Parabolic Harnack inequality for time-dependent non-symmetric Dirichlet forms

TL;DR

This paper establishes the parabolic Harnack inequality for weak solutions of the heat equation linked to time-dependent non-symmetric Dirichlet forms in metric measure spaces, ensuring local boundedness under certain conditions.

Contribution

It extends the parabolic Harnack inequality to non-symmetric, time-dependent Dirichlet forms in metric measure spaces with volume doubling and Poincaré inequality.

Findings

Weak solutions are locally bounded.

Parabolic Harnack inequality holds for these solutions.

Applicable in metric measure spaces with specific geometric properties.

Abstract

In the context of a metric measure Dirichlet space satisfying volume doubling and Poincar\'e inequality, we prove the parabolic Harnack inequality for weak solutions of the heat equation associated with local nonsymmetric bilinear forms. In particular, we show that these weak solutions are locally bounded.