Online social networks have recently become an effective and innovative channel for spreading information and influence among hundreds of millions of end users. Many prior work have carried out empirical studies and proposed diffusion models to understand the information diffusion process in online social networks. However, most of these studies focus on the information diffusion in temporal dimension, that is, how the information propagates over time. Little attempt has been given on understanding information diffusion over both temporal and spatial dimensions. In this paper, we propose a Partial Differential Equation (PDE), specifically, a Diffusive Logistic (DL) equation to model the temporal and spatial characteristics of information diffusion in online social networks. To be more specific, we develop a PDE-based theoretical framework to measure and predict the density of influenced users at a given distance from the original information source after a time period. The density of influenced users over time and distance provides valuable insight on the actual information diffusion process. We present the temporal and spatial patterns in a real dataset collected from Digg social news site, and validate the proposed DL equation in terms of predicting the information diffusion process. Our experiment results show that the DL model is indeed able to characterize and predict the process of information propagation in online social networks. For example, for the most popular news with 24,099 votes in Digg, the average prediction accuracy of DL model over all distances during the first 6 hours is 92.08%. To the best of our knowledge, this paper is the first attempt to use PDE-based model to study the information diffusion process in both temporal and spatial dimensions in online social networks.