Opuscula Math. 35, no. 5 (2015), 625-653
http://dx.doi.org/10.7494/OpMath.2015.35.5.625

 
Opuscula Mathematica

On k-summability of formal solutions for certain partial differential operators with polynomial coefficients

Kunio Ichinobe
Masatake Miyake

Abstract. We study the \(k\)-summability of divergent formal solutions for the Cauchy problem of certain linear partial differential operators with coefficients which are polynomial in \(t\). We employ the method of successive approximation in order to construct the formal solutions and to obtain the properties of analytic continuation of the solutions of convolution equations and their exponential growth estimates.

Keywords: \(k\)-summability, Cauchy problem, power series solutions, successive approximation.

Mathematics Subject Classification: 35C10, 35G10, .

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  1. W. Balser, From Divergent Power Series to Analytic Functions, Springer Lecture Notes, No. 1582, 1994.
  2. W. Balser, Summability of power-series solutions of partial differential equations with constant coefficients, J. Mathematical Sciences 124 (2004) 4, 5085-5097.
  3. K. Ichinobe, Summability of formal solution of Cauchy problem for some PDE with variable coefficients, Recent development of micro-local analysis for the theory of asymptotic analysis, RIMS Kôkyûroku Bessatsu B40 (2013), 081-094.
  4. K. Ichinobe, On a \(k\)-summability of formal solutions for a class of partial differential operators with time dependent coefficients, accepted in Journal of Differential Equations.
  5. K. Ichinobe, M. Miyake, Multisummability of Formal solutions for Some Ordinary Differential Equations, Bulletin of Aichi Univ. of Education, 61 (Natural Sciences), 15-24, March, 2012. http://repository.aichi-edu.ac.jp/dspace/bitstream/10424/4449/1/kenshi611524.pdf.
  6. K. Kitagawa, T. Sadamatsu, A remark on a necessary condition of the Cauchy-Kowalevski theorem, Publ. Research Institute for Mathematical Sciences 11 (1975/76) 2, 523-534.
  7. D. Lutz, M. Miyake, R. Schäfke, On the Borel summability of divergent solutions of the heat equation, Nagoya Math. J. 154 (1999), 1-29.
  8. S. Michalik, Analytic solutions of moment partial differential equations with constant coefficients, Funkcial. Ekvac 56 (2013), 19-50.
  9. S. Michalik, On the summability of formal solutions to some linear partial differential equations, manuscript.
  10. M. Miyake, A remark on Cauchy-Kowalevski's theorem, Publ. Research Institute for Mathematical Sciences 10 (1974/75) 1, 243-255.
  11. M. Miyake, Global and local Goursat problems in a class of holomorphic or partially holomorphic functions, J. Differential Equations 39 (1981), 445-463.
  12. M. Miyake, Borel summability of divergent solutions of the Cauchy problem to non-Kowalevskian equations, Partial Differential Equations and Their Applications (Wuhan, 1999), 225-239, World Sci. Publishing, 1999.
  13. M. Miyake, Y. Hashimoto, Newton polygons and Gevrey indices for linear partial differential operators, Nagoya Math. J. 128 (1992), 15-47.
  14. S. Mizohata, On Cauchy-Kowalevski's theorem; a necessary condition, Publ. Research Institute for Mathematical Sciences 10 (1974/75), 509-519.
  15. S. Mizohata, On Cauchy-Kowalevski's theorem, Math. Analysis and Appl., Advanced in Math., Suppl. Studies 7B (1981), 617-652.
  16. A. Shirai, Maillet type theorem for nonlinear partial differential equations and Newton polygons, J. Mathematical Society of Japan 53 (2001) 3, 565-587.
  • Kunio Ichinobe
  • Department of Mathematics Education, Aichi University of Education, Japan
  • Masatake Miyake
  • Graduate School of Mathematics, Nagoya University, Japan
  • Communicated by P.A. Cojuhari.
  • Received: 2014-02-10.
  • Revised: 2014-06-03.
  • Accepted: 2014-07-23.
  • Published online: 2015-04-27.
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Cite this article as:
Kunio Ichinobe, Masatake Miyake, On k-summability of formal solutions for certain partial differential operators with polynomial coefficients, Opuscula Math. 35, no. 5 (2015), 625-653, http://dx.doi.org/10.7494/OpMath.2015.35.5.625

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