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

Horst R. Thieme

doi:10.3934/dcds.1998.4.735

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1998, Volume 4, Issue 4: 735-764. Doi: 10.3934/dcds.1998.4.735

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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

Received: October 1997

Published: October 1998

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Abstract

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.

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Mathematics Subject Classification: 34G10, 47D06, 58F11, 92D25.

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