Abstract
We present a generation theorem for positive semigroups on an L^1 space. It provides sufficient conditions for the existence of positive and integrable solutions of initial-boundary value problems. An application to a two-phase cell cycle model is given.
Highlights
We study well-posedness of linear evolution equations on L1 of the form u (t) = Au(t), Ψ0u(t) = Ψ u(t), t > 0, u(0) = f, (1)where Ψ0, Ψ are positive and possibly unbounded linear operators on L1, the linear operator A is such that Eq (1) with Ψ = 0 generates a positive semigroup on L1, i.e., a C0-semigroup of positive operators on L1
Where Ψ0, Ψ are positive and possibly unbounded linear operators on L1, the linear operator A is such that Eq (1) with Ψ = 0 generates a positive semigroup on L1, i.e., a C0-semigroup of positive operators on L1
We present sufficient conditions for the operators A, Ψ0, and Ψ under which there is a unique positive semigroup on L1 providing solutions of the initial-boundary value problem (1)
Summary
Unbounded perturbations of the boundary conditions of a generator were studied recently in [1,2] by using extrapolated spaces and various admissibility conditions. In the proof of our perturbation theorem we apply a result about positive perturbations of resolvent positive operators [3] with non-dense domain in AL-spaces in the form given in [37, Theorem 1.4]. It is an extension of the well known perturbation result due to Desch [15] and by Voigt [41]. Our approach can be used in transport equations [8,23]
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