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July  1999, 5(3): 663-683. doi: 10.3934/dcds.1999.5.663

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, Morocco
2. 

Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany

Received  January 1998 Revised  May 1999 Published  May 1999

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.
Keywords: resolvent positive operator, Hillc-Yosida operator, Age-structured population equation, irreducibility, quasi-compact semigroup, essential spectral radius, balanced exponential growth, Miyadera perturbaion, extrapolated semigroup, positive evolution family, evolution semigroup..
Mathematics Subject Classification: Primary: 47D06, 92D25; Secondary: 34G10, 47A1.
Citation: Abdelaziz Rhandi, Roland Schnaubelt. Asymptotic behaviour of a non-autonomous population equation with diffusion in $L^1$. Discrete and Continuous Dynamical Systems, 1999, 5 (3) : 663-683. doi: 10.3934/dcds.1999.5.663
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