Recent results on thin domain problems II

Martino Prizzi, Krzysztof P. Rybakowski


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.


Thin domains; curved squeezing; evolution equations; inertial manifolds

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