Asymptotics of solutions of a reaction-diffusion system with nonlinear boundary conditions
DOI:
https://doi.org/10.31489/2026m3/156-169Keywords:
system of nonlinear equations, reaction-diffusion, global solution, asymptotic, compact support, self-similar solution, sweep method, numerical solutionAbstract
This paper investigates the asymptotic properties of a system of parabolic equations with nonlinear boundary conditions. Systems of parabolic equations describe the behavior of various processes over time, such as heat propagation and material diffusion. Nonlinear boundary conditions involve complex properties that differ from the usual local conditions at the boundaries of the system, which influence the solutions of the system. In this article, a system of parabolic differential equations associated with nonlinear boundary conditions is reduced to a system of self-similar equations by means of certain transformations using the self-similar approach, and the asymptotic behavior of solutions with compact support is found. In addition, the reaction-diffusion system is approximated using implicit difference schemes and an iterative process for the numerical solution is constructed. It is known that the main problem for the iterative process is the choice of the best initial approximation. The article proposes to use the constructed asymptotic formula to solve this problem. Computational experiments were carried out with different values of numerical parameters and the results are presented graphically. The numerical results obtained in the case of slow diffusion p>2 of a nonlinear reaction-diffusion system show that the reaction-diffusion process proceeds at a finite rate.
References
Wu, Z.Q., Zhao, J.N., Yin, J.X., & Li, H.L. (2001). Nonlinear Diffusion Equations. Singapore: World Scientific.
Levine, H.A. (1990). The role of critical exponents in blow-up theorems. SIAM Review, 32(2), 252–288. https://doi.org/10.1137/1032046
Aripov, M., & Bobokandov, M. (2022). Blow-up analysis for a doubly nonlinear parabolic nondivergence form equation with source term. Bulletin of the Institute of Mathematics, 5(4), 7–21.
Aripov, M., Bobokandov, M., & Mamatkulova, M. (2025). Analysis of a double nonlinear diffusion equation in inhomogeneous medium. Journal of Mathematical Sciences, 289(5), 657–669. https://doi.org/10.1007/s10958-024-07384-7
Aripov, M., Bobokandov, M., & Uralov, N. (2024). Analysis of Double Nonlinear Parabolic Crosswise-Diffusion Systems with Time-Dependent Nonlinearity Absorption. ICTEA: International Conference on Thermal Engineering, 1(1), 1–3.
Aripov, M., & Bobokandov, M. (2024). Analysis of a double nonlinear parabolic equation with a source in an inhomogeneous medium. Proceedings of Institute of Mathematics & Mechanics National Academy of Sciences of Azerbaijan, 50(2), 285–306. https://doi.org/10.30546/2409-4994.2024.50.2.285
Wang, Z.J., Yin, J.X., & Wang, C.P. (2007). Critical exponents of the non-Newtonian polytropic filtration equation with nonlinear boundary condition. Applied Mathematics Letters, 20(2), 142– 147. https://doi.org/10.1016/j.aml.2006.03.008
Li, Z.P., Cui, Z.J., & Mu, C.L. (2008). Critical curves for fast diffusive polytropic filtration equations coupled through boundary. Applicable Analysis, 87(9), 1041–1052. https://doi.org/10.1080/00036810802428912
Mi, Y.S., Mu, C.L., & Chen, B.T. (2011). Critical exponents for a nonlinear degenerate parabolic system coupled via nonlinear boundary flux. Journal of the Korean Mathematical Society, 48(3), 513–527. https://doi.org/10.4134/JKMS.2011.48.3.513
Li, Z.P., Mu, C.L., & Cui, Z.J. (2009). Critical curves for a fast diffusive polytropic filtration system coupled via nonlinear boundary flux. Zeitschrift fur angewandte Mathematik und Physik, 60(2), 284–298. https://doi.org/10.1007/S00033-008-7095-1
Zheng, S.N., Song, X.F., & Jiang, Z.X. (2004). Critical Fujita exponents for degenerate parabolic equations coupled via nonlinear boundary flux. Journal of Mathematical Analysis and Applications, 298(1), 308–324. https://doi.org/10.1016/j.jmaa.2004.05.017
Xiang, Z.Y., Chen, Q., & Mu, C.L. (2007). Critical curves for degenerate parabolic equations coupled via non-linear boundary flux. Applied Mathematics and Computation, 189(1), 549–559. https://doi.org/10.1016/j.amc.2006.11.130
Li, Z.P., & Mu, C.L. (2008). Critical curves for fast diffusive non-Newtonian equations coupled via nonlinear boundary flux. Journal of Mathematical Analysis and Applications, 340(2), 876– 883. https://doi.org/10.1016/j.jmaa.2007.09.009
Jiang, Z.X. (2009). Doubly degenerate parabolic equation with nonlinear inner sources or boundary flux. Doctoral Thesis. Dalian University of Technology, China.
