Open AccessA matrix Harnack inequality for semilinear heat equations
Author(s) -
Giacomo Ascione,
AUTHOR_ID,
Daniele Castorina,
Giovanni Catino,
Carlo Mantegazza,
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Publication year - 2022
Publication title -
mathematics in engineering
Language(s) - English
Resource type - Journals
ISSN - 2640-3501
DOI - 10.3934/mine.2023003
Subject(s) - harnack's inequality , mathematics , type (biology) , matrix (chemical analysis) , harnack's principle , mathematical analysis , heat equation , pure mathematics , chemistry , ecology , chromatography , biology
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.