A matrix Harnack inequality for semilinear heat equations

Giacomo Ascione; Daniele Castorina; Giovanni Catino; Carlo Mantegazza; Giacomo Ascione; Daniele Castorina; Giovanni Catino; Carlo Mantegazza

doi:10.3934/mine.2023003

Mathematics in Engineering

2023, Volume 5, Issue 1: 1-15. doi: 10.3934/mine.2023003

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A matrix Harnack inequality for semilinear heat equations

1.
Dipartimento di Matematica e Applicazioni "Renato Caccioppoli", Università degli Studi di Napoli Federico Ⅱ, Via Cintia, Monte S. Angelo 80126 Napoli, Italy
2.
Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133, Milano, Italy

Received: 28 July 2021 Revised: 26 November 2021 Accepted: 05 December 2021 Published: 10 January 2022

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.
- ancient solution,
- eternal solution,
- semilinear heat equation,
- matrix Harnack inequality
Citation: Giacomo Ascione, Daniele Castorina, Giovanni Catino, Carlo Mantegazza. A matrix Harnack inequality for semilinear heat equations[J]. Mathematics in Engineering, 2023, 5(1): 1-15. doi: 10.3934/mine.2023003

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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.

References

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[3]	R. S. Hamilton, Four-manifolds with positive curvature operator, J. Differential Geom., 24 (1986), 153–179. http://dx.doi.org/10.4310/jdg/1214440433 doi: 10.4310/jdg/1214440433
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[5]	R. S. Hamilton, A matrix Harnack estimate for the heat equation, Commun. Anal. Geom., 1 (1993), 113–126. http://dx.doi.org/10.4310/CAG.1993.v1.n1.a6 doi: 10.4310/CAG.1993.v1.n1.a6
[6]	R. S. Hamilton, The Harnack estimate for the mean curvature flow, J. Differential Geom., 41 (1995), 215–226. http://dx.doi.org/10.4310/jdg/1214456010 doi: 10.4310/jdg/1214456010
[7]	P. Li, S.-T. Yau, On the parabolic kernel of the Schrödinger operator, Acta Math., 156 (1986), 153–201. http://dx.doi.org/10.1007/BF02399203 doi: 10.1007/BF02399203
[8]	C. Mantegazza, Lecture notes on mean curvature flow, Basel: Springer, 2011. http://dx.doi.org/10.1007/978-3-0348-0145-4

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