{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T16:12:56Z","timestamp":1774627976264,"version":"3.50.1"},"reference-count":34,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2019,7,2]],"date-time":"2019-07-02T00:00:00Z","timestamp":1562025600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We first formulate the mixed backward in time problem in the context of thermoelasticity for dipolar materials. To prove the consistency of this mixed problem, our first main result is regarding the uniqueness of the solution for this problem. This is obtained based on some auxiliary results, namely, four integral identities. The second main result is regarding the temporal behavior of our thermoelastic body with a dipolar structure. This behavior is studied by means of some relations on a partition of various parts of the energy associated to the solution of the problem.<\/jats:p>","DOI":"10.3390\/sym11070863","type":"journal-article","created":{"date-parts":[[2019,7,3]],"date-time":"2019-07-03T11:14:49Z","timestamp":1562152489000},"page":"863","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":90,"title":["On the Partition of Energies for the Backward in Time Problem of Thermoelastic Materials with a Dipolar Structure"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1552-3763","authenticated-orcid":false,"given":"M.","family":"Marin","sequence":"first","affiliation":[{"name":"Department of Mathematics and Computer Science, Transilvania University of Brasov, 500093 Brasov, Romania"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8679-2579","authenticated-orcid":false,"given":"S.","family":"Vlase","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, Transilvania University of Brasov, 500093 Brasov, Romania"}]},{"given":"R.","family":"Ellahi","sequence":"additional","affiliation":[{"name":"Center for Modeling &amp; Computer Simulation, Research Institute, King Fahd University of Petroleum &amp; Minerals, Dhahran 31261, Saudi Arabia"},{"name":"Department of Mathematics &amp; Statistics, Faculty of Basic and Applied Sciences (FBAS), International Islamic University (IIUI), Islamabad 44000, Pakistan"}]},{"given":"M.M.","family":"Bhatti","sequence":"additional","affiliation":[{"name":"College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, Shandong, China"},{"name":"Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, China"}]}],"member":"1968","published-online":{"date-parts":[[2019,7,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1291","DOI":"10.1016\/0020-7225(90)90076-U","article-title":"Theory of thermo-microstretch elastic solids","volume":"28","author":"Eringen","year":"1990","journal-title":"Int. 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