English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We consider nonnegative solutions of the porous medium equation (PME) on aCartan-Hadamard manifold whose negative curvature can be unbounded. We takecompactly supported initial data because we are also interested in freeboundaries. We classify the geometrical cases we study into quasi-hyperbolic,quasi-Euclidean and critical cases, depending on the growth rate of thecurvature at infinity. We prove sharp upper and lower bounds on the long-timebehaviour of the solutions in terms of corresponding bounds on the curvature.In particular we obtain a sharp form of the smoothing effect on such manifolds.We also estimate the location of the free boundary. A global Harnack principlefollows.  We also present a change of variables that allows to transform radiallysymmetric solutions of the PME on model manifolds into radially symmetricsolutions of a corresponding weighted PME on Euclidean space and back. Thisequivalence turns out to be an important tool of the theory.

The porous medium equation on Riemannian manifolds with negative curvature. The large-time behaviour