Some recent results on thin domain problems (Q1580012)

Let \(\Omega\) be a smooth bounded domain in \({\mathbb R}^2\) and \[ \Omega_{\varepsilon}=\{(x,\varepsilon y): (x,y)\in\Omega\},\quad \varepsilon >0, \] a squeezed domain. It is considered the following problem: \(u_t=\Delta u +f(u)\) in \((0,\infty)\times\Omega-{\varepsilon}\), \(\partial_{\nu_{\varepsilon}}u=0\) on \((0,\infty)\times\partial\Omega_{\varepsilon}\). Here \(\nu_{\varepsilon}\) is the exterior normal vector field on \(\partial\Omega_{\varepsilon}\) and it is assumed that \(f\) satisfies some conditions ensuring that the corresponding semiflow on \(H^1(\Omega_{\varepsilon})\) has a global attractor \(A_{\varepsilon}\). The authors study the asymptotic behavior of \(A_{\varepsilon}\) as \(\varepsilon\to 0\).