Asymptotic analysis of a perturbed parabolic problem in a thick junction of type 3:2:2
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Dipartimento di Ingegneria dell’Informazione e Matematica Applicata, Università degli Studi di Salerno, Via Ponte don Melillo, 1, Fisciano (SA) 84084
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Dipartimento di Matematica e Applicazioni, Università degli Studi di Napoli “Federico II”, DMA “R. Caccioppoli”, Complesso Monte S. Angelo, via Cintia, 80126 Napoli
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Faculty of Mathematics & Mechanics, Taras Shevchenko National University of Kyiv, Volodymyrska str. 64, 01033 Kyiv
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Received:
01 August 2006
Revised:
01 February 2007
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Primary: 35B27, 35B40; Secondary: 35C20, 35K20, 74K30.
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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.
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[J]. Networks and Heterogeneous Media, 2007, 2(2): 255-277. doi: 10.3934/nhm.2007.2.255
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Abstract
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.
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