Asymptotic behaviour of a non-autonomous population equation with diffusion in $L^1$

Abdelaziz Rhandi; Roland Schnaubelt

doi:10.3934/dcds.1999.5.663

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1999, Volume 5, Issue 3: 663-683. Doi: 10.3934/dcds.1999.5.663

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Asymptotic behaviour of a non-autonomous population equation with diffusion in $L^1$

Abdelaziz Rhandi^1, and
Roland Schnaubelt^2,

1.
Départment de Mathématiques, Faculté des Sciences Semlalia, B.P.: S.15 40000 Marrakech, Maroc
2.
Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen

Received: December 31, 1997

Revised: April 30, 1999

Published: June 1999

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Abstract

We prove existence and uniquences of positive solutions of an age-structured population equation of McKendrick type with spatial diffusion in $L^1$. The coefficients may depend on age and position. Moreover, the mortality rate is allowed to be unbounded and the fertility rate is time dependent. In the time periodic case, we estimate the essential spectral radius of the monodromy operator which gives information on the asymptotic behaviour of solutions. Our work extends previous results in [19], [24], [30], and [31] to the non-autonomous situation. We use the theory of evolution semigroups and extrapolation spaces.

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Mathematics Subject Classification: Primary: 47D06, 92D25; Secondary: 34G10, 47A10.

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