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June  2007, 2(2): 255-277. doi: 10.3934/nhm.2007.2.255

Asymptotic analysis of a perturbed parabolic problem in a thick junction of type 3:2:2

Ciro D’Apice 1, , Umberto De Maio 2, and  T. A. Mel'nyk 3,

1. 

Dipartimento di Ingegneria dell’Informazione e Matematica Applicata, Università degli Studi di Salerno, Via Ponte don Melillo, 1, Fisciano (SA) 84084, Italy
2. 

Dipartimento di Matematica e Applicazioni, Università degli Studi di Napoli “Federico II”, DMA “R. Caccioppoli”, Complesso Monte S. Angelo, via Cintia, 80126 Napoli, Italy
3. 

Faculty of Mathematics & Mechanics, Taras Shevchenko National University of Kyiv, Volodymyrska str. 64, 01033 Kyiv, Ukraine

Received  August 2006 Revised  February 2007 Published  March 2007

We consider a perturbed initial/boundary-value problem for the heat equation in a thick multi-structure $\Omega_{\varepsilon}$ which is the union of a domain $\Omega_0$ and a large number $N$ of $\varepsilon-$periodically situated thin rings with variable thickness of order $\varepsilon = \mathcal{O}(N^{-1}).$ The following boundary condition $\partial_{\nu}u_{\varepsilon} + \varepsilon^{\alpha} k_0 u_{\varepsilon}= \varepsilon^{\beta} g_{\varepsilon}$ is given on the lateral boundaries of the thin rings; here the parameters $\alpha$ and $\beta$ are greater than or equal $1.$ The asymptotic analysis of this problem for different values of the parameters $\alpha$ and $\beta$ is made as $\varepsilon\to0.$ The leading terms of the asymptotic expansion for the solution are constructed, the corresponding estimates in the Sobolev space $L^2(0,T; H^1(\Omega_{\varepsilon}))$ are obtained and the convergence theorem is proved with minimal conditions for the right-hand sides.
Keywords: thick multi-structure, homogenization, parabolic problem, Asymptotic approximation.
Mathematics Subject Classification: Primary: 35B27, 35B40; Secondary: 35C20, 35K20, 74K3.
Citation: Ciro D’Apice, Umberto De Maio, T. A. Mel'nyk. Asymptotic analysis of a perturbed parabolic problem in a thick junction of type 3:2:2. Networks & Heterogeneous Media, 2007, 2 (2) : 255-277. doi: 10.3934/nhm.2007.2.255
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