Abstract: In this paper we consider tile initial boundary value problems for the nonlinear Schrödinger equation
iu_t+△u+f(|u|²)u=0, t ≥ 0, x∈Ω, u(0,x)=u₀(x), x∈Ω
with boundary conditions u(t,x)|∂Ω=0 or ∂u/∂v|∂Ω=0 where Ω=Rⁿ\B(0,1) is the exterior domain of the unit ball B(0,1).We prove that the smooth,radially symmetric solutions of the problem exists globally and uniquely when|f(s)|≤ Cs^(p-1)/2 and 1 ≤ p< 5 and also prove that the solutions blow up when f(s)=λs^(p-1)/2(λ>0) and p ≥ 5 under appropriate conditions on no and obtain some properties for blow-up solutions.

Key words: nonlinear Schrsödinger equation, global solution, blow-up solution