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

This paper discusses a class of inhomogeneous nonlinear Schrodinger equation {i partial derivative(t)u(t, x) = -Delta u(t, x) - V(x)vertical bar u(t, x)vertical bar(p-1)u(t, x), u(0, x) = u(0)(x), where (t, x) epsilon R x R-2, V(x) satisfies some assumptions. By a constrained variational problem, we firstly define some cross-constrained invariant sets for the inhomogeneous nonlinear Schrodinger 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 H-r(1)(R-2) with the H-1-norm of the initial data u(0) which is dominated by the minimal value of the constrained variational problem. (C) 2007 Elsevier Inc. All rights reserved.