Hölder Estimates for Singular Non-local Parabolic Equations

2011/05/17 by Sung‐Hoon Kim, Kim, Sunghoon, Ki-Ahm Lee +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1105.3286

openalex publication_date 2011/05/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we establish local Hölder estimate for non-negative solutions of the singular equation \eqrefeq-nlocal-PME-1 below, for m in the range of exponents ((n-2σ)/(n+2σ),1). Since we have trouble in finding the local energy inequality of v directly. we use the fact that the operator (-\La)^σ can be thought as the normal derivative of some extension v^∗ of v to the upper half space, \citeCS, i.e., v is regarded as boundary value of v^∗ the solution of some local extension problem. Therefore, the local Hölder estimate of v can be obtained by the same regularity of v^∗. In addition, it enables us to describe the behaviour of solution of non-local fast diffusion equation near their extinction time.

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