{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T07:47:16Z","timestamp":1777448836963,"version":"3.51.4"},"reference-count":0,"publisher":"SAGE Publications","issue":"3-4","license":[{"start":{"date-parts":[[2010,9,1]],"date-time":"2010-09-01T00:00:00Z","timestamp":1283299200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Asymptotic Analysis"],"published-print":{"date-parts":[[2010,9]]},"abstract":"<jats:p>We consider the following singularly perturbed elliptic problem<\/jats:p>\n                  <jats:p>\n                    \u03b5\n                    <jats:sup>2<\/jats:sup>\n                    \u0394\u0169+(\u0169\u2212a(y\n                    <jats:sup>\u02dc<\/jats:sup>\n                    ))(1\u2212\u0169\n                    <jats:sup>2<\/jats:sup>\n                    )=0\u2003in \u03a9,\u2003\u2003\u2202\u0169\/\u2202\u03bd=0\u2003on \u2202\u03a9,\n                  <\/jats:p>\n                  <jats:p>\n                    where \u03a9 is a bounded domain in R\n                    <jats:sup>2<\/jats:sup>\n                    with smooth boundary, \u22121&lt;a(y\n                    <jats:sup>\u02dc<\/jats:sup>\n                    )&lt;1, \u03b5 is a small parameter, \u03bd denotes the outward normal of \u2202\u03a9. Assume that \u0393={y\n                    <jats:sup>\u02dc<\/jats:sup>\n                    \u2208\u03a9:\u2009a(y\n                    <jats:sup>\u02dc<\/jats:sup>\n                    )=0} is a simple closed and smooth curve contained in \u03a9 in such a way that \u0393 separates \u03a9 into two disjoint components \u03a9\n                    <jats:sub>+<\/jats:sub>\n                    ={y\n                    <jats:sup>\u02dc<\/jats:sup>\n                    \u2208\u03a9:\u2009a(y\n                    <jats:sup>\u02dc<\/jats:sup>\n                    )&lt;0} and \u03a9\n                    <jats:sub>\u2212<\/jats:sub>\n                    ={y\n                    <jats:sup>\u02dc<\/jats:sup>\n                    \u2208\u03a9:\u2009a(y\n                    <jats:sup>\u02dc<\/jats:sup>\n                    )&gt;0} and \u2202a\/\u2202\u03bd\n                    <jats:sub>0<\/jats:sub>\n                    &gt;0 on \u0393, where \u03bd\n                    <jats:sub>0<\/jats:sub>\n                    is the outer normal of \u03a9\n                    <jats:sub>+<\/jats:sub>\n                    , pointing to the interior of \u03a9\n                    <jats:sub>\u2212<\/jats:sub>\n                    . For any fixed integer N=2m+1\u22653, we will show the existence of a clustered solution u\n                    <jats:sub>\u03b5<\/jats:sub>\n                    with N-transition layers near \u0393 with mutual distance O(\u03b5|log\u2009\u03b5|), provided that \u03b5 stays away from a discrete set of values at which resonance occurs. Moreover, u\n                    <jats:sub>\u03b5<\/jats:sub>\n                    approaches 1 in \u03a9\n                    <jats:sub>\u2212<\/jats:sub>\n                    and \u22121 in \u03a9\n                    <jats:sub>+<\/jats:sub>\n                    . Central to our analysis is the solvability of a Toda system.\n                  <\/jats:p>","DOI":"10.3233\/asy-2010-0999","type":"journal-article","created":{"date-parts":[[2019,11,29]],"date-time":"2019-11-29T19:13:20Z","timestamp":1575054800000},"page":"175-218","source":"Crossref","is-referenced-by-count":5,"title":["Toda system and cluster phase transition layers in an inhomogeneous phase transition model"],"prefix":"10.1177","volume":"69","author":[{"given":"Juncheng","family":"Wei","sequence":"first","affiliation":[{"name":"Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong, China. E-mail: wei@math.cuhk.edu.hk"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jun","family":"Yang","sequence":"additional","affiliation":[{"name":"College of Mathematics and Computational Sciences, Shenzhen University, Nanhai Ave 3688, Shenzhen, 518060 China. E-mail: jyang@szu.edu.cn"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"179","published-online":{"date-parts":[[2010,9,1]]},"container-title":["Asymptotic Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/ASY-2010-0999","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/ASY-2010-0999","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,28]],"date-time":"2026-04-28T12:39:07Z","timestamp":1777379947000},"score":1,"resource":{"primary":{"URL":"https:\/\/journals.sagepub.com\/doi\/10.3233\/ASY-2010-0999"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,9]]},"references-count":0,"journal-issue":{"issue":"3-4","published-print":{"date-parts":[[2010,9]]}},"alternative-id":["10.3233\/ASY-2010-0999"],"URL":"https:\/\/doi.org\/10.3233\/asy-2010-0999","relation":{},"ISSN":["0921-7134","1875-8576"],"issn-type":[{"value":"0921-7134","type":"print"},{"value":"1875-8576","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,9]]}}}