Galaktionov, V.A., & Levine, H.A. (1996). On critical Fujita exponents for heat equations with nonlinear flux conditions on the boundary. Israel Journal of Mathematics, 94, 125–146. https://doi.org/10.1007/BF02762700
Chen, B.T., Mi, Y.S., & Mu, C.L. (2011). Critical exponents for a doubly degenerate parabolic system coupled via nonlinear boundary flux. Journal of the Korean Mathematical Society, 48(3), 513–527. https://doi.org/10.4134/JKMS.2011.48.3.513
Quiros, F., & Rossi, J.D. (2001). Blow-up sets and Fujita type curves for a degenerate parabolic system with nonlinear boundary conditions. Indiana University Mathematics Journal, 50(1), 629–654. https://doi.org/10.1512/iumj.2001.50.1828
Xiang, Z.Y., Mu, C.L., & Wang, Y. (2009). Critical curve of the non-Newtonian polytropic filtration equations coupled via nonlinear boundary flux. Rocky Mountain Journal of Mathematics, 39(2), 689–705. https://doi.org/10.1216/RMJ-2009-39-2-689
Rakhmonov, Z.R., & Urunbayev, J.E. (2019). On a problem of cross-diffusion with nonlocal boundary conditions. Journal of Siberian Federal University. Mathematics & Physics, 12(5), 614–620. https://doi.org/10.17516/1997-1397-2019-12-5-614-620
Rakhmonov, Z.R., Khaydarov, A., & Urunbaev, J.E. (2020). Global existence and nonexistence of solutions to a cross diffusion system with nonlocal boundary conditions. Mathematics and Statistics, 8(4), 404–409. https://doi.org/10.13189/ms.2020.080405
Aripov, M.M., Rakhmonov, Z.R., & Alimov, A.A. (2024). On the behaviors of solutions of a nonlinear diffusion system with a source and nonlinear boundary conditions. Bulletin of the Karaganda University. Mathematics Series, 1(113), 28–45. https://doi.org/10.31489/2024m1/28-45
Charkaoui, A., & El Guermai, G. (2025). Periodic evolution problems driven by Leray-Lions type operators with fully nonlinear boundary conditions. Filomat, 39(27), 9473–9492. https://doi.org/10.2298/FIL2527473C
Pierre, M., & Schmitt, D. (2000). Blowup in reaction-diffusion systems with dissipation of mass. SIAM Review, 42(1), 93–106. https://doi.org/10.1137/S0036144599359735
Pierre, M. (2010). Global existence in reaction-diffusion systems with control of mass: a survey. Milan Journal of Mathematics, 78(2), 417–455. https://doi.org/10.1007/s00032-010-0133-4
Charkaoui, A., Ben-Loghfyry, A., & Zeng, S. (2024). Nonlinear parabolic double phase variable exponent systems with applications in image noise removal. Applied Mathematical Modelling, 132, 495–530. https://doi.org/10.1016/j.apm.2024.04.059
Pierre-Louis, L., & Panagiotis, E.S. (2020). The asymptotics of stochastically perturbed reaction diffusion equations and front propagation. Comptes Rendus. Math´ematique, 358(8), 931–938. https://doi.org/10.5802/crmath.117
Charkaoui, A., Kouadri, G., & Alaa, N.E. (2022). Some results on the existence of weak periodic solutions for quasilinear parabolic systems with L1 data. Boletim da Sociedade Paranaense de Matem´atica, 40, 1–15. https://doi.org/10.5269/bspm.45134
Elaassri, A., Uahabi, K.L., Charkaoui, A., Alaa, N.E., & Mesbahi, S. (2019). Existence of weak periodic solution for quasilinear parabolic problem with nonlinear boundary conditions. Annals of the University of Craiova-Mathematics and Computer Science Series, 46(1), 1–13.
Aripov, M., Matyakubov, A.S., & Imomnazarov, B. (2020). The Cauchy problem for a nonlinear degenerate parabolic system in non-divergence form. Mathematical notes of NEFU, 27(3), 27–38. https://doi.org/10.25587/SVFU.2020.93.40.003
Aripov, M., Bobokandov, M., & Mamatkulova, M. (2024). To numerical solution of the nondivergent diffusion equation in non-homogeneous medium with source or absorption. AIP Conference Proceedings, 3085(1), Article 020024. https://doi.org/10.1063/5.0194632
Wendroff, B. (1968) Difference Methods for Initial-Value Problems (Robert D. Richtmyer and K. W. Morton). SIAM Review, 10(3), 381–383. https://doi.org/10.1137/1010073
Samarskii, A.A. (2001). The Theory of Difference Schemes (1st ed.). CRC Press. https://doi.org/10.1201/9780203908518
