Global and Blow-Up Solutions for a Nonlinear Diffusion System with a Source and Nonlinear Boundary Conditions
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 homog...
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| Format: | Article |
| Language: | English |
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KamGU by Vitus Bering
2023-07-01
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| Series: | Vestnik KRAUNC: Fiziko-Matematičeskie Nauki |
| Subjects: | |
| Online Access: | https://krasec.ru/ru/alimov432023eng/ |
| _version_ | 1797785001531539456 |
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| author | Alimov, A.A. Rakhmonov, Z.R. |
| author_facet | Alimov, A.A. Rakhmonov, Z.R. |
| author_sort | Alimov, A.A. |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-13T00:47:56Z |
| format | Article |
| id | doaj.art-f2b114eab3f743e585faf25ec123649e |
| institution | Directory Open Access Journal |
| issn | 2079-6641 2079-665X |
| language | English |
| last_indexed | 2024-03-13T00:47:56Z |
| publishDate | 2023-07-01 |
| publisher | KamGU by Vitus Bering |
| record_format | Article |
| series | Vestnik KRAUNC: Fiziko-Matematičeskie Nauki |
| spelling | doaj.art-f2b114eab3f743e585faf25ec123649e2023-07-08T19:46:39ZengKamGU by Vitus BeringVestnik KRAUNC: Fiziko-Matematičeskie Nauki2079-66412079-665X2023-07-012023291910.26117/2079-6641-2023-43-2-9-1910.26117/2079-6641-2023-43-2-9-19Global and Blow-Up Solutions for a Nonlinear Diffusion System with a Source and Nonlinear Boundary ConditionsAlimov, A.A.0Rakhmonov, Z.R.1National University of Uzbekistan named after Mirzo Ulugbek; Tashkent branch of the G.V. Plekhanov Russian University of EconomicsNational University of Uzbekistan named after Mirzo UlugbekIn 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.https://krasec.ru/ru/alimov432023eng/blow-upnonlinear boundary conditioncritical exponentsnonlinear diffusion systemasymptoticобострениенелинейное краевое условиекритические показателинелинейная диффузионная системаасимптотика |
| spellingShingle | 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: Fiziko-Matematičeskie Nauki blow-up nonlinear boundary condition critical exponents nonlinear diffusion system asymptotic обострение нелинейное краевое условие критические показатели нелинейная диффузионная система асимптотика |
| title | Global and Blow-Up Solutions for a Nonlinear Diffusion System with a Source and Nonlinear Boundary Conditions |
| title_full | Global and Blow-Up Solutions for a Nonlinear Diffusion System with a Source and Nonlinear Boundary Conditions |
| title_fullStr | Global and Blow-Up Solutions for a Nonlinear Diffusion System with a Source and Nonlinear Boundary Conditions |
| title_full_unstemmed | Global and Blow-Up Solutions for a Nonlinear Diffusion System with a Source and Nonlinear Boundary Conditions |
| title_short | Global and Blow-Up Solutions for a Nonlinear Diffusion System with a Source and Nonlinear Boundary Conditions |
| title_sort | global and blow up solutions for a nonlinear diffusion system with a source and nonlinear boundary conditions |
| topic | blow-up nonlinear boundary condition critical exponents nonlinear diffusion system asymptotic обострение нелинейное краевое условие критические показатели нелинейная диффузионная система асимптотика |
| url | https://krasec.ru/ru/alimov432023eng/ |
| work_keys_str_mv | AT alimovaa globalandblowupsolutionsforanonlineardiffusionsystemwithasourceandnonlinearboundaryconditions AT rakhmonovzr globalandblowupsolutionsforanonlineardiffusionsystemwithasourceandnonlinearboundaryconditions |