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 -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 -différences-différentielles partielles est également présentée.
@article{AFST_2007_6_16_2_285_0,
author = {St\'ephane Malek},
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},
pages = {285--302},
publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques},
address = {Toulouse},
volume = {Ser. 6, 16},
number = {2},
year = {2007},
doi = {10.5802/afst.1149},
mrnumber = {2331542},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1149/}
}
TY - JOUR AU - Stéphane Malek TI - On functional linear partial differential equations in Gevrey spaces of holomorphic functions. JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2007 SP - 285 EP - 302 VL - 16 IS - 2 PB - Université Paul Sabatier, Institut de Mathématiques PP - Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1149/ DO - 10.5802/afst.1149 LA - en ID - AFST_2007_6_16_2_285_0 ER -
%0 Journal Article %A Stéphane Malek %T On functional linear partial differential equations in Gevrey spaces of holomorphic functions. %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2007 %P 285-302 %V 16 %N 2 %I Université Paul Sabatier, Institut de Mathématiques %C Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1149/ %R 10.5802/afst.1149 %G en %F AFST_2007_6_16_2_285_0
Stéphane Malek. 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. https://afst.centre-mersenne.org/articles/10.5802/afst.1149/
[1] Arendt (W.), Batty (C.), Hieber (M.), Neubrander (F.).— Vector-valued Laplace transforms and Cauchy problems. Monographs in Mathematics, 96. Birkhäuser (2001). | MR | Zbl
[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 | Zbl
[3] Balser (W.).— Formal power series and linear systems of meromorphic ordinary differential equations, Springer-Verlag, New-York (2000). | MR | Zbl
[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 -différences, Aequationes Math. 43, p. 159–176 (1993). | EuDML | MR | Zbl
[6] Di Vizio (L.), Ramis (J.-P.), Sauloy (J.), Zhang (J.).— Équations aux -différences. Gaz. Math. No. 96, p. 20–49 (2003). | MR | Zbl
[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). | EuDML | Numdam | MR | Zbl
[9] Kawagishi (M.), Yamanaka (T.).— The heat equation and the shrinking. Electron. J. Differential Equations 2003, No. 97, p.14. | EuDML | MR | Zbl
[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 | Zbl
[11] Kato (T.).— Asymptotic behavior of solutions of the functional differential equation . Delay and functional differential equations and their applications (Proc. Conf., Park City, Utah, 1972), p. 197–217. Academic Press, New York (1972). | MR | Zbl
[12] Malek (S.).— On the summability of formal solutions of linear partial differential equations, J. Dynam. Control. Syst. 11, No. 3 (2005). | MR | Zbl
[13] Malgrange (B.).— Sommation des séries divergentes. Exposition. Math. 13, no. 2-3, p. 163–222 (1995). | MR | Zbl
[14] Prüss (J.).— Evolutionary integral equations and applications. Monographs in Mathematics, 87. Birkhäuser Verlag, Basel (1993). | MR | Zbl
[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 | Zbl
[16] Ramis (J.P.).— About the growth of entire functions solutions of linear algebraic -difference equations. Ann. Fac. Sci. Toulouse Math. (6) 1, no. 1, p. 53–94 (1992). | Numdam | MR | Zbl
[17] Ramis (J.P.).— Séries divergentes et théories asymptotiques. Bull. Soc. Math. France 121, Panoramas et Syntheses, suppl., p. 74 (1993). | MR | Zbl
[18] Ramis (J.P.),Zhang (C.).— Développement asymptotique -Gevrey et fonction thêta de Jacobi. C. R. Math. Acad. Sci. Paris 335, no. 11, p. 899–902 (2002). | MR | Zbl
[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 | Zbl
[20] Yamanaka (T.).— A new higher order chain rule and Gevrey class. Ann. Global Anal. Geom. 7, no.3, p. 179–203 (1989). | MR | Zbl
[21] Zhang (C.).— Développements asymptotiques -Gevrey et séries -sommables. Ann. Inst. Fourier (Grenoble) 49, no.1, p. 227–261 (1999). | Numdam | MR | Zbl
[22] Zhang (C.).— Sur un théorème du type de Maillet-Malgrange pour les équations -différences-différentielles. Asymptot. Anal. 17, no. 4, p. 309–314 (1998). | MR | Zbl
[23] Zubelevich (O.).— Functional-differential equation with dilation and compression of argument, preprint mp arc 05-136.
Cited by Sources: