10.25673/120454">
Proceedings of International Conference on Applied Innovation in IT  ·  2025/06/27  ·  Vol. 13  ·  Issue 2  ·  pp. 339–346
Investigation of Doubly Nonlinear Parabolic Equation
Makhmud Bobokandov, Nodir Uralov, Shokhsanam Shukurova and Zilola Sultonova
📄 Download PDF DOI: <a href="http://dx.doi.org/10.25673/120454">10.25673/120454</a>
Abstract
We study the properties of solutions for a porous medium equation (PME) in non-divergent form with a source term. The PME is a fundamental model in various physical and biological processes, including fluid flow through porous media, heat transfer, and population dynamics. Unlike the classical heat equation, the PME exhibits nonlinear diffusion, leading to rich mathematical structures and solution behaviours. Our main focus is obtaining exact solutions using the separable variable method under certain parameter constraints. These solutions provide explicit representations of the evolving profile of the medium and provide insight into the dynamics of the equation. Additionally, we construct a self-similar Barenblatt-type solution, a fundamental tool for analysing long-time asymptotics and the spreading behaviour of solutions. Self-similar solutions provide insights into the scaling properties of the PME and the influence of the source term on solution evolution. Moreover, we have constructed a numerical scheme, calculated numerical results and based on numerical solutions shown graphs in some particular cases.
Keywords
Quasilinear Porous Medium Equation Separation of Variables Comparison Principle Self-Similar Solution.
References
  1. A. Gárriz, "Propagation of solutions of the Porous Medium Equation with reaction and their travelling wave behaviour," Nonlinear Analysis, vol. 195, Art. no. 111736, 2020, [Online]. Available: https://doi.org/10.1016/j.na.2019.111736.
  2. A. Matyakubov and D. Raupov, "Explicit estimate for blow-up solutions of nonlinear parabolic systems of non-divergence form with variable density," in AIP Conf. Proc., vol. 2781, no. 1, 2023, [Online]. Available: https://doi.org/10.1063/5.0144768.
  3. M. Aripov and O. Atabaev, "Qualitative behavior of solutions of doubly degenerate parabolic problem with nonlinear source and absorption terms," Bull. Inst. Math., vol. 7, no. 1, pp. 21-32, 2024.
  4. M. Aripov and O. Djabbarov, "Method WKB for solutions of double nonlinear parabolic equation with damping term and variable coefficient," in AIP Conf. Proc., vol. 2781, no. 1, 2023, [Online]. Available: https://doi.org/10.1063/5.0145383.
  5. M. Aripov, M. Bobokandov and N. Uralov, "Analysis of double nonlinear parabolic crosswise-diffusion systems with time-dependent nonlinearity absorption," ICTEA: Int. Conf. Thermal Eng., vol. 1, 2024.
  6. M. Aripov, Z.R. Rakhmonov and A.A. Alimov, "On the behaviors of solutions of a nonlinear diffusion system with a source and nonlinear boundary conditions," Bull. Karaganda Univ. Math. Ser., vol. 113, no. 1, pp. 28-45, 2024, [Online]. Available: https://doi.org/10.31489/2024m1/28-45.
  7. Z.R. Rakhmonov and A.A. Alimov, "Properties of solutions for a nonlinear diffusion problem with a gradient nonlinearity," Int. J. Appl. Math., vol. 36, no. 3, p. 405, 2023, [Online]. Available: https://doi.org/10.12732/ijam.v36i3.7.
  8. Z.R. Rakhmonov, A. Alimov and J. Urunbaev, "On the behavior of solutions for a system of multidimensional diffusion equations with nonlinear boundary conditions," in AIP Conf. Proc., vol. 3085, no. 1, Art. no. 020032, 2024, [Online]. Available: https://doi.org/10.1063/5.0194896.
  9. J.L. Díaz, "Modeling of an aircraft fire extinguishing process with a porous medium equation," SN Appl. Sci., vol. 2, pp. 1-20, 2020, [Online]. Available: https://doi.org/10.1007/s42452-020-03891-9.
  10. M. Aripov and S. Sadullaeva, Computer Simulation of Nonlinear Diffusion Processes [in Russian]. Tashkent: Press of the National University of Uzbekistan, 2020.
  11. M. Aripov, M. Bobokandov and M. Mamatkulova, "Analysis of a double nonlinear diffusion equation in inhomogeneous medium," J. Math. Sci., pp. 1-13, 2024, [Online]. Available: https://doi.org/10.1007/s10958-024-07384-7.
  12. J.L. Vázquez, The Porous Medium Equation: Mathematical Theory. Oxford: Oxford University Press, 2006, [Online]. Available: https://doi.org/10.1093/acprof:oso/9780198569039.001.0001.
  13. M. Aripov and M. Bobokandov, "Analysis of a double nonlinear parabolic equation with a source in an inhomogeneous medium," Proc. Inst. Math. Mech., vol. 50, no. 2, pp. 285-306, 2024, [Online]. Available: https://doi.org/10.30546/2409-4994.2024.50.2.285.
