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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

Horst R. Thieme 1,

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.
Keywords: growth bounds, age-structured population dynamics with another population structure., integrated semigroups, Stieltjes integral equations, asynchronous exponential growth, cumulative outputs, resolvent positive operator, positive perturbation, resolvent outputs, Semigroups, spectral bound, eventual and essential compactness.
Mathematics Subject Classification: 34G10, 47D06, 58F11, 92D2.
Citation: Horst R. Thieme. Positive perturbation of operator semigroups: growth bounds, essential compactness and asynchronous exponential growth. Discrete & Continuous Dynamical Systems, 1998, 4 (4) : 735-764. doi: 10.3934/dcds.1998.4.735
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