matmessМатематические заметки СВФУMathematical notes of NEFU2411-93262587-876XСеверо-Восточный федеральный университет имени М.К. Аммосова10.25587/SVFU.2020.93.40.003matmess-232Research ArticleМАТЕМАТИКАTHE CAUCHY PROBLEM FOR A NONLINEAR DEGENERATE PARABOLIC SYSTEM IN NON–DIVERGENCE FORMTHE CAUCHY PROBLEM FOR A NONLINEAR DEGENERATE PARABOLIC SYSTEM IN NON–DIVERGENCE FORMAripovM.AripovM.

Mersaid Aripov

National University of Uzbekistan, University street 4, Tashkent 100174, Uzbekistan 

Mersaid Aripov

National University of Uzbekistan, University street 4, Tashkent 100174, Uzbekistan 

mirsaidaripov@mail.ruMatyakubovA. S.MatyakubovA. S.

Alisher S. Matyakubov

National University of Uzbekistan, University street 4, Tashkent 100174, Uzbekistan 

Alisher S. Matyakubov

National University of Uzbekistan, University street 4, Tashkent 100174, Uzbekistan 

almasa@list.ruImomnazarovB. Kh.ImomnazarovB. Kh.

Bunyod Kh. Imomnazarov

Novosibirsk State University, 1 Pirogov Street, Novosibirsk 630090, Russia 

Bunyod Kh. Imomnazarov

Novosibirsk State University,1 Pirogov Street, Novosibirsk 630090, Russia

buned11998.07@mail.ruNational University of UzbekistanУзбекистанNational University of UzbekistanUzbekistanNovosibirsk State UniversityРоссияNovosibirsk State UniversityRussian Federation2020130420262732738Copyright © Aripov M., Matyakubov A.S., Imomnazarov B.K., 20262026Aripov M., Matyakubov A.S., Imomnazarov B.K.Aripov M., Matyakubov A.S., Imomnazarov B.K.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://matmess.elpub.ru/jour/article/view/232

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. 

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. 

a nonlinear degenerate parabolic systemnon-divergence formCauchy problema nonlinear degenerate parabolic systemnon-divergence formCauchy problemThis 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).ReferencesGao Y., Meng Q., and Guo Y., “Study of properties of solutions for quasilinear parabolic systems,” MATEC Web Conf., № 61, 1–4 (2016).Gao Y., Meng Q., and Guo Y., “Study of properties of solutions for quasilinear parabolic systems,” MATEC Web Conf., № 61, 1–4 (2016).Fan H. J., “Cauchy problem of some doubly degenerate parabolic equations with initial datum a measure,” Acta Math. Sin., 20, № 4, 663–682 (2004).Fan H. J., “Cauchy problem of some doubly degenerate parabolic equations with initial datum a measure,” Acta Math. Sin., 20, № 4, 663–682 (2004).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).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).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).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).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).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).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).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).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).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).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).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).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).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).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).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).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).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).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).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).Zhou W. S., “Some notes on a nonlinear degenerate parabolic equation,” Nonlinear Anal., 63, 601–613 (2008).Zhou W. S., “Some notes on a nonlinear degenerate parabolic equation,” Nonlinear Anal., 63, 601–613 (2008).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).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).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).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).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).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).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).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).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).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).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).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).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).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).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).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).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).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).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).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).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).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).Aripov M., Method of the standard equation for the solution of the nonlinear value problem, Fan, Tashkent (1988).Aripov M., Method of the standard equation for the solution of the nonlinear value problem, Fan, Tashkent (1988).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).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).

The authors declare that there are no conflicts of interest present.