Differential and Integral Equations

On integrated semigroups and age structured models in {$L^p$} spaces

Pierre Magal and Shigui Ruan

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In this paper, we first develop some techniques and results for integrated semigroups when the generator is not a Hille-Yosida operator and is non-densely defined. Then we establish a theorem of Da Prato and Sinestrari's type for the nonhomogeneous Cauchy problem and prove a perturbation theorem. In particular, we obtain necessary and sufficient conditions for the existence of mild solutions for non-densely defined non-homogeneous Cauchy problems. Next we extend the results to $L^{p}$ spaces and consider the semilinear and non-autonomous problems. Finally we apply the results to studying age-structured models with dynamic boundary conditions in $L^{p}$ spaces. We also demonstrate that neutral delay differential equations in $L^{p}$ can be treated as special cases of the age-structured models in an $L^{p}$ space.

Article information

Differential Integral Equations, Volume 20, Number 2 (2007), 197-239.

First available in Project Euclid: 20 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47D62: Integrated semigroups
Secondary: 34G10: Linear equations [See also 47D06, 47D09] 35K57: Reaction-diffusion equations 47D06: One-parameter semigroups and linear evolution equations [See also 34G10, 34K30] 92D25: Population dynamics (general) 92D30: Epidemiology


Magal, Pierre; Ruan, Shigui. On integrated semigroups and age structured models in {$L^p$} spaces. Differential Integral Equations 20 (2007), no. 2, 197--239. https://projecteuclid.org/euclid.die/1356039513

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