The self-improving property of higher integrability in the obstacle problem for the porous medium equation

2019 ◽  
Vol 26 (5) ◽  
Author(s):  
Yumi Cho ◽  
Christoph Scheven
Keyword(s):  
Porous Medium ◽  
Obstacle Problem ◽  
Porous Medium Equation ◽  
The Self ◽  
Higher Integrability ◽  
Medium Equation

scholarly journals Higher integrability for obstacle problem related to the singular porous medium equation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Qifan Li
Keyword(s):  
Porous Medium ◽  
Weak Solution ◽  
Weak Solutions ◽  
Obstacle Problem ◽  
Porous Medium Equation ◽  
The Self ◽  
Spatial Gradient ◽  
Higher Integrability ◽  
Medium Equation ◽  
Funct Anal

Abstract In this paper we study the self-improving property of the obstacle problem related to the singular porous medium equation by using the method developed by Gianazza and Schwarzacher (J. Funct. Anal. 277(12):1–57, 2019). We establish a local higher integrability result for the spatial gradient of the mth power of nonnegative weak solutions, under some suitable regularity assumptions on the obstacle function. In comparison to the work by Cho and Scheven (J. Math. Anal. Appl. 491(2):1–44, 2020), our approach provides some new aspects in the estimations of the nonnegative weak solution of the obstacle problem.

The obstacle problem for the porous medium equation

2015 ◽  
Vol 363 (1-2) ◽  
pp. 455-499 ◽  
Author(s):  
Verena Bögelein ◽  
Teemu Lukkari ◽  
Christoph Scheven
Keyword(s):  
Porous Medium ◽  
Obstacle Problem ◽  
Porous Medium Equation ◽  
Medium Equation

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 .

HARNACK-TYPE INEQUALITIES FOR THE POROUS MEDIUM EQUATION ON A MANIFOLD WITH NON-NEGATIVE RICCI CURVATURE

2012 ◽  
Vol 23 (04) ◽  
pp. 1250009 ◽  
Author(s):  
JEONGWOOK CHANG ◽  
JINHO LEE
Keyword(s):  
Porous Medium ◽  
Riemannian Manifold ◽  
Ricci Curvature ◽  
Porous Medium Equation ◽  
Complete Riemannian Manifold ◽  
Gradient Estimates ◽  
Medium Equation

We derive Harnack-type inequalities for non-negative solutions of the porous medium equation on a complete Riemannian manifold with non-negative Ricci curvature. Along with gradient estimates, reparametrization of a geodesic and time rescaling of a solution are key tools to get the results.

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