Time regularity for local weak solutions of the heat equation on local Dirichlet spaces

TL;DR

This paper investigates the time regularity of local weak solutions to the heat equation within local Dirichlet spaces, establishing that their time derivatives are also local weak solutions under minimal assumptions.

Contribution

It proves that, under weak conditions, the time derivatives of local weak solutions are themselves local weak solutions, extending understanding of heat equations in Dirichlet space contexts.

Findings

Time derivatives of solutions are also solutions.

Results apply to elliptic divergence form operators.

Generalizes recent results on ancient solutions.

Abstract

We study the time regularity of local weak solutions of the heat equation in the context of local regular symmetric Dirichlet spaces. Under two basic and rather minimal assumptions, namely, the existence of certain cut-off functions and a very weak $L^2$ Gaussian type upper-bound for the heat semigroup, we prove that the time derivatives of a local weak solution of the heat equation are themselves local weak solutions. This applies, for instance, to local weak solutions of parabolic equations with uniformly elliptic symmetric divergence form second order operators with measurable coefficients. We describe some applications to the structure of ancient local weak solutions of such equations which generalize recent results of [8] and [33].