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Journal of Mathematical Sciences

, Volume 73, Issue 6, pp 618–632 | Cite as

Nonlinear parabolic equations in sections of vector bundles

  • Ya. I. Belopol'skaya
Article

Abstract

We study the Cauchy problem for a nonlinear parabolic equation relative to a section of a vector bundle. The special procedure of covariant differential extension makes it possible to reduce this equation to a system of quasilinear parabolic equations, which can be studied using the theory of stochastic equations. We obtain a probabilistic representation of the solution of both the auxiliary quasilinear parabolic system and the original nonlinear equation. Bibliography: 5 titles.

Keywords

Cauchy Problem Probabilistic Representation Vector Bundle Nonlinear Equation Parabolic Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Literature Cited

  1. 1.
    Ya. I. Belopol'skaya and Yu. L. Daletskii, “A study of the Cauchy problem for a nonlinear parabolic system using operator-valued multiplicative functionals of Markov processes,”Zap. Nauch. Sem. LOMI AN SSSR,96, 23–32 (1980).MathSciNetGoogle Scholar
  2. 2.
    Ya. I. Belopol'skaya and Yu. L. Daletskii, “A study of the Cauchy problem for quasilinear parabolic equations and systems with finite and infinite numbers of arguments using Markov stochastic processes,”Izv. Vuzov, Ser. Mat., No. 6, 5–17 (1978).Google Scholar
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    Ya. I. Belopol'skaya and Yu. L. Daletskii, “Markov processes connected with nonlinear parabolic systems”,Dokl. Akad. Nauk SSSR,250, No. 3, 521–524 (1980).MathSciNetGoogle Scholar
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    Yu. L. Daletskii and Ya. I. Belopol'skaya,Stochastic Equations and Differential Geometry [in Russian], Kiev (1989).Google Scholar
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    Ya. I. Belopol'skaya and Yu. L. Daletskii, “The Ito equations and differential geometry,”Usp. Mat. Nauk,37, 95–142 (1982).MathSciNetGoogle Scholar

Copyright information

© Plenum Publishing Corporation 1995

Authors and Affiliations

  • Ya. I. Belopol'skaya

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