We consider the porous medium equation with a power-like reaction term, posed on Riemannian manifolds. Under certain assumptions on p and m in (1.1), and for small enough nonnegative initial data, we prove existence of global in time solutions, provided that the Sobolev inequality holds on the manifold. Furthermore, when both the Sobolev and the Poincaré inequalities hold, similar results hold under weaker assumptions on the forcing term. By the same functional analytic methods, we investigate global existence for solutions to the porous medium equation with source term and variable density in Rn.

Global existence of solutions and smoothing effects for classes of reaction–diffusion equations on manifolds

Grillo G.;Meglioli G.;Punzo F.
2021-01-01

Abstract

We consider the porous medium equation with a power-like reaction term, posed on Riemannian manifolds. Under certain assumptions on p and m in (1.1), and for small enough nonnegative initial data, we prove existence of global in time solutions, provided that the Sobolev inequality holds on the manifold. Furthermore, when both the Sobolev and the Poincaré inequalities hold, similar results hold under weaker assumptions on the forcing term. By the same functional analytic methods, we investigate global existence for solutions to the porous medium equation with source term and variable density in Rn.
Blow-up
Diffusions with weights
Global existence
Reaction–diffusion equations
Riemannian manifolds
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11311/1178699
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