2019/12/30 by Qi Hou, Hou, Qi, Laurent Saloff‐Coste +2
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Nonlinear Partial Differential Equations #Probability (math.PR) #Stability and Controllability of Differential Equations #math.AP #math.FA #math.PR
paper · pdf · doi:10.48550/arxiv.1912.12998
openalex publication_date 2019/12/30 · arxiv created 2020/11/13 · arxiv updated 2020/11/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 [8] and [33].