Geodesic
Notion of blowup of the~solution set of differential equations and averaging of random semigroups
Teoretičeskaâ i matematičeskaâ fizika, Tome 185 (2015) no. 2, pp. 252-271

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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, semigroup, random dynamical system, Chernoff's theorem, averaging.
Mots-clés : $\Omega$-explosion 
@article{TMF_2015_185_2_a1,
     author = {L. S. Efremova and V. Zh. Sakbaev},
     title = {Notion of blowup of the~solution set of differential equations and averaging of random semigroups},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {252--271},
     publisher = {mathdoc},
     volume = {185},
     number = {2},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2015_185_2_a1/}
}
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L. S. Efremova; V. Zh. Sakbaev. Notion of blowup of the~solution set of differential equations and averaging of random semigroups. Teoretičeskaâ i matematičeskaâ fizika, Tome 185 (2015) no. 2, pp. 252-271. http://geodesic.mathdoc.fr/item/TMF_2015_185_2_a1/