Semilinear Schrödinger equation with time dependent coefficients

In this paper we study local and global well-posedness in L2 and H1 of the Cauchy problem for the following nonlinear Schr¨odinger equations iu_t + a(t)Δu = ±|u|^{γ−1}u, in the space R^{1+n}, with n ≥ 2. The coefficient a(t) is assumed to be positive, and possibly vanishing with finite order on a discrete set; we find the relationship between the critical powers γ for the well-posedness and the order of the zeroes of a.