{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:44:11Z","timestamp":1760060651401,"version":"build-2065373602"},"reference-count":27,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2025,9,15]],"date-time":"2025-09-15T00:00:00Z","timestamp":1757894400000},"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 establish global decay and scattering in the energy space H2(Rd), for d\u22655, of radial solutions to the damped nonlinear biharmonic Schr\u00f6dinger equation with general complex-valued, time-dependent damping coefficients. Assuming radial data exploits O(d)-symmetry and strengthens Morawetz-type controls through spherical averaging, we introduce new Morawetz-type identities and localized inequalities adapted to the fourth-order dispersive flow and compatible with this symmetry. As a consequence, and under explicit conditions for the damping coefficients that include slowly decaying or oscillatory profiles, we prove that solutions decay in Lebesgue norms and scatter to free biharmonic evolutions.<\/jats:p>","DOI":"10.3390\/sym17091541","type":"journal-article","created":{"date-parts":[[2025,9,15]],"date-time":"2025-09-15T11:51:43Z","timestamp":1757937103000},"page":"1541","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Scattering in the Energy Space for the Damped Nonlinear Fourth-Order Schr\u00f6dinger Equation in Higher Dimensions Under Spherical Symmetry"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5397-590X","authenticated-orcid":false,"given":"Mirko","family":"Tarulli","sequence":"first","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"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,9,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"44","DOI":"10.1007\/s00009-025-02813-6","article-title":"Energy scattering for the unsteady damped nonlinear Schr\u00f6dinger equation","volume":"22","author":"Hamouda","year":"2025","journal-title":"Mediterr. J. Math."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"2365","DOI":"10.3934\/cpaa.2023069","article-title":"Global existence and scattering for nonlinear Schr\u00f6dinger equations with time-dependent damping","volume":"22","author":"Bamri","year":"2023","journal-title":"Commun. Pure Appl. Anal."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1437","DOI":"10.1137\/S0036139901387241","article-title":"Self-focusing with fourth order dispersion","volume":"62","author":"Fibich","year":"2002","journal-title":"SIAM J. Appl. Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1336","DOI":"10.1103\/PhysRevE.53.R1336","article-title":"Stabilization of soliton instabilities by higher-order dispersion: Fourth order nonlinear Schr\u00f6dinger-type equations","volume":"53","author":"Karpman","year":"1996","journal-title":"Phys. Rev. E"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"194","DOI":"10.1016\/S0167-2789(00)00078-6","article-title":"Stability of soliton described by nonlinear Schr\u00f6dinger-type equations with higher-order dispersion","volume":"144","author":"Karpman","year":"2000","journal-title":"Phys. D"},{"key":"ref_6","first-page":"841","article-title":"Well-posedness for the fourth-order nonlinear Schr\u00f6dinger type equation related to the vortex filament","volume":"16","author":"Segata","year":"2003","journal-title":"Differ. Integral Equ."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.jde.2004.09.005","article-title":"The Cauchy problem for the fourth-order nonlinear Schr\u00f6dinger equation related to the vortex filament","volume":"214","author":"Huo","year":"2005","journal-title":"J. Differ. Equ."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1493","DOI":"10.1080\/03605300701629385","article-title":"A refined well-posedness for the fourth-order nonlinear Schr\u00f6dinger equation related to the vortex filament","volume":"32","author":"Huo","year":"2007","journal-title":"Commun. Partial Differ. Equ."