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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)
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