Exponential dichotomies for elliptic PDE on radial domains
Exponential dichotomies for elliptic PDE on radial domains
复制标题
径向域上椭圆偏微分方程的指数二分法
DOI：
10.1007/978-3-030-47174-3
复制
发表时间：
2020
期刊：
影响因子：
--
通讯作者：
M. Beck, G. Cox
中科院分区：
文献类型：
--
作者：
M. Beck, G. Cox
作者最近证明了半线性椭圆方程可以用收缩单参数域族上的边界数据表示为无穷维动力系统。由此产生的系统是不适定的，在这个意义上，解决方案通常不存在向前或向后的时间。本文考虑一个径向域族，证明了线性化系统存在指数型二分性，其不稳定子空间对应于线性偏微分方程弱解的边界数据。这推广了空间动力学方法，它适用于无限圆柱（通道）域，也概括了以前的工作径向域，因为我们没有施加对称性假设的方程或其解决方案。
It was recently shown by the authors that a semilinear elliptic equation can be represented as an infinite-dimensional dynamical system in terms of boundary data on a shrinking one-parameter family of domains. The resulting system is ill-posed, in the sense that solutions do not typically exist forward or backward in time. In this paper we consider a radial family of domains and prove that the linearized system admits an exponential dichotomy, with the unstable subspace corresponding to the boundary data of weak solutions to the linear PDE. This generalizes the spatial dynamics approach, which applies to infinite cylindrical (channel) domains, and also generalizes previous work on radial domains as we impose no symmetry assumptions on the equation or its solutions.