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

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 the local weak solutions of a uniformly elliptic symmetric divergence form second order operator 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 [31].

Time analyticity of the biharmonic heat equation, the heat equation with potentials and some nonlinear heat equations

“…Remark 3.3. By the method of Steklov average, or to be more precise, by Theorem 4.1 of Q.Hou&L.Saloff-Coste [10] which states if the heat kernel Γ V of (1.2) satisfies the L 2 Gaussian type upper bound and, for any weak solution u of (1.2), ∂ l t u is also a weak solution of (1.2) for any l = 1, 2, • • • . on the one hand, if V ≥ 0 and if Γ is the heat kernel of heat equation on the same manifold M, then by maximal principle, 0 ≤ Γ V ≤ Γ, which means Γ V satisfies this Gaussian type upper bound condition considering (4.2) and the mean value inequality.…”