We propose a method for modelling spatially dependent functional data, based on regression with differential regularization. The regularizing term enables to include problem-specific information about the spatio-temporal variation of phenomenon under study, formalized in terms of a time-dependent partial differential equation. The method is implemented using a discretization based on finite elements in space and finite differences in time. This non-tensor product basis allows to efficiently handle data distributed over complex domains and where the shape of the domain influences the phenomenon behavior. Moreover, the method can comply with specific conditions at the boundary of the domain of interest. Simulation studies compare the proposed model to available techniques for spatio-temporal data. The method is also illustrated via an application to the study of blood-flow velocity field in a carotid artery affected by atherosclerosis, starting from echo-color doppler and magnetic resonance imaging data.