Pullback and forward dynamics of nonautonomous Laplacian lattice systems on weighted spaces

Abstract

<p style='text-indent:20px;'>A nonautonomous lattice system with discrete Laplacian operator is revisited in the weighted space of infinite sequences <inline-formula><tex-math id="M1">\begin{document}$ {{\ell_{\rho}^2}} $\end{document}</tex-math></inline-formula>. First the existence of a pullback attractor in <inline-formula><tex-math id="M2">\begin{document}$ {{\ell_{\rho}^2}} $\end{document}</tex-math></inline-formula> is established by utilizing the dense inclusion of <inline-formula><tex-math id="M3">\begin{document}$ \ell^2 \subset {{\ell_{\rho}^2}} $\end{document}</tex-math></inline-formula>. Moreover, the pullback attractor is shown to consist of a singleton trajectory when the lattice system is uniformly strictly contracting. Then forward dynamics is investigated in terms of the existence of a nonempty compact forward omega limit set. A general class of weights for the sequence space are adopted, instead of particular types of weights often used in the literature. The analysis presented in this work is more direct compare with previous studies.</p>