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

We study heat equations ∂ₜ u + Lu = 0 on bounded Lipschitz domains Ω, where L=-div(A∇\;·\;) is a second-order uniformly elliptic operator with generalised Robin boundary conditions of the form ν· A∇ u + Bu=0, where B(L²(∂Ω)) 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 (e-tL)t≥ 0. 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.