{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:36:53Z","timestamp":1760236613120,"version":"build-2065373602"},"reference-count":26,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2021,12,16]],"date-time":"2021-12-16T00:00:00Z","timestamp":1639612800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Science and Technology Research Program of Chongqing Municipal Educational Commission","award":["KJQN202000518"],"award-info":[{"award-number":["KJQN202000518"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This work mainly focuses on the continuity and analyticity for the generalized Benjamin\u2013Ono (g-BO) equation. From the local well-posedness results for g-BO equation, we know that its solutions depend continuously on their initial data. In the present paper, we further show that such dependence is not uniformly continuous in Sobolev spaces Hs(R) with s&gt;3\/2. We also provide more information about the stability of the data-solution map, i.e., the solution map for g-BO equation is H\u00f6lder continuous in Hr-topology for all 0\u2264r&lt;s with exponent \u03b1 depending on s and r. Finally, applying the generalized Ovsyannikov type theorem and the basic properties of Sobolev\u2013Gevrey spaces, we prove the Gevrey regularity and analyticity for the g-BO equation. In addition, by the symmetry of the spatial variable, we obtain a lower bound of the lifespan and the continuity of the data-to-solution map.<\/jats:p>","DOI":"10.3390\/sym13122435","type":"journal-article","created":{"date-parts":[[2021,12,16]],"date-time":"2021-12-16T21:32:40Z","timestamp":1639690360000},"page":"2435","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Continuity and Analyticity for the Generalized Benjamin\u2013Ono Equation"],"prefix":"10.3390","volume":"13","author":[{"given":"Xiaolin","family":"Pan","sequence":"first","affiliation":[{"name":"College of Mathematics Science, Chongqing Normal University, Chongqing 401331, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Bin","family":"Wang","sequence":"additional","affiliation":[{"name":"Chongqing Fengmingshan High School, Chongqing 401331, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Rong","family":"Chen","sequence":"additional","affiliation":[{"name":"Personnel Department, Chongqing Normal University, Chongqing 401331, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,12,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"559","DOI":"10.1017\/S002211206700103X","article-title":"Internal waves of permanent form in fluids of great depth","volume":"29","author":"Benjamin","year":"1967","journal-title":"J. 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