Abstract

We consider the mixed problem on the exterior of the unit ball in ${\mathbb{R}}^n,n\ge 2,$ for a defocusing Schrödinger equation with a power nonlinearity $|u{|}^{p-1}u,$ with zero boundary data. Assuming that the initial data are non-radial, sufficiently small perturbations of large radial initial data, we prove that for all powers $p>n+6$ the solution exists for all times, its Sobolev norms do not inflate, and the solution is unique in the energy class.

中文翻译：

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关于球外部的超临界薛定谔方程

摘要

我们考虑单元球外部的混合问题 ${\mathbb{电阻}}^n,n\ge 2,$ 对于具有功率非线性的散焦薛定谔方程 $|你{|}^{磷-1}你,$零边界数据。假设初始数据是非径向的，大径向初始数据的扰动足够小，我们证明对于所有幂$磷>n+6$ 该解一直存在，其 Sobolev 范数不会膨胀，并且该解在能量类中是唯一的。