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Inertial Manifolds on Squeezed Domains

Authors:

Affiliation:

(1) Universität Rostock, Fachbereich Mathematik, Universitätsplatz 1, 18055 Rostock, Germany;(2) Dipartimento di Scienze Matematiche, Università degli Studi di Trieste, via Valerio 12/b, 34100 Trieste, Italy

Abstract:

Let be an arbitrary smooth bounded domain in $mathbb{R}^2$ and > 0 be arbitrary. Squeeze by the factor in the y-direction to obtain the squeezed domain = {(x,y) mid

(x,y)

}. In this paper we study the family of reaction-diffusion equations

$begin{gathered} ;;;u_t = Delta u + f(u),{text{ }}t > 0,{text{ (}}x,y{text{)}} in Omega _varepsilon hfill partial _{v_varepsilon } u = 0,{text{ }}t > 0,{text{ (}}x,y{text{)}} in partial Omega _varepsilon ,{text{ (}}E_varepsilon {text{)}} hfill end{gathered}$

where f is a dissipative nonlinearity of polynomial growth. In a previous paper we showed that, as rarr

0, the equations (E) have a limiting equation which is an abstract semilinear parabolic equation defined on a closed linear subspace of H¹(). We also proved that the family $$A_varepsilon $$

of the corresponding attractors is upper semicontinuous at = 0. In this paper we prove that, if satisfies some natural assumptions, then there is a family $$M_varepsilon $$

of inertial C¹-manifolds for (E) of some fixed finite dimension . Moreover, as rarr

0, the flow on $$M_varepsilon $$

converges in the C¹-sense to the limit flow on $$M_0 $$

Keywords:

reaction-diffusion equations thin domains inertial manifolds

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