On non-autonomous parabolic equations with measure-valued right hand sides and applications to optimal control

Abstract

The main aim of this paper is to develop a theory for non-autonomous parabolic equations with time-dependent measures on the spatial domain appearing as right hand sides. Restricting these measures to ones which have their supports on 'curves' or 'surfaces' -- the latter understood in the sense of geometric measure theory -- we succeed in interpreting them as distributional objects from a (negatively indexed) Sobolev--Slobodetskii space with differentiability index close to minus one. For these indices a tailor suited parabolic theory is established, based on previous results. It is also demonstrated that the proposed frame work is well-suited for optimal control with controls acting on sub-manifolds.