Characterization of the limit of some higher dimensional thin domain problems

Thomas Elsken, Martino Prizzi

DOI: http://dx.doi.org/10.12775/TMNA.2002.031

Abstract


A reaction-diffusion equation on a family of three dimensional thin
domains, collapsing onto a two dimensional subspace, is considered.
In [< i> The effect of domain squeezing upon the dynamics
of reaction-diffusion equations< /i> , J. Differential Equations
< b> 173< /b> (2001), 271–320] it was proved that, as the thickness of the domains
tends to zero, the
solutions of the equations converge in a strong sense to the solutions of
an abstract semilinear parabolic equation living in a closed subspace of
$H^1$. Also, existence and upper semicontinuity of the attractors was
proved. In this work, for a specific class of domains, the limit problem
is completely characterized as a system of two-dimensional
reaction-diffusion equations, coupled by mean of compatibility and balance
boundary conditions.

Keywords


Reaction-diffusion equations; thin domains

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