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On the summability of formal solutions of linear partial differential equations. (English) Zbl 1085.35043
Linear partial differential equations (PDEs) with holomorphic coefficients are under consideration. The goal of the paper is to provide a general approach to the construction of Borel summable formal series solution of above mentioned PDEs. Sufficient conditions for the existence and uniqueness of formal series in variable $$t$$ are given, and the $$k$$-summability properties of these formal series are investigated.

##### MSC:
 35C10 Series solutions to PDEs 35C20 Asymptotic expansions of solutions to PDEs 40G10 Abel, Borel and power series methods
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##### References:
 [1] 1. W. Arendt, C. Batty, M. Hieber, and F. Neubrander, Vector-valued Laplace transforms and Cauchy problems. Monogr. Math. 96 (2001). · Zbl 0978.34001 [2] 2. W. Balser, Formal power series and linear systems of meromorphic ordinary differential equations. Springer-Verlag, New York (2000). · Zbl 0942.34004 [3] 3. W. Balser, Multisummability of formal power series solutions of partial differential equations with constant coefficients. J. Differential Equations 201 (2004), No. 1, 63–74. · Zbl 1052.35048 [4] 4. W. Balser and S. Malek, Formal solutions of the complex heat equation in higher spatial dimensions. Ulmer Seminar (2003). [5] 5. F. Bergeron and U. Sattler, Constructible differentially finite algebraic series in several variables. Theor. Comput. Sci. 144 (1995), No. 1–2, 59–65. · Zbl 0874.68010 [6] 6. M. Bousquet-Mélou and M. Petkovsek, Linear recurrences with constant coefficients: the multivariate case. Discrete Math. 225 (2000), No. 1–3, 51–75. · Zbl 0963.05005 [7] O. Costin and S. Tanveer, On the existence and uniqueness of solutions of nonlinear evolution systems of PDEs in $$\mathbb{R}$$+ {$$\times$$} $$\mathbb{C}$$d, their asymptotic and Borel summability properties. Manuscript (2003). · Zbl 1236.35019 [8] 8. O. Costin and S. Tanveer, Complex singularity analysis for a nonlinear PDE. Preprint (2003). · Zbl 1136.35332 [9] 9. J. Écalle, Les fonctions résurgentes. Publ. Math. Orsay (1981). · Zbl 0499.30034 [10] 10. R. Gérard and H. Tahara, Formal power series solutions of nonlinear first order partial differential equations. Funkcial. Ekvac. 41 (1998), 133–166. · Zbl 1142.35310 [11] 11. D. Gourdin and M. Mechab, Solutions globales d’un problème de Cauchy linéaire. J. Funct. Anal. 202 (2003), No. 1, 123–146. · Zbl 1035.35002 [12] 12. Y. Hamada and A. Takeuchi, Sur un domaine d’existence et un prolongement analytique des solutions de problèmes de Goursat. J. Math. Pures Appl. (9) 75 (1996), No. 5, 469–483. · Zbl 0863.35018 [13] 13. M. Hibino, Borel summability of divergent solutions for singular first order linear partial differential equations with polynomial coefficients. J. Math. Sci. Univ. Tokyo 10 (2003), 279–309. · Zbl 1036.35051 [14] 14. L. Hörmander, Linear partial differential operators. Springer-Verlag, Berlin–New York (1976). [15] 15. D. A. Lutz, M. Miyake, and R. Schaefke, On the Borel summability of divergent solutions of the heat equation. Nagoya Math. J. 154 (1999), 1–29. · Zbl 0958.35061 [16] 16. S. Malek, On the multisummability of formal solutions of partial differential equations. Ulmer Seminar (2003). [17] 17. S. Malek, Hypergeometric functions and parabolic partial differential equations. Preprint (2003). Submitted to JDCS. [18] 18. B. Malgrange, Sommation des séries divergentes. Exposition. Math. 13 (1995), No. 2–3, 163–222. · Zbl 0836.40004 [19] 19. M. Miyake, Borel summability of divergent solutions of the Cauchy problem to non-Kowalewskian equations. In: Partial Differential Equations and Their Applications (C. Hua and L. Rodino, eds.), Singapore, World Scientific (1999), pp. 225–239. · Zbl 0990.35005 [20] 20. M. Miyake, Global and local Goursat problems in a class of holomorphic or partially holomorphic functions. J. Differential Equations 39 (1981), No. 3, 445–463. · Zbl 0495.35002 [21] 21. M. Miyake and Y. Hashimoto, Newton polygons and Gevrey indices for linear partial differential operators. Nagoya Math. J. 128 (1992), 15–47. · Zbl 0815.35007 [22] 22. S. Ouchi, Multisummability of formal solutions of some linear partial differential equations. J. Differential Equations 185 (2002), 513–549. · Zbl 1020.35018 [23] 23. S. Ouchi, Formal solutions with Gevrey type estimates of nonlinear partial differential equations. J. Math. Sci. Univ. Tokyo 1 (1994), No. 1, 205–237. · Zbl 0810.35006 [24] 24. J. Prüss, Maximal regularity for evolution equations in Lp-spaces. Conf. Semin. Mat. Univ. Bari 285 (2002), 1–39. [25] 25. J. P. Ramis, Dévissage Gevrey. Astérisque 59–60 (1978), 173–204. [26] 26. A. Shirai, Maillet type theorem for nonlinear partial differential equations and Newton polygons. J. Math. Soc. Japan 53 (2001), No. 3, 565–587. · Zbl 0995.35002 [27] 27. R. J. Swift, A stochastic predator-prey model. Irish Math. Soc. Bull. 48 (2002), 57–63. · Zbl 1267.60084
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