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Spectral convergence and nonlinear dynamics of reaction-diffusion equations under perturbations of the domain. (English) Zbl 1058.35028
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.

MSC:
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35B41 Attractors
35K57 Reaction-diffusion equations
37L15 Stability problems for infinite-dimensional dissipative dynamical systems
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