## Nagoya Mathematical Journal

### On the Borel summability of divergent solutions of the heat equation

#### Abstract

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.

#### Article information

Source
Nagoya Math. J., Volume 154 (1999), 1-29.

Dates
First available in Project Euclid: 27 April 2005

https://projecteuclid.org/euclid.nmj/1114631218

Mathematical Reviews number (MathSciNet)
MR1689170

Zentralblatt MATH identifier
0958.35061

Subjects
Primary: 35K05: Heat equation
Secondary: 35A20: Analytic methods, singularities 35B99: None of the above, but in this section

#### Citation

Lutz, D. A.; Miyake, M.; Schäfke, R. On the Borel summability of divergent solutions of the heat equation. Nagoya Math. J. 154 (1999), 1--29. https://projecteuclid.org/euclid.nmj/1114631218

#### References

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