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 QuirósEmail author
  • Noemí Wolanski


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.


Initial Data Asymptotic Behavior Fundamental Solution Heat Equation Exterior Domain 
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  1. 1.
    Andreu, F., Mazón, J.M., Rossi, J.D., Toledo, J.: Nonlocal Diffusion Problems. Mathematical Surveys and Monographs, 165. American Mathematical Society, Providence; Real Sociedad Matemática Española, Madrid, 2010Google Scholar
  2. 2.
    Bates P.W., Chen F.: Spectral analysis of traveling waves for nonlocal evolution equations. SIAM J. Math. Anal. 38(1), 116–126 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bates P.W., Chmaj A.: An integrodifferential model for phase transitions: stationary solutions in higher dimensions. J. Stat. Phys. 95(5–6), 1119–1139 (1999)MathSciNetADSzbMATHCrossRefGoogle Scholar
  4. 4.
    Bates P.W., Chmaj A.: A discrete convolution model for phase transitions. Arch. Rational Mech. Anal. 150(4), 281–305 (1999)MathSciNetADSzbMATHCrossRefGoogle Scholar
  5. 5.
    Bates P.W., Zhao G.: Existence, uniqueness and stability of the stationary solution to a nonlocal evolution equation arising in population dispersal. J. Math. Anal. Appl. 332(1), 428–440 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Brändle C., Chasseigne E., Ferreira R.: Unbounded solutions of the nonlocal heat equation. Commun. Pure Appl. Anal. 10(6), 1663–1686 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Brändle C., Quirós F., Vázquez J.L.: Asymptotic behaviour of the porous media equation in domains with holes. Interfaces Free Bound. 9(2), 211–232 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Carrillo C., Fife P.: Spatial effects in discrete generation population models. J. Math. Biol. 50(2), 161–188 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Chasseigne E., Chaves M., Rossi J.D.: Asymptotic behavior for nonlocal diffusion equations. J. Math. Pures Appl. (9) 86(3), 271–291 (2006)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Cortázar C., Elgueta, M., Rossi, J.D., Wolanski, N.: How to approximate the heat equation with Neumann boundary conditions by nonlocal diffusion problems. Arch. Rational Mech. Anal. 187(1), 137–156 (2008)ADSzbMATHCrossRefGoogle Scholar
  11. 11.
    Fife P.: Some nonclassical trends in parabolic and parabolic-like evolutions Trends in Nonlinear Analysis, 153–191. Springer, Berlin (2003)Google Scholar
  12. 12.
    Gilboa G., Osher S.: Nonlocal operators with application to image processing. Multiscale Model. Simul. 7(3), 1005–1028 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Herraiz L.A.: A nonlinear parabolic problem in an exterior domain. J. Differ. Equ. 142(2), 371–412 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Iagar R.G., Vázquez J.L.: Asymptotic analysis of the p-Laplacian flow in an exterior domain. Ann. Inst. H. Poincaré Anal. Non Linéaire 26(2), 497–520 (2009)ADSzbMATHCrossRefGoogle Scholar
  15. 15.
    Ignat L.I., Rossi J.D.: Refined asymptotic expansions for nonlocal diffusion equations. J. Evol. Equ. 8(4), 617–629 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Lederman C., Wolanski N.: Singular perturbation in a nonlocal diffusion model. Commun. Partial Differ. Equ. 31(1–3), 195–241 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Terra J., Wolanski N.: Large time behavior for a nonlocal diffusion equation with absorption and bounded initial data. Discrete Contin. Dyn. Syst. 31(2), 581–605 (2011)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Carmen Cortázar
    • 1
  • Manuel Elgueta
    • 1
  • Fernando Quirós
    • 2
    Email author
  • 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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