A matrix Harnack inequality for semilinear heat equations

2021/07/29 by Ascione, Giacomo, Castorina, Daniele, Catino, Giovanni +1
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2107.13846

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~\citehamilton7 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.

Related

• Semilinear Li & Yau inequalities
• On an alternate proof of Hamilton's matrix Harnack inequality for the Ricci flow
• Differential Harnack inequalities on Riemannian manifolds I : linear heat equation
• An improved Hamilton matrix estimates for the heat equation
• Backward Harnack inequality and Hamilton estimates for heat type equations