  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 $).

Functional Analysis