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

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 (negative indexed) Sobolev-Slobodetskii space Ws,2(Ω) with s close to −1. For these indices s a tailor suited parabolic theory is established, based on results of Disser et al. [Ann. Sci. Norm. Super. Pisa, Cl. Sci. 17 (2017) 65-79] and Haller-Dintelmann et al. [Ann. Mat. Pura Appl. 198 (2019) 1227-1241]. The proposed frame work is well-suited for optimal control problems with controls acting on sub-manifolds.