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Positive perturbation of operator semigroups: growth bounds, essential compactness and asynchronous exponential growth
If $B$ is the generator of an increasing
locally Lipschitz continuous integrated semigroup on an
abstract L space $X$ and $C: D(B) \to X$ perturbs $B$ positively, then
$A = B + C$ is again the generator of an increasing l.L.c. integrated
semigroup. In this paper we study the growth bound and the compactness
properties of the $C_0$ semigroup $S_\circ$
that is generated by the part of $A$
in $X_\circ = \overline {D(B)}$.
We derive conditions in terms of
the resolvent outputs $ F(\lambda) = C (\lambda - B)^{-1} $
for the semigroup
$S_\circ$ to be eventually compact or essentially compact and to exhibit
asynchronous exponential growth.
We apply our results to age-structured population models with
additional structures. We consider an age-structured model with
spatial diffusion and an age-size-structured model.