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

2019/05/26 by Daniele Andreucci, Andreucci, Daniele, Anatoli F. Tedeev +1
Computer Science · Mathematics · #35B40 #35K55 #35K65 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1905.10803

openalex publication_date 2019/05/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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