{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:22:16Z","timestamp":1760059336986,"version":"build-2065373602"},"reference-count":19,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2025,6,6]],"date-time":"2025-06-06T00:00:00Z","timestamp":1749168000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>We will show, in any space dimension d\u22653, the decay and scattering in the energy space for the solution to the damped nonlinear Schr\u00f6dinger equation posed on Rd\u00d7T and initial data in H1(Rd\u00d7T). We will also derive new bilinear Morawetz identities and corresponding localized Morawetz estimates.<\/jats:p>","DOI":"10.3390\/axioms14060447","type":"journal-article","created":{"date-parts":[[2025,6,6]],"date-time":"2025-06-06T06:11:08Z","timestamp":1749190268000},"page":"447","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Scattering in the Energy Space for Solutions of the Damped Nonlinear Schr\u00f6dinger Equation on Rd\u00d7T"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0009-0008-1826-0931","authenticated-orcid":false,"given":"Taim","family":"Saker","sequence":"first","affiliation":[{"name":"Institute of Mathematics and Informatics, Bulgarian Academy of Science, Acad. Georgi Bonchev Str., Block 8, 1113 Sofia, Bulgaria"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5397-590X","authenticated-orcid":false,"given":"Mirko","family":"Tarulli","sequence":"additional","affiliation":[{"name":"Institute of Mathematics and Informatics, Bulgarian Academy of Science, Acad. Georgi Bonchev Str., Block 8, 1113 Sofia, Bulgaria"},{"name":"Mathematics and Science Department, American University in Bulgaria, 1 Georgi Izmirliev Sq., 2700 Blagoevgrad, Bulgaria"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7163-3508","authenticated-orcid":false,"given":"George","family":"Venkov","sequence":"additional","affiliation":[{"name":"Department of Mathematical Analysis and Differential Equations, Faculty of Applied Mathematics and Informatics, Technical University of Sofia, 1756 Sofia, Bulgaria"}]}],"member":"1968","published-online":{"date-parts":[[2025,6,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Cazenave, T. (2003). 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Appl."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"126533","DOI":"10.1016\/j.jmaa.2022.126533","article-title":"Decay in energy space for the solution of fourth-order Hartree-Fock equations with general non-local interactions","volume":"516","author":"Tarulli","year":"2022","journal-title":"J. Math. Anal. Appl."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1149","DOI":"10.1007\/s00028-020-00621-x","article-title":"Decay and scattering in energy space for the solution of weakly coupled Schr\u00f6dinger-Choquard and Hartree-Fock equations","volume":"21","author":"Tarulli","year":"2021","journal-title":"J. Evol. Equ."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1163","DOI":"10.4171\/rmi\/911","article-title":"Well-posedness and scattering for nonlinear Schr\u00f6dinger equations on \u211dd \u00d7 \ud835\udd4b in the energy space","volume":"32","author":"Tzvetkov","year":"2016","journal-title":"Rev. Mat. 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