### Recent results on thin domain problems II

#### 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.

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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