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Research Article | Open Access

A matrix Harnack inequality for semilinear heat equations

Giacomo Ascione1, Daniele Castorina1( ), Giovanni Catino2, Carlo Mantegazza1
Dipartimento di Matematica e Applicazioni "Renato Caccioppoli", Università degli Studi di Napoli Federico Ⅱ, Via Cintia, Monte S. Angelo 80126 Napoli, Italy
Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133, Milano, Italy
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Abstract

We derive a matrix version of Li & Yau–type estimates for positive solutions of semilinear heat equations on Riemannian manifolds with nonnegative sectional curvatures and parallel Ricci tensor, similarly to what R. Hamilton did in [5] for the standard heat equation. We then apply these estimates to obtain some Harnack–type inequalities, which give local bounds on the solutions in terms of the geometric quantities involved.

Keywords

ancient solutioneternal solutionsemilinear heat equationmatrix Harnack inequality

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Mathematics in Engineering
Volume 5 Issue 1,
February 2023
Pages 1-15
DOI: 10.3934/mine.2023003

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Cite this article:
Ascione G, Castorina D, Catino G, et al. A matrix Harnack inequality for semilinear heat equations. Mathematics in Engineering, 2023, 5(1): 1-15. https://doi.org/10.3934/mine.2023003

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Received: 28 July 2021
Revised: 26 November 2021
Accepted: 05 December 2021
Published: 15 February 2023
©2023 the Author(s), licensee AIMS Press.

This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)