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The objective of this paper is twofold. First, we conduct a careful study ofvarious functional inequalities involving the fractional Laplacian operators,including nonlocal Sobolev-Poincar\'e, Nash, Super Poincar\'e and logarithmicSobolev type inequalities, on complete Riemannian manifolds satisfying somemild geometric assumptions. Second, based on the derived nonlocal functionalinequalities, we analyze the asymptotic behavior of the solution to thefractional 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 globalwell-posedness of the equation on an arbitrary complete Riemannian manifold.

evolution equations and control theory

Functional Inequalities involving Nonlocal Operators on Complete  Riemannian Manifolds and Their Applications to The Fractional Porous Medium  Equation