×

zbMATH — the first resource for mathematics

A generalisation of the Cauchy-Kovalevskaïa theorem. (English) Zbl 1353.35010
Summary: We prove that time evolution of a linear analytic initial value problem leads to sectorial holomorphic solutions in time.
MSC:
35A10 Cauchy-Kovalevskaya theorems
35K10 Second-order parabolic equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Audin, M.: Souvenirs sur Sofia Kovalevskaya. Calvage & Mounet, Paris (2008) · Zbl 1154.01009
[2] Balser, W.: From Divergent Power Series to Analytic Functions. Lecture Notes in Mathematics, vol. 1582. Springer, Berlin (1994) · Zbl 0810.34046
[3] Baouendi, MS; Goulaouic, C, Remarks on the abstract form of nonlinear Cauchy-kovalevsky theorems, Commun. Partial Differ. Equ., 2, 1151-1162, (1977) · Zbl 0391.35006
[4] Boutet de Monvel, L., Krée, P.: Pseudo differential operators and Gevrey classes. Annales de l’Institut Fourier 17(1), 295-323 (1967) · Zbl 0195.14403
[5] Candelpergher, B., Nosmas, J.C., Pham, F.: Approche de la résurgence. Hermann, Paris (1993) · Zbl 0791.32001
[6] Dineen, S.: Complex Analysis on Locally Convex Spaces. North Holland Mathematical Studies, vol. 57 (1981) · Zbl 0484.46044
[7] Domrin, AV; Domrina, AV, On the divergence of the Kontsevich-Witten series, Russ. Math. Surv., 109, 773-775, (2008) · Zbl 1179.37091
[8] Écalle, J.: Les fonctions résurgentes, vol. 1, algèbres de fonctions résurgentes. Pub. Math. Orsay, 247 (1981) · Zbl 0499.30034
[9] Euler, L.: De seriebus divergentibus. Novi Commentarii academiae scientiarum Petropolitanae 5, 205-237 (1760) [German translation available on http://www.eulerarchive.org/, E247] · Zbl 1179.37091
[10] Fernández-Fernández, MC; Castro-Jiménez, FJ, Gevrey solutions of the irregular hypergeometric system associated with an affine monomial curve, Trans. Am. Math. Soc., 363, 923-948, (2011) · Zbl 1219.32007
[11] Garay, M.: On the Gevrey convergence of some characteristic Cauchy problems (2008). arXiv:0710.1753v3 (unpublished) · Zbl 1179.37091
[12] Gevrey, Sur les fonctions indéfiniments dérivables de classe donnée et leur rôle dans la théorie des équations partielles. Comptes rendus à l’Académie des Sciences, tome 157 num 23, 1913, 1121-1124 · JFM 44.0433.01
[13] Gevrey, M, Sur la nature analytique des solutions des équations aux dérivées partielles, Annales scientifiques de l’École Normale Supérieure, 35, 129-190, (1918) · JFM 46.0721.01
[14] Grothendieck, A.: Espaces vectoriels topologiques. Instituto de Matemàtica Pura e Aplicada, Universidade de São Paulo (1954) [English Translation: Topological Vector Spaces. Gordon and Breach (1973)] · Zbl 0058.33401
[15] Grothendieck, A.: Produits tensoriels topologiques et espaces nucléaires. Mem. Am. Math. Soc. 16 (1955) · Zbl 0123.30301
[16] Hadamard, J.: Le problème de Cauchy et les équations aux dérivées partielles hyperboliques. Hermann et Cie, Paris (1932) · Zbl 0006.20501
[17] Landau, L., Lifschitz, E.: Mécanique quantique, Physique Théorique, vol. III. MIR, Moscow (1967) · Zbl 0719.35018
[18] Lutz, DA; Miyake, M; Schäfke, R, On the Borel summability of divergent solutions of the heat equation, Nagoya Math. J., 154, 1-29, (1999) · Zbl 0958.35061
[19] Łysik, G, Non-analyticity in time of solutions to the KdV equation, Z. Anal. Anwendungen, 23, 67-93, (2004) · Zbl 1059.35120
[20] Łysik, G; Michalik, S, Formal solutions of semilinear heat equations, J. Math. Anal. Appl., 341, 372-385, (2008) · Zbl 1139.35035
[21] Malgrange, B, Sommation des séries divergentes, Expositiones Mathematicae, 13, 163-222, (1995) · Zbl 0836.40004
[22] Nagumo, M, Über das anfangswertproblem partieller differentialgleichungen, Jpn. J. Math., 18, 41-47, (1942) · Zbl 0061.21107
[23] Nirenberg, L, An abstract form of the nonlinear Cauchy-kowalewski theorem, J. Differ. Geom., 6, 561-576, (1972) · Zbl 0257.35001
[24] Nishida, T, A note on a theorem of Nirenberg, J. Differ. Geom., 12, 629-633, (1977) · Zbl 0368.35007
[25] Ouchi, S, Characteristic Cauchy problems and solutions of formal power series, Ann. Inst. Fourier, 33, 131-176, (1983) · Zbl 0494.35017
[26] Ovsyannikov, IV, A singular operator in a scale of Banach spaces, Sov. Math. Dokl., 6, 1025-1028, (1965) · Zbl 0144.39003
[27] Ramis, JP; Iagolnitzer, D (ed.), LES series \(κ \)-sommables et leurs applications, 178-199, (1980), Berlin · Zbl 1251.32008
[28] Tahara, H, Gevrey regularity in time of solutions to nonlinear partial differential equations, J. Math. Sci. Univ. Tokyo, 18, 67-137, (2011) · Zbl 1255.35070
[29] Tahara, H.: Maillet type theorem and Gevrey Regularity in Time of Solutions to Nonlinear Partial Differential Equations. Formal and Analytic Solutions of Differential and Difference Equations, vol. 97. Banach Center Publications, Banach center pp. 125-149 (2011) · Zbl 1272.35062
[30] Kowalevsky, S, Zur theorie der partiellen differentialgleichungen, Journal für reine und angewandte Mathematik, 80, 1-32, (1875) · JFM 07.0201.01
[31] Yonemura, A, Newton polygons and formal Gevrey classes, Publ. RIMS, 26, 197-204, (1990) · Zbl 0719.35018
[32] Zinn-Justin, J.: Quantum Field Theory and Critical Phenomena. Oxford Science Publications, Oxford (1993) · Zbl 0865.00014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.