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On the Borel summability of divergent solutions of the heat equation

Published online by Cambridge University Press:  22 January 2016

D. A. Lutz ,
M. Miyake  and
R. Schäfke
D. A. Lutz
Affiliation:
Department of Mathematical Science, San Diego State University, San Diego, CA, 92182-7720, USA
M. Miyake
Affiliation:
Graduate School of Mathematics, Nagoya University, Nagoya, 464-8602, Japan
R. Schäfke
Affiliation:
Départment de Mathématique, Universitée de Strasbourg, 7, rue Renée-Descartes, 67084, Strasbourg, France
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Abstract

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In recent years, the theory of Borel summability or multisummability of divergent power series of one variable has been established and it has been proved that every formal solution of an ordinary differential equation with irregular singular point is multisummable. For partial differential equations the summability problem for divergent solutions has not been studied so well, and in this paper we shall try to develop the Borel summability of divergent solutions of the Cauchy problem of the complex heat equation, since the heat equation is a typical and an important equation where we meet diveregent solutions. In conclusion, the Borel summability of a formal solution is characterized by an analytic continuation property together with its growth condition of Cauchy data to infinity along a stripe domain, and the Borel sum is nothing but the solution given by the integral expression by the heat kernel. We also give new ways to get the heat kernel from the Borel sum by taking a special Cauchy data.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 154 , 1999 , pp. 1 - 29
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1999

References

[Bal] Balser, W., From divergent power series to analytic functions, Lecture Notes in Mathematics 1582, Springer, Germany, 1994.Google Scholar
[Doe] Doetsch, G., Handbuch der Laplace-Transformation Vol II, Birkhäauser Verlag Basel und Stuttgart, Ulm, Germany, 1955.Google Scholar
[Ec] Ecalle, J., Les fonctions résurgentes I–III, Publication mathématiques d’Orsay, Paris, France, 1981, 1985.Google Scholar
[Ger-Tah] Géerard, R. and Tahara, H., Singular nonlinear partial differential equations, Vieweg, 1996.Google Scholar
[Kow] Kowalevski, S., Zur Theorie der partiellen Differentialeichungen, J. Reine Angew. Math., 80 (1875), 1–32.Google Scholar
[Mal] Malgrange, N., Sommation des Séries divergentes, Expositiones Mathematicae, 13 (1995), Heidelberg, Germany, 163–222.Google Scholar
[Mal-Ram] Malgrange, B. and Ramis, J. P., Fonctions multisommables, Ann. Inst. Fourier, 41 (1991), 1–16.Google Scholar
[Miy 1] Miyake, M., A remark on Cauchy-Kowalevski’s theorem, Publ. Res. Inst. Sci., 10 (1974), 243–255.CrossRefGoogle Scholar
[Miy 2] Miyake, M., Global and local Goursat problems in a class of analytic or partially analytic functions, J. Differential Equations, 39 (1981), 445–463.Google Scholar
[Miy 3] Miyake, M., Newyon polygons and formal Gevrey indices in the Cauchy-Goursat- Fuchs type equations, J. Math. Soc. Japan, 41 (1991), 305–330.Google Scholar
[Miy-Has] Miyake, M. and Hashimoto, Y., Newton polygons and Gevrey indices for partial differential operators, Nagoya Math. J., 128 (1992), 15–47.Google Scholar
[Miy-Yos 1] Miyake, M. and Yoshino, M., Wiener-Hopf equation and Fredholm property of the Goursat problem in Gevrey space, Nagoya Math. J., 135 (1994), 165–196.Google Scholar
[Miy-Yos 2] Miyake, M. and Yoshino, M., Fredholm property for differential operators on formal Gevrey space and Toeplitz operator method, C. R. Acad. Bulgare des Sci., 47 (1994), 21–26.Google Scholar
[Nev] Nevanlinna, F., Zur Theorie der asymptotischen Potenzreihen, Ann. Acad. Sci. Fenn. Ser. A1, Math. Dissertationes 12, 1918.Google Scholar
[Ōuc] Ōuchi, S., Characteristic Cauchy problems and solutions of formal power series, Ann. Inst. Fourier, 33 (1983), 131–176.CrossRefGoogle Scholar
[Pic] Picard, E., Leon sur quelques types simples d’équations aux déerivéees partielles avec des applications à la physique mathématique, Gauthier-Villars, Paris, 1927.Google Scholar
[Ram 1] Ramis, J. P., Théorèmes d’indices Gevrey pour les équations différentielles ordinaire, Mem. Amer. Math. Soc. 48, No. 296, 1984.Google Scholar
[Ram 2] Ramis, J. P., Les séries k-sommables et leurs applications, Springer Notes in Physics, 126 (1980), 178–209.Google Scholar
[Was] Wasow, W., Asymptotic expansions for ordinary differential equations, Interscience Publ. John Wiley and Sons, Inc., New York, London, Sydney, 1965.Google Scholar
[Yon] Yonemura, A., Newton polygons and formal Gevrey classes, Publ. Res. Inst. Math. Sci., 26 (1990), 197–204.Google Scholar