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Unique solvability of coupling equations in holomorphic functions. (English) Zbl 1372.35380

Summary: The theory of coupling equations was introduced by the third author [Publ. Res. Inst. Math. Sci. 43, No. 3, 535–583 (2007; Zbl 1140.35002)], as a theory of a class of transformations between some nonlinear partial differential equations in complex domains. There, he constructed, to the initial value problem of a coupling equation, a formal power series solution of a special form in infinitely many variables, satisfying suitable estimates. It would be desirable, from several aspects, to study the coupling equations and their solvability as functional equations for ‘holomorphic functions’.
In this report, we consider coupling equations for partial differential equations of normal form in the \(t\) variable. After preparing and recalling some notions of holomorphy on infinite dimensional spaces, we announce our recent result on the unique solvability of the initial value problem of a coupling equation, using the contraction mapping principle.

MSC:

35R50 PDEs of infinite order
35A22 Transform methods (e.g., integral transforms) applied to PDEs
35A10 Cauchy-Kovalevskaya theorems
46E50 Spaces of differentiable or holomorphic functions on infinite-dimensional spaces

Citations:

Zbl 1140.35002
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