Author(s):
Khaydarov Abdugappar
PhD. , Associate Professor
Position: Professor
Office: National university of Uzbekistan named after Mirzo Ulugbek
Address: 100174, Uzbekistan, Tashkent, University Street, 4th House
Phone Office: (998) 71 227 12 24
E-mail: abdugapparxaydarov1953@gmail.com

Begulov Utkir Uktam ugli
Position: Student
Office: National university of Uzbekistan named after Mirzo Ulugbek
Address: 100174, Uzbekistan, Tashkent, University Street, 4th House
Phone Office: (998) 71 227 12 24
E-mail: begulov0108@gmail.com

References:
[1] Samarskii A.A., Galaktionov V.A., Kurdyumov S.P., Mikhailov A.P. Blow-up in quasilinear parabolic equations. Berlin; N.Y.: Walter de Gruyter; 1995: 535. DOI:10.1515/9783110889864.

[2] Aripov M.M., Sadullaeva Sh.A. Computer modeling of nonlinear diffusion processes. Monography. Tashkent: University; 2020: 670. (In Uzbek.)

[3] Aripov M.M., Matyakubov A.S., Imomnazarov B.Kh. The Cauchy problem for a nonlinear degenerate parabolic system in non-divergence form. Mathematical Notes of NEFU. 2020; 27(3):27–38. DOI:10.25587/SVFU.2020.93.40.003. (In Russ.)

[4] Aripov M.M., Matyakubov A.S., Bobokandov M. Cauchy problem for the heat dissipation equation in non-homogeneous medium. AIP Conference Proceedings. 2023; (2781):020027. DOI:10.1063/5.0144807.

[5] Aripov M.M., Bobokandov M. The Cauchy problem for a doubly nonlinear parabolic equation with variable density and nonlinear time-dependent absorption. Journal of Mathematical Sciences. 2023; 3(277):020055. DOI:10.1007/s10958-023-06840-0.

[6] Aripov M.M., Bobokandov M., Mamatkulova M. To numerical solution of the nondivergent diffusion equation in non-homogeneous medium with source or absorption. AIP Conference Proceedings. 2024; (3085):020024. DOI:10.1063/5.0194632.

[7] Aripov M.M., Sayfullayeva M., Kabiljanova F., Bobokandov M. About one exact solution to the nonlinear problem of a biological population with absorption in a heterogeneous medium. AIP Conference Proceedings. 2024; (3085):020031. DOI:10.1063/5.0194751.

[8] Galaktionov V.A., King J.R. On the behaviour of blow-up interfaces for an inhomogeneous filtration equation. IMA Journal of Applied Mathematics. 1996; 57(1):53–77. DOI:10.1093/imamat/57.1.53.

[9] Martynenko A.V., Tedeev A.F., Shramenko V.N. The Cauchy problem for a degenerate parabolic equation with inhomogeneous density and source in the class of slowly decaying initial data. Izvestiya: Mathematics. 2012; 76(3):563–580. DOI:10.1070/IM2012v076n03ABEH002595. (In Russ.)

[10] Martynenko A.V., Tedeev A.F. Cauchy problem for a quasilinear parabolic equation with a source term and an inhomogeneous density. Computational Mathematics and Mathematical Physics. 2007; (47):238–248. DOI:10.1134/S096554250702008X. (In Russ.)

[11] Aripov M.M., Rakhmonov Z.R. On the asymptotic behavior of the self-similar solutions of a nonlinear problem of polytropic filtration with a nonlinear boundary condition. Vychislitel’nye Tekhnologii. 2013; 18(4):50–55. (In Russ.)

[12] 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. 2018; 9(3):323–329. DOI:10.17586/2220-8054. (In Russ.)

[13] Galaktionov V.A., Kurdyumov S.P., Samarskii A.A. On asymptotic “eigenfunctions” of the Cauchy problem for a nonlinear parabolic equation. Mathematics of the USSR — Sbornik. 1986; 54(2):421–455. DOI:10.1070/SM1986v054n02ABEH002979. (In Russ.)

[14] Aripov M.M., Sadullaeva Sh.A. To properties of solutions to reaction-diffusion equation with double nonlinearity with distributed parameters. Journal of Siberian Federal University. Mathematics & Physics. 2013; 6(2):157–167. Available at: https://cyberleninka.ru/article/n/to-properties-of-solutions-to-reaction-diusion-equation-with-double-nonlinearity-with-distributed-parameters. (In Russ.)

