Classical solutions of nonlinear Schrödinger equations (Q1078757)

The author considers the initial value problem for the nonlinear Schrödinger equation \[ iu_ t-\Delta u+q(| u|^ 2)u=0, \] in \((t,x)\in {\mathbb{R}}\times {\mathbb{R}}^ n\), for q(s) a smooth real-valued function on \({\mathbb{R}}\) having the growth order \[ | q(s)| \leq C(1+| s|^{(p-1)/2}),\quad 1<p<(n+2)/(n-2). \] He proves, for \(6\leq n\leq 11\), the existence of global classical solutions u, with some additional conditions imposed on q and its derivatives. The method is to estimate the time derivatives of time-local solutions in suitable Sobolev spaces.