  14. D.G. Aronson, "The porous medium equation," in Nonlinear Diffusion Problems, Lecture Notes in Mathematics, vol. 1224, Springer, Berlin, Heidelberg, 1986, [Online]. Available: https://doi.org/10.1007/BFb0072687.
  15. F. Filomena, J.L. Vázquez and B. Volzone, "Anisotropic fast diffusion equations," Nonlinear Analysis, vol. 233, Art. no. 113298, 2023, [Online]. Available: https://doi.org/10.1016/j.na.2023.113298.
  16. Z.R. Rakhmonov, J.E. Urunbaev and A.A. Alimov, "Properties of solutions of a system of nonlinear parabolic equations with nonlinear boundary conditions," in AIP Conf. Proc., vol. 2637, no. 1, 2022, [Online]. Available: https://doi.org/10.1063/5.0119747.
  17. M. Winkler, "Universal bounds for global solutions of a forced porous medium equation," Nonlinear Analysis: Theory, Methods & Applications, vol. 57, no. 3, pp. 349-362, 2004, [Online]. Available: https://doi.org/10.1016/j.na.2004.02.019.
  18. R.G. Iagar and D.R. Munteanu, "A porous medium equation with spatially inhomogeneous absorption. Part I: Self-similar solutions," J. Math. Anal. Appl., vol. 543, no. 1, Art. no. 128965, 2025, [Online]. Available: https://doi.org/10.1016/j.jmaa.2024.128965.
  19. R.G. Iagar, M. Latorre and A. Sánchez, "Blow-up patterns for a reaction-diffusion equation with weighted reaction in general dimension," Adv. Differ. Equ., vol. 29, no. 7/8, pp. 515-574, 2024, [Online]. Available: https://doi.org/10.57262/ade029-0708-515.
  20. H. Fujita, "On the blowing up of solutions of the Cauchy problem," J. Fac. Sci. Univ. Tokyo, vol. 13, pp. 109-124, 1966.
  21. P.G. Estevez, Ch. Qu and Sh. Zhang, "Separation of variables of a generalized porous medium equation with nonlinear source," J. Math. Anal. Appl., vol. 275, no. 1, pp. 44-59, 2002, [Online]. Available: https://doi.org/10.1016/S0022-247X(02)00214-7.
  22. G. Meglioli and F. Punzo, "Blow-up and global existence for solutions to the porous medium equation with reaction and fast decaying density," Nonlinear Analysis, vol. 203, Art. no. 112187, 2021, [Online]. Available: https://doi.org/10.1016/j.na.2020.112187.
  23. B.B. Hernández, R.G. Iagar, P.R. Gordoa, A. Pickering and A. Sánchez, "Equivalence and finite time blow-up of solutions and interfaces for two nonlinear diffusion equations," J. Math. Anal. Appl., vol. 482, no. 1, Art. no. 123503, 2020, [Online]. Available: https://doi.org/10.1016/j.jmaa.2019.123503.
  24. V.L. Natyaganov and Y.D. Skobennikova, "A solution to heat equation with exacerbation and stopped heat wave," Moscow Univ. Mech. Bull., vol. 77, pp. 151-153, 2022, [Online]. Available: https://doi.org/10.3103/S0027133022050016.
  25. Y. Belaud and A. Shishkov, "Extinction in a finite time for solutions of a class of quasilinear parabolic equations," Asymptotic Anal., vol. 127, no. 1-2, pp. 97-119, 2021, [Online]. Available: https://doi.org/10.3233/ASY-211674.
  26. S. Kamin and P. Rosenau, "Propagation of thermal waves in an inhomogeneous medium," Commun. Pure Appl. Math., vol. 34, no. 6, pp. 831-852, 1981, [Online]. Available: https://doi.org/10.1002/cpa.3160340605.
  27. V.A. Galaktionov, S.P. Kurdyumov and A.A. Samarskii, "On asymptotic stability of invariant solutions of nonlinear heat equations with a source," Differents. Uravn., vol. 20, no. 4, pp. 614-632, 1984.
  28. A.D. Polyanin and A.I. Zhurov, Separation of Variables and Exact Solutions to Nonlinear PDEs. New York: Chapman and Hall/CRC, 2021, [Online]. Available: https://doi.org/10.1201/9781003042297.
  29. P.M. Senik, "Inversion of the incomplete beta function," Ukrainian Math. J., vol. 21, no. 3, pp. 271-278, 1969, [Online]. Available: https://doi.org/10.1007/BF01085368.
  30. A.A. Samarskii, V.A. Galaktionov, S.P. Kurdyumov and A.P. Mikhailov, Blow-Up in Quasilinear Parabolic Equations. Berlin, New York: De Gruyter, 1995, [Online]. Available: https://doi.org/10.1515/9783110889864.

Proceedings of the International Conference on Applied Innovations in IT by Anhalt University of Applied Sciences is licensed under CC BY-SA 4.0  ·  This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License

ICAIIT 2026
International Conference on Applied Innovation in IT
Navigation
Publisher
ISSN2199-8876
Publisher Edition Hochschule Anhalt
Location Anhalt University of Applied Sciences
Email leiterin.hsb@hs-anhalt.de
Phone +49 (0) 3496 67 5611
Address Building 01, Room 425
Bernburger Str. 55
D-06366 Köthen, Germany
Open Access License
CC BY-SA 4.0

All works are licensed under the Creative Commons Attribution-ShareAlike 4.0 International License (CC BY-SA 4.0), unless otherwise noted.

Published by ICAIIT in cooperation with Anhalt University of Applied Sciences.

© 2026 ICAIIT — International Conference on Applied Innovations in IT. Anhalt University of Applied Sciences, Köthen, Germany.
Visitors: site traffic counter