An integrated semigroup approach for age structured equations with diffusion and non-homogeneous boundary conditions

TL;DR

This paper develops a new integrated semigroup approach to analyze linear age-structured equations with spatial diffusion and non-homogeneous boundary conditions, providing a novel theoretical framework and applications.

Contribution

It introduces a new result on the closedness of a sum of non-densely defined operators using integrated semigroups, applied to age-structured PDEs with boundary conditions.

Findings

Established a new abstract result on operator sum closedness.

Associated a suitable integrated semigroup to age-structured problems.

Characterized the infinitesimal generator of the semigroup.

Abstract

In this work, we consider a linear age-structured problem with diffusion and non-homogeneous boundary conditions both for the age and the space variables. We handle this linear problem by re-writing it as a non-densely defined abstract Cauchy problem. To that aim we develop a new result on the closedness of a commutative sum of two non-densely defined operators by using the theory of integrated semigroups. As an application of this abstract result, we are able to associate a suitable integrated semigroup to some age-structured problem with spatial diffusion and equipped with non-homogeneous boundary conditions. This integrated semigroup is characterized by the description of its infinitesimal generator. Further applications of our abstract result are also given to the commutative sum of two almost sectorial operators, for which we derive a closedness results.