A unified framework for pointwise convergence to the initial data of heat equations in metric measure spaces

Abstract

Given a metric measure space $(\mathcal{X}, d, \mu)$ satisfying the volume doubling condition, we consider a semigroup $\{S_t\}$ and the associated heat operator. We propose general conditions on the heat kernel so that the solutions of the associated heat equations attain the initial data pointwise. We demonstrate that these conditions are satisfied by a broad class of operators, including the Laplace operators perturbed by a gradient, fractional Laplacian, mixed local-nonlocal operators, Laplacian on Riemannian manifolds, Dunkl Laplacian and many more. In addition, we consider the Laplace operator in $\mathbb{R}^n$ with the Hardy potential and establish a characterization for the pointwise convergence to the initial data. We also prove similar results for the nonhomogeneous equations and showcase an application for the power-type nonlinearities.