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

Nikolaos Roidos; Yuanzhen Shao

doi:10.3934/eect.2021026

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2022, Volume 11, Issue 3: 793-825. Doi: 10.3934/eect.2021026

This issue Previous Article The lifespan of solutions for a viscoelastic wave equation with a strong damping and logarithmic nonlinearity Next Article On a class of differential quasi-variational-hemivariational inequalities in infinite-dimensional Banach spaces

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

Nikolaos Roidos^1, and
Yuanzhen Shao^2, ,

1.
Department of Mathematics, University of Patras, 26504 Rio Patras, Greece
2.
Department of Mathematics, The University of Alabama, Tuscaloosa, AL 35487-0350, USA

^* Corresponding author: Yuanzhen Shao
^* Corresponding author: Yuanzhen Shao

Received: August 31, 2020

Revised: March 31, 2021

Published: June 2022

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Abstract

The objective of this paper is twofold. First, we conduct a careful study of various functional inequalities involving the fractional Laplacian operators, including nonlocal Sobolev-Poincaré, Nash, Super Poincaré and logarithmic Sobolev type inequalities, on complete Riemannian manifolds satisfying some mild geometric assumptions. Second, based on the derived nonlocal functional inequalities, we analyze the asymptotic behavior of the solution to the fractional porous medium equation, $ \partial_t u +(-\Delta)^\sigma (|u|^{m-1}u ) = 0 $ with $ m>0 $ and $ \sigma\in (0, 1) $. In addition, we establish the global well-posedness of the equation on an arbitrary complete Riemannian manifold.

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Mathematics Subject Classification: Primary: 26A33, 35R11, 76S05; Secondary: 35K65, 35K67, 35R01, 39B62.

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