Convergence results for a class of nonlinear fractional heat equations

摘要

In this article we study various convergence results for a class of nonlinear fractional heat equations of the form $$left{ begin{gathered} u_t (t,x) - mathcal{I}[u(t, cdot )](x) = f(t,x),(t,x) in (0,T) times mathbb{R}^n , hfill u(0,x) = u_0 (x),x in mathbb{R}^n , hfill end{gathered} right.$$ where I is a nonlocal nonlinear operator of Isaacs type. Our aim is to study the convergence of solutions when the order of the operator changes in various ways. In particular, we consider zero order operators approaching fractional operators through scaling and fractional operators of decreasing order approaching zero order operators. We further give rate of convergence in cases when the solution of the limiting equation has appropriate regularity assumptions. 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Silvestre, On the differentiability of the solution to the Hamilton-Jacobi equation with critical fractional diffusion, Advances in Mathematics 226 (2011), 2020–2039.MathSciNetCrossRefMATH About this Article Title Convergence results for a class of nonlinear fractional heat equations Journal Israel Journal of Mathematics Volume 198, Issue 1 , pp 1-34 Cover Date2013-11 DOI 10.1007/s11856-013-0008-9 Print ISSN 0021-2172 Online ISSN 1565-8511 Publisher Springer US Additional Links Register for Journal Updates Editorial Board About This Journal Manuscript Submission Topics Mathematics, general Algebra Group Theory and Generalizations Analysis Applications of Mathematics Theoretical, Mathematical and Computational Physics Industry Sectors Finance, Business & Banking IT & Software Telecommunications Authors Patricio Felmer (1) Erwin Topp (1) Author Affiliations 1. Departamento de Ingeniería Matemática and CMM (UMI 2807 CNRS), Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile Continue reading... To view the rest of this content please follow the download PDF link above.