# American Institute of Mathematical Sciences

October  1998, 4(4): 735-764. doi: 10.3934/dcds.1998.4.735

## Positive perturbation of operator semigroups: growth bounds, essential compactness and asynchronous exponential growth

 1 Department of Mathematics, Arizona State University, Tempe, AZ 85287-1804, United States

Received  October 1997 Published  July 1998

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.
Citation: Horst R. Thieme. Positive perturbation of operator semigroups: growth bounds, essential compactness and asynchronous exponential growth. Discrete & Continuous Dynamical Systems - A, 1998, 4 (4) : 735-764. doi: 10.3934/dcds.1998.4.735
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