Singular large diffusivity and spatial homogenization in a non homogeneous linear parabolic problem

Aníbal Rodríguez-Bernal; Robert Willie

doi:10.3934/dcdsb.2005.5.385

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2005, Volume 5, Issue 2: 385-410. Doi: 10.3934/dcdsb.2005.5.385

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Singular large diffusivity and spatial homogenization in a non homogeneous linear parabolic problem

Aníbal Rodríguez-Bernal^1, and
Robert Willie^2,

1.
Departamento de Matemática Aplicada, Universidad Complutense de Madrid, Madrid 28040
2.
Universidad Complutense de Madrid, Facultad de Ciencias Matemática, Matemática Aplicada, 28040, Madrid

Received: November 30, 2002

Revised: June 30, 2004

Published: April 2008

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Abstract

We make precise the sense in which spatial homogenization to a constant function in space is attained in a linear parabolic problem when large diffusion in all parts of the domain is assumed. Also interaction between diffusion and boundary flux terms is considered. Our starting point is a detailed analysis of the large diffusion effects on the associated elliptic and eigenvalue problems. Here convergence is shown in the energy space $H^1(\Omega)$ and in the space of continuous functions $C(\overline{\Omega})$. In the parabolic case we prove convergence in the functional space $L^\infty ((0,T),L^2(\Omega)) \bigcap L^2 ((0,T),H^1(\Omega)).

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Mathematics Subject Classification: 35Bxx,35B25, 35B40, 35B45.

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