American Institute of Mathematical Sciences

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

 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.
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 and Heterogeneous Media, 2007, 2 (2) : 255-277. doi: 10.3934/nhm.2007.2.255
 [1] Grégoire Allaire, Alessandro Ferriero. Homogenization and long time asymptotic of a fluid-structure interaction problem. Discrete and Continuous Dynamical Systems - B, 2008, 9 (2) : 199-220. doi: 10.3934/dcdsb.2008.9.199 [2] Andro Mikelić, Giovanna Guidoboni, Sunčica Čanić. Fluid-structure interaction in a pre-stressed tube with thick elastic walls I: the stationary Stokes problem. Networks and Heterogeneous Media, 2007, 2 (3) : 397-423. doi: 10.3934/nhm.2007.2.397 [3] R.G. Duran, J.I. Etcheverry, J.D. Rossi. Numerical approximation of a parabolic problem with a nonlinear boundary condition. Discrete and Continuous Dynamical Systems, 1998, 4 (3) : 497-506. doi: 10.3934/dcds.1998.4.497 [4] T. A. Shaposhnikova, M. N. Zubova. Homogenization problem for a parabolic variational inequality with constraints on subsets situated on the boundary of the domain. Networks and Heterogeneous Media, 2008, 3 (3) : 675-689. doi: 10.3934/nhm.2008.3.675 [5] Aníbal Rodríguez-Bernal, Robert Willie. Singular large diffusivity and spatial homogenization in a non homogeneous linear parabolic problem. Discrete and Continuous Dynamical Systems - B, 2005, 5 (2) : 385-410. doi: 10.3934/dcdsb.2005.5.385 [6] G. Acosta, Julián Fernández Bonder, P. Groisman, J.D. Rossi. Numerical approximation of a parabolic problem with a nonlinear boundary condition in several space dimensions. Discrete and Continuous Dynamical Systems - B, 2002, 2 (2) : 279-294. doi: 10.3934/dcdsb.2002.2.279 [7] Yarui Duan, Pengcheng Wu, Yuying Zhou. Penalty approximation method for a double obstacle quasilinear parabolic variational inequality problem. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022017 [8] Salim Meddahi, David Mora. Nonconforming mixed finite element approximation of a fluid-structure interaction spectral problem. Discrete and Continuous Dynamical Systems - S, 2016, 9 (1) : 269-287. doi: 10.3934/dcdss.2016.9.269 [9] V. Balaji, I. Biswas and D. S. Nagaraj. Principal bundles with parabolic structure. Electronic Research Announcements, 2001, 7: 37-44. [10] Fanghua Lin, Xiaodong Yan. A type of homogenization problem. Discrete and Continuous Dynamical Systems, 2003, 9 (1) : 1-30. doi: 10.3934/dcds.2003.9.1 [11] Yao Xu, Weisheng Niu. Periodic homogenization of elliptic systems with stratified structure. Discrete and Continuous Dynamical Systems, 2019, 39 (4) : 2295-2323. doi: 10.3934/dcds.2019097 [12] Thomas Blanc, Mihai Bostan, Franck Boyer. Asymptotic analysis of parabolic equations with stiff transport terms by a multi-scale approach. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 4637-4676. doi: 10.3934/dcds.2017200 [13] David Gérard-Varet, Alexandre Girodroux-Lavigne. Homogenization of stiff inclusions through network approximation. Networks and Heterogeneous Media, 2022, 17 (2) : 163-202. doi: 10.3934/nhm.2022002 [14] Fujun Zhou, Junde Wu, Shangbin Cui. Existence and asymptotic behavior of solutions to a moving boundary problem modeling the growth of multi-layer tumors. Communications on Pure and Applied Analysis, 2009, 8 (5) : 1669-1688. doi: 10.3934/cpaa.2009.8.1669 [15] Renata Bunoiu, Claudia Timofte. Homogenization of a thermal problem with flux jump. Networks and Heterogeneous Media, 2016, 11 (4) : 545-562. doi: 10.3934/nhm.2016009 [16] Sunghan Kim, Ki-Ahm Lee, Henrik Shahgholian. Homogenization of the boundary value for the Dirichlet problem. Discrete and Continuous Dynamical Systems, 2019, 39 (12) : 6843-6864. doi: 10.3934/dcds.2019234 [17] Jean Louis Woukeng. $\sum$-convergence and reiterated homogenization of nonlinear parabolic operators. Communications on Pure and Applied Analysis, 2010, 9 (6) : 1753-1789. doi: 10.3934/cpaa.2010.9.1753 [18] Junde Wu, Shangbin Cui. Asymptotic behavior of solutions for parabolic differential equations with invariance and applications to a free boundary problem modeling tumor growth. Discrete and Continuous Dynamical Systems, 2010, 26 (2) : 737-765. doi: 10.3934/dcds.2010.26.737 [19] Nguyen Thi Hoai. Asymptotic approximation to a solution of a singularly perturbed linear-quadratic optimal control problem with second-order linear ordinary differential equation of state variable. Numerical Algebra, Control and Optimization, 2021, 11 (4) : 495-512. doi: 10.3934/naco.2020040 [20] Alexei Pokrovskii, Oleg Rasskazov. Structure of index sequences for mappings with an asymptotic derivative. Discrete and Continuous Dynamical Systems, 2007, 17 (3) : 653-670. doi: 10.3934/dcds.2007.17.653

2021 Impact Factor: 1.41