Blow-up and global existence for the porous medium equation with\n reaction on a class of Cartan-Hadamard manifolds

2018/04/05 by Gabriele Grillo, Grillo, Gabriele, Matteo Muratori +3
Mathematics · #Advanced Mathematical Physics Problems #Nonlinear Partial Differential Equations #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.1804.02078

Abstract

We consider the porous medium equation with power-type reaction terms u^p\non negatively curved Riemannian manifolds, and solutions corresponding to\nbounded, nonnegative and compactly supported data. If p>m, small data give\nrise to global-in-time solutions while solutions associated to large data blow\nup in finite time. If p<m, large data blow up at worst in infinite time, and\nunder the stronger restriction p\∈(1,(1+m)/2] all data give rise to\nsolutions existing globally in time, whereas solutions corresponding to large\ndata blow up in infinite time. The results are in several aspects significantly\ndifferent from the Euclidean ones, as has to be expected since negative\ncurvature is known to give rise to faster diffusion properties of the porous\nmedium equation.\n

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