# American Institute of Mathematical Sciences

September  2014, 13(5): 2059-2093. doi: 10.3934/cpaa.2014.13.2059

## Attractors for the nonlinear elliptic boundary value problems and their parabolic singular limit

 1 Institute for Information Transmission Problems, Russian Academy of Sciences, Bolshoy Karetniy 19, Moscow 127994, GSP-4, Russian Federation 2 Department of Mathematics, University of Surrey, Guildford, GU2 7XH

Received  November 2012 Revised  November 2013 Published  June 2014

We apply the dynamical approach to the study of the second order semi-linear elliptic boundary value problem in a cylindrical domain with a small parameter $\varepsilon$ at the second derivative with respect to the variable $t$ corresponding to the axis of the cylinder. We prove that, under natural assumptions on the nonlinear interaction function $f$ and the external forces $g(t)$, this problem possesses the uniform attractor $\mathcal A_\varepsilon$ and that these attractors tend as $\varepsilon \to 0$ to the attractor $\mathcal A_0$ of the limit parabolic equation. Moreover, in case where the limit attractor $\mathcal A_0$ is regular, we give the detailed description of the structure of the uniform attractor $\mathcal A_\varepsilon$, if $\varepsilon>0$ is small enough, and estimate the symmetric distance between the attractors $\mathcal A_\varepsilon$ and $\mathcal A_0$.
Citation: Mark I. Vishik, Sergey Zelik. Attractors for the nonlinear elliptic boundary value problems and their parabolic singular limit. Communications on Pure & Applied Analysis, 2014, 13 (5) : 2059-2093. doi: 10.3934/cpaa.2014.13.2059
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##### References:
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