Vestnik КRAUNC. Fiz.-Mat. nauki. 2023. vol. 43. no. 2. P. 9-19. ISSN 2079-6641

Research Article
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MSC 35B44, 35C06, 35K51, 35K61

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Global and Blow-Up Solutions for a Nonlinear Diffusion System with a Source and Nonlinear Boundary Conditions

A. A. Alimov¹²^\ast, Z. R. Rakhmonov¹

¹National University of Uzbekistan named after Mirzo Ulugbek, 100174, Tashkent, Universitetskaya, 4, Uzbekistan
²Tashkent branch of the G.V. Plekhanov Russian University of Economics, 100164, Tashkent, Shakhriobod, 3, Uzbekistan

Abstract. In this paper, we study the global solvability and unsolvability of a nonlinear diffusion system with nonlinear boundary conditions in the case of slow diffusion. The conditions for the global existence of the solution in time and the unsolvability of the solution of the diffusion problem in a homogeneous medium are found on the basis of comparison principle and self-similar analysis. We obtain the critical exponent of the Fujita type and the critical global existence exponent, which plays an important role in the study of the qualitative properties of nonlinear models of reaction-diffusion, heat transfer, filtration and other physical, chemical, biological processes. In the global solvability case the principal terms of the asymptotic of solutions are obtained. It is well known that iterative methods require the presence of a suitable initial approximation, resulting in a rapid convergence to the exact solution and preserving qualitative properties of nonlinear processes under study, it is a major challenge for the numerical solution of nonlinear problems. This difficulty, depending on the value of the numerical parameters of the equation is overcome by a successful choice of initial approximations, which are mainly in the calculations suggested taking asymptotic formula.

Key words: blow-up, nonlinear boundary condition, critical exponents, nonlinear diffusion system, asymptotic

Received: 21.05.2023; Revised: 09.06.2023; Accepted: 30.06.2023; First online: 03.07.2023

For citation. Alimov A. A., Rakhmonov Z. R. Global and blow-up solutions for a nonlinear diffusion system with a source and nonlinear boundary conditions. Vestnik KRAUNC. Fiz.-mat. nauki. 2023, 43: 2, 9-19. EDN: XJQODE.

Funding. The work was carried out without the support of funds.

Competing interests. There are no conflicts of interest regarding authorship and publication.

Contribution and Responsibility. Авторы участвовали в написании статьи и полностью несут ответственность за
предоставление окончательной версии статьи в печать.

^\astCorrespondence: E-mail:

The content is published under the terms of the Creative Commons Attribution 4.0 International License

© Alimov A. A., Rakhmonov Z. R., 2023

© Institute of Cosmophysical Research and Radio Wave Propagation, 2023 (original layout, design, compilation)


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Information about authors

Alimov Akram Abdurashidovich – Senior Lecturer of the Dep. of Information Systems and Mathematical Disciplines, Tashkent branch of the G.V. Plekhanov Russian University of Economics, ORCID 0009-0009-8518-4366

Rakhmonov Zafar Ravshanovich – D. Sci. (Phys. & Math.), Dean of the Faculty of Applied Mathematics and Intellectual Technologies of the National University of Uzbekistan, Tashkent, Uzbekistan., ORCID 0000-0002-4190-7069.