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• W1990314360 abstract "Restricted accessMoreSectionsView PDF ToolsAdd to favoritesDownload CitationsTrack Citations ShareShare onFacebookTwitterLinked InRedditEmail Cite this article Hayashi Nakao, Kaikina Elena I and Zavala J. L. Guardado 2003On the boundary‐value problem for the Korteweg–de Vries equationProc. R. Soc. Lond. A.4592861–2884http://doi.org/10.1098/rspa.2003.1147SectionRestricted accessOn the boundary‐value problem for the Korteweg–de Vries equation Nakao Hayashi Nakao Hayashi Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560‐0043, Japan () Google Scholar Find this author on PubMed Search for more papers by this author , Elena I Kaikina Elena I Kaikina Departamento de Ciencias Básicas, Instituto Tecnológico de Morelia, CP 58120, Morelia, Michoacán, México () Google Scholar Find this author on PubMed Search for more papers by this author and J. L. Guardado Zavala J. L. Guardado Zavala Programa de Graduados e Investigación en Ingeniería Eléctrica, Instituto Tecnológico de Morelia, CP 58120, Morelia, Michoacán, México () Google Scholar Find this author on PubMed Search for more papers by this author Nakao Hayashi Nakao Hayashi Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560‐0043, Japan () Google Scholar Find this author on PubMed Search for more papers by this author , Elena I Kaikina Elena I Kaikina Departamento de Ciencias Básicas, Instituto Tecnológico de Morelia, CP 58120, Morelia, Michoacán, México () Google Scholar Find this author on PubMed Search for more papers by this author and J. L. Guardado Zavala J. L. Guardado Zavala Programa de Graduados e Investigación en Ingeniería Eléctrica, Instituto Tecnológico de Morelia, CP 58120, Morelia, Michoacán, México () Google Scholar Find this author on PubMed Search for more papers by this author Published:08 November 2003https://doi.org/10.1098/rspa.2003.1147AbstractWe consider the initial-boundary-value problem on the half-line for the Korteweg-de Vries equation ut+uuxu+uxxx=0,t>0,x>0,u(x,0)=u0(x),x>0,u(0,t)=0,t>0.}We prove that if the initial data u0∈H10,2∩H20,3/2and the norm||u0||H10,2+||u0||H20,3/2 are sufficiently small, where Hps,k={f∈L2;||f||Hps,k=||〈x〉k〈i∂x〉sf||Lp<∞},〈x〉1 + x2, then there exists a unique solutionu∈C([0,∞),H20,2)∩L∞(0,∞,H20,3/2)∩C((0,∞),H22,0)∩L∞(0,∞,H23,0) of the initial-boundary-value problem. Moreover, we proved that there exists a constant Bsuch that the solution has the following asymptotics, u(x,t)=t-1Bxt3Ai(xt3)+o(t-1-δ/3max(1,xt3))for t→∞ uniformly with respect to x > 0, where 0 <δ, Ai(q) is the Airy function defined by Ai(q)=∫-i∞i∞e-z3+zqdz. Previous ArticleNext Article VIEW FULL TEXT DOWNLOAD PDF FiguresRelatedReferencesDetailsCited by Cavalcanti M, Domingos Cavalcanti V, Komornik V and Rodrigues J Global well-posedness and exponential decay rates for a KdV–Burgers equation with indefinite damping, Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 10.1016/j.anihpc.2013.08.003, 31:5, (1079-1100) Cavalcanti M, Domingos Cavalcanti V, Faminskii A and Natali F (2011) Decay of Solutions to Damped Korteweg–de Vries Type Equation, Applied Mathematics & Optimization, 10.1007/s00245-011-9156-7, 65:2, (221-251), Online publication date: 1-Apr-2012. Rosier L and Zhang B (2009) Control and stabilization of the Korteweg-de Vries equation: recent progresses, Journal of Systems Science and Complexity, 10.1007/s11424-009-9194-2, 22:4, (647-682), Online publication date: 1-Dec-2009. Cardiel-Cervantes R and Naumkin P Neumann problem for a nonlinear nonlocal equation on a half-line, SUT Journal of Mathematics, 10.55937/sut/1248706004, 45:1 Leach J and Needham D (2008) The large-time development of the solution to an initial-value problem for the Korteweg–de Vries equation: I. Initial data has a discontinuous expansive step, Nonlinearity, 10.1088/0951-7715/21/10/010, 21:10, (2391-2408), Online publication date: 1-Oct-2008. Guardado L, Beníte F, Kaikina E, Ruiz F and Hernández M (2007) An asymptotic solution to non-linear transmission lines, Nonlinear Analysis: Real World Applications, 10.1016/j.nonrwa.2006.05.005, 8:3, (715-724), Online publication date: 1-Jul-2007. Shen J, Wu J and Yuan J (2007) Eventual periodicity for the KdV equation on a half-line, Physica D: Nonlinear Phenomena, 10.1016/j.physd.2007.02.003, 227:2, (105-119), Online publication date: 1-Mar-2007. Hayashi N and Kaikina E (2006) Neumann problem for the Korteweg–de Vries equation, Journal of Differential Equations, 10.1016/j.jde.2006.01.017, 225:1, (168-201), Online publication date: 1-Jun-2006. CARDIEL R, KAIKINA E and NAUMKIN P (2011) ASYMPTOTICS FOR NONLINEAR NONLOCAL EQUATIONS ON A HALF-LINE, Communications in Contemporary Mathematics, 10.1142/S021919970600209X, 08:02, (189-217), Online publication date: 1-Apr-2006. Kaikina E (2004) Nonlinear nonlocal Whitham equation on a segment, Nonlinear Analysis: Theory, Methods & Applications, 10.1016/j.na.2004.07.002, 59:1-2, (55-83), Online publication date: 1-Oct-2004. This Issue08 November 2003Volume 459Issue 2039 Article InformationDOI:https://doi.org/10.1098/rspa.2003.1147Published by:Royal SocietyPrint ISSN:1364-5021Online ISSN:1471-2946History: Published online08/11/2003Published in print08/11/2003 License: Citations and impact KeywordsKorteweg‐de Vries equationnonlinear evolution equationlarge‐time asymptoticshalf‐line" @default.
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