Reaction-diffusion equations on unbounded thin domains (Q696172)

The authors consider the family of reaction-diffusion Neumann problems \[ u_t=\Delta u - \lambda u + f(u) +g\quad\text{in} \Omega_\varepsilon\times\mathbb{R}_+,\qquad \frac{\partial u}{\partial\nu_\varepsilon}=0\quad\text{in} \partial\Omega_\varepsilon\times{\mathbb R}_+, \] where \(\nu_\varepsilon\) is the outward normal to \(\partial\Omega_\varepsilon\), \(\Omega_\varepsilon=\{(x, \varepsilon y)\in \mathbb{R}^{N+P}: (x, y)\in\Omega\}\). The main result is the proof of existence and upper semicontinuity of attractors.