Asymptotic properties of solutions to the Cauchy problem for degenerate parabolic equations with inhomogeneous density on manifolds

TL;DR

This paper analyzes the long-term behavior of solutions to degenerate parabolic equations with inhomogeneous density on noncompact Riemannian manifolds, linking solution properties to geometric and density asymptotics.

Contribution

It provides a classification of solution behaviors based on the density and manifold geometry, introducing a universal geometric characteristic function.

Findings

Solutions stabilize to zero over time

Finite speed of propagation observed

Universal bounds and interface blow-up established

Abstract

We consider the Cauchy problem for doubly nonlinear degenerate parabolic equations with inhomogeneous density on noncompact Riemannian manifolds. We give a qualitative classification of the behavior of the solutions of the problem depending on the behavior of the density function at infinity and the geometry of the manifold, which is described in terms of its isoperimetric function. We establish for the solutions properties as: stabilization of the solution to zero for large times, finite speed of propagation, universal bounds of the solution, blow up of the interface. Each one of these behaviors of course takes place in a suitable range of parameters, whose definition involves a universal geometrical characteristic function, depending both on the geometry of the manifold and on the asymptotics of the density at infinity.