### Characterization of the limit of some higher dimensional thin domain problems

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.

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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