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On the global solvability of the Cauchy problem in spaces of ramified analytic functions. (English. Russian original) Zbl 0806.35002
Sov. Math., Dokl. 44, No. 3, 836-838 (1992); translation from Dokl. Akad. Nauk SSSR 321, No. 5, 910-913 (1991).
The Cauchy problem for a differential operator $$\widehat H = H(x, \partial /\partial x)$$ of order $$m$$ with entire coefficients in $$\mathbb{C}^ n$$ and Cauchy data on an irreducible analytic set $$X$$ of codimension 1 is studied under the assumption that the highest symbol $$H_ m (p)$$ of $$\widehat H$$ is independent of $$x$$ and that the set $$X$$ is not everywhere characteristic relative to $$H_ m(p)$$. As the main result the authors give an expression for a solution of the Cauchy problem in terms of an elementary solution of the operator $$\widehat H$$. This solution is defined in the whole space $$\mathbb{C}^ n$$ as a multivalued analytic function with singularities on some analytic set in $$\mathbb{C}^ n$$, which can be effectively computed.
Reviewer: J.Fuka (Praha)
##### MSC:
 35A08 Fundamental solutions to PDEs 35C15 Integral representations of solutions to PDEs 58J47 Propagation of singularities; initial value problems on manifolds 32C30 Integration on analytic sets and spaces, currents