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https://hdl.handle.net/1822/21428
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Campo DC | Valor | Idioma |
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dc.contributor.author | Panthee, Mahendra | - |
dc.contributor.author | Scialom, Marcia | - |
dc.date.accessioned | 2012-12-11T17:21:21Z | - |
dc.date.available | 2012-12-11T17:21:21Z | - |
dc.date.issued | 2013 | - |
dc.date.submitted | 2012 | - |
dc.identifier.issn | 1021-9722 | por |
dc.identifier.issn | 1420-9004 | por |
dc.identifier.uri | https://hdl.handle.net/1822/21428 | - |
dc.description.abstract | For the initial value problem (IVP) associated to the generalized Korteweg-de Vries (gKdV) equation with supercritical nonlinearity, \begin{equation*} u_{t}+\partial_x^3u+\partial_x(u^{k+1}) =0,\qquad k\geq 5, \end{equation*} numerical evidence [Bona J.L., Dougalis V.A., Karakashian O.A., McKinney W.R.: Conservative, high-order numerical schemes for the generalized Korteweg–de Vries equation. Philos. Trans. Roy. Soc. London Ser. A 351, 107–164 (1995) ] shows that, there are initial data $\phi\in H^1(\mathbb{R})$ such that the corresponding solution may blow-up in finite time. Also, with the evidence from numerical simulation [Abdullaev F.K., Caputo J.G., Kraenkel R.A., Malomed B.A.: Controlling collapse in Bose–Einstein condensates by temporal modulation of the scattering length. Phys. Rev. A 67, 012605 (2003) and Konotop V.V., Pacciani P.: Collapse of solutions of the nonlinear Schrödinger equation with a time dependent nonlinearity: application to the Bose–Einstein condensates. Phys. Rev. Lett. 94, 240405 (2005) ], it has been claimed that a periodic time dependent coefficient in the nonlinearity would disturb the blow-up solution, either accelerating or delaying it. In this work, we investigate the IVP associated to the gKdV equation \begin{equation*} u_{t}+\partial_x^3u+g(\omega t)\partial_x(u^{k+1}) =0, \end{equation*} where $g$ is a periodic function and $k\geq 5$ is an integer. We prove that, for given initial data $\phi \in H^1(\mathbb{R})$, as $|\omega|\to \infty$, the solution $u_{\omega}$ converges to the solution $U$ of the initial value problem associated to \begin{equation*} U_{t}+\partial_x^3U+m(g)\partial_x(U^{k+1}) =0, \end{equation*} with the same initial data, where $m(g)$ is the average of the periodic function $g$. Moreover, if the solution $U$ is global and satisfies $\|U\|_{L_x^5L_t^{10}}<\infty$, then we prove that the solution $u_{\omega}$ is also global provided $|\omega|$ is sufficiently large. | por |
dc.description.sponsorship | M. P. was partially supported by the Research Center of Mathematics of the University of Minho, Portugal through the FCT Pluriannual Funding Program, and through the project PTDC/MAT/109844/2009, and M. S. was partially supported by FAPESP Brazil. | por |
dc.language.iso | eng | por |
dc.publisher | Springer | por |
dc.relation | info:eu-repo/grantAgreement/FCT/5876-PPCDTI/109844/PT | - |
dc.rights | openAccess | por |
dc.subject | Korteweg-de vries equation | por |
dc.subject | Cauchy problem | por |
dc.subject | Local and global well-posedness | por |
dc.title | On the supercritical KDV equation with time-oscillating nonlinearity | por |
dc.type | article | por |
dc.peerreviewed | yes | por |
dc.relation.publisherversion | http://link.springer.com/ | por |
sdum.publicationstatus | published | por |
oaire.citationStartPage | 1191 | por |
oaire.citationEndPage | 1212 | por |
oaire.citationIssue | 3 | por |
oaire.citationTitle | NoDEA : Nonlinear Differential Equations and Applications | por |
oaire.citationVolume | 20 | por |
dc.identifier.doi | 10.1007/s00030-012-0204-z | por |
dc.subject.wos | Science & Technology | por |
sdum.journal | Nodea : Nonlinear Differential Equations and Applications | por |
Aparece nas coleções: | CMAT - Artigos em revistas com arbitragem / Papers in peer review journals |
Ficheiros deste registo:
Ficheiro | Descrição | Tamanho | Formato | |
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g-KdV 27-09-2012.pdf | 377,06 kB | Adobe PDF | Ver/Abrir |