{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T07:53:31Z","timestamp":1777449211688,"version":"3.51.4"},"reference-count":0,"publisher":"SAGE Publications","issue":"3-4","license":[{"start":{"date-parts":[[2015,3,1]],"date-time":"2015-03-01T00:00:00Z","timestamp":1425168000000},"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":[[2015,3]]},"abstract":"<jats:p>We investigate the boundedness and large time behavior of solutions of the Cauchy\u2013Dirichlet problem for the one-dimensional degenerate parabolic equation with gradient nonlinearity:<\/jats:p>\n                  <jats:p>\n                    u\n                    <jats:sub>t<\/jats:sub>\n                    =(|u\n                    <jats:sub>x<\/jats:sub>\n                    |\n                    <jats:sup>p\u22122<\/jats:sup>\n                    u\n                    <jats:sub>x<\/jats:sub>\n                    )\n                    <jats:sub>x<\/jats:sub>\n                    +|u\n                    <jats:sub>x<\/jats:sub>\n                    |\n                    <jats:sup>q<\/jats:sup>\n                    \u2003in (0,\u221e)\u00d7(0,1),q&gt;p&gt;2.\n                  <\/jats:p>\n                  <jats:p>\n                    We prove that: either u\n                    <jats:sub>x<\/jats:sub>\n                    blows up in finite time, or u is global and converges in W\n                    <jats:sup>1,\u221e<\/jats:sup>\n                    (0,1) to the unique steady state. This in particular eliminates the possibility of global solutions with unbounded gradient. For that purpose a Lyapunov functional is constructed by the approach of Zelenyak.\n                  <\/jats:p>","DOI":"10.3233\/asy-141263","type":"journal-article","created":{"date-parts":[[2019,11,29]],"date-time":"2019-11-29T19:38:20Z","timestamp":1575056300000},"page":"233-251","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":4,"title":["Boundedness of global solutions of a p-Laplacian evolution equation with a nonlinear gradient term"],"prefix":"10.1177","volume":"91","author":[{"given":"Amal","family":"Attouchi","sequence":"first","affiliation":[{"name":"Universit\u00e9 Paris 13, Sorbonne Paris Cit\u00e9, Laboratoire Analyse, G\u00e9om\u00e9trie et Applications, CNRS, UMR 7539, 93430 Villetaneuse, France. E-mail: attouchi@math.univ-paris13.fr"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"179","published-online":{"date-parts":[[2015,3,1]]},"container-title":["Asymptotic Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/ASY-141263","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/ASY-141263","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,28]],"date-time":"2026-04-28T12:40:22Z","timestamp":1777380022000},"score":1,"resource":{"primary":{"URL":"https:\/\/journals.sagepub.com\/doi\/10.3233\/ASY-141263"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,3]]},"references-count":0,"journal-issue":{"issue":"3-4","published-print":{"date-parts":[[2015,3]]}},"alternative-id":["10.3233\/ASY-141263"],"URL":"https:\/\/doi.org\/10.3233\/asy-141263","relation":{},"ISSN":["0921-7134","1875-8576"],"issn-type":[{"value":"0921-7134","type":"print"},{"value":"1875-8576","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,3]]}}}