Local behavior of solutions of quasilinear parabolic equations on metric spaces

TL;DR

This paper develops a framework for quasilinear parabolic equations on metric measure spaces, establishing local boundedness and Harnack inequalities for solutions, with applications to maximum principles and pointwise estimates.

Contribution

It introduces a novel notion of quasilinear parabolic equations on metric spaces and proves fundamental regularity results under sharp structural conditions.

Findings

Local weak solutions are locally bounded

Solutions satisfy the parabolic Harnack inequality

Applications include maximum principles and pointwise estimates

Abstract

We introduce a notion of quasilinear parabolic equations over metric measure spaces. Under sharp structural conditions, we prove that local weak solutions are locally bounded and satisfy the parabolic Harnack inequality. Applications include the parabolic maximum principle and pointwise estimates for weak solutions.