A matrix Harnack inequality for semilinear heat equations

TL;DR

This paper develops a matrix Harnack inequality for positive solutions of semilinear heat equations on certain Riemannian manifolds, extending classical estimates and providing local bounds based on geometric properties.

Contribution

It introduces a matrix version of Li & Yau-type estimates for semilinear heat equations on manifolds with nonnegative sectional curvature and parallel Ricci tensor, generalizing Hamilton's work.

Findings

Derived matrix Harnack inequalities for semilinear heat equations

Obtained local solution bounds in terms of geometric quantities

Extended classical estimates to a broader geometric setting

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~\cite{hamilton7} 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.