Open AccessExistence and approximation of attractors for nonlinear coupled lattice wave equations
Author(s) -
Lianbing She,
M. M. Freitas,
Mauricio S. Vinhote,
Renhai Wang
Publication year - 2022
Publication title -
discrete and continuous dynamical systems. series b
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.864
H-Index - 53
eISSN - 1553-524X
pISSN - 1531-3492
DOI - 10.3934/dcdsb.2021272
Subject(s) - attractor , mathematics , semigroup , order (exchange) , compact space , integer (computer science) , lattice (music) , combinatorics , discrete mathematics , pure mathematics , mathematical analysis , physics , finance , computer science , acoustics , economics , programming language
This paper is concerned with the asymptotic behavior of solutions to a class of nonlinear coupled discrete wave equations defined on the whole integer set. We first establish the well-posedness of the systems in \begin{document}$ E: = \ell^2\times\ell^2\times\ell^2\times\ell^2 $\end{document} . We then prove that the solution semigroup has a unique global attractor in \begin{document}$ E $\end{document} . We finally prove that this attractor can be approximated in terms of upper semicontinuity of \begin{document}$ E $\end{document} by a finite-dimensional global attractor of a \begin{document}$ 2(2n+1) $\end{document} -dimensional truncation system as \begin{document}$ n $\end{document} goes to infinity. The idea of uniform tail-estimates developed by Wang (Phys. D, 128 (1999) 41-52) is employed to prove the asymptotic compactness of the solution semigroups in order to overcome the lack of compactness in infinite lattices.