Smoothing effect for nonlinear Schrödinger equations in exterior domains
We consider the following nonlinear Schrödinger equations in exterior domains: \[ (*)\quad i\partial_ tu+\Delta u=| u|^ 2u,\quad (t,x)\in {\mathbb{R}}\times D, \] \[ u(0,x)=\phi (x),\quad x\in D,\quad u(t,x)=0\quad (or\quad \partial u/\partial \nu =0),\quad (t,x)\in {\mathbb{R}}\times \partial D, \] where \(D=\{x\in {\mathbb{R}}^ n\); \(| x| >R\}\), \(\partial D=\{x\in {\mathbb{R}}^ n\); \(| x| =R\}\), \(R>0\), and \(\nu\) denotes the outward normal unit vector at \(x\in \partial D\). In this paper we prove that the radially symmetric solutions of (*) have a smoothing property.
scientific article; zbMATH DE number 4159301
scientific article