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Conditions for global existence of solutions of ordinary differential, stochastic differential, and parabolic equations

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  • Yuri E. Gliklikh
  • Lora A. Morozova

Abstract

First, we prove a necessary and sufficient condition for global in time existence of all solutions of an ordinary differential equation (ODE). It is a condition of one-sided estimate type that is formulated in terms of so-called proper functions on extended phase space. A generalization of this idea to stochastic differential equations (SDE) and parabolic equations (PE) allows us to prove similar necessary and sufficient conditions for global in time existence of solutions of special sorts: L 1 -complete solutions of SDE (this means that they belong to a certain functional space of L 1 type) and the so-called complete Feller evolution families giving solutions of PE. The general case of equations on noncompact smooth manifolds is under consideration.

Suggested Citation

  • Yuri E. Gliklikh & Lora A. Morozova, 2004. "Conditions for global existence of solutions of ordinary differential, stochastic differential, and parabolic equations," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2004, pages 1-12, January.
    • Handle: RePEc:hin:jijmms:475816
    DOI: 10.1155/S016117120430503X
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