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In this work, we consider a linear age-structured problem with diffusion andnon-homogeneous boundary conditions both for the age and the space variables.We handle this linear problem by re-writing it as a non-densely definedabstract Cauchy problem. To that aim we develop a new result on the closednessof a commutative sum of two non-densely defined operators by using the theoryof integrated semigroups. As an application of this abstract result, we areable to associate a suitable integrated semigroup to some age-structuredproblem with spatial diffusion and equipped with non-homogeneous boundaryconditions. This integrated semigroup is characterized by the description ofits infinitesimal generator. Further applications of our abstract result arealso given to the commutative sum of two almost sectorial operators, for whichwe derive a closedness results.

An integrated semigroup approach for age structured equations with diffusion and non-homogeneous boundary conditions