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We construct the fundamental solution of the Porous Medium Equation posed inthe hyperbolic space $H^n$ and describe its asymptotic behaviour as$t\to\infty$. We also show that it describes the long time behaviour ofintegrable nonnegative solutions, and very accurately if the solutions are alsoradial and compactly supported. By radial we mean functions depending on thespace variable only through the geodesic distance $r$ from a given point $O\inH^n$. We also construct an exact generalized traveling wave solution. We showthat the location of the free boundary of compactly supported solutions growslogarithmically for large times, in contrast with the well-known power-likegrowth of the PME in the Euclidean space. Very slow propagation at longdistances is a feature of porous medium flow in hyperbolic space.

Fundamental solution and long time behaviour of the Porous Medium Equation in Hyperbolic Space