{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T07:32:32Z","timestamp":1777447952122,"version":"3.51.4"},"reference-count":25,"publisher":"SAGE Publications","issue":"1-2","license":[{"start":{"date-parts":[[2019,3,6]],"date-time":"2019-03-06T00:00:00Z","timestamp":1551830400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"content-domain":{"domain":["journals.sagepub.com"],"crossmark-restriction":true},"short-container-title":["Asymptotic Analysis"],"published-print":{"date-parts":[[2019,3,6]]},"abstract":"<jats:p>We study the large time behavior of solutions of first-order convex Hamilton\u2013Jacobi Equations of Eikonal type [Formula: see text], set in the whole space [Formula: see text]. We assume that l is bounded from below but may have arbitrary growth and therefore the solutions may also have arbitrary growth. A complete study of the structure of solutions of the ergodic problem [Formula: see text] is provided: contrarily to the periodic setting, the ergodic constant is not anymore unique, leading to different large time behavior for the solutions. We establish the ergodic behavior of the solutions of the Cauchy problem (i) when starting with a bounded from below initial condition and (ii) for some particular unbounded from below initial condition, two cases for which we have different ergodic constants which play a role. When the solution is not bounded from below, an example showing that the convergence may fail in general is provided.<\/jats:p>","DOI":"10.3233\/asy-181488","type":"journal-article","created":{"date-parts":[[2019,3,8]],"date-time":"2019-03-08T10:31:42Z","timestamp":1552041102000},"page":"1-22","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":1,"title":["Large time behavior of unbounded solutions of first-order Hamilton\u2013Jacobi equations in\u00a0RN"],"prefix":"10.1177","volume":"112","author":[{"given":"Guy","family":"Barles","sequence":"first","affiliation":[{"name":"LMPT, F\u00e9d\u00e9ration Denis Poisson, Universit\u00e9 Fran\u00e7ois-Rabelais Tours, France. E-mail:\u00a0"}]},{"given":"Olivier","family":"Ley","sequence":"additional","affiliation":[{"name":"IRMAR, INSA Rennes, France"}]},{"given":"Thi-Tuyen","family":"Nguyen","sequence":"additional","affiliation":[{"name":"Dipartimento di Matematica, Universit\u00e0 di Padova, Italy. E-mail:\u00a0"}]},{"given":"Thanh Viet","family":"Phan","sequence":"additional","affiliation":[{"name":"Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam. E-mail:\u00a0"}]}],"member":"179","published-online":{"date-parts":[[2019,3,6]]},"reference":[{"key":"ref001","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-36433-4"},{"key":"ref002","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-8176-4755-1"},{"key":"ref003","unstructured":"G.\u00a0Barles, Solutions de Viscosit\u00e9 des \u00c9quations de Hamilton\u2013Jacobi, Springer-Verlag, Paris, 1994."},{"key":"ref004","doi-asserted-by":"publisher","DOI":"10.1007\/s13373-013-0036-0"},{"key":"ref005","doi-asserted-by":"publisher","DOI":"10.1080\/03605302.2016.1244208"},{"key":"ref006","doi-asserted-by":"publisher","DOI":"10.1080\/03605300500361461"},{"key":"ref007","doi-asserted-by":"publisher","DOI":"10.1137\/S0036141099350869"},{"key":"ref008","doi-asserted-by":"publisher","DOI":"10.1080\/03605309908820745"},{"key":"ref009","doi-asserted-by":"publisher","DOI":"10.1090\/S0273-0979-1992-00266-5"},{"key":"ref010","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1983-0690039-8"},{"key":"ref011","doi-asserted-by":"publisher","DOI":"10.1081\/PDE-120002788"},{"key":"ref012","doi-asserted-by":"publisher","DOI":"10.1137\/050621955"},{"key":"ref013","doi-asserted-by":"publisher","DOI":"10.1016\/S0764-4442(98)80144-4"},{"key":"ref014","doi-asserted-by":"publisher","DOI":"10.1007\/s00030-007-2047-6"},{"key":"ref015","doi-asserted-by":"publisher","DOI":"10.1007\/s00222-003-0323-6"},{"key":"ref016","doi-asserted-by":"publisher","DOI":"10.1007\/s00526-004-0271-z"},{"key":"ref017","doi-asserted-by":"publisher","DOI":"10.4310\/MAA.2008.v15.n2.a8"},{"key":"ref018","doi-asserted-by":"publisher","DOI":"10.1007\/s00205-008-0170-0"},{"key":"ref019","doi-asserted-by":"publisher","DOI":"10.1512\/iumj.1984.33.33038"},{"key":"ref020","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1987-0884461-3"},{"key":"ref021","doi-asserted-by":"publisher","DOI":"10.1016\/j.anihpc.2006.09.002"},{"key":"ref022","unstructured":"H.\u00a0Ishii, Asymptotic solutions of Hamilton\u2013Jacobi equations for large time and related topics, in: ICIAM 07\u20146th International Congress on Industrial and Applied Mathematics, Eur. 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Soc., Z\u00fcrich, 2009, pp.\u00a0193\u2013217."},{"key":"ref023","doi-asserted-by":"publisher","DOI":"10.57262\/ade\/1357141855"},{"key":"ref024","unstructured":"P.L.\u00a0Lions, B.\u00a0Papanicolaou and S.R.S.\u00a0Varadhan, Homogenization of Hamilton\u2013Jacobi equations,\n                      Unpublished\n                      (1986)."},{"key":"ref025","doi-asserted-by":"publisher","DOI":"10.1080\/03605309908821451"}],"container-title":["Asymptotic Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/ASY-181488","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/full-xml\/10.3233\/ASY-181488","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/ASY-181488","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,28]],"date-time":"2026-04-28T12:36:22Z","timestamp":1777379782000},"score":1,"resource":{"primary":{"URL":"https:\/\/journals.sagepub.com\/doi\/10.3233\/ASY-181488"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,3,6]]},"references-count":25,"journal-issue":{"issue":"1-2","published-print":{"date-parts":[[2019,3,6]]}},"alternative-id":["10.3233\/ASY-181488"],"URL":"https:\/\/doi.org\/10.3233\/asy-181488","relation":{},"ISSN":["0921-7134","1875-8576"],"issn-type":[{"value":"0921-7134","type":"print"},{"value":"1875-8576","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,3,6]]}}}