## Differential and Integral Equations

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

#### Abstract

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

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

Dates
First available in Project Euclid: 20 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356039513

Mathematical Reviews number (MathSciNet)
MR2294465

Zentralblatt MATH identifier
1212.35238

#### Citation

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