{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,18]],"date-time":"2026-03-18T02:50:45Z","timestamp":1773802245301,"version":"3.50.1"},"reference-count":10,"publisher":"World Scientific Pub Co Pte Lt","issue":"08","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2014,8]]},"abstract":"<jats:p> We consider singularly perturbed reaction\u2013diffusion equations with singularly perturbed Neumann boundary conditions. We establish the existence of a time-periodic solution u(x, t, \u03b5) with boundary layers and derive conditions for their asymptotic stability. The boundary layer part of u(x, t, \u03b5) is of order one, which distinguishes our case from the case of regularly perturbed Neumann boundary conditions, where the boundary layer is of order \u03b5. Another peculiarity of our problem is that \u2014 in contrast to the case of Dirichlet boundary conditions \u2014 it may have several asymptotically stable time-periodic solutions, where these solutions differ only in the description of the boundary layers. Our approach is based on the construction of sufficiently precise lower and upper solutions. <\/jats:p>","DOI":"10.1142\/s0218127414400197","type":"journal-article","created":{"date-parts":[[2014,8,29]],"date-time":"2014-08-29T08:35:53Z","timestamp":1409301353000},"page":"1440019","source":"Crossref","is-referenced-by-count":11,"title":["Periodic Solutions with a Boundary Layer of Reaction\u2013Diffusion Equations with Singularly Perturbed Neumann Boundary Conditions"],"prefix":"10.1142","volume":"24","author":[{"given":"Valentin F.","family":"Butuzov","sequence":"first","affiliation":[{"name":"Faculty of Physics, Department of Mathematics, Moscow State University, Leninskie Gori, 119992 Moscow, Russia"}]},{"given":"Nikolay N.","family":"Nefedov","sequence":"additional","affiliation":[{"name":"Faculty of Physics, Department of Mathematics, Moscow State University, Leninskie Gori, 119992 Moscow, Russia"}]},{"given":"Lutz","family":"Recke","sequence":"additional","affiliation":[{"name":"Institute of Mathematics, Humboldt University of Berlin, Unter den Linden 6, D-10099 Berlin, Germany"}]},{"given":"Klaus R.","family":"Schneider","sequence":"additional","affiliation":[{"name":"Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstra\u00dfe 39, D-10117 Berlin, Germany"}]}],"member":"219","published-online":{"date-parts":[[2014,8,29]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.1016\/j.na.2013.01.013"},{"key":"rf2","series-title":"Pitman Research Notes in Math. Series","volume-title":"Periodic-Parabolic Boundary Value Problems and Positivity","volume":"247","author":"Hess P.","year":"1991"},{"key":"rf3","first-page":"1142","volume":"31","author":"Nefedov N. N.","year":"1995","journal-title":"Diff. Uravn."},{"key":"rf4","first-page":"262","volume":"36","author":"Nefedov N. N.","year":"2000","journal-title":"Diff. Uravn."},{"key":"rf5","doi-asserted-by":"publisher","DOI":"10.1134\/S1560354710020231"},{"key":"rf6","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2013.03.051"},{"key":"rf7","first-page":"1051","volume":"37","author":"Nesterov A. V.","year":"1997","journal-title":"Comput. Math. Math. Phys."},{"key":"rf8","volume-title":"Nonlinear Parabolic and Elliptic Equations","author":"Pao C. V.","year":"1992"},{"key":"rf10","doi-asserted-by":"publisher","DOI":"10.1134\/S0081543810010189"},{"key":"rf11","doi-asserted-by":"publisher","DOI":"10.1134\/S0965542506040075"}],"container-title":["International Journal of Bifurcation and Chaos"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218127414400197","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,6]],"date-time":"2019-08-06T20:32:51Z","timestamp":1565123571000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0218127414400197"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,8]]},"references-count":10,"journal-issue":{"issue":"08","published-online":{"date-parts":[[2014,8,29]]},"published-print":{"date-parts":[[2014,8]]}},"alternative-id":["10.1142\/S0218127414400197"],"URL":"https:\/\/doi.org\/10.1142\/s0218127414400197","relation":{},"ISSN":["0218-1274","1793-6551"],"issn-type":[{"value":"0218-1274","type":"print"},{"value":"1793-6551","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014,8]]}}}