Opuscula Math. 46, no. 1 (2026), 5-40
https://doi.org/10.7494/OpMath.202512271

 
Opuscula Mathematica

Parametric formal Gevrey asymptotic expansions in two complex time variable problems

Guoting Chen
Alberto Lastra
Stéphane Malek

Abstract. The analytic and formal solutions to a family of singularly perturbed partial differential equations in the complex domain involving two complex time variables are considered. The analytic continuation properties of the solution of an auxiliary problem in the Borel plane overcomes the absence of adequate domains which would guarantee summability of the formal solution. Moreover, several exponential decay rates of the difference of analytic solutions with respect to the perturbation parameter at the origin are observed, leading to several asymptotic levels relating the analytic and the formal solution.

Keywords: singularly perturbed, formal solution, several complex variables, Cauchy problem.

Mathematics Subject Classification: 35C10, 35R10, 35C15, 35C20.

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  1. W. Balser, Multisummability of complete formal solutions for non-linear systems of meromorphic ordinary differential equations, Complex Var. Theory Appl. 34 (1997), 19-24. https://doi.org/10.1080/17476939708815034
  2. W. Balser, Formal Power Series and Linear Systems of Meromorphic Ordinary Differential Equations, Universitext, Springer-Verlag, New York, 2000. https://doi.org/10.1007/b97608
  3. W. Balser, Multisummability of formal power series solutions of partial differential equations with constant coefficients, J. Differential Equations 201 (2004), 63-74. https://doi.org/10.1016/j.jde.2004.02.002
  4. W. Balser, B. Braaksma, J.-P. Ramis, Y. Sibuya, Multisummability of formal power series solutions of linear ordinary differential equations, Asymptot. Anal. 5 (1991), 27-45. https://doi.org/10.3233/asy-1991-5102
  5. B. Braaksma, Multisummability of formal power series solutions of nonlinear meromorphic differential equations, Ann. Inst. Fourier (Grenoble) 42 (1992), 517-540. https://doi.org/10.5802/aif.1301
  6. G. Chen, A. Lastra, S. Malek, Parametric Gevrey asymptotics in two complex time variables through truncated Laplace transforms, Adv. Differ. Equ. 2020 (2020), Article no. 307. https://doi.org/10.1186/s13662-020-02773-z
  7. O. Costin, S. Tanveer, Short time existence and Borel summability in the Navier-Stokes equation in \(\mathbb{R}^{3}\), Comm. Partial Differential Equations 34 (2009), 785-817. https://doi.org/10.1080/03605300902892469
  8. P. Hsieh, Y. Sibuya, Basic Theory of Ordinary Differential Equations, Universitext, Springer-Verlag, New York, 1999. https://doi.org/10.1007/978-1-4612-1506-6
  9. J. Jiménez-Garrido, S. Kamimoto, A. Lastra, J. Sanz, Multisummability in Carleman ultraholomorphic classes by means of nonzero proximate orders, J. Math. Anal. Appl. 472 (2019), 627-686. https://doi.org/10.1016/j.jmaa.2018.11.043
  10. A. Lastra, S. Malek, On parametric Gevrey asymptotics for some nonlinear initial value Cauchy problems, J. Differential Equations 259 (2015), 5220-5270. https://doi.org/10.1016/j.jde.2015.06.020
  11. A. Lastra, S. Malek, On parametric multisummable formal solutions to some nonlinear initial value Cauchy problems, Adv. Differ. Equ. 2015 (2015), Paper no. 200. https://doi.org/10.1186/s13662-015-0541-4
  12. A. Lastra, S. Malek, On parametric Gevrey asymptotics for some initial value problems in two asymmetric complex time variables, Results Math. 73 (2018), Article no. 155. https://doi.org/10.1007/s00025-018-0914-6
  13. A. Lastra, S. Malek, Multiscale Gevrey asymptotics in boundary layer expansions for some initial value problem with merging turning points, Adv. Differ. Equ. 24 (2019), 69-136. https://doi.org/10.57262/ade/1544497235
  14. A. Lastra, S. Malek, Boundary layer expansions for initial value problems with two complex time variables, Adv. Differ. Equ. 2020 (2020), Paper no. 20. https://doi.org/10.1186/s13662-020-2496-3
