The heat flow in an optimal Fréchet space of unbounded initial data in R d
The heat flow in an optimal Fréchet space of unbounded initial data in R d
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R d 中无界初始数据的最佳 Fréchet 空间中的热流
DOI：
10.1016/j.jde.2020.07.017
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发表时间：
2020
影响因子：
2.4
通讯作者：
Robinson J
中科院分区：
文献类型：
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作者：
Robinson J
本文证明了用热核给出的热方程的解定义了Fréchet空间L0 p（Rd）族上的半群，Fréchet空间L0 p（Rd）是函数空间L ε p（Rd）的交（在所有ε> 0上），使得|X| 2| f（x）|p d x<∞。这些空间由“无穷大”的函数组成，L 0 1（R d）是最大的空间，其中可以使用热核来获得热方程的全局定义的解。证明了这类半群从L0 p（Rd）到L0 q（Rd）（q≥ p）的适当估计.然后，我们考虑在与这些函数空间对偶的空间中的热半群，即由非常快速减少的函数构成的空间L− ε p（Rd），使得|X| 2| f（x）|p d x<∞。我们证明了（Lp ε p（Rd））′= L− q ε q（Rd）（其中1< p<∞且（p，q）共轭），并且对于适当的（依赖于时间的）δ选择，L ε p（Rd）上的热流是L− δ q（Rd）上的热流的伴随。
In this paper we show that solutions of the heat equation that are given in terms of the heat kernel define semigroups on the family of Fréchet spaces L 0 p (R d), the intersection (over all ε> 0) of the spaces L ε p (R d) of functions such that∫ R d e− ε| x| 2| f (x)| p d x<∞. These spaces consist of functions that are ‘large at infinity’, and L 0 1 (R d) is the maximal space in which one can use the heat kernel to obtain globally-defined solutions of the heat equation. We prove suitable estimates from L 0 p (R d) into L 0 q (R d), q≥ p, for these semigroups. We then consider the heat semigroup posed in spaces that are dual to these spaces of functions, namely the spaces L− ε p (R d) of very-rapidly decreasing functions such that∫ R d e ε| x| 2| f (x)| p d x<∞. We show that (L p ε p (R d))′= L− q ε q (R d)(with 1< p<∞ and (p, q) conjugate), and that the heat flow on L ε p (R d) is the adjoint of the flow on L− δ q (R d) for an appropriate (time-dependent) choice of δ.