热方程考希问题解的初始条件本地化

Pub Date : 2024-04-22 DOI:10.1134/s0965542524030096

A. N. Konenkov

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引用次数: 0

摘要

摘要研究了右边为零的热方程的 Cauchy 问题。假定初始函数属于调和分布空间。研究了从某个固定时间 \(T >0\)的解值确定初始函数支持的问题。得到了支撑位于给定凸紧凑集的必要条件和充分条件。这些条件是根据解在无穷远处的衰减率提出的。为关于快速衰减解不存在的 Landis-Oleinik 猜想找到了指数中的一个尖锐常数。

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Localizing the Initial Condition for Solutions of the Cauchy Problem for the Heat Equation

Abstract

The Cauchy problem for the heat equation with zero right-hand side is considered. The initial function is assumed to belong to the space of tempered distributions. The problem of determining the support of the initial function from solution values at some fixed time \(T > 0\) is studied. Necessary and sufficient conditions for the support to lie in a given convex compact set are obtained. These conditions are formulated in terms of the solution’s decay rate at infinity. A sharp constant in the exponential for the Landis–Oleinik conjecture on the nonexistence of fast decaying solutions is found.

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