Continuity of attractors for net-shaped thin domains
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Reaction-diffusion equations, thin net shaped domain, continuity of attractorsAbstrakt
Consider a reaction-diffusion equation $u_t=\triangle u+f(u)$ on a family of net-shaped thin domains $\Omega_\varepsilon$ converging to a one dimensional set as $\varepsilon\downarrow 0$. With suitable growth and dissipativeness conditions on $f$ these equations define global semiflows which have attractors $\mathcal{A}_\varepsilon$. In [Th. Elsken, < i> A reaction-diffusion equation on a net-shaped thin domain< /i> , Studia Math. < b> 165< /b> (2004), 159–199] it has been shown that there is a limit problem which also defines a semiflow having an attractor $\mathcal{A}_0$, and the family of attractors is upper-semi-continuous at $\varepsilon=0$. Here we show that under a stronger dissipativeness condition the family of attractors $\mathcal{A}_\varepsilon$, $\varepsilon\ge 0$, is actually continuous at $\varepsilon=0$.Pobrania
Opublikowane
2005-12-01
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ELSKEN, Thomas. Continuity of attractors for net-shaped thin domains. Topological Methods in Nonlinear Analysis [online]. 1 grudzień 2005, T. 26, nr 2, s. 315–354. [udostępniono 22.7.2024].
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