Pointwise and Oscillation Estimates via Riesz Potentials for Mixed Local and Nonlocal Parabolic Equations

TL;DR

This paper develops pointwise and oscillation estimates for solutions to mixed local and nonlocal parabolic equations using caloric Riesz potentials, extending previous tail estimates and applicable to SOLA solutions.

Contribution

It introduces new local Hölder estimates with an optimal $L^q$-Tail for homogeneous problems, capturing both local and nonlocal features of double phase parabolic equations.

Findings

Established pointwise bounds for weak solutions.

Extended $L^ ext{infty}$-Tail to $L^q$-Tail for solutions.

Results valid for SOLA solutions.

Abstract

We establish a class of pointwise estimates for weak solutions to mixed local and nonlocal parabolic equations involving measure data and merely measurable coefficients via caloric Riesz potentials. Such estimates effectively bound the sizes and oscillations of weak solutions, respectively. The proof relies on demonstrating a new local H\"{o}lder estimate with an optimal $L^q$-Tail for weak solutions to the corresponding homogeneous problem, which remarkably extends the $L^\infty$-Tail in previous work. It is worth mentioning that our main results capture both local and nonlocal features of the double phase parabolic equations and, more importantly, remain valid for SOLA (Solutions Obtained by Limit of Approximations).