{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T11:25:35Z","timestamp":1787225135259,"version":"build-2736575974"},"reference-count":23,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","funder":[{"name":"National Key Research and Development Program of China","award":["2023YFA1009100"],"award-info":[{"award-number":["2023YFA1009100"]}]},{"name":"National Key Research and Development Program of China","award":["2024YFA1012300"],"award-info":[{"award-number":["2024YFA1012300"]}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12071430"],"award-info":[{"award-number":["12071430"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Key Laboratory of Collaborative Sensing and Autonomous Unmanned Systems of Zhejiang Province"},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11771074"],"award-info":[{"award-number":["11771074"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12371421"],"award-info":[{"award-number":["12371421"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"National Key Research and Development Program of China","award":["2020YFA0714102"],"award-info":[{"award-number":["2020YFA0714102"]}]},{"name":"National Key Research and Development Program of China","award":["2021YFA1003400"],"award-info":[{"award-number":["2021YFA1003400"]}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12371421"],"award-info":[{"award-number":["12371421"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Appl. Math."],"published-print":{"date-parts":[[2025,8,31]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>In this paper, we show for the first time the stability of the inverse source problem for the three-dimensional Helmholtz equation with a variable damping coefficient. The stability estimate consists of the Lipschitz-type data discrepancy and the high wavenumber tail of the source function, where the latter decreases as the upper bound of the wavenumber increases. The stability is shown to have an exponential dependence on the upper bound of the damping coefficient. The analysis employs scattering theory and semiclassical analysis to obtain an analytic domain and an upper bound for the resolvent of the elliptic operator.<\/jats:p>","DOI":"10.1137\/24m1675035","type":"journal-article","created":{"date-parts":[[2025,7,14]],"date-time":"2025-07-14T18:07:26Z","timestamp":1752516446000},"page":"1438-1457","source":"Crossref","is-referenced-by-count":0,"title":["Stability for the Inverse Source Problem of the Helmholtz Equation with Damping"],"prefix":"10.1137","volume":"85","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9699-780X","authenticated-orcid":true,"given":"Tianjiao","family":"Wang","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences, Zhejiang University, Hangzhou 310058, China."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0626-7609","authenticated-orcid":true,"given":"Xiang","family":"Xu","sequence":"additional","affiliation":[{"name":"1. School of Mathematical Sciences, Zhejiang University, Hangzhou, 310058, China. 2. Center for Interdisciplinary Applied Mathematics, Zhejiang University, Hangzhou 310058, China."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ganghua","family":"Yuan","sequence":"additional","affiliation":[{"name":"KLAS, School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin 130024, China."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yue","family":"Zhao","sequence":"additional","affiliation":[{"name":"Corresponding author. School of Mathematics and Statistics, and Key Lab NAA\u2013MOE, Central China Normal University, Wuhan 430079, China."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2025,7,15]]},"reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/28\/7\/075013"},{"key":"ref2","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/22\/3\/018"},{"key":"ref3","doi-asserted-by":"publisher","DOI":"10.1016\/j.jde.2010.08.013"},{"key":"ref4","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/31\/9\/093001"},{"key":"ref5","doi-asserted-by":"publisher","DOI":"10.1016\/j.matpur.2019.06.006"},{"key":"ref6","doi-asserted-by":"publisher","DOI":"10.3934\/ipi.2009.3.275"},{"key":"ref7","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-03537-5"},{"key":"ref8","volume-title":"Appl. Math. Sci. 144","author":"Nedelec J.-C.","year":"2001"},{"key":"ref9","doi-asserted-by":"publisher","DOI":"10.1137\/040610337"},{"key":"ref10","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4614-4942-3"},{"key":"ref11","doi-asserted-by":"publisher","DOI":"10.1016\/j.jde.2015.11.030"},{"key":"ref12","doi-asserted-by":"publisher","DOI":"10.1063\/1.1522135"},{"key":"ref13","doi-asserted-by":"publisher","DOI":"10.1137\/060658618"},{"key":"ref14","doi-asserted-by":"publisher","DOI":"10.1090\/gsm\/200"},{"key":"ref15","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/25\/11\/115005"},{"key":"ref16","volume-title":"Inverse Problems for Partial Differential Equations","author":"Isakov V.","year":"2006"},{"key":"ref17","doi-asserted-by":"publisher","DOI":"10.1137\/17M1112704"},{"key":"ref18","doi-asserted-by":"publisher","DOI":"10.2969\/jmsj\/04740617"},{"key":"ref19","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.3038"},{"key":"ref20","doi-asserted-by":"publisher","DOI":"10.1137\/S0036141003433577"},{"key":"ref21","doi-asserted-by":"publisher","DOI":"10.1137\/20M1334267"},{"key":"ref22","doi-asserted-by":"publisher","DOI":"10.1016\/S0022-0396(02)00101-8"},{"key":"ref23","doi-asserted-by":"publisher","DOI":"10.1090\/gsm\/138"}],"container-title":["SIAM Journal on Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/24M1675035","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T10:52:24Z","timestamp":1787223144000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/24M1675035"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,15]]},"references-count":23,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2025,8,31]]}},"alternative-id":["10.1137\/24M1675035"],"URL":"https:\/\/doi.org\/10.1137\/24m1675035","relation":{},"ISSN":["0036-1399","1095-712X"],"issn-type":[{"value":"0036-1399","type":"print"},{"value":"1095-712X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,7,15]]}}}