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

Abstract

<p style='text-indent:20px;'>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, <inline-formula><tex-math id="M1">\begin{document}$ \partial_t u +(-\Delta)^\sigma (|u|^{m-1}u ) = 0 $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M2">\begin{document}$ m>0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ \sigma\in (0, 1) $\end{document}</tex-math></inline-formula>. In addition, we establish the global well-posedness of the equation on an arbitrary complete Riemannian manifold.</p>