Ukrainian Mathematical Journal

, Volume 63, Issue 12, pp 1855–1882 | Cite as

Homogenization of a quasilinear parabolic problem with different alternating nonlinear fourier boundary conditions in a two-level thick junction of the type 3:2:2

  • T. A. Mel’nyk
  • D. Yu. Sadovyi
Article
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We investigate the asymptotic behavior of a solution of a quasilinear parabolic boundary-value problem in a two-level thick junction of the type 3:2:2. This junction consists of a cylinder on which thin disks of variable thickness are ε-periodically threaded. The thin disks are divided into two levels, depending on their geometric structure and the conditions imposed on their boundaries. In this problem, we consider different alternating, inhomogeneous, nonlinear Fourier conditions; moreover, the Fourier conditions depend on additional perturbation parameters. We prove theorems on the convergence of a solution of this problem as ε → 0 for different values of these parameters.

Keywords

Weak Solution Thin Disk Integral Identity Homogenize Problem Junction Zone 
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References

  1. 1.
    E. Ya. Khruslov, “On resonance phenomena in a diffraction problem,” Teor. Funkts. Funkts. Anal. Prilozhen., 10, 113–120 (1968).Google Scholar
  2. 2.
    V. P. Kotlyarov and E. Ya. Khruslov, “On the limit boundary condition of one Neumann problem,” Teor. Funkts. Funkts. Anal. Prilozhen., 10, 83–96 (1970).MATHGoogle Scholar
  3. 3.
    V. A. Marchenko and E. Ya. Khruslov, Boundary-Value Problems in Domains with Fine-Grained Boundary [in Russian], Naukova Dumka, Kiev (1974).Google Scholar
  4. 4.
    T. A. Mel’nik and S. A. Nazarov, “Asymptotic structure of the spectrum in the problem of harmonic vibrations of a nave with heavy spokes,” Dokl. Ros. Akad. Nauk, 333, 13–15 (1993).Google Scholar
  5. 5.
    T. A. Mel’nik and S. A. Nazarov, “Asymptotics of a solution of the spectral Neumann problem in a domain of the ‘thick-comb’ type,” Tr. Sem. Im. I. G. Petrovskogo, 19, 138–174 (1996); English translation: J. Math. Sci., 85, No. 6, 2326–2346 (1997).MathSciNetCrossRefGoogle Scholar
  6. 6.
    T. A. Mel’nyk, “Homogenization of the Poisson equation in a thick periodic junction,” Z. Anal. Anwendungen., 18, No. 4, 953–975 (1999).MathSciNetMATHGoogle Scholar
  7. 7.
    T. A. Mel’nik and S. A. Nazarov, “Asymptotic analysis of a Neumann problem on a junction of a body with thin heavy rods,” Alg. Analiz, 12, No. 2, 188–238 (2000); English translation: 12, No. 2, 317–351 (2001).MathSciNetGoogle Scholar
  8. 8.
    T. A. Mel’nyk, “Homogenization of a singularly perturbed parabolic problem in a thick periodic junction of the type 3:2:1,” Ukr. Mat. Zh., 52, No. 11, 1524–1533 (2000); English translation: Ukr. Math. J., 52, No. 11, 1737–1748 (2000).MathSciNetCrossRefGoogle Scholar
  9. 9.
    S. A. Nazarov, “Junctions of singularly degenerating domains of different limit dimensions. I, II,” Tr. Sem. Im. I. G. Petrovskogo, 18, 1–78 (1995); 20, 155–196 (2000).Google Scholar
  10. 10.
    T. A. Mel’nyk, “Homogenization of a boundary value problem with a nonlinear boundary condition in a thick junction of type 3:2:1,” Math. Models Meth. Appl. Sci., 31, 1005–1027 (2008); http://www.interscience.wiley.com/DOI: 10.1002/mma.951.MathSciNetMATHGoogle Scholar
  11. 11.
    D. Blanchard, A. Gaudiello, and T. A. Mel’nyk, “Boundary homogenization and reduction of dimension in a Kirchhoff–Love plate,” SIAM J. Math. Anal., 39, No. 6, 1764–1787 (2008).MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    D. Blanchard, A. Gaudiello, and G. Griso, “Junction of a periodic family of elastic rods with 3d plate. I, II,” J. Math. Pures Appl., 88, No. 9, 1–33 (2007); 88, No. 9, 149–190 (2007).MathSciNetMATHGoogle Scholar
  13. 13.
    T. A. Mel’nik and G. A. Chechkin, “Asymptotic analysis of boundary-value problems in thick three-dimensional multilevel junctions,” Mat. Sb., 200, No. 3, 49–74 (2009).MathSciNetGoogle Scholar
  14. 14.
    T. A. Mel’nyk, “Eigenmodes and pseudo-eigenmodes of thick multi-level junctions,” in: Proceedings of the International Conference “Days on Diffraction-2004” (2004), pp. 51–52.Google Scholar
  15. 15.
    U. De Maio, T. Durante, and T. A. Mel’nyk, “Asymptotic approximation for the solution to the Robin problem in a thick multi-level junction,” Math. Models Meth. Appl. Sci., 15, No. 12, 1897–1921 (2005).MATHCrossRefGoogle Scholar
  16. 16.
    T. A. Mel’nik and P. S. Vashchuk, “Homogenization of the boundary-value problem with changeable type of boundary conditions in a thick junction,” Differents. Uravn., 43, No. 5, 677–685 (2007).MathSciNetGoogle Scholar
  17. 17.
    T. Durante and T. A. Mel’nyk, “Asymptotic analysis of a parabolic problem in a thick two-level junction,” J. Math. Phys., Anal., Geometry, 3, No. 3, 313–341 (2007).MathSciNetMATHGoogle Scholar
  18. 18.
    U. De Maio and T. A. Mel’nyk, “Homogenization of the Robin problem for the Poisson equation in a thick multi-structure of type 3:2:2,” Asympt. Anal., 41, 161–177 (2005).MATHGoogle Scholar
  19. 19.
    C. D’Apice, U. De Maio, and T. A. Mel’nyk, “Asymptotic analysis of a perturbed parabolic problem in a thick junction of type 3:2:2,” Networks Heterogeneous Media, 2, No. 2, 255–277 (2007).MATHCrossRefGoogle Scholar
  20. 20.
    H. Gajewski, K. Gröger, and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Akademie, Berlin (1974).MATHGoogle Scholar
  21. 21.
    T. A. Mel’nik and E. A. Sivak, “Asymptotic analysis of a parabolic semilinear problem with nonlinear boundary multiphase interactions in a perforated domain,” Probl. Mat. Analiz., 43, 107–128 (2009).Google Scholar
  22. 22.
    V. P. Mikhailov, Partial Differential Equations [in Russian], Nauka, Moscow (1983).Google Scholar
  23. 23.
    R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, American Mathematical Society, Providence, RI (1977).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2012

Authors and Affiliations

  • T. A. Mel’nyk
    • 1
  • D. Yu. Sadovyi
    • 1
  1. 1.KyivUkraine

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