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

Abstract

We construct the fundamental solution of the Porous Medium Equation posed in the hyperbolic space $H^n$ and describe its asymptotic behaviour as $t\to\infty$. We also show that it describes the long time behaviour 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 $O\in H^n$. We also construct an exact generalized traveling wave solution. 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.