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

Abstract

Available online xxxx MSC: 35K65 58J35 35K55 35B40 Keywords: Porous Medium Equation Hyperbolic space Fundamental solutions Asymptotic behaviorWe construct the fundamental solution of the Porous Medium Equation posed in the hyperbolic space H n 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 ∈ H n . 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. © 2015 Elsevier Masson SAS. All rights reserved. r é s u m é On construit la solution fondamentale de l'équation des milieux poreux posée dans l'espace hyperbolique H n et on décrit son comportement asymptotique quand t → ∞. On montre également que cette solution spéciale décrit le comportement sur le long terme de toutes les solutions à données intégrables et non négatives, et très précisément si les solutions sont aussi radiales et à support compact. Par radiale on entend des fonctions qui dépendent de la variable d'espace via la distance géodésique mesurée à partir d'un point donné O ∈ H n .De plus, on montre que la frontière libre des solutions à support compact est située à une distance logarithmique pour les grands temps, contrairement à la propriété de croissance de type puissance en temps qui est bien connue pour la PME dans l'espace euclidien. On établit que la propagation très lente pour de longues distances est une caractéristique du flux de type milieux poreux dans l'espace hyperbolique. On présente également un exemple de non-unicité pour le problème de Cauchy basé sur la construction d'une solution exacte du type onde progressive généralisée, qui a pour origine un point de l'horizon infini. Cette solution exacte aide à expliquer les propriétés asymptotiques des solutions fondamentales.

The porous medium equation on Riemannian manifolds with negative curvature. The large-time behaviour

“…In the precise case of H n we recover the estimates obtained in [24], although a more accurate analysis was carried out there in order to get a sharp estimate over (1.4), namely (1.5) R(t) ∼ log t (m − 1)(n − 1) (where the curvature is taken as −1). On the other hand, in n-dimensional Euclidean space it is well known (see e.g.…”

“…In [24], the last of the present authors constructed and studied the fundamental tool for studying the asymptotic behaviour of general solutions of (1.1) on H n , namely the Barenblatt solutions, which are solutions of (1.1) corresponding to a Dirac delta as initial datum. Two of the most important results of [24] can be summarized as follows:…”

“…To start with, we assume that both bounds behave like −r 2µ as r → ∞, with µ ∈ (−1, 1) and r = dist(x, o), o being a given pole and dist being Riemannian distance. We call this range for convenience the quasi-hyperbolic range, since the results bear a resemblance with the study of the equation on the hyperbolic space H n done in [24]. In that case our results imply that positive solutions of equation (1.1) starting from bounded and compactly supported initial data satisfy for all large times Moreover, if we denote by R(t) the radius of the smallest ball that contains the support of the solution at time t (or, equivalently, the biggest ball contained in the support), then (1.4) R(t) ≈ (log t)…”

“…i) The case of the hyperbolic space treated in [24] basically corresponds to the choice µ 1 = µ 2 = 0.…”

“…Such a transformation, which by the way applies to a larger class of diffusion problems, has been introduced by [24,Section 6], and was an essential tool in the study of the asymptotics of solutions of the porous medium equation in the special case of hyperbolic space. Here, we aim at extending it to a quite general class of quasi-hyperbolic (according to the terminology adopted in Section 1.1) or super-hyperbolic (namely, corresponding to µ > 1) Riemannian models of dimension n ≥ 3.…”