{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,23]],"date-time":"2026-07-23T15:06:29Z","timestamp":1784819189181,"version":"3.55.0"},"reference-count":21,"publisher":"Elsevier BV","license":[{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/tdm\/userlicense\/1.0\/"},{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/legal\/tdmrep-license"},{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-017"},{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-037"},{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-012"},{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-029"},{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-004"}],"funder":[{"DOI":"10.13039\/501100004731","name":"Zhejiang Province Natural Science Foundation","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100004731","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["elsevier.com","sciencedirect.com"],"crossmark-restriction":true},"short-container-title":["Applied Mathematics Letters"],"published-print":{"date-parts":[[2026,11]]},"DOI":"10.1016\/j.aml.2026.110054","type":"journal-article","created":{"date-parts":[[2026,7,3]],"date-time":"2026-07-03T15:01:36Z","timestamp":1783090896000},"page":"110054","update-policy":"https:\/\/doi.org\/10.1016\/elsevier_cm_policy","source":"Crossref","is-referenced-by-count":0,"special_numbering":"C","title":["Growth of Sobolev norms for a coupled system of wave equations with harmonic potential"],"prefix":"10.1016","volume":"182","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9041-6805","authenticated-orcid":false,"given":"Qidi","family":"Zhang","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ting","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"78","reference":[{"key":"10.1016\/j.aml.2026.110054_b1","doi-asserted-by":"crossref","first-page":"463","DOI":"10.1103\/RevModPhys.71.463","article-title":"Theory of Bose\u2013Einstein condensation in trapped gases","volume":"71","author":"Dalfovo","year":"1999","journal-title":"Rev. Modern Phys."},{"key":"10.1016\/j.aml.2026.110054_b2","article-title":"Derivation and analysis of the wave equation with harmonic potential","volume":"53","author":"Gal","year":"2012","journal-title":"J. Math. Phys."},{"key":"10.1016\/j.aml.2026.110054_b3","series-title":"Optical Solitons: From Fibers To Photonic Crystals","author":"Kivshar","year":"2003"},{"issue":"37","key":"10.1016\/j.aml.2026.110054_b4","doi-asserted-by":"crossref","first-page":"1897","DOI":"10.1155\/S1073792804133321","article-title":"Long-time existence for small data nonlinear Klein\u2013Gordon equations on tori and spheres","volume":"2004","author":"Delort","year":"2004","journal-title":"Int. Math. Res. Not."},{"key":"10.1016\/j.aml.2026.110054_b5","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1007\/BF02843163","article-title":"On the growth in time of higher Sobolev norms of smooth solutions of Hamiltonian PDE","volume":"72","author":"Bourgain","year":"1997","journal-title":"J. Anal. Math."},{"issue":"1","key":"10.1016\/j.aml.2026.110054_b6","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1007\/s00222-010-0242-2","article-title":"Transfer of energy to high frequencies in the cubic defocusing nonlinear Schr\u00f6dinger equation","volume":"181","author":"Colliander","year":"2010","journal-title":"Invent. Math."},{"issue":"1","key":"10.1016\/j.aml.2026.110054_b7","doi-asserted-by":"crossref","first-page":"71","DOI":"10.4171\/jems\/499","article-title":"Growth of Sobolev norms in the cubic defocusing nonlinear Schr\u00f6dinger equation","volume":"17","author":"Gu\u00e0rdia","year":"2015","journal-title":"J. Eur. Math. Soc. (JEMS)"},{"issue":"2","key":"10.1016\/j.aml.2026.110054_b8","doi-asserted-by":"crossref","first-page":"601","DOI":"10.4171\/jems\/676","article-title":"Erratum to Growth of Sobolev norms in the cubic defocusing nonlinear Schr\u00f6dinger equation","volume":"19","author":"Gu\u00e0rdia","year":"2017","journal-title":"J. Eur. Math. Soc. (JEMS)"},{"issue":"4","key":"10.1016\/j.aml.2026.110054_b9","first-page":"997","article-title":"Growth of Sobolev norms in quasi-integrable quantum systems","volume":"58","author":"Bambusi","year":"2025","journal-title":"Ann. Sci. \u00c9c. Norm. Sup\u00e9r. (4)"},{"issue":"1","key":"10.1016\/j.aml.2026.110054_b10","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1080\/03605302.2013.843428","article-title":"Growth of Sobolev norms for solutions of time dependent Schr\u00f6dinger operators with harmonic oscillator potential","volume":"39","author":"Delort","year":"2014","journal-title":"Comm. Partial Differential Equations"},{"issue":"2","key":"10.1016\/j.aml.2026.110054_b11","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1007\/s10884-012-9244-7","article-title":"On growth of Sobolev norms in linear Schr\u00f6dinger equations with time dependent Gevrey potential","volume":"24","author":"Fang","year":"2012","journal-title":"J. Dynam. Differential Equations"},{"issue":"4","key":"10.1016\/j.aml.2026.110054_b12","doi-asserted-by":"crossref","first-page":"1405","DOI":"10.4171\/rmi\/1371","article-title":"Growth of Sobolev norms for 2d NLS with harmonic potential","volume":"39","author":"Planchon","year":"2023","journal-title":"Rev. Mat. Iberoam."