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Transactions of the American Mathematical Society

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An abstract nonlinear Cauchy-Kovalevska theorem

Author: François Trèves
Journal: Trans. Amer. Math. Soc. 150 (1970), 77-92
MSC: Primary 35.03
MathSciNet review: 0274911
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Abstract: A nonlinear version of Ovcyannikov's theorem is proved. If $ F(u,t)$ is an analytic function of $ t$ real or complex and of $ u$ varying in a scale of Banach spaces, valued in a scale of Banach spaces, the Cauchy problem $ {u_t} = F(u,t),u(0) = {u_0}$, 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.

References [Enhancements On Off] (What's this?)

  • [O1] 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-T1] S. Steinberg and F. Treves, Pseudo-Fokker Planck equations and hyperdifferential operators, (to appear).
  • [T1] 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)
  • [T2] -, 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

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