# zbMATH — the first resource for mathematics

Exponential-type Nagumo norms and summability of formal solutions of singular partial differential equations. (English. French summary) Zbl 1252.30025
Authors’ abstract: In this paper, we study a class of first order nonlinear degenerate partial differential equations with singularity at $$(t,x)=(0,0)\in {\mathbb C}^2$$. Using exponential-type Nagumo norm approach, the Gevrey asymptotic analysis is extended to case of holomorphic parameters in a natural way. A sharp condition is then established to deduce the $$k$$-summability of the formal solutions. Furthermore, analytical solutions in conical domains are found for each type of these nonlinear singular PDEs.

##### MSC:
 30E15 Asymptotic representations in the complex plane 32D15 Continuation of analytic objects in several complex variables 35C10 Series solutions to PDEs 35C20 Asymptotic expansions of solutions to PDEs
Full Text:
##### References:
 [1] Balser, W., Formal power series and linear systems of meromorphic ordinary differential equations, (2000), Springer-Verlag, New York · Zbl 0942.34004 [2] Balser, W., Multisummability of formal power series solutions of partial differential equations with constant coefficients, J. Differential Equations, 201, 63-74, (2004) · Zbl 1052.35048 [3] Braaksma, B. L. J., Multisummability of formal power series solutions of nonlinear meromorphic differential equations, Ann. Inst. Fourier, 42, 517-540, (1992) · Zbl 0759.34003 [4] Canalis-Durand, M.; Ramis, J.-P.; Schäfke, R.; Sibuya, Y., Gevrey solutions of singularly perturbed differential equations, J. Reine Angew. Math., 518, 95-129, (2000) · Zbl 0937.34075 [5] Chen, H.; Luo, Z., On the holomorphic solution of non-linear totally characteristic equations with several space variables, Acta Mathematica Scientia, 22B, 393-403, (2002) · Zbl 1003.35005 [6] Chen, H.; Luo, Z.; Tahara, H., Formal solution of nonlinear first order totally characteristic type PDE with irregular singularity, Ann. Inst. Fourier, 51, 1599-1620, (2001) · Zbl 0993.35003 [7] Chen, H.; Luo, Z.; Zhang, C., On the summability of formal solutions for a class of nonlinear singular PDEs with irregular singularity, Contemporary of Mathematics, 400, 53-64, (2006), Amer. Math. Soc. · Zbl 1098.35006 [8] Chen, H.; Tahara, H., On totally characteristic type non-linear differential equations in the complex domain, Publ. RIMS, Kyoto Univ., 35, 621-636, (1999) · Zbl 0961.35002 [9] Chen, H.; Tahara, H., On the holomorphic solution of non-linear totally characteristic equations, Mathematische Nachrichten, 219, 85-96, (2000) · Zbl 1017.35006 [10] Costin, O.; Tanveer, S., Existence and uniqueness for a class of nonlinear higher-order partial differential equations in the complex plane, Comm. Pure and Appl. Math., LIII, 0001-0026, (2000) [11] Di Vizio, L., An ultrametric version of the maillet-malgrange theorem for non linear q-difference equations, Proc. Amer. Math. Soc., 136, 2803-2814, (2008) · Zbl 1152.33011 [12] Fife, P. C.; McLeod, J. B., The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Ration. Mech. Anal., 65, 335-361, (1977) · Zbl 0361.35035 [13] Gérard, R.; Tahara, H., Singular nonlinear partial differential equations, (1996), Vieweg Verlag · Zbl 0874.35001 [14] Gevrey, M., Sur LES équations aux dérivées partielles du type parabolique, J. de Mathématique, 9, 305-476, (1913) · JFM 44.0431.03 [15] Hagan, P. S.; Ockendon, J. R., Half-range analysis of a counter-current separator, J. Math. Anal. Appl., 160, 358-378, (1991) · Zbl 0753.76190 [16] Hibino, M., Borel summability of divergent solutions for singular first order linear partial differential equations with polynomial coefficients, J. Math. Sci. Univ. Tokyo, 10, 279-309, (2003) · Zbl 1036.35051 [17] Hibino, M., Borel summability of divergence solutions for singular first-order partial differential equations with variable coefficients. I, J. Differential Equations, 227, 499-533, (2006) · Zbl 1147.35016 [18] Hibino, M., Borel summability of divergent solutions for singular first-order partial differential equations with variable coefficients. II, J. Differential Equations, 227, 534-563, (2006) · Zbl 1147.35017 [19] Hörmander, L., An introduction to complex analysis in several variables, (1990), North-Holland Publishing Co., Amsterdam · Zbl 0271.32001 [20] Luo, Z.; Chen, H.; Zhang, C., On the summability of the formal solutions for some PDEs with irregular singularity, C.R. Acad. Sci. Paris, Sér. I, 336, 219-224, (2003) · Zbl 1028.35006 [21] Luo, Z.; Zhang, C., On the Borel summability of divergent power series respective to two variables, (2010) [22] Lutz, D. A.; 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 [23] Malgrange, B., Sur le théorème de maillet, Asymptot. Anal., 2, 1-4, (1989) · Zbl 0693.34004 [24] Martinet, J.; Ramis, J.-P., Problèmes de modules pour des équations différentielles non linéaires du premier ordre, Publ. Math., Inst. Hautes Études Sci., 55, 63-164, (1982) · Zbl 0546.58038 [25] Martinet, J.; Ramis, J.-P., Elementary acceleration and multisummability I, Annales de l’I.H.P. Physique théorique, 54, 331-401, (1991) · Zbl 0748.12005 [26] Nagumo, M., Über das anfangswertproblem partieller differentialgleichungen, Jap. J. Math., 18, 41-47, (1942) · Zbl 0061.21107 [27] Ouchi, S., Multisummability of formal solutions of some linear partial differential equations, J. Differential Equations, 185, 513-549, (2002) · Zbl 1020.35018 [28] Ouchi, S., Borel summability of formal solutions of some first order singular partial differential equations and normal forms of vector fields, J. Math. Soc. Japan, 57, 415-460, (2005) · Zbl 1082.35046 [29] Ouchi, S., Multisummability of formal power series solutions of nonlinear partial differential equations in complex domains, Asymptot. Anal., 47, 187-225, (2006) · Zbl 1152.35015 [30] Pagani, C. D.; Talenti, G., On a forward-backward parabolic equation, Ann. Mat. Pura. Appl., 90, 1-57, (1971) · Zbl 0238.35043 [31] Ramis, J.-P., LES séries $$k$$-sommables et leurs applications, Complex Analysis, Microlocal Calculus and Relativistic Quantum Theory, 126, 178-199, (1980), Springer-Verlag, New York · Zbl 1251.32008 [32] Tougeron, J.-C., Sur LES ensembles semi-analytiques avec conditions Gevrey au bord, Ann. Sci. École Norm. Sup., 27, 173-208, (1994) · Zbl 0803.32003 [33] Zhang, C., Sur un théorème du type de maillet-malgrange pour LES équations $$q$$-différences-différentielles, Asymptot. Anal., 17, 309-314, (1998) · Zbl 0938.34064
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.