Non-Positivity of the heat equation with non-local Robin boundary conditions

TL;DR

This paper investigates heat equations with non-local Robin boundary conditions on Lipschitz domains, showing that solutions can lose positivity but still exhibit ultracontractivity and eventual positivity under mild assumptions.

Contribution

It extends the analysis of heat equations with non-local Robin boundary conditions to include operators that destroy positivity, demonstrating ultracontractivity and eventual positivity.

Findings

Semigroup remains ultracontractive despite non-positivity-preserving boundary operators.

Certain boundary operators lead to solutions that are eventually positive.

The study broadens understanding of boundary conditions affecting heat equation positivity.

Abstract

We study heat equations $\partial_t u - \operatorname{div}(A\nabla u) = 0$ on bounded Lipschitz domains $\Omega$, where $-\operatorname{div}(A\nabla\,\cdot\,)$ is a second-order uniformly elliptic operator with generalised Robin boundary conditions. These boundary conditions are formally given by $\nu\cdot A\nabla u + Bu=0$, where $B\in\mathcal{L}(L^2(\partial\Omega))$ is a general operator. In contrast to large parts of the literature on non-local Robin boundary conditions, we also allow for operators $B$ that destroy the positivity preserving property of the solution semigroup. Nevertheless, we obtain ultracontractivity of the semigroup under quite mild assumptions on $B$. For a certain class of operators $B$ we demonstrate that the semigroup is in fact eventually positive rather than positivity preserving.