Time Regularity for Local Weak Solutions of the Heat Equation on Local Dirichlet Spaces

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 L2 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 Colding and Minicozzi (Duke Math. J., 170(18), 4171–4182 2021) and Zhang (Proc. Amer. Math. Soc., 148(4), 1665–1670 2020).