Fundamental solution and long time behavior of the Porous Medium Equation in hyperbolic space

Abstract

We construct the fundamental solution of the Porous Medium Equation posed in the hyperbolic space Hn and describe its asymptotic behavior as t→∞. We also show that it describes the long time behavior of integrable nonnegative solutions, and very accurately if the solutions are also radial and compactly supported. By radial we mean functions depending on the space variable only through the geodesic distance r from a given point O∈Hn. We show that the location of the free boundary of compactly supported solutions grows logarithmically for large times, in contrast with the well-known power-like growth of the PME in the Euclidean space. Very slow propagation at long distances is a feature of porous medium flow in hyperbolic space. We also present a non-uniqueness example for the Cauchy Problem based on the construction of an exact generalized traveling wave solution that originates from one point of the infinite horizon. This explicit solutions gives a clue to the asymptotic properties of the fundamental solutions.