On the asymptotic behaviour of solutions to the fractional porous medium equation with variable density

TL;DR

This paper investigates the long-term behavior of solutions to the fractional porous medium equation with variable density, revealing different asymptotic states depending on the decay rate of the density at infinity.

Contribution

It establishes the asymptotic limits of solutions based on the decay rate of the variable density, connecting to Barenblatt solutions and fractional elliptic equations.

Findings

Solutions approach Barenblatt-type solutions with slow density decay.

Solutions converge to fractional elliptic solutions with rapid density decay.

The decay rate of density determines the asymptotic behavior of solutions.

Abstract

We are concerned with the long time behaviour of solutions to the fractional porous medium equation with a variable spatial density. We prove that if the density decays slowly at infinity, then the solution approaches the Barenblatt-type solution of a proper singular fractional problem. If, on the contrary, the density decays rapidly at infinity, we show that the minimal solution multiplied by a suitable power of the time variable converges to the minimal solution of a certain fractional sublinear elliptic equation.