Compactness of the Solution Operator for a Linear Evolution Equation with Distributed Measures

The main goal of the present paper is to define the solution operator (ξ, g) 7→ u associated to the evolution equation du = (Au)dt + dg, u(0) = ξ, where A generates a C0-semigroup in a Banach space X, ξ ∈ X, g ∈ BV ([ a, b ];X), and to study its main properties, such as regularity, compactness, and continuity. Some necessary and/or sufficient conditions for the compactness of the solution operator extending some earlier results due to the author and to Baras, Hassan, Veron, as well as some applications to the existence of certain generalized solutions to a semilinear equation involving distributed, or even spatial, measures, are also included. Two concrete examples of elliptic and parabolic partial differential equations subjected to impulsive dynamic conditions on the boundary illustrate the effectiveness of the abstract results.