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

Abstract

We study heat equations ${\partial }_{t}u-\mathrm{div}(A\nabla u)=0$ on bounded Lipschitz domains $\Omega$, where $-\mathrm{div}(A\nabla \cdot )$ 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.

32 pages, 1 figure