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

Abstract

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

Non-linear heat equation on the Hyperbolic space: Global existence and finite-time Blow-up

“…It is beyond our scope to give a detailed account of the results for Porous medium equation. Instead, we refer the interested readers to the manuscripts [9,13,31,38] and the references quoted therein for more detailed treatment in this area.…”

“…Acknowledgments. We are grateful to G. Grillo for mentioning the references [9,10,13] to us and explaining some of their results. D. Ganguly is partially supported by the INSPIRE faculty fellowship (IFA17-MA98).…”

Global existence and blow-up of solutions to the nonlinear porous medium equation

“…The physical applications of the porous medium equation describe widely processes involving fluid flow, heat transfer or diffusion, and its other applications in different fields such as mathematical biology, lubrication, boundary layer theory, and etc. There is a huge literature dealing with an existence and nonexistence of solutions to problem (1.1) for the reaction term u p in the case m = 1 and m > 1, for example, [1,2,4,5,6,7,8,9,10,11,12,13,14,15,16,18,20,24,25,26]. By using the concavity method, Schaefer [27] established a condition on the initial data of a Dirichlet type initial-boundary value problem for the porous medium equation with a power function reaction term when blow-up of the solution in finite time occurs and a global existence of the solution holds.…”

The generalized porous medium equation on graphs: existence and uniqueness of solutions with $\ell^1$ data

“…The GPME has a long story and we invite the interested reader to look at the seminal book by J. L. Vázquez [65] for a detailed and exhaustive account. In recent years, research interest about properties of solutions of the GPME has focused on the Riemannian setting as can be seen from the increasing number of related works, see for example [7,51,25,64,26,31,29,30,6,27,23,28,24,52] and references therein for an overview of the most significant developments.…”