多调和热流的一些局部估计及唯一性结果

数学物理学报 ›› 2026, Vol. 46 ›› Issue (6): 2117-2139.

多调和热流的一些局部估计及唯一性结果

高敏^*()

澳门大学科技学院数学系中国澳门 999078

收稿日期:2025-08-12 修回日期:2025-10-21 出版日期:2026-12-26 发布日期:2026-08-14
通讯作者: 高敏 E-mail:yc47474@um.edu.mo
基金资助:
国家自然科学基金(11223344)

Some Local Estimates and a Uniqueness Result for the Entire Poly-Harmonic Heat Flow

Min Gao^*()

Department of Mathematics, Faculty of Science and Technology, University of Macau, Taipa, Macau 999078

Received:2025-08-12 Revised:2025-10-21 Online:2026-12-26 Published:2026-08-14
Contact: Min Gao E-mail:yc47474@um.edu.mo
Supported by:
NSFC(11223344)

摘要/Abstract

摘要：

该文研究了 $\mathbb R^n\times (0,T]$ 上多调和热流方程 $\partial_t u+\Delta^{2m}u=0$ 的光滑解. 主要关注一类满足特定全局时间控制条件的解, 即存在常数 $k_0>0$, 使得 $|\Delta u(x,t)|^2\leq k_0t^{-1/m}$ 成立. 该文首先建立了此类解的局部估计, 并阐明了这些估计如何推导出光滑解的唯一性. 最后, 构造了一个具体的示例, 论证了上述控制条件的自然性和必要性.

关键词: 多调和热流, 局部估计, 唯一性.

Abstract:

We consider smooth solutions to the poly-harmonic heat equation $\partial_tu+\Delta^{2m}u=0$ on $\mathbb{R} ^n\times (0,T]$. We focus on solutions satisfying a global growth condition: $|\Delta u|(x,t)^2\leq k_0t^{-1/m}$ for some constant $k_0>0$. We establish local estimates (in space and time) for such solutions and demonstrate how these estimates imply uniqueness of smooth solutions within this class. Finally, we provide an example to illustrate the naturalness of this control condition.

Key words: poly-harmonic heat flow, local estimates, uniqueness

中图分类号:

O175.2

高敏. 多调和热流的一些局部估计及唯一性结果[J]. 数学物理学报, 2026, 46(6): 2117-2139.

Min Gao. Some Local Estimates and a Uniqueness Result for the Entire Poly-Harmonic Heat Flow[J]. Acta mathematica scientia,Series A, 2026, 46(6): 2117-2139.

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