On the global Cauchy problem for some nonlinear Schrödinger equations (Q1061315)

The authors study the Cauchy problem for a class of nonlinear Schrödinger equations in space time dimension \(n+1\). They look for solutions which belong to the class \(C({\mathbb{R}},H^ k({\mathbb{R}}^ n))\), \(k>n/2\). Under some suitable assumptions on the nonlinearity the authors prove the existence of a (global) solution for \(n\leq 7\). The global existence proof breaks down for \(n\geq 8\).