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Mat. Sb., 2004, Volume 195, Number 1, Pages 103–128 (Mi msb795)  

This article is cited in 8 scientific papers (total in 8 papers)

Global attractor of a contact parabolic problem in a thin two-layer domain

A. M. Rekalo, I. D. Chueshov

V. N. Karazin Kharkiv National University

Abstract: A semilinear parabolic equation is considered in the union of two bounded thin cylindrical domains $\Omega_{1,\varepsilon}=\Gamma\times(0,\varepsilon)$ and $\Omega_{2,\varepsilon}=\Gamma\times(-\varepsilon,0)$ adjoining along their bases, where $\Gamma$ is a domain in $\mathbb R^d$, $d\leqslant3$. The unknown functions are related by means of an interface condition on the common base $\Gamma$. This problem can serve as a reaction-diffusion model describing the behaviour of a system of two components interacting at the boundary. The intensity of the reaction is assumed to depend on $\varepsilon$ and the thickness of the domains, and to be of order $\varepsilon^\alpha$.
Under investigation are the limiting properties of the evolution semigroup $S_{\alpha,\varepsilon}(t)$, generated by the original problem as $\varepsilon\to0$ (that is, as the domain becomes ever thinner). These properties are shown to depend essentially on the exponent $\alpha$. Depending on whether $\alpha$ is equal to, greater than, or smaller than 1, the original system can have three distinct systems of equations on $\Gamma$ as its asymptotic limit. The continuity properties of the global attractor of the semigroup $S_{\alpha,\varepsilon}(t)$ as $\varepsilon\to0$ are established under natural assumptions.

DOI: https://doi.org/10.4213/sm795

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English version:
Sbornik: Mathematics, 2004, 195:1, 97–119

Bibliographic databases:

UDC: 517.94
MSC: 35K57, 35B40
Received: 15.01.2003

Citation: A. M. Rekalo, I. D. Chueshov, “Global attractor of a contact parabolic problem in a thin two-layer domain”, Mat. Sb., 195:1 (2004), 103–128; Sb. Math., 195:1 (2004), 97–119

Citation in format AMSBIB
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\pages 103--128
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Caraballo T., Chueshov I.D., Kloeden P.E., “Synchronization of a stochastic reaction-diffusion system on a thin two-layer domain”, SIAM J. Math. Anal., 38:5 (2006), 1489–1507  crossref  mathscinet  zmath  isi
    2. Naboka O., “Synchronization of nonlinear oscillations of two coupling Berger plates”, Nonlinear Anal., 67:4 (2007), 1015–1026  crossref  mathscinet  zmath  isi
    3. Naboka O., “Synchronization phenomena in the system consisting of m coupled Berger plates”, J. Math. Anal. Appl., 341:2 (2008), 1107–1124  crossref  mathscinet  zmath  isi
    4. Naboka O., “On partial synchronization of nonlinear oscillations of two Berger plates coupled by internal subdomains”, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods, 71:12 (2009), 6299–6311  crossref  mathscinet  zmath  isi
    5. Naboka O., “On synchronization of oscillations of two coupled Berger plates with nonlinear interior damping”, Commun. Pure Appl. Anal., 8:6 (2009), 1933–1956  crossref  mathscinet  zmath  isi
    6. Sergey A. Nazarov, Andrey S. Slutskij, Guido H. Sweers, “Korn Inequalities for a Reinforced Plate”, J Elast, 2010  crossref  mathscinet  isi
    7. José M. Arrieta, Alexandre N. Carvalho, Marcone C. Pereira, Ricardo P. Silva, “Semilinear parabolic problems in thin domains with a highly oscillatory boundary”, Nonlinear Analysis: Theory, Methods & Applications, 2011  crossref  mathscinet  isi
    8. Hu Ch., “Global strong solutions of Navier–Stokes equations with interface boundary in three-dimensional thin domains”, Nonlinear Analysis-Theory Methods & Applications, 74:12 (2011), 3964–3997  crossref  mathscinet  zmath  isi
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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