Large time behavior of periodic viscosity solutions for uniformly parabolic integro-differential equations

  • Guy Barles
  • Emmanuel Chasseigne
  • Adina CiomagaEmail author
  • Cyril Imbert


In this paper, we study the large time behavior of solutions of a class of parabolic fully nonlinear integro-differential equations in a periodic setting. In order to do so, we first solve the ergodic problem (or cell problem), i.e. we construct solutions of the form \(\lambda t + v(x).\) We then prove that solutions of the Cauchy problem look like those specific solutions as time goes to infinity. We face two key difficulties to carry out this classical program: (1) the fact that we handle the case of “mixed operators” for which the required ellipticity comes from a combination of the properties of the local and nonlocal terms and (2) the treatment of the superlinear case (in the gradient variable). Lipschitz estimates previously proved by the authors (2012) and Strong Maximum principles proved by the third author (2012) play a crucial role in the analysis.

Mathematics Subject Classification (1991)

35B40 35R09 35D40 35D10 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Guy Barles
    • 1
  • Emmanuel Chasseigne
    • 1
  • Adina Ciomaga
    • 2
    Email author
  • Cyril Imbert
    • 3
  1. 1.Laboratoire de Mathématiques et Physique Théorique, CNRS UMR 7350, Fédération Denis PoissonUniversité François RabelaisToursFrance
  2. 2.Department of MathematicsUniversity of ChicagoChicagoUSA
  3. 3.CNRS, Laboratoire d’Analyse et de Mathématiques Appliquées, UMR 8050Université Paris-Est CréteilCréteil cedexFrance

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