Global and blow-up solutions for a nonlinear diffusion system with a source and nonlinear boundary conditions

A. A. Alimov   ; Z. R. Rakhmonov

Vestnik KRAUNC. Fiziko-matematičeskie nauki, Tome 43 (2023) no. 2, pp. 9-19

Résumé

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.

Détail

Keywords: blow-up, nonlinear boundary condition, critical exponents, nonlinear diffusion system, asymptotic.

Bibliographie

[1] Samarskii A. A., Galaktionov V. A., Kurdyumov S. P., Mikhailov A. P., Blow-up in Quasilinear Parabolic Equations, Walter de Gruyter, Berlin, 1995 | MR | Zbl

[2] Wu Z. Q., Zhao J. N., Yin J. X., Li H. L., Nonlinear Diffusion Equations, World Scientific Publishing Co Inc., River Edge NJ, 2001 | MR | Zbl

[3] Kalashnikov A. S., “Some problems of the qualitative theory of nonlinear degenerate parabolic equations of second order”, Uspekhi Mat Nauk, 42 (1987), 135–176 | MR | Zbl

[4] Deng K., Levine H. A., “The role of critical exponents in blow-up theorems.”, The sequel J Math Anal Appl., 243 (2000), 85-126 | DOI | MR | Zbl

[5] Galaktionov V. A., Levine H. A., “On critical Fujita exponents for heat equations with nonlinear flux conditions on the boundary”, Israel J Math, 94 (1996), 125-146 | DOI | MR | Zbl

[6] Rakhmonov Z. R., Tillaev A. I., “On the behavior of the solution of a nonlinear polytropic filtration problem with a source and multiple nonlinearities.”, Nanosystems: physics chemistry mathematics, 9:3 (2018), 1-7

[7] Rakhmonov Z. R., Khaydarov A. T., Urunbaev J. E., “Global Existence and Nonexistence of Solutions to a Cross Diffusion System with Nonlocal Boundary Conditions.”, Mathematics and Statistics, 8:4 (2020), 404 - 409 | DOI | MR

[8] Rakhmonov Z., “On the properties of solutions of multidimensional nonlinear filtration problem with variable density and nonlocal boundary condition in the case of fast diffusion”, Journal of Siberian Federal University. Mathematics Physics, 9 (2016), 236-245

[9] Zhaoyin X., Chunlai M., Yulan W., “Critical curve of the non-Newtonian polytropic filtration equations coupled via nonlinear boundary flux”, Rocky mountain journal of mathematics, 2:39 (2009) | MR | Zbl

[10] Levine H. A., “The role of critical exponents in blow up theorems”, SIAM Rev., 32 (1990), 262-288 | DOI | MR | Zbl

[11] Li Z. P., Mu C. L., Cui Z. J., “Critical curves for a fast diffusive polytropic filtration system coupled via nonlinear boundary flux”, Z. Angew Math. Phys., 60 (2009), 284-296 | DOI | MR

[12] Chen B., Mi. Y., Mu Ch., “Global existence and nonexistence for a doubly degenerate parabolic system coupled via nonlinear boundary flux”, Acta Mathematica Scientia, 31B(2) (2011), 681-693 | MR

[13] Yongsheng M., Chunlai M., Botao Ch., “Critical exponents for a doubly degenerate parabolic system coupled via nonlinear boundary flux”, J. Korean Math, 48:3 (2011), 513-527 | DOI | MR | Zbl

[14] Aripov M. M., Matyakubov A. S., “To the qualitative properties of solution of system equations not in divergence form of polytrophic filtration in variable density”, Nanosystems: Physics Chemistry Mathematics, 8(3) (2017), 317-322 | DOI

[15] Rakhmonov Z., Urunbaev J., Alimov A., “Properties of solutions of a system of nonlinear parabolic equations with nonlinear boundary conditions”, AIP Conference Proceedings, 2637:040008 (2022)

[16] Rakhmonov Z., Parovik R., Alimov A., “Global existence and nonexistence for a multidimensional system of parabolic equations with nonlinear boundary conditions.”, AIP Conference Proceedings, 2365:060022 (2021)

[17] Aripov M., Matyakubov A., Bobokandov M., “Cauchy problem for the heat dissipation equation in non-homogeneous medium”, AIP Conference Proceedings, 2781 (2023)