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Strongly regular multi-level solutions of singularly perturbed linear partial differential equations. (English) Zbl 1353.35033

Summary: We study the asymptotic behavior of the solutions related to a family of singularly perturbed partial differential equations in the complex domain. The analytic solutions are asymptotically represented by a formal power series in the perturbation parameter. The geometry of the problem and the nature of the elements involved in it give rise to different asymptotic levels related to the so-called strongly regular sequences. The result leans on a novel version of a multi-level Ramis-Sibuya theorem.

MSC:

35B25 Singular perturbations in context of PDEs
35C10 Series solutions to PDEs
35C20 Asymptotic expansions of solutions to PDEs
40C10 Integral methods for summability
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[1] Balser, W.: From divergent power series to analytic functions. Theory and application of multisummable power series. Lecture Notes in Mathematics. Springer, Berlin, x+108 pp (1994) · Zbl 0810.34046
[2] Balser, W.: Formal power series and linear systems of meromorphic ordinary differential equations. Universitext. Springer, New York, xviii+299 pp (2000) · Zbl 0942.34004
[3] Balser, W.; Yoshino, M., Gevrey order of formal power series solutions of inhomogeneous partial differential equations with constant coefficients, Funkcialaj Ekvac., 53, 411-434, (2010) · Zbl 1237.34149
[4] Canalis-Durand, M.; Mozo-Fernández, J.; Schäfke, R., Monomial summability and doubly singular differential equations, J. Differ. Equ., 233, 485-511, (2007) · Zbl 1166.34334
[5] Chen, H., Rodino, L.: General theory of PDE and Gevrey classes. General theory of partial differential equations and microlocal analysis (Trieste, 1995), N.81, Pitman Res. Notes Math. Ser., vol. 349. Longman, Harlow (1996)
[6] Costin, O.; Kruskal, M., Optimal uniform estimates and rigorous asymptotics beyond all orders for a class of ordinary differential equations, Proc. R. Soc. Lond. Ser. A, 452, 1057-1085, (1996) · Zbl 0866.34047
[7] Djakov, P.; Mityagin, B., Smoothness of solutions of nonlinear ODE, Math. Ann., 324, 225-254, (2002) · Zbl 1020.34083
[8] Djakov, P.; Mityagin, B., Smoothness of solutions of a nonlinear ODE, Integr. Equ. Oper. Theory, 44, 149-171, (2002) · Zbl 1029.34004
[9] Hörmander, L., A counterexample of Gevrey class to the uniqueness of the Cauchy problem, Math. Res. Lett., 7, 615-624, (2000) · Zbl 0974.35002
[10] Hsieh P., Sibuya Y.: Basic Theory of Ordinary Differential Equations, Universitext. Springer, New York (1999) · Zbl 0924.34001
[11] Immink, G.K.: Exact asymptotics of nonlinear difference equations with levels 1 and 1+. Ann. Fac. Sci. Toulouse T. XVII(2), 309-356 (2008) · Zbl 1160.39003
[12] Immink, G.K., Accelero-summation of the formal solutions of nonlinear difference equations, Ann. Inst. Fourier (Grenoble), 61, 1-51, (2011) · Zbl 1225.39005
[13] Jiménez, J., Sanz, J.: Strongly regular sequences, proximate orders and kernels of summability (2015, in preparation) · Zbl 0586.35060
[14] Kamimoto, S.: On the exact WKB analysis of singularly perturbed ordinary differential equations at an irregular singular point (2013, preprint RIMS-1779)
[15] Lastra, A., Malek, S.: Multi-level Gevrey solutions of singularly perturbed linear partial differential equations (submitted). http://arxiv.org/abs/1407.2008 · Zbl 1375.35080
[16] Lastra, A., Malek, S.: On parametric Gevrey asymptotics for some nonlinear initial value Cauchy problems (submitted). http://arxiv.org/abs/1403.2350 · Zbl 1326.35072
[17] Lastra, A.; Malek, S.; Sanz, J., On Gevrey solutions of threefold singular nonlinear partial differential equations, J. Differ. Equ., 255, 3205-3232, (2013) · Zbl 1320.35150
[18] Lastra, A., Malek, S., Sanz, J.: Summability in general Carleman ultraholomorphic classes (submitted). http://arxiv.org/abs/1402.1669. · Zbl 1327.30004
[19] Malek, S.: On Gevrey functions solutions of partial differential equations with fuchsian and irregular singularities. J. Dyn. Control Syst. 15, 2 (2009) · Zbl 1203.35068
[20] Malek, S., On the summability of formal solutions for doubly nonlinear partial differential equations, J. Dyn. Control Syst., 18, 45-82, (2012) · Zbl 1250.35062
[21] Michalik, S., Analytic solutions of moment partial differential equations with constant coefficients, Funkcialaj Ekvac., 56, 19-50, (2013) · Zbl 1282.35129
[22] Michalik, S., Summability of formal solutions of linear partial differential equations with divergent initial data, J. Math. Anal. Appl., 406, 243-260, (2013) · Zbl 1306.35007
[23] Sanz, J., Flat functions in Carleman ultraholomorphic classes via proximate orders, J. Math. Anal. Appl., 415, 623-643, (2014) · Zbl 1308.30051
[24] Sibuya, Y., Formal power series solutions in a parameter, J. Differ. Equ., 190, 559-578, (2003) · Zbl 1029.34079
[25] Tahara, H., Asymptotic expansions of solutions of Fuchsian hyperbolic equations in spaces of functions of Gevrey classes, Proc. Japan Acad. Ser. A Math. Sci., 61, 255-258, (1985) · Zbl 0592.35091
[26] Tahara, H., Cauchy problems for Fuchsian hyperbolic equations in spaces of functions of Gevrey classes, Proc. Japan Acad. Ser. A Math. Sci., 61, 63-65, (1985) · Zbl 0586.35060
[27] Tahara, H.; Yamazawa, H., Multisummability of formal solutions to the Cauchy problem for some linear partial differential equations, J. Differ. Equ., 255, 3592-3637, (2013) · Zbl 1320.35151
[28] Thilliez, V., Smooth solutions of quasianalytic or ultraholomorphic equations, Monatsh. Math., 160, 443-453, (2010) · Zbl 1215.26013
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