{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T07:32:27Z","timestamp":1777447947894,"version":"3.51.4"},"reference-count":22,"publisher":"SAGE Publications","issue":"1","license":[{"start":{"date-parts":[[2018,12,10]],"date-time":"2018-12-10T00:00:00Z","timestamp":1544400000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"content-domain":{"domain":["journals.sagepub.com"],"crossmark-restriction":true},"short-container-title":["Asymptotic Analysis"],"published-print":{"date-parts":[[2019,1,1]]},"abstract":"<jats:p>We study the homogeneous Dirichlet problem for the class of nonlinear parabolic equations with variable nonlinearity [Formula: see text] in the cylinder [Formula: see text] with given nonnegative weights [Formula: see text], [Formula: see text], measurable bounded exponents [Formula: see text], [Formula: see text] and a globally Lipschitz function [Formula: see text]. Sufficient conditions of existence and uniqueness of weak and strong solutions are derived. We find conditions on the exponents [Formula: see text], [Formula: see text] which guarantee that the associated semigroup has a compact global attractor in [Formula: see text]. It is shown that in case the exponents [Formula: see text] and [Formula: see text] do not meet the sufficient conditions of existence of a nontrivial global attractor and [Formula: see text] is sufficiently small, then every solution with bounded [Formula: see text] either vanishes in a finite time, or decays exponentially as [Formula: see text].<\/jats:p>","DOI":"10.3233\/asy-181486","type":"journal-article","created":{"date-parts":[[2018,12,11]],"date-time":"2018-12-11T11:58:19Z","timestamp":1544529499000},"page":"43-68","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":0,"title":["Asymptotic behavior for a class of parabolic equations in weighted variable Sobolev spaces"],"prefix":"10.1177","volume":"111","author":[{"given":"Sergey","family":"Shmarev","sequence":"first","affiliation":[{"name":"Departamento de Matem\u00e1ticas, Universidad de Oviedo, c\/Calvo Sotelo, s\/n, 33007, Oviedo, Spain. E-mail:\u00a0"}]},{"given":"Jacson","family":"Simsen","sequence":"additional","affiliation":[{"name":"Instituto de Matem\u00e1tica e Computa\u00e7\u00e3o, Universidade Federal de Itajub\u00e1, Av. BPS n. 1303, Bairro Pinheirinho, 37500-903, Itajub\u00e1, MG, Brasil. E-mails:\u00a0,\u00a0"}]},{"given":"Mariza","family":"Stefanello Simsen","sequence":"additional","affiliation":[{"name":"Instituto de Matem\u00e1tica e Computa\u00e7\u00e3o, Universidade Federal de Itajub\u00e1, Av. BPS n. 1303, Bairro Pinheirinho, 37500-903, Itajub\u00e1, MG, Brasil. E-mails:\u00a0,\u00a0"}]},{"given":"Marcos Roberto T.","family":"Primo","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1tica, Universidade Estadual de Maring\u00e1, 87020-900, Maring\u00e1, Paran\u00e1, Brasil. 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