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

在这项工作中，我们考虑了一个线性年龄结构问题，该问题具有年龄和空间变量的扩散和非均匀边界条件。我们通过将其重写为非密集定义的抽象柯西问题来处理这个线性问题。为了这个目的，我们通过使用集成半群理论开发了一个关于两个非密集定义算子的可交换和的封闭性的新结果。作为这个抽象结果的应用，我们能够将一个合适的集成半群与一些具有空间扩散和配备非齐次边界条件的年龄结构问题相关联。这个集成半群的特征在于它的无穷小生成元的描述。我们的抽象结果的进一步应用也被用于两个几乎扇形的算子的交换和，