Skip to main content
SpringerLink
Log in

On Some Properties of the Blow-Up Solutions of a Nonlinear Parabolic System Non-divergent Form with Cross-Diffusion

  • Conference paper
  • First Online:
Technological Advancements in Construction

Part of the book series: Lecture Notes in Civil Engineering ((LNCE,volume 180))

Abstract

In this paper, studies the blow-up properties of solutions of nonlinear parabolic systems of non-divergent equations with a source. In solving the initial boundary value problems for the equations of mathematical physics by the difference method important issue is the choice of the order of approximation with respect to the spatial coordinates and the choice of a function as an initial approximation, as well as the search of an efficient algorithm for solving the systems of difference equations. We construct the scheme of high accuracy for nonlinear system of not in divergence form. Computational experiments confirming the high order accuracy scheme were conducted.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 229.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 299.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 299.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Samarskiy AA (2005) Introduction to numerical methods. Textbook, 3rd ed. Lan, MSU, SPb, p 288

    Google Scholar 

  2. Woodford C, Phillips C (2020) Numerical methods with worked examples: Matlab Edition. SpringerLink, Bücher, Springer Netherlands, pp 1–16

    Google Scholar 

  3. Aripov M, Sadullaeva SA (2015) Qualitative properties of solutions of a doubly nonlinear reaction-diffusion system with a source. J Appl Math Phys 3(9):1090–1099

    Article  Google Scholar 

  4. Zhou W, Yao Z (2010) Cauchy problem for a degenerate parabolic equation with non-divergence form. Acta Math Sci 30B(5):1679–1686

    Google Scholar 

  5. Jin C, Yin J (2013) Self-similar solutions for a class of non-divergence form equations. Nonlinear Differ Equ Appl Nodea 20(3):873–893

    Article  MathSciNet  Google Scholar 

  6. Aripov M, Raimbekov J (2019) The critical curves of a doubly nonlinear parabolic equation in non-divergent form with a source and nonlinear boundary flux. J Siberian Fed Univ Math Phys 12(1):112–124

    Article  MathSciNet  Google Scholar 

  7. Sadullaeva SA (2016) Numerical investigation of solutions to a reaction-diffusion system with variable density. J Siberian Fed Univ Math Phys 9(1):90–101

    Google Scholar 

  8. Aripov M, Rakhmonov Z (2016) On the behavior of the solution of a nonlinear multidimensional polytrophic filtration problem with a variable coefficient and nonlocal boundary condition. Contemp Anal Appl Math 4(1):23–32

    MathSciNet  MATH  Google Scholar 

  9. Aripov M, Matyakubov AS (2016) To the qualitative properties of solution of system equations not in divergence form. Int J Innov Sci Eng Technol 3(8):533–537

    Google Scholar 

  10. Aripov M, Matyakubov AS, Imomnazarov BK (2020) The Cauchy problem for a nonlinear degenerate parabolic system in non-divergence form. Math Notes NEFU 27(3):27–38

    Google Scholar 

  11. Matyakubov AS (2017) 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

    Article  Google Scholar 

  12. Wang M (2020) Some degenerate and quasilinear parabolic systems not in divergence form. J Math Anal Appl 274:424–436

    Article  MathSciNet  Google Scholar 

  13. Wang MX, Xie CH (2004) A degenerate strongly coupled quasilinear parabolic system not in divergence form. Z Angew Math Phys 55:741–755

    Article  MathSciNet  Google Scholar 

  14. Wang M, Wei Y (2008) Blow-up properties for a degenerate parabolic system with nonlinear localized sources. J Math Anal Appl 343:621–635

    Article  MathSciNet  Google Scholar 

  15. Duan Z, Zhou L (2000) Global and blow-up solutions for nonlinear degenerate parabolic systems with crosswise-diffusion. J Math Anal Appl 244:263–278

    Article  MathSciNet  Google Scholar 

  16. Lu H (2009) Global existence and blow-up analysis for some degenerate and quasilinear parabolic systems. Electron J Qual Theory Differ Equ 49:1–14

    Article  MathSciNet  Google Scholar 

  17. Deng W, Li Y, Xie Ch (2003) Global existence and nonexistence for a class of degenerate parabolic systems. Nonlinear Anal Theory Methods Appl 55(3):233–244

    Article  MathSciNet  Google Scholar 

  18. Aripov M, Matyakubov AS (2017) To the qualitative properties of solution of system equations not in divergence form of polytrophic filtration in variable density. Nanosyst Phys Chem Math 8(3):317–322

    Google Scholar 

  19. Aripov M, Matyakubov AS (2017) To the properties of the solutions of a cross-diffusion parabolic system not in divergence form. Univ J Comput Math 5(1):1–7

    Article  Google Scholar 

  20. Aripov M, Matyakubov AS (2017) Self-similar solutions of a cross-diffusion parabolic system with variable density: explicit estimates and asymptotic behaviour. Nanosyst Phys Chem Mathe 8(1):5–12

    Google Scholar 

  21. Rakhmonov ZR, Tillaev AI (2018) On the behavior of the solution of a nonlinear polytropic filtration problem with a source and multiple nonlinearities. Nanosyst Phys Chem Math 9(3):323–329

    Google Scholar 

  22. Rakhmonov ZR, Urunbayev JE (2019) On a problem of cross-diffusion with nonlocal boundary conditions. J Siberian Fed Univ Math Phys 5:614–620

    Article  MathSciNet  Google Scholar 

  23. Rakhmonov ZR, Khaydarov A, Urunbaev JE (2020) Global existence and nonexistence of solutions to a cross diffusion system with nonlocal boundary conditions. Math Stat 8(4):404–409

    Article  Google Scholar 

  24. Parovik R, Rakhmonov Z, Zunnunov R (2020) Modeling of fracture concentration by Sel’kov fractional dynamic system. E3S Web Conf 196:02018

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. National University of Uzbekistan, Tashkent, Uzbekistan

    Alisher Matyakubov & Dilmurod Raupov

Authors
  1. Alisher Matyakubov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dilmurod Raupov
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Moscow State University of Civil Engineering, Moscow, Russia

    Angela Mottaeva

Rights and permissions

Reprints and permissions

Copyright information

© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Matyakubov, A., Raupov, D. (2022). On Some Properties of the Blow-Up Solutions of a Nonlinear Parabolic System Non-divergent Form with Cross-Diffusion. In: Mottaeva, A. (eds) Technological Advancements in Construction. Lecture Notes in Civil Engineering, vol 180. Springer, Cham. https://doi.org/10.1007/978-3-030-83917-8_27

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-83917-8_27

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-83916-1

  • Online ISBN: 978-3-030-83917-8

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation