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.
Citation
Pierre Magal. Shigui Ruan. "On integrated semigroups and age structured models in {$L^p$} spaces." Differential Integral Equations 20 (2) 197 - 239, 2007. https://doi.org/10.57262/die/1356039513
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