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International Journal of Mathematics and Mathematical Sciences
Volume 2004, Issue 17, Pages 901-912
http://dx.doi.org/10.1155/S016117120430503X

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

Mathematics Faculty and Research Institute of Mathematics, Voronezh State University, Universitetskaya pl. 1, Voronezh 394006, Russia

Received 6 May 2003

Copyright © 2004 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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: L1-complete solutions of SDE (this means that they belong to a certain functional space of L1 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.