Advertisement

Properties of Solutions of the Cauchy Problem for Degenerate Nonlinear Cross Systems with Convective Transfer and Absorption

  • Sh. A. Sadullaeva
  • M. B. Khojimurodova
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 264)

Abstract

In this paper the Cauchy problem for nonlinear systems is considered. The conditions of existence of the solutions on time for the problem Cauchy are given. Moreover the properties of the finite velocity of a propagation and localization of the disturbance, an asymptotic of self-similar solutions will be defined. The results of numerical solutions will be carried out and on the basis of calculations some necessary statements will be given.

Keywords

Nonlinear systems Finite velocity Localization Asymptotic Self-similar 

References

  1. 1.
    Aripov, M.: Asymptotic of the solutions of the non-Newton polytrophic filtration equation. ZAMM 80(3), 767–768 (2000)Google Scholar
  2. 2.
    Aripov, M.: One invariant group method for the quasilinear equations and their system. In: Proceedings of the International Conference on Mathematics and its Applications in the New Millennium. Malaysia, pp. 535–543 (2000)Google Scholar
  3. 3.
    Aripov, M., Muhammadiev, J.: Asymptotic behavior of auto model solutions for one system of quasilinear equations of parabolic type. Buletin Stiintific-Universitatea din Pitesti, Seria Matematica si Informatica. 3, 19–40 (1999)Google Scholar
  4. 4.
    Aripov, M., Sadullaeva, ShA: An asymptotic analysis of a self-similar solution for the double nonlinear reaction-diffusion system. J. Nanosyst. Phys. Chem. Math. 6(6), 793–802 (2015)CrossRefGoogle Scholar
  5. 5.
    Aripov, M., Sadullaeva, ShA: Qualitative properties of solutions of a doubly nonlinear reaction-diffusion system with a source. J. Appl. Math. Phys. 3, 1090–1099 (2015)CrossRefGoogle Scholar
  6. 6.
    Cho, Chien-Hong: On the computation of the numerical blow-up time. Jpn. J. Ind. Appl. Math. 30(2), 331–349 (2013)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Deng, K., Levine, H.A.: The role of critical exponents in blow-up theorems: The sequel. J. Math. Anal. Appl. 243, 85–126 (2000)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Ferreira, Rael, Perez-Llanos, Mayte: Blow-up for the non-local -Laplacian equation with a reaction term. Nonlinear Anal.: Theory Methods Appl. 75(14), 5499–5522 (2012)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Jiang, Z.X., Zheng, S.N.: Doubly degenerate parabolic equation with nonlinear inner sources or boundary flux. Doctor Thesis, Dalian University of Technology, In China, (2009)Google Scholar
  10. 10.
    Martynenko, A.V., Tedeev, A.F.: The Cauchy problem for a quasilinear parabolic equation with a source and inhomogeneous density. Comput. Math. Math. Phys. 47(2), 238–248 (2007)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Martynenko, A.V., Tedeev, A.F.: On the behavior of solutions to the Cauchy problem for a degenerate parabolic equation with inhomogeneous density and a source. Comput. Math. Math. Phys. 48(7), 1145–1160 (2008)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Mu, C., Zheng, P.: Dengming Liu, Localization of solutions to a doubly degenerate parabolic equation with a strongly nonlinear source. Commun. Contemp. Math. 14, 1250018 [18 pages].  https://doi.org/10.1142/S0219199712500186MathSciNetCrossRefGoogle Scholar
  13. 13.
    Sadullaeva, ShA: Numerical investigation of solutions to a reaction-diffusion system with variable density. Journal Sib. Fed. Univ. Math. Phys., J. Sib. Fed. Univ. Math. Phys. 9(1), 90–101 (2016)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Samarskii, A.A., Galaktionov, V.A., Kurduomov, S.P., Mikhailov, A.P.: Blowe-up in quasilinear parabolic equations, vol. 4, p. 535. Walter de Grueter, Berlin (1995)CrossRefGoogle Scholar
  15. 15.
    Tedeyev, A.F.: Conditions for the existence and nonexistence of a compact support in time of solutions of the Cauchy problem for quasilinear degenerate parabolic equations. Sib. Math. Jour. 45(1), 189–200 (2004)Google Scholar
  16. 16.
    Vazquez, J.L.: The Porous Medium Equation. Mathematical Theory, Oxford Mathematical Monographs, p. 430. The Clarendon Press, Oxford University Press, Oxford (2007)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Tashkent University of Information Technologies, TUITTashkentUzbekistan

Personalised recommendations