Recent results on thin domain problems II

Martino Prizzi, Krzysztof P. Rybakowski

DOI: http://dx.doi.org/10.12775/TMNA.2002.010

Abstract


In this paper we survey some recent results
on parabolic equations on curved squeezed domains. More
specifically, consider the family of semilinear
Neumann boundary value problems
$$
\alignedat2 u_t &= \Delta u + f(u), &\quad&t> 0,\ x\in \Omega_\varepsilon,
\\ \partial_{\nu_\varepsilon}u&= 0, &\quad& t> 0,\ x\in \partial \Omega_\varepsilon
\endaligned \leqno{(\text{\rm E}_\varepsilon)}
$$
where, for $\varepsilon> 0$ small, the set
$\Omega_\varepsilon$ is a thin domain in $\mathbb R^\ell$, possibly with holes,
which collapses, as $\varepsilon\to0^+$, onto
a (curved) $k$-dimensional submanifold $\mathcal M$ of $\mathbb R^\ell$.
If $f$ is dissipative, then equation (E$_\varepsilon$) has a global attractor
${\mathcal A}_\varepsilon$.
We identify a ``limit'' equation for the family (E$_\varepsilon$), establish an upper semicontinuity result
for the family ${\mathcal A}_\varepsilon$
and prove an inertial manifold theorem in case $\mathcal M$ is a $k$-sphere.

Keywords


Thin domains; curved squeezing; evolution equations; inertial manifolds

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