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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

${\stackrel{˙}{u}}_{m}+\left(2{u}_{m}-{u}_{m-1}-{u}_{m+1}\right)+{\lambda }_{m}{u}_{m}+{f}_{m}\left({u}_{m}\right)={g}_{m},\phantom{\rule{1.em}{0ex}}m\in ℤ,$

which is perturbed by the $\epsilon$-small two order term

$\epsilon {\stackrel{¨}{u}}_{m}+{\stackrel{˙}{u}}_{m}+\left(2{u}_{m}-{u}_{m-1}-{u}_{m+1}+{\lambda }_{m}{u}_{m}+{f}_{m}\left({u}_{m}\right)={g}_{m},\phantom{\rule{1.em}{0ex}}m\in ℤ·$

Under certain conditions on ${f}_{m}$, ${\lambda }_{m}$ and ${g}_{m}$, the original systems and the $\epsilon$-small perturbed systems have global attractors $𝒜$ in ${\ell }^{2}$ and ${𝒜}_{\epsilon }$ in ${\ell }^{2}×{\ell }^{2}$, respectively, and $𝒜$ can be naturally embedded into a compact set ${𝒜}_{0}$ in ${\ell }^{2}×{\ell }^{2}$. We prove the upper semicontinuity of ${𝒜}_{0}$ with respect to the attractors ${𝒜}_{\epsilon }$ at zero by showing that for any neighborhood $𝒪\left({𝒜}_{0}\right)$ of ${𝒜}_{0}$, ${𝒜}_{\epsilon }$ enters $𝒪\left({𝒜}_{0}\right)$ if $\epsilon$ is small enough.

##### MSC:
 34A33 Lattice differential equations 34E15 Asymptotic singular perturbations, general theory (ODE) 34D45 Attractors