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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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