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arXiv · 2505.08449

Nonnegative solutions to nonlocal parabolic equations

Naian Liao·Marvin Weidner
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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.

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Naian Liao, Marvin Weidner. 2025-05-13. Nonnegative solutions to nonlocal parabolic equations. https://arxiv.org/abs/2505.08449

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