[15] Sadullaeva S.A., Khaydarov A.T., Kabilianova F.A. Modeling of multidimensional problems in nonlinear heat conductivity in non-divergence case. 3rd International Symposium on Multidisciplinary Studies and Innovative Technologies, ISMSIT 2019 — Proceedings. 2019: 8932954. Available at: https://www.elibrary.ru/item.asp?id=43232677&pff=1.

[16] Barenblatt G.I., Vazquez J.L. A new free boundary problem for unsteady flows in porous media. European Journal of Applied Mathematics. 1998; 9(1):37–54. DOI:10.1017/S0956792597003331.

[17] Bertsch M., Kersner R., Peletier L.A. Sur le comportement de la fronti`ere libre dans une ´equation en th´eorie de la filtration. [On the behavior of the free boundary for an equation in filtration theory]. Comptes Rendus des Seances de l’Academie des Sciences. Vie Academique; 1982; 295(2):63–66. Available at: http://mathscinet.ams.org/mathscinet-getitem?mr=676363.

[18] Fujita H. On the blowing up of solutions to the Cauchy problem for 𝑢1 + Δ𝑢+𝑢1+𝛼. Journal of the Faculty of Science, University of Tokyo. 1966; 13(2):109–124. Available at: https://repository.dl.itc.u-tokyo.ac.jp/records/39882.

[19] Gilding B.H., Peletier L.A. On a class of similarity solutions of the porous media equation II. Journal of Mathematical Analysis and Applications. 1977; 57(3):522–538. DOI:10.1016/0022-247X(77)90244-X.

[20] Galaktionov V.A. Conditions for global non-existence and localization of solutions of the Cauchy problem for a class of non-linear parabolic equations. USSR Computational Mathematics and Mathematical Physics. 1983; 23(6):36–44. DOI:10.1016/S0041-5553(83)80073-1.

[21] Kersner R. On the behavior when t growths to infinity of generalized solutions of the generate quasilinear parabolic equations. Acta Mathematica Academiae Scientiarum Hungaricae. 1979; (34):157–163.

[22] Rakhmonov Z., Parovik R., Alimov A. Global existence and nonexistence for a multidimensional system of parabolic equations with nonlinear boundary conditions. AIP Conference Proceedings. 2021; (2365):060022. DOI:10.1063/5.0057139.

[23] Rakhmonov Z.R., Urunbaev J.E., Alimov A.A. Properties of solutions of a system of nonlinear parabolic equations with nonlinear boundary conditions. AIP Conference Proceedings. 2022; (2637):040008. DOI:10.1063/5.0119747.

[24] Rakhmonov Z.R., Alimov A.A. Properties of solutions for a nonlinear diffusion problem with a gradient nonlinearity. International Journal of Applied Mathematics. 2023; 36(3):405–424. DOI:10.12732/ijam.v36i3.7.

[25] Khaydarov A., Mamatov A. Modeling of double nonlinear thermal conductivity processes in two-dimensional domains using solutions of an approximately self-similar. AIP Conference Proceedings. 2023; (2781):020067. DOI:10.1063/5.0144746.

[26] Sadullaeva Sh.A., Anarova Sh., Khaydarov A., Fayzullayeva Z., Nazirova D., Mirkhodzhaeva N. Numerical modelling of the system of doubly nonlinear diffusion equations with convective transfer and damping. Proceedings of the Cognitive Models and Artificial Intelligence Conference. 2024: 182–185. DOI:10.1145/3660853.3660909.

[27] Xaydarov A.T., Begulov U.U. Eksponensial uzgaruvchan zichlikli chiziqsiz muhitda issiqlik tarqalish jarayonini matematik modellashtirish [Mathematical modelling of the process of heat dissipation in an exponentially variable density non-linear environment]. ACTA NUUz. 2023; 1(1):246–253. (In Uzbek.)

[28] Matyakubov A.S., Raupov D. Explicit estimate for blow-up solutions of nonlinear parabolic systems of non-divergence form with variable density. AIP Conference Proceedings. 2023; (2781):020055. DOI:10.1063/5.0144768.

[29] Begulov U.U., Khaydarov A.T., Salimov J.I. Mathematical modeling of the double nonlinear exsponential inhomogeneous density heat dissipation process. ACTA NUUz. 2024; (2/1):34–42. Available at: https://journalsnuu.uz/index.php/actanuuz/issue/view/163/626.

[30] Andreucci D., Tedeev A.F. Sharp estimates and finite speed of propagation for a Neumann problem in domains narrowing at infinity. Advances in Differential Equations. 2000; 5(7–9):833–860. DOI:10.57262/ade/1356651289.