Online social networks such as Facebook and Twitter have played a significant role in information spreading. The availability of unprecedented amounts of digital data has accelerated research on information diffusion. 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. Most of dynamical models arising from online social networks only involve ordinary differential equations. In a number of recent works, we propose to use partial differential equation models to study the temporal and spatial characteristics of information diffusion in online social networks. In this talk, I will examine several partial differential equation models for online social networks. Free boundary value problems and bifurcation problems arising from online social networks will be discussed.