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**The non-classical models of mathematical physics the multipoint initial-final value condition.**
*(Russian.
English summary)*
Zbl 1492.35005

Summary: The article contains a review of the results obtained by the authors in the field of non-classical models of mathematical physics, for which we consider the multipoint initial-final value conditions that generalize Cauchy conditions and Showalter-Sidorov conditions. Recall that non-classical models of mathematical physics are models, whose representations in the form of equations or systems of equations in partial derivatives do not fit within the framework of one of the classical types: elliptic, parabolic or hyperbolic.

Abstract results are illustrated by concrete multipoint initial-final value problems for partial differential equations in various statements appeared recently in applications. Among them, we consider the non-autonomous Chen-Gurtin model with complex coefficients, the stochastic evolutionary Davis model, the macro model of transport flow at the crossroads based on the Oskolkov equations considered in the system of geometric graphs, taking into account the condition of continuity, balance of flows and the condition of the ban on traffic.

Abstract results are illustrated by concrete multipoint initial-final value problems for partial differential equations in various statements appeared recently in applications. Among them, we consider the non-autonomous Chen-Gurtin model with complex coefficients, the stochastic evolutionary Davis model, the macro model of transport flow at the crossroads based on the Oskolkov equations considered in the system of geometric graphs, taking into account the condition of continuity, balance of flows and the condition of the ban on traffic.

### MSC:

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

35K70 | Ultraparabolic equations, pseudoparabolic equations, etc. |

60H30 | Applications of stochastic analysis (to PDEs, etc.) |

### Keywords:

Sobolev-type equations; degenerate \(C_0\)-semiflow; resolving (semi)groups of operators; relatively spectral projectors; multipoint initial-final value condition; non-autonomous Chen-Gurtin model; stochastic Davis model; macro model of traffic flow at crossroad
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\textit{S. A. Zagrebina} and \textit{A. S. Konkina}, Vestn. Yuzhno-Ural. Gos. Univ., Ser. Mat. Model. Program. 15, No. 1, 60--83 (2022; Zbl 1492.35005)

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