Spectral convergence and nonlinear dynamics of reaction-diffusion equations under perturbations of the domain
Under consideration are the reaction-diffusion equations \[ (*)_\varepsilon\quad\quad u_t=\Delta u+f(x,u) \quad\text{in}\quad \Omega_\varepsilon \] accompanying with the Neumann boundary condition \(\partial u/\partial n=0\) in \(\partial\Omega_\varepsilon,\) where \(\Omega_\varepsilon\) (\(0\leq\varepsilon\leq 1\)) are bounded Lipschitz domains in \({\mathbb R}^N\) (\(N\geq 2\)). Under the usual conditions on the nonlinearity \(f(x,u)\) which ensure the existence of an attractor \({\mathcal A}_\varepsilon\) for each \((*)_\varepsilon\) with respect to the \(H^1\)-topology, the authors show the convergence of the attractors \({\mathcal A}_\varepsilon\) as \(\varepsilon\to 0\) to the attractor \({\mathcal A}_0\) for the unperturbed problem \((*)_0\) (i.e., the case \(\varepsilon=0\)) if, besides an easy condition on the domains, the following spectral condition is satisfied: Every equilibrium of the unperturbed problem \((*)_0\) is hyperbolic.
scientific article; zbMATH DE number 2105464
scientific article