{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,10]],"date-time":"2026-05-10T00:35:48Z","timestamp":1778373348878,"version":"3.51.4"},"reference-count":16,"publisher":"SAGE Publications","issue":"1-2","license":[{"start":{"date-parts":[[2018,10,9]],"date-time":"2018-10-09T00:00:00Z","timestamp":1539043200000},"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":[[2018,10,9]]},"abstract":"<jats:p>In this paper we study the singular vanishing-viscosity limit of a gradient flow in a finite dimensional Euclidean space, focusing on the so-called delayed loss of stability of stationary solutions. We find a class of time-dependent energy functionals and initial conditions for which we can explicitly calculate the first discontinuity time [Formula: see text] of the limit. For our class of functionals, [Formula: see text] coincides with the blow-up time of the solutions of the linearized system around the equilibrium, and is in particular strictly greater than the time [Formula: see text] where strict local minimality with respect to the driving energy gets lost. Moreover, we show that, in a right neighborhood of [Formula: see text], rescaled solutions of the singularly perturbed problem converge to heteroclinic solutions of the gradient flow. Our results complement the previous ones by Zanini [ Discrete Contin. Dyn. Syst. Ser. A 18 ( 2007 ), 657\u2013675], where the situation we consider was excluded by assuming the so-called transversality conditions, and the limit evolution consisted of strict local minimizers of the energy up to a negligible set of times.<\/jats:p>","DOI":"10.3233\/asy-181475","type":"journal-article","created":{"date-parts":[[2018,10,9]],"date-time":"2018-10-09T15:13:00Z","timestamp":1539097980000},"page":"1-19","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":3,"title":["Delayed loss of stability in singularly perturbed finite-dimensional gradient flows"],"prefix":"10.1177","volume":"110","author":[{"given":"Giovanni","family":"Scilla","sequence":"first","affiliation":[{"name":"Department of Mathematics and Applications \u201cR. Caccioppoli\u201d, University of Naples Federico II, Via\u00a0Cintia, Monte S. Angelo \u2013 80126 Naples, Italy. E-mails:\u00a0,\u00a0"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Francesco","family":"Solombrino","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Applications \u201cR. Caccioppoli\u201d, University of Naples Federico II, Via\u00a0Cintia, Monte S. Angelo \u2013 80126 Naples, Italy. E-mails:\u00a0,\u00a0"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"179","published-online":{"date-parts":[[2018,10,9]]},"reference":[{"key":"ref001","doi-asserted-by":"publisher","DOI":"10.3934\/dcds.2012.32.1125"},{"key":"ref002","doi-asserted-by":"publisher","DOI":"10.1016\/j.jde.2017.08.027"},{"key":"ref003","doi-asserted-by":"publisher","DOI":"10.1007\/s12215-014-0184-4"},{"key":"ref004","unstructured":"L.\u00a0Ambrosio, N.\u00a0Gigli and G.\u00a0Savar\u00e9, Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd edn, Lectures in Mathematics ETH Zurich, Birkh\u00e4user Verlag, Basel, 2008."},{"key":"ref005","doi-asserted-by":"publisher","DOI":"10.1023\/B:JOTH.0000021571.21423.52"},{"key":"ref006","doi-asserted-by":"crossref","unstructured":"J.\u00a0Hale and H.\u00a0Ko\u00e7ak, Dynamics and Bifurcation, Texts in Applied Mathematics, Vol.\u00a03, Springer Verlag, New York, 1991.","DOI":"10.1007\/978-1-4612-4426-4_1"},{"key":"ref007","doi-asserted-by":"publisher","DOI":"10.1371\/journal.pone.0093183"},{"key":"ref008","doi-asserted-by":"publisher","DOI":"10.3934\/dcds.2009.25.585"},{"key":"ref009","doi-asserted-by":"publisher","DOI":"10.4171\/jems\/639"},{"key":"ref010","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4939-2706-7_2"},{"key":"ref011","doi-asserted-by":"publisher","DOI":"10.1007\/s00205-011-0460-9"},{"key":"ref012","doi-asserted-by":"publisher","DOI":"10.1051\/cocv\/2014004"},{"issue":"12","key":"ref013","first-page":"2060","volume":"23","author":"Ne\u01d0shtadt A.I.","year":"1987","journal-title":"Differentsial\u2019nye Uravneniya"},{"issue":"2","key":"ref014","first-page":"226","volume":"24","author":"Ne\u01d0shtadt A.I.","year":"1988","journal-title":"Differentsial\u2019nye Uravneniya"},{"key":"ref015","doi-asserted-by":"publisher","DOI":"10.1090\/gsm\/140"},{"key":"ref016","doi-asserted-by":"publisher","DOI":"10.3934\/dcds.2007.18.657"}],"container-title":["Asymptotic Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/ASY-181475","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/full-xml\/10.3233\/ASY-181475","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/ASY-181475","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,28]],"date-time":"2026-04-28T12:36:18Z","timestamp":1777379778000},"score":1,"resource":{"primary":{"URL":"https:\/\/journals.sagepub.com\/doi\/10.3233\/ASY-181475"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,10,9]]},"references-count":16,"journal-issue":{"issue":"1-2","published-print":{"date-parts":[[2018,10,9]]}},"alternative-id":["10.3233\/ASY-181475"],"URL":"https:\/\/doi.org\/10.3233\/asy-181475","relation":{},"ISSN":["0921-7134","1875-8576"],"issn-type":[{"value":"0921-7134","type":"print"},{"value":"1875-8576","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,10,9]]}}}