Evolution equations with eventually positive solutions

Abstract

We discuss linear autonomous evolution equations on function spaces which have the property that a positive initial value leads to a solution which initially changes sign, but then becomes -and stays -positive again for sufficiently large times. This eventual positivity phenomenon has recently been discovered for various classes of differential equations, but so far a general theory to explain this type of behaviour exists only under additional spectral assumptions.

Pointwise eigenvector estimates by landscape functions: some variations on the Filoche--Mayboroda--van den Berg bound

“…Because E ρ is in fact an AM space -indeed, it is isometrically Banach lattice isomorphic to C(K; R) for some compact Hausdorff space K (with ρ 1 K ), the infimum in (1.5) is attained, i.e., |f | ≤ f ρ ρ for all f ∈ E ρ . We will occasionally also make use of some ideas from the theory of eventually positive C 0 -semigroups, see [28] for a gentle introduction. We will also discuss the properties of (nonlinear) maximal monotone operators: two standard references are [11] for the general theory of such operators, and [7,14] for their interplay with the theory of Banach lattices.…”