Hölder estimates for singular non-local parabolic equations |
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Authors: | Sunghoon Kim Ki-Ahm Lee |
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Institution: | School of Mathematical Sciences, Seoul National University, San56-1, Shinrim-dong, Kwanak-gu, Seoul 151-747, South Korea |
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Abstract: | In this paper, we establish local Hölder estimate for non-negative solutions of the singular equation (M.P) below, for m in the range of exponents . Since we have trouble in finding the local energy inequality of v directly, we use the fact that the operator σ(−Δ) can be thought as the normal derivative of some extension v? of v to the upper half space (Caffarelli and Silvestre, 2007 5]), 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 behavior of solution of non-local fast diffusion equation near their extinction time. |
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Keywords: | Fractional Laplacian Extension problem Fully non-linear parabolic equations Porous medium equation Fast diffusion equation |
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