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Apriori estimates for evolution systems


Presented by Taras Shevchenko National University of Kyiv


Authors:Anikushin Andriy Valeriovych, Hulianytskyi Andrii Leonidovych and Zatula Dmytro Vasylovych 


In this work, we study the solvability of several types of differential equations with integral terms, fractional differential equations and evolution systems with a random component.

Basing on theoretical research, we establish the generalized solvability of elliptic, parabolic, pseudoparabolic, and hyperbolic integro-differential equations, the equations with a nonnegative-definite integral operator (in particular, of high order). These results yield the existence of optimal control (in particular, impulse and pointwise) for the corresponding systems. For the aforementioned types of integro-differential equations and the time-fractional diffusion equation, we obtain the convergence of Galerkin approximations. We also prove weak solvability and construct numerical methods for a variable-order subdiffusion equation. For evolution systems with a random component we prove the existence and uniqueness of a solution. The method is based on the estimation of the random component in various spaces of random variables, namely the estimates of the distributions of Hölder semi-norms for random processes from Fψ(Ω), Lp(Ω) and Orlicz spaces, combined with the approximation of such processes by real-valued functions.

The basic conceptual foundations for the study of the multidimensional thermoelasticity problem for complex materials, taking into account the history of the process, are formed.

The results of the work are proposed as the theoretical basis for the simulation of a broad class of physical, biological and other systems with memory, as well as the optimal control of these systems.

Comparison with world counterparts. At the level with world counterparts. In some aspects, it exceeds the known world analogues.

Number of publications:64, including 1 monograph, 2 textbooks, 23 articles (6 - foreign publications). According to the Google Scholar database, the total number of citations on authors' publications, presented in the work, is 10, h-index (for the work) = 5.