The porous medium equation (PME) is a nonlinear partial differential equation that describes the flow of a fluid through a porous medium, where the medium's permeability and the fluid's properties can lead to complex behaviors. It is commonly used in various fields such as hydrology, geology, and materials science to model processes like groundwater flow, diffusion of gases in soils, and heat conduction in porous materials.
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The porous medium equation with is a nonlinear partial differential equation whose diffusivity vanishes at . Unlike the heat equation, it has compactly supported solutions with finite propagation speed.