Nonnegative solutions to nonlocal parabolic equations

TL;DR

This paper investigates nonnegative solutions to a broad class of nonlocal parabolic equations, establishing fundamental theorems, bounds, and estimates that extend existing results to more general operators.

Contribution

It proves a Widder-type theorem, derives sharp bounds for the fundamental solution, and establishes novel Harnack-type estimates for nonlocal parabolic equations.

Findings

Proved a Widder-type theorem for nonlocal operators.

Established sharp two-sided bounds for the fundamental solution.

Derived new Harnack-type estimates for fractional heat equations.

Abstract

We aim to study nonnegative, global solutions to a general class of nonlocal parabolic equations with bounded measurable coefficients. First, we prove a Widder-type theorem. Such a result has previously been studied only for certain translation invariant operators, and new ideas are needed in our general setting. Second, we establish sharp two-sided bounds for the fundamental solution via purely variational techniques, entirely bypassing tools from semigroup theory, Dirichlet forms, and stochastic analysis. Third, we derive sharp Harnack-type estimates that are novel even for the fractional heat equation.