Advertisement

Gevrey Properties and Summability of Formal Power Series Solutions of Some Inhomogeneous Linear Cauchy-Goursat Problems

  • Pascal RemyEmail author
Article
  • 2 Downloads

Abstract

In this article, we investigate the Gevrey and summability properties of the formal power series solutions of some inhomogeneous linear Cauchy-Goursat problems with analytic coefficients in a neighborhood of \((0,0)\in \mathbb {C}^{2}\). In particular, we give necessary and sufficient conditions under which these solutions are convergent or are k-summable, for a convenient positive rational number k, in a given direction.

Keywords

Linear partial differential equation Linear integro-differential equation Divergent power series Newton polygon Gevrey order Gevrey asymptotic Summability 

Mathematics Subject Classification (2010)

35C10 35C20 40B05 

Notes

References

  1. 1.
    Balser W. Divergent solutions of the heat equation: on an article of Lutz, Miyake and Schäfke. Pacific J Math 1999;188(1):53–63.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Balser W. Formal Power Series and Linear Systems of Meromorphic Ordinary Differential Equations. New-York: Springer; 2000.zbMATHGoogle Scholar
  3. 3.
    Balser W. Multisummability of formal power series solutions of partial differential equations with constant coefficients. J Differential Equations 2004;201(1):63–74.MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Balser W, Loday-Richaud M. Summability of solutions of the heat equation with inhomogeneous thermal conductivity in two variables. Adv Dyn Syst Appl 2009;4 (2):159–177.MathSciNetGoogle Scholar
  5. 5.
    Balser W, Miyake M. Summability of formal solutions of certain partial differential equations. Acta Sci Math (Szeged) 1999;65(3-4):543–551.MathSciNetzbMATHGoogle Scholar
  6. 6.
    Balser W, Yoshino M. Gevrey order of formal power series solutions of inhomogeneous partial differential equations with constant coefficients. Funkcial Ekvac 2010;53:411–434.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Canalis-Durand M, Ramis J-P, Schäfke R, Sibuya Y. Gevrey solutions of singularly perturbed differential equations. J Reine Angew Math 2000;518:95–129.MathSciNetzbMATHGoogle Scholar
  8. 8.
    Costin O, Park H, Takei Y. Borel summability of the heat equation with variable coefficients. J Differential Equations 2012;252(4):3076–3092.MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Hibino M. Borel summability of divergence solutions for singular first-order partial differential equations with variable coefficients. I & II. J Differential Equations 2006;227(2):499–563.MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Hibino M. On the summability of divergent power series solutions for certain first-order linear PDEs. Opuscula Math 2015;35(5):595–624.MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Ichinobe K. On k-summability of formal solutions for a class of partial differential operators with time dependent coefficients. J Differential Equations 2014;257(8):3048–3070.MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Lastra A, Malek S. On parametric Gevrey asymptotics for some nonlinear initial value Cauchy problems. J Differential Equations 2015;259:5220–5270.MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Lastra A, Malek S. On parametric multisummable formal solutions to some nonlinear initial value Cauchy problems. Adv Differ Equ 2015;2015:200.MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Lastra A, Malek S, Sanz J. On Gevrey solutions of threefold singular nonlinear partial differential equations. J Differential Equations 2013;255:3205–3232.MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Loday-Richaud M. Stokes phenomenon, multisummability and differential Galois groups. Ann Inst Fourier (Grenoble) 1994;44(3):849–906.MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Loday-Richaud M. Divergent Series, Summability and Resurgence II. Simple and Multiple Summability, vol 2154 of Lecture Notes in Math. Berlin: Springer; 2016.zbMATHGoogle Scholar
  17. 17.
    Lutz DA, Miyake M, Schäfke R. On the Borel summability of divergent solutions of the heat equation. Nagoya Math J 1999;154:1–29.MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Malek S. On the summability of formal solutions of linear partial differential equations. J Dyn Control Syst 2005;11(3):389–403.MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Malek S. On singularly perturbed partial integro-differential equations with irregular singularity. J Dyn Control Syst 2007;13(3):419–449.MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Malek S. On the summability of formal solutions of nonlinear partial differential equations with shrinkings. J Dyn Control Syst 2007;13(1):1–13.MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Malek S. On the Stokes phenomenon for holomorphic solutions of integrodifferential equations with irregular singularity. J Dyn