This paper generalizes considerably earlier investigations of the authors, constituting the first systematic and detailed study of abstract parabolic evolution equations involving measures that depend nonlinearly on the solution. These measures can be initial, Neumann, or Dirichlet data, or e. g. represent flow across a moving membrane, or a number of moving point sources located in the interior of the domain of existence. In some situations the theory applies for distributions more singular than measures.
The paper contains a number of existence and continuous dependence on data results (in suitable spaces), and a collection of examples illustrating how the theory applies. The methods used stem from semigroup theory and functional analysis.