Optimal estimates for fractional fast diffusion equations
type: Item
instance of: scholarly article
title: Optimal estimates for fractional fast diffusion equations
zbMATH Open document ID: 1327.35417
author: Bruno Volzone
author: Juan Luis Vázquez
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 2015-01-23T00:00:00Z
full work available at URL: https://arxiv.org/abs/1310.3218
review text: Using parabolic comparison results from a previous work [ibid. (9) 101, No. 5, 553--582 (2014; Zbl 1290.35317)], the authors obtain precise a priori estimates with best constants for the solutions of the fractional diffusion equation \(\partial_t u+(-\Delta)^{\sigma/2}A(u)=f\), \(0<\sigma <2\), \(0<m\leq 1\). Some extra regularity conditions are accepted as needed, like \(A\) being smooth with \(A'(u) >0\) for all \(u >0\). The problem is posed in the whole space \(\mathbb{R}^N\). Special attention is paid to cases of the form \(A(u) =u^m\) with \(m >0\); the equation is then called the fractional heat equation when \(m =1\), the fractional porous medium equation if \(m >1\), and the fractional fast diffusion equation if \(m <1\). The linear equation \(\partial_t+(-\Delta)^{\sigma/2}A(u)=0\) is a model of the so-called anomalous diffusion, a much studied topic in physics. Classical results for \(\sigma=2\) are recovered in the limit.
Mathematics Subject Classification ID: 35R11
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 35K55
zbMATH DE Number: 6390092
zbMATH Keywords: fractional Laplacian
zbMATH Keywords: symmetrization
zbMATH Keywords: nonlocal parabolic equations
zbMATH Keywords: comparison theorems
zbMATH Keywords: best constants
reviewed by: Boris V. Loginov
MaRDI profile type: Publication
OpenAlex ID: W2964300420
arXiv ID: 1310.3218
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DOI: 10.1016/J.MATPUR.2014.07.002
description: scientific article