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

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

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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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
2. Naboka O., “Synchronization of nonlinear oscillations of two coupling Berger plates”, Nonlinear Anal., 67:4 (2007), 1015–1026
3. Naboka O., “Synchronization phenomena in the system consisting of m coupled Berger plates”, J. Math. Anal. Appl., 341:2 (2008), 1107–1124
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
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
6. Sergey A. Nazarov, Andrey S. Slutskij, Guido H. Sweers, “Korn Inequalities for a Reinforced Plate”, J Elast, 2010
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
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
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