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Boundary and Extremum Problems for the Nonlinear Reaction–Diffusion–Convection Equation under the Dirichlet Condition

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Abstract

The global solvability of a boundary value problem for the reaction–diffusion–convection equation in which the reaction coefficient nonlinearly depends on the solution is proved. An inhomogeneous Dirichlet boundary condition for concentration is considered. In this case, the nonlinearity caused by the reaction coefficient is not monotonic in the entire domain. The solvability of a control problem with boundary, distributed, and multiplicative controls is proved. In the case when the reaction coefficient and quality functionals are Fréchet differentiable, optimality systems for extremum problems are derived. Based on their analysis, a stationary analogue of the bang–bang principle for specific control problems is established.

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Funding

The first author’s work was carried out within a state assignment for the Institute of Applied Mathematics of the Far Eastern Branch of the Russian Academy of Sciences (topic no. 075-01095-20-00), and the second author acknowledges the support of the Ministry of Science and Higher Education of the Russian Federation (project no. 075-02-2020-1482-1, additional agreement of April 21, 2020).

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Authors and Affiliations

  1. Institute of Applied Mathematics, Far Eastern Branch, Russian Academy of Sciences, 690041, Vladivostok, Russia

    R. V. Brizitskii

  2. Far Eastern Federal University, 690950, Vladivostok, Russia

    P. A. Maksimov

Authors
  1. R. V. Brizitskii
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  2. P. A. Maksimov
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Correspondence to R. V. Brizitskii or P. A. Maksimov.

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Translated by E. Chernokozhin

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Brizitskii, R.V., Maksimov, P.A. Boundary and Extremum Problems for the Nonlinear Reaction–Diffusion–Convection Equation under the Dirichlet Condition. Comput. Math. and Math. Phys. 61, 974–986 (2021). https://doi.org/10.1134/S0965542521060038

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