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In this paper, we study the properties of self-similar solutions of the Cauchy problem to degenerate type double nonlinear parabolic equation with variable density and absorption. The property a FSP of solution of the Cauchy problem for a nonlinear parabolic equation is established. Based on self-similar analysis of solution the condition of Fujita type global solvability of the Cauchy problem for double nonlinear degenerate type parabolic equation with variable density is proved. The estimates for weak solution depending on grows of density, value of the numerical parameters is established. The critical cases are studied.

  • Internet ҳавола
  • DOI
  • UzSCI тизимида яратилган сана 14-06-2021
  • Ўқишлар сони 445
  • Нашр санаси 20-04-2020
  • Мақола тилиIngliz
  • Саҳифалар сони5-11
Русский

В данной работе мы изучаем свойства автомодельных решений задачи Коши для вырожденного параболического уравнения с двойной нелинейностью и с переменными плотностью и поглощением. Установлено свойство конечной скорости распространения решения задачи Коши для нелинейного параболического уравнения. На основе автомодельного анализа решения доказано условие глобальной разрешимости типа Фуджиты задачи Коши для параболического уравнения с двойной нелинейностью и с переменными плотностью и поглощением. Установлены оценки для слабого решения в зависимости от роста плотности, значения числовых параметров. Также в статье исследуются критические случаи.

English

In this paper, we study the properties of self-similar solutions of the Cauchy problem to degenerate type double nonlinear parabolic equation with variable density and absorption. The property a FSP of solution of the Cauchy problem for a nonlinear parabolic equation is established. Based on self-similar analysis of solution the condition of Fujita type global solvability of the Cauchy problem for double nonlinear degenerate type parabolic equation with variable density is proved. The estimates for weak solution depending on grows of density, value of the numerical parameters is established. The critical cases are studied.

№ Муаллифнинг исми Лавозими Ташкилот номи
1 Aripov M.M. профессор Ўзбекистон Миллий университети
2 Alanezi M.. Ўзбекистон Миллий университети
№ Ҳавола номи
1 Samarskii A.A., Galaktionov V.A., Kurdyomov S.P., Mikhailov A.P. 1995. Blow-up in quasilinear parabolic equations. Berlin: Walter de Grueter. 535 p.
2 Xiang Zh., Mu Ch., Hu X. 2008. Support properties of solutions to a degenerate equation with absorption and variable density. Nonlinear analysis. 68:1940-1953.
3 Qi Y.M., Wang M.X. 2001. The self-similar profi les of solutions of generalized KPZ equation. Pacifi c J. Math. 201:223-240.
4 Wang M.X., Xie C.H. 2004. A degenerate strongly coupled quasilinear parabolic system not in divergence form. Z. Angew. Math. Phys. 55:741-755.
5 Lu H. 2009. Global existence and blow-up analysis for some degenerate and quasilinear parabolic systems. Electronic Journal of Qualitative Theory of Diff erential Equations. 49:1-14.
6 Zhou W., Yao Z. 2010. Cauchy problem for a degenerate parabolic equation with non-divergence form. Acta Mathematica Scientia. 30(5):1679-1686.
7 Chunhua J., Jingxue Y. 2013. Self-similar solutions for a class of non-divergence form equations. Nonlinear Diff er. Equ. Appl. Nodea. 20(3):873-893.
8 Gao Y., Meng Q., Guo Y. 2016. Study of properties of solutions for quasilinear parabolic systems. MATEC Web of Conferences. 61(1):1-4.
9 Aripov M., Sadullaeva Sh.A. 2015. Qualitative Properties of Solutions of a Doubly Nonlinear Reaction-Diffusion System with a Source. Journal of Applied Mathematics and Physics. 3:1090-1099.
10 Martynenko A.V., Tedeev A.F. 2007. The Cauchy Problem for a Quasilinear Parabolic Equation with a Source and Inhomogeneous Density. Computational Mathematics and Mathematical Physics. 47:238-248.
11 Martynenko A.V., Shramenko V.N. 2010. Estimate of solutions of the Cauchy Problem near the Time of Exacerbation for a Quasilinear Parabolic Equation with a Source and a Variable Density. Nonlinear Boundary Value Problems. 20:104-115.
12 Zhan H. 2012. The Self-Similar Solutions of a Diffusion Equation. WSEAS Transaction on Mathematics. 4(11):345-356.
13 Aripov M., Sadullaeva Sh. 2013. To properties of the equation of reaction diffusion with double nonlinearity and distributed parameters. Journal of Siberian Federal University. Mathematics & Physics. 3:13-22.
14 Aripov M. 2017. The Fujita and Secondary Type Critical Exponents in Nonlinear Parabolic Equations and Systems. Differential Equations and Dynamical Systems. 9-24.
15 Samarskii A.A., Sobol I.M. 1963. Examples of numerical calculation of temperature waves. Jour. Vychisl. Math and Math. Phis. 3(4):702-719.
16 Chen X.F., Qi Y.W., Wang M.X. 2000. Self-similar singular parabolic equations with absorption. Electronic J. Diff . Equ. 67:1-22.
17 Aripov M. 1988. Methods of standard equations for solving of nonlinear boundary value problems. Tashkent: Fan. 138 p.
18 Vazquez J.L., Galaktionov V.A. 1989. Asymptotic behavior of solutions of the nonlinear diffusion equation with absorption at a critical exponent. Report of the Academy of Sciences of USSR. 6:5.
19 Aripov M., Mukimov A., Mirzayev B. 2019. To asymptotic of the solution of the heat conduction problem with double nonlinearity with absorption at a critical parameter. Mathematics and Statistics. 7(5):205-217.
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