Long time dynamics of non-radial solutions to inhomogeneous nonlinear Schrödinger equations

Abstract

We study long time dynamics of non-radial solutions to the focusing inhomogeneous nonlinear Schrödinger equation. By using the concentration/compactness and rigidity method, we establish a scattering criterion for non-radial solutions to the equation. We also prove a non-radial blow-up criterion for the equation whose proof makes use of localized virial estimates. As a byproduct of these criteria, we study long time dynamics of non-radial solutions to the equation with data lying below, at, and above the ground state threshold. In addition, we provide a new argument showing the existence of finite time blow-up solution to the equation with cylindrically symmetric data. The ideas developed in this paper are robust and can be applicable to other types of nonlinear Schrödinger equations.

35 pages, to appear in SIAM Journal of Mathematical Analysis