Positive irreducible semigroups and their long-time behaviour

Abstract

The notion Perron–Frobenius theory usually refers to the interaction between three properties of operator semigroups: positivity, spectrum and long-time behaviour. These interactions gives rise to a profound theory with plenty of applications. By a brief walk-through of the field and with many examples, we highlight two aspects of the subject, both related to the long-time behaviour of semigroups: (i) The classical question how positivity of a semigroup can be used to prove convergence to an equilibrium as t → ∞. (ii) The more recent phenomenon that positivity itself sometimes occurs only for large t , while being absent for smaller times. This article is part of the theme issue ‘Semigroup applications everywhere’.

Some Study of Semigroups ofh-Bi-Ideals of Semirings

“…As a generalization of k -clifford semirings, Bhuniya introduced the left k -clifford semiring [ 27 ]. New research in semigroups and semirings is advancing day by day in different fields of asymptotics, control theory, biology, and medicine [ 28 – 31 ].…”

Convergence to equilibrium for linear parabolic systems coupled by matrix‐valued potentials

“…As the matrix W is also irreducible, we can hence apply the perturbation result in [5, Proposition C‐III‐3.3] to see that the semigroup generated by $$\mathcal {B}_2 + V$$ on $$L^p(\Omega ;\mathbb {C}^N)$$ is also irreducible; this argument is taken from [24, Proposition in Section 8]. The irreducibility of the semigroup implies that the limit operator is either 0 or has rank 1; this follows from classical arguments in Perron–Frobenius theory, see, for instance, [4, Proposition 3.1(c)] for a detailed explanation. Since the semigroups generated by $$\mathcal {B}_p + V$$ act consistently on the $$L^p$$‐scale, the same is true for the limit operator.…”

“…Due to the irreducibility in the preceding remark, even a bit more can actually be said about the limit operator P . We refrain from discussing this in detail here and refer to the general result explained in [4, Proposition 3.1(c)] instead. We conclude this subsection with a simple example.…”

Local uniform convergence and eventual positivity of solutions to biharmonic heat equations

“…The papers [17,18] contain the foundation of the theory, and there have since been various refinements and extensions [14,15,16,9]. In [6,Sections 7,8] the reader will find a snapshot of some applications of the theory of eventual positivity. Further applications to the analysis of partial differential equations can, for instance, be found in [19,Section 7], while the reader may consult [25,Section 6] and [11,Section 5] for recent applications to the study of differential operators on graphs.…”