On functional linear partial differential equations in Gevrey spaces of holomorphic functions.
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 16 (2007) no. 2, p. 285-302

We investigate existence and unicity of global sectorial holomorphic solutions of functional linear partial differential equations in some Gevrey spaces. A version of the Cauchy-Kowalevskaya theorem for some linear partial q-difference-differential equations is also presented.

Nous étudions l’existence et l’unicité de solutions globales holomorphes sectorielles d’équations fonctionnelles linéaires aux dérivées partielles dans certains espaces de fonctions Gevrey. Une version du théorème de Cauchy-Kowalevskaya pour des équations linéaires aux q-différences-différentielles partielles est également présentée.

@article{AFST_2007_6_16_2_285_0,
     author = {Malek, St\'ephane},
     title = {On functional linear partial differential equations in Gevrey spaces of holomorphic functions.},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 16},
     number = {2},
     year = {2007},
     pages = {285-302},
     doi = {10.5802/afst.1149},
     mrnumber = {2331542},
     zbl = {pre05236227},
     language = {en},
     url = {http://www.numdam.org/item/AFST_2007_6_16_2_285_0}
}
Malek, Stéphane. On functional linear partial differential equations in Gevrey spaces of holomorphic functions.. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 16 (2007) no. 2, pp. 285-302. doi : 10.5802/afst.1149. http://www.numdam.org/item/AFST_2007_6_16_2_285_0/

[1] Arendt (W.), Batty (C.), Hieber (M.), Neubrander (F.).— Vector-valued Laplace transforms and Cauchy problems. Monographs in Mathematics, 96. Birkhäuser (2001). | MR 1886588 | Zbl 0978.34001

[2] Augustynowicz (A.), Leszczyński (H.), Walter (W.).— Cauchy-Kovalevskaya theory for equations with deviating variables. Dedicated to Janos Aczel on the occasion of his 75th birthday. Aequationes Math. 58, no. 1-2, p. 143–156 (1999). | MR 1714328 | Zbl 0929.35004

[3] Balser (W.).— Formal power series and linear systems of meromorphic ordinary differential equations, Springer-Verlag, New-York (2000). | MR 1722871 | Zbl 0942.34004

[4] Balser (W.), Malek (S.).— Formal solutions of the complex heat equation in higher spatial dimensions, RIMS, 1367, p. 95-102 (2004).

[5] Bézivin (J.P.).— Sur les équations fonctionnelles aux q-différences, Aequationes Math. 43, p. 159–176 (1993). | MR 1158724 | Zbl 0757.39002

[6] Di Vizio (L.), Ramis (J.-P.), Sauloy (J.), Zhang (J.).— Équations aux q-différences. Gaz. Math. No. 96, p. 20–49 (2003). | MR 1988639 | Zbl 1063.39015

[7] Écalle (J.).— Les fonctions résurgentes. Publications Mathématiques d’Orsay (1981).

[8] Fruchard (A.), Zhang (C.).— Remarques sur les développements asymptotiques. Ann. Fac. Sci. Toulouse Math. (6) 8, no. 1, p. 91–115 (1999). | Numdam | MR 1721570 | Zbl 1157.30322 | Zbl 01491215

[9] Kawagishi (M.), Yamanaka (T.).— The heat equation and the shrinking. Electron. J. Differential Equations 2003, No. 97, p.14. | MR 2000693 | Zbl 1039.35048

[10] Kawagishi (M.), Yamanaka (T.).— On the Cauchy problem for non linear PDEs in the Gevrey class with shrinkings. J. Math. Soc. Japan 54, no. 3, p. 649–677 (2002). | MR 1900961 | Zbl 1032.35059

[11] Kato (T.).— Asymptotic behavior of solutions of the functional differential equation y (x)=ay(λx)+by(x). Delay and functional differential equations and their applications (Proc. Conf., Park City, Utah, 1972), p. 197–217. Academic Press, New York (1972). | MR 390432 | Zbl 0278.34070

[12] Malek (S.).— On the summability of formal solutions of linear partial differential equations, J. Dynam. Control. Syst. 11, No. 3 (2005). | MR 2147192 | Zbl 1085.35043

[13] Malgrange (B.).— Sommation des séries divergentes. Exposition. Math. 13, no. 2-3, p. 163–222 (1995). | MR 1346201 | Zbl 0836.40004

[14] Prüss (J.).— Evolutionary integral equations and applications. Monographs in Mathematics, 87. Birkhäuser Verlag, Basel (1993). | MR 1238939 | Zbl 0784.45006

[15] Ramis (J.-P.).— Dévissage Gevrey. Journées Singulières de Dijon (Univ. Dijon, 1978), p. 4, 173-204, Astérisque, 59-60, Soc. Math. France, Paris (1978). | MR 542737 | Zbl 0409.34018

[16] Ramis (J.P.).— About the growth of entire functions solutions of linear algebraic q-difference equations. Ann. Fac. Sci. Toulouse Math. (6) 1, no. 1, p. 53–94 (1992). | Numdam | MR 1191729 | Zbl 0796.39005

[17] Ramis (J.P.).— Séries divergentes et théories asymptotiques. Bull. Soc. Math. France 121, Panoramas et Syntheses, suppl., p. 74 (1993). | MR 1272100 | Zbl 0830.34045

[18] Ramis (J.P.),Zhang (C.).— Développement asymptotique q-Gevrey et fonction thêta de Jacobi. C. R. Math. Acad. Sci. Paris 335, no. 11, p. 899–902 (2002). | MR 1952546 | Zbl 1025.39014

[19] Rossovskii (L.E.).— On the boundary value problem for the elliptic functional-differential equation with contractions. International Conference on Differential and Functional Differential Equations (Moscow, 1999). Funct. Differ. Equ. 8, no. 3-4, p. 395–406 (2001). | MR 1950983 | Zbl 1054.35117

[20] Yamanaka (T.).— A new higher order chain rule and Gevrey class. Ann. Global Anal. Geom. 7, no.3, p. 179–203 (1989). | MR 1039118 | Zbl 0719.58002

[21] Zhang (C.).— Développements asymptotiques q-Gevrey et séries Gq-sommables. Ann. Inst. Fourier (Grenoble) 49, no.1, p. 227–261 (1999). | Numdam | MR 1688144 | Zbl 0974.39009

[22] Zhang (C.).— Sur un théorème du type de Maillet-Malgrange pour les équations q-différences-différentielles. Asymptot. Anal. 17, no. 4, p. 309–314 (1998). | MR 1656811 | Zbl 0938.34064

[23] Zubelevich (O.).— Functional-differential equation with dilation and compression of argument, preprint mp arc 05-136.