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

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.


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