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

2014; American Institute of Mathematical Sciences; Volume: 13; Issue: 5 Linguagem: Inglês

10.3934/cpaa.2014.13.2059

1553-5258

Mark Vishik, Sergey Zelik,

Advanced Mathematical Physics Problems

We apply the dynamical approach to the study of the second ordersemi-linear elliptic boundary value problem in a cylindrical domainwith a small parameter $\varepsilon$ at the second derivative with respect tothe variable $t$ corresponding to the axis of the cylinder.We prove that, under natural assumptions on the nonlinear interactionfunction $f$ and the external forces $g(t)$, this problem possessesthe uniform attractor $\mathcal A_\varepsilon$ and that these attractors tendas $\varepsilon \to 0$ to the attractor $\mathcal A_0$ of the limit parabolicequation. Moreover, in case where the limit attractor $\mathcal A_0$ isregular, we give the detailed description of the structure ofthe uniform attractor $\mathcal A_\varepsilon$, if $\varepsilon>0$ is small enough, andestimate the symmetric distance between the attractors $\mathcal A_\varepsilon$and $\mathcal A_0$.