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Uniqueness of $L^p$ subsolutions to the heat equation on Finsler measure spaces

Part of: Partial differential equations on manifolds; differential operators Global differential geometry

Published online by Cambridge University Press:  05 June 2023

Qiaoling Xia Open the ORCID record for Qiaoling Xia [Opens in a new window]
Qiaoling Xia*
Affiliation:
Department of Mathematics, School of Sciences, Hangzhou Dianzi University, Hangzhou, Zhejiang Province 310018, P.R. China
*
e-mail: xiaqiaoling@hdu.edu.cn
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Abstract

Let $(M, F, m)$ be a forward complete Finsler measure space. In this paper, we prove that any nonnegative global subsolution in $L^p(M)(p>1)$ to the heat equation on $\mathbb R^+\times M$ is uniquely determined by the initial data. Moreover, we give an $L^p(0<p\leq 1)$ Liouville-type theorem for nonnegative subsolutions u to the heat equation on $\mathbb R\times M$ by establishing the local $L^p$ mean value inequality for u on M with Ric$_N\geq -K(K\geq 0)$.

Keywords

Finsler measure spaceweighted Ricci curvatureheat equationmean value inequality

MSC classification

Primary: 53C60: Finsler spaces and generalizations (areal metrics)
Secondary: 58J35: Heat and other parabolic equation methods 53C23: Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
Type
Article
Information
Canadian Mathematical Bulletin , Volume 67 , Issue 1 , March 2024 , pp. 166 - 175
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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Footnotes

This paper is supported by the NNSFC (Grant No. 12071423) and the Scientific Research Foundation of HDU (Grant No. KYS075621060).

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