  15. A. Lastra, S. Malek, On parametric Gevrey asymptotics for some nonlinear initial value problems in symmetric complex time variables, Asymptotic Anal. 118 (2020), 49-79. https://doi.org/10.3233/asy-191568
  16. A. Lastra, S. Michalik, M. Suwińska, Multisummability of formal solutions for a family of generalized singularly perturbed moment differential equations, Results Math. 78 (2023), Paper no. 49. https://doi.org/10.1007/s00025-022-01828-9
  17. M. Loday-Richaud, Stokes phenomenon, multisummability and differential Galois groups, Ann. Inst. Fourier (Grenoble) 44 (1994), 849-906. https://doi.org/10.5802/aif.1419
  18. M. Loday-Richaud, Divergent Series, Summability and Resurgence II. Simple and Multiple Summability, Lecture Notes in Mathematics, vol. 2154, Springer, 2016. https://doi.org/10.1007/978-3-319-29075-1
  19. S. Malek, On a partial \(q\)-analog of a singularly perturbed problem with fuchsian and irregular time singularities, Abstr. Appl. Anal. 2020 (2020), Article ID 7985298. https://doi.org/10.1155/2020/7985298
  20. S. Malek, Double-scale expansions for a logarithmic type solution to a \(q\)-analog of a singular initial value problem, Abstr. Appl. Anal. 2023 (2023), Article ID 3025513. https://doi.org/10.20944/preprints202310.1781.v1
  21. B. Malgrange, J.-P. Ramis, Fonctions multisommables, Ann. Inst. Fourier (Grenoble) 42 (1992), 353-368. https://doi.org/10.5802/aif.1295
  22. S. Michalik, On the multisummability of divergent solutions of linear partial differential equations with constant coefficients, J. Differential Equations 249 (2010), 551-570. https://doi.org/10.1016/j.jde.2010.03.018
  23. S. Michalik, Multisummability of formal solutions of inhomogeneous linear partial differential equations with constant coefficients, J. Dyn. Control Syst. 18 (2012), 103-133. https://doi.org/10.1007/s10883-012-9136-5
  24. S. Ouchi, Multisummability of formal power series solutions of nonlinear partial differential equations in complex domains, Asymptotic Anal. 47 (2006), 187-225. https://doi.org/10.3233/asy-2006-745
  25. J.-P. Ramis, Y. Sibuya, A new proof of multisummability of formal solutions of nonlinear meromorphic differential equations, Ann. Inst. Fourier (Grenoble) 44 (1994), 811-848. https://doi.org/10.5802/aif.1418
  26. H. Tahara, Asymptotic existence theorem for formal solutions with singularities of nonlinear partial differential equations via multisummability, J. Math. Soc. Japan 75 (2023), 1055-1117. https://doi.org/10.2969/jmsj/88248824
  27. H. Tahara, H. Yamazawa, Multisummability of formal solutions to the Cauchy problem for some linear partial differential equations, J. Differential Equations 255 (2013), 3592-3637. https://doi.org/10.1016/j.jde.2013.07.061
  28. Y. Takei, On the multisummability of WKB solutions of certain singularly perturbed linear ordinary differential equations, Opuscula Math. 35 (2015), 775-802. https://doi.org/10.7494/opmath.2015.35.5.775
  29. H. Yamazawa, On multisummability of formal solutions with logarithmic terms for some linear partial differential equations, Funkc. Ekvacioj 60 (2017), 371-406. https://doi.org/10.1619/fesi.60.371
  30. M. Yoshino, Parametric Borel summability of partial differential equations of irregular singular type, [in:] Analytic, Algebraic and Geometric Aspects of Differential Equations, Trends Math., Birkhäuser-Springer, Cham, 2017, 455-471. https://doi.org/10.1007/978-3-319-52842-7_15
  • Communicated by Yoshishige Haraoka.
  • Received: 2025-09-08.
  • Revised: 2025-12-12.
  • Accepted: 2025-12-27.
  • Published online: 2026-01-27.
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Cite this article as:
Guoting Chen, Alberto Lastra, Stéphane Malek, Parametric formal Gevrey asymptotic expansions in two complex time variable problems, Opuscula Math. 46, no. 1 (2026), 5-40, https://doi.org/10.7494/OpMath.202512271

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