Smoothing effect for nonlinear Schrödinger equations in exterior domains

Abstract

We consider the following nonlinear Schrödinger equations in exterior domains: i∂ tu + 1 2 Δu = ¦u¦ 2 u, (t,x)∈R x D , u(0, x) = π(x), x ϵ D , (∗) u(t, x) = 0 ( or ∂u t6v = 0), (t,x)∈R x ∂D , where D={ x ∈ R n ;| x|> R}, ∂D= R > 0, and v denotes the outward normal unit vector at x ϵ ∂D. In this paper we prove the radially symmetric solutions of (∗) have a smoothing property.