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Archive for Rational Mechanics and Analysis

, Volume 205, Issue 2, pp 673–697 | Cite as

Asymptotic Behavior for a Nonlocal Diffusion Equation in Domains with Holes

  • Carmen Cortázar
  • Manuel Elgueta
  • Fernando Quirós
  • Noemí Wolanski
Article

Abstract

The paper deals with the asymptotic behavior of solutions to a non-local diffusion equation, u t  = J*uu := Lu, in an exterior domain, Ω, which excludes one or several holes, and with zero Dirichlet data on \({\mathbb{R}^N\setminus\Omega}\) . When the space dimension is three or more this behavior is given by a multiple of the fundamental solution of the heat equation away from the holes. On the other hand, if the solution is scaled according to its decay factor, close to the holes it behaves like a function that is L-harmonic, Lu = 0, in the exterior domain and vanishes in its complement. The height of such a function at infinity is determined through a matching procedure with the multiple of the fundamental solution of the heat equation representing the outer behavior. The inner and the outer behaviors can be presented in a unified way through a suitable global approximation.

Keywords

Initial Data Asymptotic Behavior Fundamental Solution Heat Equation Exterior Domain 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Carmen Cortázar
    • 1
  • Manuel Elgueta
    • 1
  • Fernando Quirós
    • 2
  • Noemí Wolanski
    • 3
  1. 1.Departamento de MatemáticaPontificia Universidad Católica de ChileSantiagoChile
  2. 2.Departamento de MatemáticasUniversidad Autónoma de MadridMadridSpain
  3. 3.Departamento de MatemáticaFCEyN, UBA, and IMAS, CONICETBuenos AiresArgentina

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