Global existence and Scattering for nonlinear Schrödinger equations with time-dependent damping

We establish new results of global existence and scattering for the nonlinear Schrödinger equation i ∂ₜv+Δ v+λ h(t)|v|^α v = 0 on RN with oscillating data. In particular, we give a relation between the range of the allowed values of α for the scattering and the decay of the time-dependent coefficient $ h(t). $ We reveal that the more the potential h decreases at infinity, the more the range of the allowed values of α for the scattering expands. We also consider the related nonlinear Schrödinger equation with linear time-dependent damping i ∂ₜu+Δ u+1/2i a(t) u+λ|u|^α u = 0 with limt→∞ ta(t) = γ≥0. We show that for α>2/(N+γ) scattering, in some sense, holds while for α<2/(N+γ) and λ is a non-zero real number, solutions do not scatter. We prove that the critical value α = 2/(N+γ) is unchanged through an L¹-perturbation of $ a(t) $. In addition, we provide examples of $ a(t) $ where the critical value belongs to the scattering or to the non-scattering case.