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On the solvability of multidimensional Fuchsian equations in classes of resurgent functions. (English. Russian original) Zbl 0908.35025
Dokl. Math. 54, No. 3, 877-880 (1996); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 351, No. 3, 309-312 (1996).
The purpose of this work is the construction of special asymptotic solutions to the multidimensional Fuchsian equations of the form \[ H\Biggl(x, x{\partial\over\partial x}\Biggr) u(\alpha)= f(x), \] where \(x\in \mathbb{R}^n\), \(H(x,p)\) is a polynomial of the variable \(p\) whose coefficients are smooth with respect to \(x\), and \[ x{\partial\over\partial x}= \Biggl(x^1{\partial\over\partial x^1},\dots, x^n{\partial\over\partial x^n}\Biggr). \] We construct asymptotic solutions in a neighborhood of a singularity, using the technique of the resurgent analysis for functions of several independent variables \((x^1,\dots, x^n)\), which we developed in [B. Yu. Sternin and V. E. Shatalov, Math. Nachr. 171, 283-301 (1995; Zbl 0842.32006)].

35F20 Nonlinear first-order PDEs
35A20 Analyticity in context of PDEs
35C20 Asymptotic expansions of solutions to PDEs