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Evol. Equ."],"published-print":{"date-parts":[[2020,6]]},"abstract":"<jats:title>Abstract<\/jats:title>\n<jats:p>In this work, we examine first-order lattice dynamical systems, which are discretized versions of reaction\u2013diffusion equations on the real line. We prove the existence of a global attractor in <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\ell ^2$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:msup><mml:mi>\u2113<\/mml:mi><mml:mn>2<\/mml:mn><\/mml:msup><\/mml:math><\/jats:alternatives><\/jats:inline-formula>, and using the method by Chueshov and Lasiecka (Dynamics of quasi-stable dissipative systems, Springer, Berlin, 2015; Memoirs of the American Mathematical Society, vol 195(912), AMS, 2008), we estimate its fractal dimension. We also show that the global attractor is contained in a finite-dimensional exponential attractor. The approach relies on the interplay between the discretized diffusion and reaction, which has not been exploited as yet for the lattice systems. Of separate interest is a\u00a0characterization of positive definiteness of the discretized Schr\u00f6dinger operator, which refers to the well-known Arendt and Batty\u2019s result (Differ Int Equ 6:1009\u20131024, 1993).<\/jats:p>","DOI":"10.1007\/s00028-019-00535-3","type":"journal-article","created":{"date-parts":[[2019,9,16]],"date-time":"2019-09-16T17:02:47Z","timestamp":1568653367000},"page":"485-515","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Lattice dynamical systems: dissipative mechanism and fractal dimension of global and exponential attractors"],"prefix":"10.1007","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1874-9477","authenticated-orcid":false,"given":"Jan W.","family":"Cholewa","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2245-2916","authenticated-orcid":false,"given":"Rados\u0142aw","family":"Czaja","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2019,9,16]]},"reference":[{"key":"535_CR1","doi-asserted-by":"publisher","first-page":"217","DOI":"10.1016\/j.jmaa.2007.06.054","volume":"339","author":"AY Abdallah","year":"2008","unstructured":"A.Y.\u00a0Abdallah, Exponential attractors for first-order lattice dynamical systems, J. 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