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

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

Abstract

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 

References

  1. 1.
    Applebaum, D.: Lévy Processes and Stochastic Calculus, vol. 116, 2nd edn. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2009)Google Scholar
  2. 2.
    Arisawa, M.: Homogenization of a class of integro-differential equations with Lévy operators. Commun. Partial Differ. Equ. 34, 617–624 (2009)Google Scholar
  3. 3.
    Arisawa, M., Lions, P.-L.: On ergodic stochastic control. Commun. Partial Differ. Equ. 23, 333–358 (1998)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Barles, G.: Asymptotic behavior of viscosity solutions of first Hamilton Jacobi equations. Ricerche Mat. 34, 227–260 (1985)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Barles, G., Chasseigne, E., Ciomaga, A., Imbert, C.: Lipschitz regularity of solutions for mixed integro-differential equations. J. Differ. Equ. 252, 6012–6060 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Barles, G., Chasseigne, E., Imbert, C.: Hölder continuity of solutions of second-order non-linear elliptic integro-differential equations. J. Eur. Math. Soc. (JEMS) 13, 1–26 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Barles, G., Da Lio, F.: On the boundary ergodic problem for fully nonlinear equations in bounded domains with general nonlinear Neumann boundary conditions. Ann. Inst. H. Poincaré Anal. Non Linéaire 22, 521–541 (2005)CrossRefzbMATHGoogle Scholar
  8. 8.
    Barles, G., Da Lio, F., Lions, P.-L., Souganidis, P.E.: Ergodic problems and periodic homogenization for fully nonlinear equations in half-space type domains with Neumann boundary conditions. Indiana Univ. Math. J. 57, 2355–2375 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Barles, G., Imbert, C.: Second-order elliptic integro-differential equations: viscosity solutions’ theory revisited. Ann. Inst. H. Poincaré Anal. Non Linéaire 25, 567–585 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Barles, G., Ishii, H., Mitake, H.: On the large time behavior of solutions of Hamilton-Jacobi equations associated with nonlinear boundary conditions. Arch. Ration. Mech. Anal. 204, 515–558 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Barles, G., Mitake, H.: A PDE approach to large-time asymptotics for boundary-value problems for nonconvex Hamilton–Jacobi equations. Commun. Partial Differ. Equ. 37, 136–168 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Barles, G., Porretta, A., Tchamba, T.T.: On the large time behavior of solutions of the Dirichlet problem for subquadratic viscous Hamilton–Jacobi equations. J. Math. Pures Appl. 94(9), 497–519 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Barles, G., Souganidis, P.E.: On the large time behavior of solutions of Hamilton–Jacobi equations. SIAM J. Math. Anal. 31, 925–939 (2000) (electronic)Google Scholar
  14. 14.
    Barles, G., Souganidis, P.E.: Some counterexamples on the asymptotic behavior of the solutions of Hamilton–Jacobi equations. C. R. Acad. Sci. Paris Sér. I Math 330, 963–968 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Barles, G., Souganidis, P.E.: Space-time periodic solutions and long-time behavior of solutions to quasi-linear parabolic equations. SIAM J. Math. Anal. 32, 1311–1323 (2001) (electronic)Google Scholar
  16. 16.
    Bensoussan, A., Lions, J.-L., Papanicolaou, G.: Asymptotic Analysis for Periodic Structures, vol. 5 of Studies in Mathematics and its Applications. North-Holland Publishing Co., Amsterdam (1978)Google Scholar
  17. 17.
    Chasseigne, E., Chaves, M., Rossi, J.D.: Asymptotic behavior for nonlocal diffusion equations. J. Math. Pures Appl. 86(9), 271–291 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Ciomaga, A.: On the strong maximum principle for second order nonlinear parabolic integro-differential equations. Adv. Differ. Equ. 17, 635–671 (2012)zbMATHMathSciNetGoogle Scholar
  19. 19.
    Da Lio, F.: Large time behavior of solutions to parabolic equations with Neumann boundary conditions. J. Math. Anal. Appl. 339, 384–398 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Davini, A., Siconolfi, A.: A generalized dynamical approach to the large time behavior of solutions of Hamilton–Jacobi equations. SIAM J. Math. Anal. 38, 478–502 (2006) (electronic)Google Scholar
  21. 21.
