scholarly journals Functional inequalities involving nonlocal operators on complete Riemannian manifolds and their applications to the fractional porous medium equation

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Nikolaos Roidos ◽  
Yuanzhen Shao
Keyword(s):  
Porous Medium ◽  
Riemannian Manifolds ◽  
Porous Medium Equation ◽  
Functional Inequalities ◽  
Nonlocal Operators ◽  
Medium Equation ◽  
Complete Riemannian Manifolds

Harnack estimates for a kind of porous medium equation

2021 ◽  
Vol 115 ◽  
pp. 106978
Author(s):  
Feida Jiang ◽  
Xinyi Shen ◽  
Hui Wu
Keyword(s):  
Porous Medium ◽  
Porous Medium Equation ◽  
Medium Equation ◽  
Harnack Estimates

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

2021 ◽  
Author(s):  
Gabriele Grillo ◽  
Giulia Meglioli ◽  
Fabio Punzo
Keyword(s):  
Porous Medium ◽  
Global Existence ◽  
Source Term ◽  
Reaction Diffusion ◽  
Porous Medium Equation ◽  
Variable Density ◽  
Reaction Diffusion Equations ◽  
Medium Equation ◽  
Global Existence Of Solutions ◽  
Smoothing Effects

AbstractWe 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 $${{\mathbb {R}}}^n$$ R n .

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