In this paper, we establish local H\"older estimate for non-negative solutions of the singular equation \eqref{ eq-nlocal-PME-1} below, for $m$ in the range of exponents $(\frac{ n-2\sigma}{ n+2\sigma},1)$. Since we have trouble in finding the local energy inequality of $v$ directly. we use the fact that the operator $(-\La)^{ \sigma}$ can be thought as the normal derivative of some extension $v^{ \ast}$ of $v$ to the upper half space, \cite{ CS}, i.e., $v$ is regarded as boundary value of $v^{ \ast}$ the solution of some local extension problem. Therefore, the local H\"older estimate of $v$ can be obtained by the same regularity of $v^{ \ast}$. In addition, it enables us to describe the behaviour of solution of non-local fast diffusion equation near their extinction time.
Categories: math.AP