# Long time behavior of solutions of degenerate parabolic equations with inhomogeneous density on manifolds

## TL;DR

This paper studies the long-term behavior of solutions to degenerate parabolic equations with inhomogeneous density on manifolds, revealing how geometry and coefficients influence solution decay, propagation, and blow-up phenomena.

## Contribution

It establishes new estimates on solution decay, propagation speed, and blow-up conditions for degenerate parabolic equations on manifolds with inhomogeneous densities.

## Key findings

- Solutions exhibit finite speed of propagation in subcritical ranges.
- Universal bounds and finite-time blow-up are proven in supercritical ranges.
- The geometry of the manifold and decay of the weight function critically affect solution behavior.

## Abstract

We consider the Cauchy problem for doubly non-linear degenerate parabolic equations on Riemannian manifolds of infinite volume, or in $\R^N$. The equation contains a weight function as a capacitary coefficient which we assume to decay at infinity. We connect the behavior of non-negative solutions to the interplay between such coefficient and the geometry of the manifold, obtaining, in a suitable subcritical range, estimates of the vanishing rate for long times and of the finite speed of propagation. In supercritical ranges we obtain universal bounds and prove blow up in a finite time of the (initially bounded) support of solutions.