On semilinear Cauchy problems with non-dense domain

Abstract

We are interested in studying semilinear Cauchy problems in which the closed linear operator is not Hille-Yosida and its domain is not densely defined. Using integrated semigroup theory, we study the positivity of solutions to the semilinear problem, the Lipschitz perturbation of the problem, differentiability of the solutions with respect to the state variable, time differentiability of the solutions, and the stability of equilibria. The obtained results can be used to study several types of differential equations, including delay differential equations, age-structure models in population dynamics, and evolution equations with nonlinear boundary conditions.