THE CAUCHY PROBLEM FOR A NONLINEAR DEGENERATE PARABOLIC SYSTEM IN NON–DIVERGENCE FORM
https://doi.org/10.25587/SVFU.2020.93.40.003
Аннотация
We deal with degenerate quasilinear parabolic systems in the non-divergence form under positive initial conditions. An asymptotic behavior of self-similar solutions in the case of slow diffusion is established. Depending on values of the numerical parameters and the initial value, the existence of the global solutions of the Cauchy problem is proved. In addition, the asymptotic representation of the solution is obtained.
При поддержке: This work is supported by the Russian–Uzbek project MRT–OT–81–2018 and the Russian Foundation for Basic Research (grants 18–51–41002 and 18–31–00120).
Об авторах
M. Aripov
National University of Uzbekistan
Узбекистан
Узбекистан
Mersaid Aripov
National University of Uzbekistan, University street 4, Tashkent 100174, Uzbekistan
A. S. Matyakubov
National University of Uzbekistan
Узбекистан
Узбекистан
Alisher S. Matyakubov
National University of Uzbekistan, University street 4, Tashkent 100174, Uzbekistan
B. Kh. Imomnazarov
Novosibirsk State University
Россия
Россия
Bunyod Kh. Imomnazarov
Novosibirsk State University, 1 Pirogov Street, Novosibirsk 630090, Russia
Список литературы
1. Gao Y., Meng Q., and Guo Y., “Study of properties of solutions for quasilinear parabolic systems,” MATEC Web Conf., № 61, 1–4 (2016).
2. Fan H. J., “Cauchy problem of some doubly degenerate parabolic equations with initial datum a measure,” Acta Math. Sin., 20, № 4, 663–682 (2004).
3. Li J., “Cauchy problem and initial trace for a doubly degenerate parabolic equation with strongly nonlinear sources,” J. Math. Anal. Appl., 264, № 1, 49–67 (2001).
4. Zheng P., Mu Ch., Liu D., Yao X., and Zhou Sh., “Blow-up analysis for a quasilinear degenerate parabolic equation with strongly nonlinear source,” Abstract Appl. Anal., 2012, 1–19 (2012).
5. Zhou Sh. and Zheng S., “Asymptotic behavior of solutions to a degenerate parabolic equation with a gradient source term,” Asymptotic Anal., 91, № 2, 91–102 (2015).
6. Lu H., “Global existence and blow-up analysis for some degenerate and quasilinear parabolic systems,” Electron. J. Qualit. Theory Differ. Equ., 49, 1–14 (2009).
7. Chunhua J. and Jingxue Y., “Self-similar solutions for a class of non-divergence form equations,” Nonlinear Differ. Equ. Appl., 20, № 3, 873–893 (2013).
8. Raimbekov J. R., “The properties of the solutions for Cauchy problem of nonlinear parabolic equations in non-divergent form with density,” J. Sib. Fed. Univ., Math. Phys., 8, № 2, 192– 200 (2015).
9. Sun Y., Shi Y., and Wu M., “Nonexistence and longtime behaviors of solutions to a class of nonlinear degenerate parabolic equations not in divergence form,” Acta Math. Appl. Sin., English Ser., 32, № 2, 327–332 (2016).
10. Zhou S., Tang X., and Yang C., “A doubly degenerate diffusion equation not in divergence form with gradient term,” Boundary Value Probl., 126, 1–19 (2016).
11. Zhong G. and Tian L., “Blow-up for a parabolic system with nonlocal sources and nonlocal boundary conditions,” Boundary Value Probl., 61, 1–14 (2015).
12. Wang J., “Behaviors of solutions to a class of nonlinear degenerate parabolic equations not in divergence form,” Appl. Math. Lett., 24, № 2, 191–195 (2011).
13. Zhou W. S., “Some notes on a nonlinear degenerate parabolic equation,” Nonlinear Anal., 63, 601–613 (2008).
14. Zhou W. S. and Wu Z. Q., “Some results on a class of degenerate parabolic equations not in divergence form,” Nonlinear Anal., 60, 863–886 (2005).
15. Yao Z. A. and Zhou W. S., “Some results on a degenerate and singular diffusion equation,” Acta Math. Sci., Ser. B, 27, № 3, 581–601 (2007).
16. Cao C. L., Lin B. X., and Wang H. M., “Existence of solutions to a class of nonlinear degenerate parabolic equations not in divergence form,” Northeast. Math. J., 24, № 2, 118–128 (2008).
17. Zhou W. S. and Wu Z. Q., “Some results on a class of nonlinear degenerate parabolic equations not in divergence form,” Northeast. Math. J., № 4, 291–294 (2003).
18. Wang S., Wang M. X., and Xie C. H., “A nonlinear degenerate diffusion equation not in divergence form,” Z. Angew. Math. Phys., 51, 149–159 (2000).
19. Aripov M. and Matyakubov A. S., “To the qualitative properties of solution of system equations not in divergence form,” Int. J. Innov. Sci., Eng. Technol., 3, № 8, 533–537 (2016).
20. Aripov M. and Matyakubov A. S., “To the properties solutions of a cross-diffusion parabolic system not in divergence form,” Univ. J. Comput. Math., 5, № 1, 1–7 (2017).
21. Aripov M. and Matyakubov A. S., “Self-similar solutions of a cross-diffusion parabolic system with variable density: explicit estimates and asymptotic behavior,” Nanosystems: Phys., Chem., Math., 8, № 1, 43–50 (2017).
22. Matyakubov A. S., “Finite speed of the perturbation distribution and asymptotic behavior of the solutions of a parabolic system not in divergence form,” Univ. J. Comput. Math., 5, № 3, 57–67 (2017).
23. Aripov M. and Matyakubov A. S., “To the qualitative properties of solution of system equations not in divergence form of polytrophic filtration in variable density,” Nanosystems: Phys., Chem., Math., 8, № 3, 317–322 (2017).
24. Aripov M. and Sadullaeva Sh. A., “Qualitative properties of solutions of a doubly nonlinear reaction-diffusion system with a source,” J. Appl. Math. Phys., 3, 1090–1099 (2015).
25. Aripov M., Method of the standard equation for the solution of the nonlinear value problem, Fan, Tashkent (1988).
26. Samarskii A. A., Galaktionov V. A., Kurdyumov S. P., and Mikhailov A. P., Blow-Up in Quasilinear Parabolic Equations, De Gruyter, Berlin (2011) (De Gruyter Expo. Math., vol. 19).
Рецензия
Для цитирования:
Aripov M., Matyakubov A.S., Imomnazarov B.Kh. THE CAUCHY PROBLEM FOR A NONLINEAR DEGENERATE PARABOLIC SYSTEM IN NON–DIVERGENCE FORM. Математические заметки СВФУ. 2020;27(3):27-38. https://doi.org/10.25587/SVFU.2020.93.40.003
For citation:
Aripov 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. https://doi.org/10.25587/SVFU.2020.93.40.003
Просмотров:
26
JATS XML
Послать статью по эл. почте 