    Dirr, N., Souganidis, P.E.: Large-time behavior for viscous and nonviscous Hamilton-Jacobi equations forced by additive noise. Siam J Math. Anal. 37, 777–796 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Fathi, A.: Sur la convergence du semi-groupe de Lax-Oleinik. C. R. Acad. Sci. Paris Sér. I Math 327, 267–270 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Forcadel, N., Imbert, C., Monneau, R.: Homogenization of some particle systems with two-body interactions and of the dislocation dynamics. Discrete Contin. Dyn. Syst. 23, 785–826 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Franke, B.: A functional non-central limit theorem for jump-diffusions with periodic coefficients driven by stable Lévy-noise. J. Theoret. Probab. 20, 1087–1100 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Franke, B.: Homogenization of random transport along periodic two-dimensional flows. Stoch. Process. Appl. 119, 327–346 (2009)Google Scholar
  26. 26.
    Fujiwara, T., Tomisaki, M.: Martingale approach to limit theorems for jump processes. Stoch. Stoch. Rep. 50, 35–64 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Giga, Y., Liu, Q., Mitake, H.: Large-time asymptotics for one-dimensional Dirichlet problems for Hamilton-Jacobi equations with noncoercive Hamiltonians. J. Differ. Equ. 252, 1263–1282 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Horie, M., Inuzuka, T., Tanaka, H.: Homogenization of certain one-dimensional discontinuous Markov processes. Hiroshima Math. J. 7, 629–641 (1977)zbMATHMathSciNetGoogle Scholar
  29. 29.
    Ichihara, N., Ishii, H.: Long-time behavior of solutions of Hamilton–Jacobi equations with convex and coercive Hamiltonians. Arch. Ration. Mech. Anal. 194, 383–419 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Imbert, C.: A non-local regularization of first order Hamilton–Jacobi equations. J. Differ. Equ. 211, 218–246 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Imbert, C., Monneau, R., Rouy, E.: Homogenization of first order equations with \((u/\epsilon )\)-periodic Hamiltonians. II. Application to dislocations dynamics. Commun. Partial Differ. Equ. 33, 479–516 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Ishii, H.: Asymptotic solutions for large time of Hamilton–Jacobi equations in Euclidean \(n\) space. Ann. Inst. H. Poincaré Anal. Non Linéaire 25, 231–266 (2008)CrossRefzbMATHGoogle Scholar
  33. 33.
    Ishii, H.: Long-time asymptotic solutions of convex Hamilton–Jacobi equations with Neumann type boundary conditions. Calc. Var. Partial Differ. Equ. 42, 189–209 (2011)Google Scholar
  34. 34.
    Lions, P.-L.: Generalized Solutions of Hamilton–Jacobi Equations, vol. 69 of Research Notes in Mathematics. Pitman (Advanced Publishing Program), Boston, (1982)Google Scholar
  35. 35.
    Lions, P.-L., Papanicolaou, G., Varadhan, S.: Homogenization of Hamilton–Jacobi equations. (1987) (Preprint)Google Scholar
  36. 36.
    Mitake, H.: Asymptotic solutions of Hamilton–Jacobi equations with state constraints. Appl. Math. Optim. 58, 393–410 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  37. 37.
    Namah, G., Roquejoffre, J.-M.: Convergence to periodic fronts in a class of semilinear parabolic equations. NoDEA Nonlinear Differ. Equ. Appl. 4, 521–536 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  38. 38.
    Namah, G., Roquejoffre, J.-M.: Remarks on the long time behaviour of the solutions of Hamilton–Jacobi equations. Commun. Partial Differ. Equ. 24, 883–893 (1999)Google Scholar
  39. 39.
    Rhodes, R., Vargas, V.: Scaling limits for symmetric Itô-Lévy processes in random medium. Stoch. Process. Appl. 119, 4004–4033 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  40. 40.
    Roquejoffre, J.-M.: Convergence to steady states or periodic solutions in a class of Hamilton–Jacobi equations. J. Math. Pures Appl. 80(9), 85–104 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  41. 41.
    Schwab, R.W.: Periodic homogenization for nonlinear integro-differential equations. SIAM J. Math. Anal. 42, 2652–2680 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  42. 42.
    Tabet Tchamba, T.: Large time behavior of solutions of viscous Hamilton–Jacobi equations with superquadratic Hamiltonian. Asymptot. Anal. 66, 161–186 (2010)zbMATHMathSciNetGoogle Scholar
  43. 43.
    Tomisaki, M.: Homogenization of càdlàg processes. J. Math. Soc. Japan 44, 281–305 (1992)CrossRefzbMATHMathSciNetGoogle Scholar

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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