Locally eventually positive operator semigroups

Abstract

We initiate a theory of locally eventually positive operator semigroups on Banach lattices. Intuitively this means: given a positive initial datum, the solution of the corresponding Cauchy problem becomes (and stays) positive in a part of the domain, after a sufficiently large time. A drawback of the present theory of eventually positive C0-semigroups is that it is applicable only when the leading eigenvalue of the semigroup generator has a strongly positive eigenvector. We weaken this requirement and give sufficient criteria for individual and uniform local eventual positivity of the semigroup. This allows us to treat a larger class of examples by giving us more freedom on the domain when dealing with function spaces − for instance, the square of the Laplace operator with Dirichlet boundary conditions on L2 and the Dirichlet bi-Laplacian on Lp-spaces. Besides, we establish various spectral and convergence properties of locally eventually positive semigroups.