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"055205","DOI":"10.1088\/1751-8113\/42\/5\/055205","article-title":"A small initial data criterion of global existence for the damped nonlinear Schr\u00f6dinger equation","volume":"42","author":"Chen","year":"2009","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"945","DOI":"10.1063\/1.863074","article-title":"Dimensionality and dissipation in Langmuir collapse","volume":"29","author":"Goldman","year":"1980","journal-title":"Phys. Fluids"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"763","DOI":"10.1090\/proc\/14276","article-title":"Asymptotic behavior of the nonlinear damped Schr\u00f6dinger equation","volume":"147","author":"Inui","year":"2019","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"599","DOI":"10.3934\/eect.2020082","article-title":"Blow-up criteria for linearly damped nonlinear Schr\u00f6dinger equations","volume":"10","author":"Dinh","year":"2021","journal-title":"Evol. Equ. Control Theory"},{"key":"ref_13","first-page":"291","article-title":"H2-scattering for Systems of Weakly Coupled Fourth-order NLS Equations in Low Space Dimensions","volume":"51","author":"Tarulli","year":"2019","journal-title":"Potential Anal."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"919","DOI":"10.4310\/MRL.2009.v16.n5.a14","article-title":"On the decay of solutions to a class of defocusing NLS","volume":"16","author":"Visciglia","year":"2009","journal-title":"Math. Res. Lett."},{"key":"ref_15","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_16","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_17","doi-asserted-by":"crossref","unstructured":"Cazenave, T. (2003). Semilinear Schr\u00f6dinger Equations, New York University Courant Institute of Mathematical Sciences. Courant Lecture Notes in Mathematics, 10.","DOI":"10.1090\/cln\/010"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"197","DOI":"10.4310\/DPDE.2007.v4.n3.a1","article-title":"Global well-posedness for energy critical fourth-order Schr\u00f6dinger equations in the radial case","volume":"4","author":"Pausader","year":"2007","journal-title":"Dyn. Partial Differ. Equ."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"2473","DOI":"10.1016\/j.jfa.2008.11.009","article-title":"The cubic fourth-order Schr\u00f6dinger equation","volume":"256","author":"Pausader","year":"2009","journal-title":"J. Funct. Anal."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"651","DOI":"10.1142\/S0219891610002256","article-title":"The mass-critical fourth-order Schr\u00f6dinger equation in high dimensions","volume":"7","author":"Pausader","year":"2010","journal-title":"J. Hyp. Differ. Equ."},{"key":"ref_21","first-page":"1","article-title":"Defocusing fourth-order coupled nonlinear Schr\u00f6dinger equations","volume":"2016","author":"Ghanimi","year":"2016","journal-title":"Electron. J. Differ. Equ."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"6643","DOI":"10.3934\/dcds.2019289","article-title":"Scattering of radial data in the focusing NLS and generalized Hartree equations","volume":"39","author":"Arora","year":"2019","journal-title":"Discrete Contin. Dyn. Syst."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"798","DOI":"10.1002\/mana.201400012","article-title":"Scattering theory below energy for the cubic fourth-order Schr\u00f6dinger equation","volume":"288","author":"Miao","year":"2015","journal-title":"Math. Nachrichten"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"479","DOI":"10.4310\/MAA.2000.v7.n3.a5","article-title":"Time decay for the nonlinear Beam equation","volume":"7","author":"Levandosky","year":"2000","journal-title":"Methods Appl. Anal."},{"key":"ref_25","first-page":"363","article-title":"Scattering theory in the energy space for a class of nonlinear Schr\u00f6dinger equations","volume":"64","author":"Ginibre","year":"1985","journal-title":"J. Math. Pures Appl."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1016\/s0294-1449(16)30428-0","article-title":"The concentration-compactness principle in the calculus of variations. The locally compact case, part 1","volume":"1","author":"Lions","year":"1984","journal-title":"Ann. De L\u2019Institut Henri Poincar\u00c9 (C) Anal. Non Lineaire"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1016\/s0294-1449(16)30422-x","article-title":"The concentration-compactness principle in the calculus of variations. The locally compact case, part 2","volume":"1","author":"Lions","year":"1984","journal-title":"Ann. De L\u2019Institut Henri Poincar\u00c9 (C) Anal. Non Lineaire"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/9\/1541\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:45:48Z","timestamp":1760035548000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/9\/1541"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,9,15]]},"references-count":27,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2025,9]]}},"alternative-id":["sym17091541"],"URL":"https:\/\/doi.org\/10.3390\/sym17091541","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2025,9,15]]}}}