On functional linear partial differential equations in Gevrey spaces of holomorphic functions.

Malek, Stéphane

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 16 (2007) no. 2, pp. 285-302.

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.

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.

[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 $q$-différences, Aequationes Math. 43, p. 159–176 (1993). | EuDML | MR | Zbl

[6] Di Vizio (L.), Ramis (J.-P.), Sauloy (J.), Zhang (J.).— Équations aux $q$-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 $y^{\prime }\left(x\right)=ay\left(\lambda x\right)+by\left(x\right)$. 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). | Numdam | MR | Zbl

[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 | 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 $q$-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 $q$-Gevrey et séries $Gq$-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 $q$-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.