Optimal existence classes and nonlinear-like dynamics in the linear heat equation in R d
Optimal existence classes and nonlinear-like dynamics in the linear heat equation in R d
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R d 线性热方程中的最优存在类和类非线性动力学
DOI：
10.1016/j.aim.2018.06.009
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发表时间：
2018
影响因子：
1.7
通讯作者：
Robinson J
中科院分区：
文献类型：
--
作者：
Robinson J
本文分析了R d中的线性热方程的解的性质，其初始数据为Radon测度类M ε（R d），且R d e− ε| X| 2个d|单位0| <∞。我们证明了这些类对于局部和整体非负解的存在性是最优的：特别是M0（Rd）：= ε ε> 0 M ε（Rd）由那些初始数据组成，对于这些初始数据，可以使用热核表示公式给出热方程的解。我们证明了存在性，唯一性和规律性的结果，这样的初始数据，可以迅速增长到无穷大，然后表明，它们产生的属性更经常与非线性模型。我们证明了有限时间爆破的解决方案，显示的一组爆破点是一个凸集的补充，并给予任何封闭的凸集有一个初始条件，其解决方案保持有界精确在这个集合上的“爆破时间”。我们还证明了当t→∞时，从非负初始数据出发，有可能出现剧烈振荡，并且可以将u（0，t）的行为定义为[0，∞）上的任意实解析函数γ（t）.
We analyse the behaviour of solutions of the linear heat equation in R d for initial data in the classes M ε (R d) of Radon measures with∫ R d e− ε| x| 2 d| u 0|<∞. We show that these classes are optimal for local and global existence of non-negative solutions: in particular M 0 (R d):=∩ ε> 0 M ε (R d) consists of those initial data for which a solution of the heat equation can be given for all time using the heat kernel representation formula. We prove existence, uniqueness, and regularity results for such initial data, which can grow rapidly at infinity, and then show that they give rise to properties associated more often with nonlinear models. We demonstrate the finite-time blowup of solutions, showing that the set of blowup points is the complement of a convex set, and that given any closed convex set there is an initial condition whose solutions remain bounded precisely on this set at the ‘blowup time’. We also show that wild oscillations are possible from non-negative initial data as t→∞ and that one can prescribe the behaviour of u (0, t) to be any real-analytic function γ (t) on [0,∞).