On integrated semigroups and age structured models in \(L^p\) spaces. (English) Zbl 1212.35238

Magal, Pierre; Ruan, Shigui

Differ. Integral Equ. 20, No. 2, 197-239 (2007).

Summary: 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.

MSC:

35K57	Reaction-diffusion equations
47H20	Semigroups of nonlinear operators
92D30	Epidemiology

Keywords:

integrated semigroup; non-densely defined generator; nonhomogeneous Cauchy problem; \(L^p\)-space