},{"issue":"1","key":"10.1016\/j.aml.2026.110054_b13","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1215\/S0012-7094-97-08604-X","article-title":"On the growth of high Sobolev norms of solutions for KdV and Schr\u00f6dinger equations","volume":"86","author":"Staffilani","year":"1997","journal-title":"Duke Math. J."},{"issue":"4","key":"10.1016\/j.aml.2026.110054_b14","doi-asserted-by":"crossref","first-page":"10","DOI":"10.1007\/s00030-022-00776-1","article-title":"Global existence for the semi-linear wave\/Klein\u2013Gordon equation associated to the harmonic oscillator in low dimensions","volume":"29","author":"Xue","year":"2022","journal-title":"NoDEA Nonlinear Differential Equations Appl."},{"key":"10.1016\/j.aml.2026.110054_b15","first-page":"1","article-title":"Almost global solutions of 1D nonlinear Klein\u2013Gordon equations with small weakly decaying initial data","author":"Hou","year":"2025","journal-title":"Trans. Amer. Math. Soc."},{"issue":"12","key":"10.1016\/j.aml.2026.110054_b16","doi-asserted-by":"crossref","first-page":"2713","DOI":"10.1007\/s11425-023-2219-1","article-title":"Almost global smooth solutions of the 3D quasilinear Klein\u2013Gordon equations on the product space R2\u00d7T","volume":"67","author":"Li","year":"2024","journal-title":"Sci. China Math."},{"key":"10.1016\/j.aml.2026.110054_b17","series-title":"The critical power of short pulse initial data on the global existence or blowup of smooth solutions to 3-D semilinear Klein\u2013Gordon equations","author":"Shen","year":"2025"},{"key":"10.1016\/j.aml.2026.110054_b18","series-title":"Almost global existence for Hamiltonian PDEs on compact manifolds","author":"Bambusi","year":"2025"},{"key":"10.1016\/j.aml.2026.110054_b19","doi-asserted-by":"crossref","first-page":"e15","DOI":"10.1017\/fmp.2025.7","article-title":"On blow-up for the supercritical defocusing nonlinear wave equation","volume":"13","author":"Shao","year":"2025","journal-title":"Forum Math. Pi"},{"key":"10.1016\/j.aml.2026.110054_b20","series-title":"Applied Mathematical Sciences","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4757-4187-2","article-title":"Partial differential equations II: Qualitative studies of linear equations","volume":"Vol. 116","author":"Taylor","year":"1996"},{"issue":"1103","key":"10.1016\/j.aml.2026.110054_b21","article-title":"Quasi-linear perturbations of Hamiltonian Klein\u2013Gordon equations on spheres","volume":"234","author":"Delort","year":"2015","journal-title":"Mem. Amer. Math. Soc."}],"container-title":["Applied Mathematics Letters"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0893965926001850?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0893965926001850?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2026,7,23]],"date-time":"2026-07-23T14:49:07Z","timestamp":1784818147000},"score":1,"resource":{"primary":{"URL":"https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S0893965926001850"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,11]]},"references-count":21,"alternative-id":["S0893965926001850"],"URL":"https:\/\/doi.org\/10.1016\/j.aml.2026.110054","relation":{},"ISSN":["0893-9659"],"issn-type":[{"value":"0893-9659","type":"print"}],"subject":[],"published":{"date-parts":[[2026,11]]},"assertion":[{"value":"Elsevier","name":"publisher","label":"This article is maintained by"},{"value":"Growth of Sobolev norms for a coupled system of wave equations with harmonic potential","name":"articletitle","label":"Article Title"},{"value":"Applied Mathematics Letters","name":"journaltitle","label":"Journal Title"},{"value":"https:\/\/doi.org\/10.1016\/j.aml.2026.110054","name":"articlelink","label":"CrossRef DOI link to publisher maintained version"},{"value":"article","name":"content_type","label":"Content Type"},{"value":"\u00a9 2026 Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and similar technologies.","name":"copyright","label":"Copyright"}],"article-number":"110054"}}