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

Abstract

We establish new results of global existence and scattering for the nonlinear Schrödinger equation $ i \partial_{t}v+\Delta v+\lambda h(t)|v|^\alpha v = 0 $ on $ {\mathbb{R}}^N $ with oscillating data. In particular, we give a relation between the range of the allowed values of $ \alpha $ 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 $ \alpha $ for the scattering expands. We also consider the related nonlinear Schrödinger equation with linear time-dependent damping $ i \partial_{t}u+\Delta u+\frac{1}{2}i a(t) u+\lambda|u|^\alpha u = 0 $ with $ \lim_{t\to\infty} ta(t) = \gamma\geq0. $ We show that for $ \alpha>{2}/({N+\gamma}) $ scattering, in some sense, holds while for $ \alpha<{2}/({N+\gamma}) $ and $ \lambda $ is a non-zero real number, solutions do not scatter. We prove that the critical value $ \alpha = {2}/({N+\gamma}) $ is unchanged through an $ L^1 $-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.

Locations

• Communications on Pure &amp Applied Analysis - View - PDF