{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T07:50:21Z","timestamp":1777449021633,"version":"3.51.4"},"reference-count":0,"publisher":"SAGE Publications","issue":"2","license":[{"start":{"date-parts":[[2013,3,1]],"date-time":"2013-03-01T00:00:00Z","timestamp":1362096000000},"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":[[2013,3]]},"abstract":"<jats:p>We prove an asymptotic energy equipartition result for abstract damped wave equations of the form<\/jats:p>\n                  <jats:p>\n                    u\n                    <jats:sub>tt<\/jats:sub>\n                    +2F(S)u\n                    <jats:sub>t<\/jats:sub>\n                    +S\n                    <jats:sup>2<\/jats:sup>\n                    u=0,\n                  <\/jats:p>\n                  <jats:p>\n                    where S is a strictly positive self-adjoint operator and the damping operator F(S) is \u201csmall\u201d. This means that under certain assumptions, the ratio of suitably modified kinetic and potential energies, K\n                    <jats:sup>\u02dc<\/jats:sup>\n                    (t)\/P\n                    <jats:sup>\u02dc<\/jats:sup>\n                    (t), tends to 1 as t\u2192\u221e for all nonzero solutions u(t) of the equation. Here, K\n                    <jats:sup>\u02dc<\/jats:sup>\n                    (t) and P\n                    <jats:sup>\u02dc<\/jats:sup>\n                    (t) are conveniently weighted versions of the usual kinetic and potential energies of the associated undamped equation. Previous results, concerning the undamped case and the scalar-damped one, are particular cases. We propose an extension of the concepts of hyperbolicity and unitarity that allows one to consider the equipartition property in a more general setting. Some examples involving PDEs, as well as pseudo-differential equations, are given.\n                  <\/jats:p>","DOI":"10.3233\/asy-2012-1124","type":"journal-article","created":{"date-parts":[[2019,11,29]],"date-time":"2019-11-29T19:26:20Z","timestamp":1575055580000},"page":"171-187","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":2,"title":["Asymptotic equipartition of operator-weighted energies in damped wave equations"],"prefix":"10.1177","volume":"81","author":[{"given":"Jerome A.","family":"Goldstein","sequence":"first","affiliation":[{"name":"University of Memphis, Memphis, TN, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Guillermo","family":"Reyes","sequence":"additional","affiliation":[{"name":"Universidad Polit\u00e9cnica de Madrid, Madrid, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"179","published-online":{"date-parts":[[2013,3,1]]},"container-title":["Asymptotic Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/ASY-2012-1124","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/ASY-2012-1124","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,28]],"date-time":"2026-04-28T12:39:46Z","timestamp":1777379986000},"score":1,"resource":{"primary":{"URL":"https:\/\/journals.sagepub.com\/doi\/10.3233\/ASY-2012-1124"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,3]]},"references-count":0,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2013,3]]}},"alternative-id":["10.3233\/ASY-2012-1124"],"URL":"https:\/\/doi.org\/10.3233\/asy-2012-1124","relation":{},"ISSN":["0921-7134","1875-8576"],"issn-type":[{"value":"0921-7134","type":"print"},{"value":"1875-8576","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013,3]]}}}