Attractors for reaction-diffusion equations on thin domains whose linear part is non-self-adjoint
The author considers semiflows and attractors of reaction-diffusion equations (RDEs) with nonsymmetrical linear part on thin domains, i.e., on sets of the from \(\Omega_{\varepsilon}:=\{(x,y)\in{\mathbb R}^{N_x}\times{\mathbb R}^{N_y}: (x,y/\varepsilon)\in\Omega\}\) with a bounded \(C^2\)-domain \(\Omega\). These RDEs are of the form \[ v_t = \sum_{k=1}^{N}\frac{\partial}{\partial z_k} \left( \sum_{l=1}^{N} \alpha_{l,k}(z)\frac{\partial v}{\partial z_l} \right) - \sum_{l=1}^{N}\beta_l(z)\frac{\partial v}{\partial z_l} - \gamma(z)v + f(z,v) \tag \(*\) \] for \(z\in\Omega_{\varepsilon}\) and boundary condition \[ 0=\sum_{k,l=1}^{N}\alpha_{k,l}(z)\frac{\partial v}{\partial z_l}\nu_{\varepsilon,k} \] for \(z\in\partial\Omega_{\varepsilon}\), where \(t>0\), \(N=N_x+N_y\), \(\nu_\varepsilon\) is the outer normal to \(\partial\Omega_\varepsilon\), \(\alpha_{k,l}\), \(\beta_l\), \(\gamma\), \(f\) are smooth functions, and the \(\alpha_{k,l}\) are uniformly elliptic. Under suitable growth and dissipativity conditions on \(f\), the abstract evolution equation \(u_t=-A_{\varepsilon}u+f_{\varepsilon}(u)\) corresponding to \((\ast)\) transformed to the fixed domain \(\Omega\) (i.e., \(u(x,y)=v(x,\varepsilon y)\)), defines a global semiflow \(\pi_{\varepsilon}\) with attractor \({\mathcal A}_{\varepsilon}\) for \(\varepsilon\in(0,1]\). As main result it is shown under quite natural conditions that for \(\varepsilon\searrow 0\), \(\pi_{\varepsilon}\) converges to a limit semiflow \(\pi_0\) solving a limit equation in an appropriate phase space. Moreover, the attractors \({\mathcal A}_\varepsilon\) are upper-semicontinuous.
scientific article; zbMATH DE number 2116219