Control Syst 2008;14(3):371–408.MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Malek S. On Gevrey functions solutions of partial differential equations with fuchsian and irregular singularities. J Dyn Control Syst 2009;15(2):277–305.MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Malek S. On Gevrey asymptotic for some nonlinear integro-differential equations. J Dyn Control Syst 2010;16(3):377–406.MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Malek S. On the summability of formal solutions for doubly singular nonlinear partial differential equations. J Dyn Control Syst 2012;18(1):45–82.MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Malgrange B. Sommation des séries divergentes. Expo Math 1995;13:163–222.zbMATHGoogle Scholar
  26. 26.
    Malgrange B, Ramis J-P. Fonctions multisommables. Ann Fourier (Grenoble) 1992;42:353–368.MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Michalik S. Summability of divergent solutions of the n-dimensional heat equation. J Differential Equations 2006;229:353–366.MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Michalik S. On the multisummability of divergent solutions of linear partial differential equations with constant coefficients. J Differential Equations 2010;249:551–570.MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Michalik S. Summability and fractional linear partial differential equations. J Control Syst 2010;16(4):557–584.MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Michalik S. Multisummability of formal solutions of inhomogeneous linear partial differential equations with constant coefficients. J Dyn Control Syst 2012;18(1):103–133.MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Michalik S. Corrigendum to “On the multisummability of divergent solutions of linear partial differential equations with constant coefficients” [j. differential equations 249 (3) (2010) 551-570]. J Differential Equations 2013;255:2400–2401.MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Miyake M. Newton polygons and formal Gevrey indices in the Cauchy-Goursat-Fuchs type equations. J Math Soc Newton Japan 1991;43(2):305–330.MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Miyake M. Borel summability of divergent solutions of the Cauchy problem to non-Kovaleskian equations. Partial differential equations and their applications (Wuhan, 1999). River Edge: World Sci. Publ.; 1999. p. 225–239.Google Scholar
  34. 34.
    Miyake M, Hashimoto Y. Newton polygons and Gevrey indices for linear partial differential operators. Nagoya Math J 1992;128:15–47.MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Nagumo M. ÜBer das Anfangswertproblem partieller Differentialgleichungen. Jap J Math 1942;18:41–47.MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Ouchi S. Multisummability of formal solutions of some linear partial differential equations. J Differential Equations 2002;185(2):513–549.MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Ouchi S. Borel summability of formal solutions of some first order singular partial differential equations and normal forms of vector fields. J Math Soc Japan 2005;57(2): 415–460.MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Pliś ME, Ziemian B. Borel resummation of formal solutions to nonlinear L,aplace equations in 2 variables. Ann Polon Math 1997;67(1):31–41.MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Ramis J-P. Dévissage Gevrey. Astérisque Soc Math France, Paris 1978;59-60: 173–204.zbMATHGoogle Scholar
  40. 40.
    Ramis J-P. Les séries k-sommables et leurs applications. Complex analysis, microlocal calculus and relativistic quantum theory (Proc Internat Colloq, Centre Phys, Les Houches, 1979), vol 126 of Lecture Notes in Phys. Berlin: Springer; 1980. p. 178–199.Google Scholar
  41. 41.
    Ramis J-P. Théorèmes d’indices Gevrey pour les équations différentielles ordinaires. Mem Amer Math Soc 1984;48:viii+ 95.zbMATHGoogle Scholar
  42. 42.
    Remy P. Gevrey order and summability of formal series solutions of some classes of inhomogeneous linear partial differential equations with variable coefficients. J Dyn Gevrey Control Syst 2016;22(4):693–711.MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Remy P. Gevrey order and summability of formal series solutions of certain classes of inhomogeneous linear integro-differential equations with variable coefficients. J Dyn Gevrey Control Syst 2017;23(4):853–878.MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Tahara H, Yamazawa H. Multisummability of formal solutions to the C,auchy problem for some linear partial differential equations. J Differential Equations 2013;255 (10):3592–3637.MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Yonemura A. Newton polygons and formal Gevrey classes. Publ Res Inst Math Newton Sci 1990;26:197–204.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques de VersaillesUniversité de Versailles Saint-QuentinVersailles CedexFrance

Personalised recommendations