# Regularity properties of some perturbations of non-densely defined operators with applications

## TL;DR

This paper investigates how bounded perturbations affect the regularity and growth properties of semigroups generated by non-densely defined operators, with applications to age-structured population models.

## Contribution

It generalizes existing results on Hille-Yosida operators to a broader class of non-densely defined operators and applies these findings to population dynamics models.

## Key findings

- Bounded perturbations can preserve regularity properties of semigroups.
- The growth bounds of semigroups are characterized under perturbations.
- Applications to age-structured population models demonstrate practical relevance.

## Abstract

This paper is to study some conditions on semigroups, generated by some class of non-densely defined operators in the closure of its domain, in order that certain bounded perturbations preserve some regularity properties of the semigroup such as norm continuity, compactness, differentiability and analyticity. Furthermore, we study the critical and essential growth bound of the semigroup under bounded perturbations. The main results generalize the corresponding results in the case of Hille-Yosida operators. As an illustration, we apply the main results to study the asymptotic behaviors of a class of age-structured population models in $ L^p $ spaces ($ 1 \leq p < \infty $).