The fractional porous medium equation on the hyperbolic space

Authors: Elvise Berchio, Matteo Bonforte (ICMAT-UAM), Debdip Ganguly and Gabriele Grillo

Source: Calculus of Variations and Partial Differential Equations

Date of publication: 15 September 2020



In this article, the authors consider a nonlinear degenerate parabolic equation of porous medium type, i.e. with a convex nonlinearity, whose diffusion is driven by the (spectral) fractional Laplacian on the hyperbolic space, which is the s-power (in the spectral sense) of the standard Laplace-Beltrami operator on that space. This equation is used in classical Porous Medium Equation (PME) models of slow diffusion processes, which are realistic models for density-driven diffusions. Nowadays, the PME has many applications, that stem from physics to engineering, and more recently biology and even finance.

In a previous paper of 2008, two of the authors showed that the negative curvature of the ambient-space somehow accelerates the diffusion processes. Later, Vázquez showed in 2014 that solutions to the PME on the hyperbolic space have faster speed (but still finite) of propagation, precisely they accelerate by a logarithmic factor. In this case, the diffusion is made even faster as a side effect due to the fractional operators, and the speed of propagation becomes infinite. Roughly speaking, the particles processes underlying the fractional or non-local operators allow the particles to jump; hence, the diffusion becomes really fast, since the particles have a nontrivial probability of jumping at infinity. Fractional derivatives (which are integro-differential operators) are often called nonlocal derivatives or operators, because in order to compute the nonlocal derivative of a function at some point, it is necessary to know the value of the function at all points.

In this new article, the fractional porous medium equation has been introduced for the first time on non-compact, geometrically non-trivial examples. This is the simplest nontrivial examples in which the different geometric quantities, like negative curvature, exponential growth of the volume of balls, etc. enter into play in a significant way and provide important differences with respect to the Euclidean case.

In fact, the researchers provide the existence and uniqueness of solutions for this equation in an appropriate weak sense, for data belonging either to the usual Lp spaces or to larger (weighted) spaces, determined either in terms of a ground state of the Laplace-Beltrami operator on the Hyperbolic space, or of the (fractional) Green’s function.  These quantities are strictly connected with the operator and somehow provide a natural environment, larger than the usual one. For example, slowly (polynomially) decreasing data can be included in this theory, instead of the fast (exponentially) decreasing ones. However, one must keep in mind that the hyperbolic measure grows exponentially, hence integrable function must decrease exponentially fast.