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On the Cauchy problem for differential equations in spaces of resurgent functions. (English. Russian original) Zbl 0804.35019
Russ. Acad. Sci., Izv., Math. 40, No. 1, 67-94 (1993); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 56, No. 1, 75-104 (1992).
Summary: A new method is presented for finding asymptotic expansions of solutions of Cauchy problems of a wide class of equations of the type \[ \left[ - {\partial^ m \over \partial t \partial \xi^{m - 1}} + H \left( x, - {\partial \over \partial x}, {\partial \over \partial\xi} \right) \right] u(x,t,\xi) = 0, \quad x \in \mathbb{C}^ n,\;t \in \mathbb{C},\;\xi \in \mathbb{C}, \] where \(H(x, \cdot, \cdot)\) is a polynomial of order \(m\). The method takes into consideration subdominant (i.e. exponentially small) terms, and it is based on the study of problems in special spaces of resurgent multivalued analytic functions.
35C20 Asymptotic expansions of solutions to PDEs
35G25 Initial value problems for nonlinear higher-order PDEs
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