NOTION OF BLOWUP OF THE SOLUTION SET OF DIFFERENTIAL EQUATIONS AND AVERAGING OF RANDOM SEMIGROUPS

作者：Efremova L S^*; Sakbaev V Zh

来源：Theoretical and Mathematical Physics, 2015, 185(2): 1582-1598.

DOI：10.1007/s11232-015-0366-z

• boundary-value problem
• blowup
• dynamical system
• Omega-explosion
• semigroup
• random dynamical system
• Chernoff's theorem
• averaging

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.

• 出版日期2015-11