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

作者：Grillo Gabriele; Muratori Matteo; Luis Vazquez Juan

来源：Advances in Mathematics, 2017, 314: 328-377.

DOI：10.1016/j.aim.2017.04.023

• Porous medium equation
• Flows on manifolds
• Asymptotic behaviour
• Finite propagation

We consider nonnegative solutions of the porous medium equation (PME) on Cartan-Hadamard manifolds whose negative curvature can be unbounded. We take compactly supported initial data because we are also interested in free boundaries. We classify the geometrical cases we study into quasi-hyperbolic, quasi-Euclidean and critical cases, depending on the growth rate of the curvature at infinity. We prove sharp upper and lower bounds on the long-time behaviour of the solutions in terms of corresponding bounds on the curvature. In particular we estimate the location of the free boundary. A global Harnack principle follows.
We also present a change of variables that allows to transform radially symmetric solutions of the PME on model manifolds into radially symmetric solutions of a corresponding weightedPME on Euclidean space and back. This equivalence turns out to be an important tool of the theory.

• 出版日期2017-7-9