An abstract nonlinear Cauchy-Kovalevska theorem
Abstract: A nonlinear version of Ovcyannikov's theorem is proved. If is an analytic function of real or complex and of varying in a scale of Banach spaces, valued in a scale of Banach spaces, the Cauchy problem , has a unique analytic solution. This is an abstract version of the Cauchy-Kovalevska theorem which can be applied to equations other than partial-differential, e.g. to certain differential-convolution or, more generally, differential-pseudodifferential equations.
- [O1] L. V. Ovsjannikov, Singular operator in the scale of Banach spaces, Dokl. Akad. Nauk SSSR 163 (1965), 819–822 (Russian). MR 0190754
- [S-T1] S. Steinberg and F. Treves, Pseudo-Fokker Planck equations and hyperdifferential operators, (to appear).
- [T1] François Trèves, On the theory of linear partial differential operators with analytic coefficients, Trans. Amer. Math. Soc. 137 (1969), 1–20. MR 0247267, https://doi.org/10.1090/S0002-9947-1969-0247267-8
- [T2] François Trèves, Ovcyannikov theorem and hyperdifferential operators, Notas de Matemática, No. 46, Instituto de Matemática Pura e Aplicada, Conselho Nacional de Pesquisas, Rio de Janeiro, 1968. MR 0290202
- L. V. Ovsjannikov, Singular operators in Banach spaces scales, Dokl. Akad. Nauk SSSR 163 (1965), 819-822 = Soviet Math. Dokl. 6 (1965), 1025-1028. MR 32 #8164. MR 0190754 (32:8164)
- S. Steinberg and F. Treves, Pseudo-Fokker Planck equations and hyperdifferential operators, (to appear).
- F. Treves, On the theory of linear partial differential equations with analytic coefficients, Trans. Amer. Math. Soc. 137 (1969), 1-20. MR 0247267 (40:536)
- -, Ovcyannikov theorem and hyperdifferential operators, Instituto de Matematica Pura e Aplicada, Rio de Janeiro, 1968. MR 0290202 (44:7386)
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Keywords: Cauchy-Kovalevska, nonlinear, Cauchy problem, scale of Banach spaces, analytic, Ovcyannikov, Banach algebras, Fourier transform, analytic functionals, Gevrey class
Article copyright: © Copyright 1970 American Mathematical Society