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Upper semicontinuity of attractors for lattice systems under singular perturbations. (English) Zbl 1189.34021
Summary: Consider the following first order lattice system $$\dot u_m+(2u_m-u_{m-1}-u_{m+1})+\lambda_mu_m+f_m(u_m)=g_m,\quad m\in\Bbb Z,$$ which is perturbed by the $\varepsilon$-small two order term $$\varepsilon\ddot u_m+\dot u_m+(2u_m-u_{m-1}-u_{m+1}+\lambda_mu_m+f_m(u_m)=g_m,\quad m\in \Bbb Z.$$ Under certain conditions on $f_m$, $\lambda_m$ and $g_m$, the original systems and the $\varepsilon$-small perturbed systems have global attractors $\cal A$ in $\ell^2$ and ${\cal A}_\varepsilon$ in $\ell^2\times \ell^2$, respectively, and $\cal A$ can be naturally embedded into a compact set ${\cal A}_0$ in $\ell^2\times \ell^2$. We prove the upper semicontinuity of ${\cal A}_0$ with respect to the attractors ${\cal A}_\varepsilon$ at zero by showing that for any neighborhood ${\cal O}({\cal A}_0)$ of ${\cal A}_0$, ${\cal A}_\varepsilon$ enters ${\cal O}({\cal A}_0)$ if $\varepsilon$ is small enough.
MSC:
34A33Lattice differential equations
34E15Asymptotic singular perturbations, general theory (ODE)
34D45Attractors
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References:
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