Online social networks such as Twitter and Facebook have gained tremendous popularity for information exchange. The availability of unprecedented amounts of digital data has accelerated research on information diffusion in online social networks. However, the mechanism of information spreading in online social networks remains elusive due to the complexity of social interactions and rapid change of online social networks. Much of prior work on information diffusion over online social networks has based on empirical and statistical approaches. The majority of dynamical models arising from information diffusion over online social networks involve ordinary differential equations which only depend on time. In a number of recent papers, the authors propose to use partial differential equations(PDEs) to characterize temporal and spatial patterns of information diffusion over online social networks. Built on intuitive cyber-distances such as friendship hops in online social networks, the reaction-diffusion equations take into account influences from various external out-of-network sources, such as the mainstream media, and provide a new analytic framework to study the interplay of structural and topical influences on information diffusion over online social networks. In this survey, we discuss a number of PDE-based models that are validated with real datasets collected from popular online social networks such as Digg and Twitter. Some new developments including the conservation law of information flow in online social networks and information propagation speeds based on traveling wave solutions are presented to solidify the foundation of the PDE models and highlight the new opportunities and challenges for mathematicians as well as computer scientists and researchers in online social networks.