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TMF, 2015, Volume 185, Number 2, Pages 252–271 (Mi tmf8835)  

This article is cited in 4 scientific papers (total in 4 papers)

Notion of blowup of the solution set of differential equations and averaging of random semigroups

L. S. Efremovaa, V. Zh. Sakbaevb

a Lobachevsky State University of Nizhny Novgorod, Nizhny Novgorod, Russia
b Moscow Institute of Physics and Technology (State University), Dolgoprudny, Moscow Oblast, Russia

Abstract: We propose a unique approach to studying the violation of the well-posedness of initial boundary-value problems for differential equations. The blowup of the set of solutions of a problem for a differential equation is defined as a discontinuity of a multivalued map associating an initial boundary-value problem with the set of solutions of this problem. We show that such a definition not only describes effects of the solution destruction or its nonuniqueness but also permits prescribing a procedure for extending the solution through the singularity origination instant by using an appropriate random process. Considering the initial boundary-value problems whose solution sets admit singularities of the blowup type and a neighborhood of these problems in the space of problems permits associating the initial problem with the set of limit points of a sequence of solutions of the approximating problems. Endowing the space of problems with the structure of a space with measure, we obtain a random semigroup generated by the initial problem. We study the properties of the mathematical expectations (means) of a random semigroup and their equivalence in the sense of Chernoff to semigroups with averaged generators.

Keywords: boundary-value problem, blowup, dynamical system, $\Omega$-explosion, semigroup, random dynamical system, Chernoff's theorem, averaging

Funding Agency Grant Number
Russian Science Foundation 14-11-00687
Ministry of Education and Science of the Russian Federation 10-14


DOI: https://doi.org/10.4213/tmf8835

Full text: PDF file (548 kB)
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English version:
Theoretical and Mathematical Physics, 2015, 185:2, 1582–1598

Bibliographic databases:

Document Type: Article
Received: 05.12.2014
Revised: 13.04.2015

Citation: L. S. Efremova, V. Zh. Sakbaev, “Notion of blowup of the solution set of differential equations and averaging of random semigroups”, TMF, 185:2 (2015), 252–271; Theoret. and Math. Phys., 185:2 (2015), 1582–1598

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. V. Zh. Sakbaev, “On the law of large numbers for compositions of independent random semigroups”, Russian Math. (Iz. VUZ), 60:10 (2016), 72–76  mathnet  crossref  mathscinet  isi  elib  elib
    2. L. S. Efremova, “Dynamics of skew products of interval maps”, Russian Math. Surveys, 72:1 (2017), 101–178  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    3. V. Zh. Sakbaev, “Averaging of random walks and shift-invariant measures on a Hilbert space”, Theoret. and Math. Phys., 191:3 (2017), 886–909  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    4. Sakbaev V.Zh., “Averaging of Random Flows of Linear and Nonlinear Maps”, European Conference - Workshop Nonlinear Maps and Applications, Journal of Physics Conference Series, 990, IOP Publishing Ltd, 2018, UNSP 012012  crossref  isi  scopus
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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