Global existence and blow up of solutions for the inhomogeneous nonlinear Schrödinger equation in R2

This paper discusses a class of inhomogeneous nonlinear Schrödinger equation i∂t u(t, x)=− u(t, x)− V (x) u(t, x) p−1 u(t, x), u(0, x) = u0(x), where (t, x) ∈ R×R2, V (x) satisfies some assumptions. By a constrained variational problem, we firstly define some cross-constrained invariant sets for the inhomogeneous nonlinear Schrödinger equation, then we obtain some sharp conditions for global existence and blow up of solutions. As a consequence it is shown that the solution is globally well-posed in H1 r (R2) with the H1-norm of the initial data u0 which is dominated by the minimal value of the constrained variational problem. © 2007 Elsevier Inc. All rights reserved.