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

We study heat equations ${\mathrm{\partial }}_{t}u-\mathrm{div}(A\mathrm{\nabla }u)=0$ on bounded Lipschitz domains $\mathrm{\Omega }$, where $-\mathrm{div}(A\mathrm{\nabla }\cdot )$ is a second-order uniformly elliptic operator with generalised Robin boundary conditions. These boundary conditions are formally given by $\nu \cdot A\mathrm{\nabla }u+Bu=0$, where $B\in \mathcal{L}(L^2(\mathrm{\partial }\mathrm{\Omega }))$ 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.