5] in space‐time integrability properties of solutions. Particularly, we prove that the existence and uniqueness of solutions can be described exclusively in terms of Lebesgue spaces (not Lorentz spaces as in 5]) as far as the space integrability of solutions. We also discuss the equation with coefficient of an explicit homogeneous function and describe the associated Strichartz estimate and contraction mapping argument.