We propose a new method for the analysis of functional data defined over spatio-temporal domains. These data can be interpreted as time evolving surfaces or spatially dependent curves. The proposed method is based on regression with differential regularization. We are in particular interested to the case when prior knowledge on the phenomenon under study is available. The prior knowledge is described in terms of a time-dependent Partial Differential Equation (PDE) that jointly models the spatial and temporal variation of the phenomenon. We consider various samplings designs, including geo-statistical and areal data. We show that the corresponding estimation problem are well posed and can be discretized in space by means of the Finite Element method, and in time by means of the Finite Difference method. The model can handle data distributed over spatial domains having complex shapes, such as domains with strong concavities and holes. Moreover, various types of boundary conditions can be considered. The proposed method is compared to existing techniques for the analysis of spatio-temporal models, including space-time kriging and methods based on thin plate splines and soap film smoothing. As a motivating example, we study the blood flow velocity field in the common carotid artery, using data from Echo-Color Doppler.