On the critical KdV equation with time-oscillating nonlinearity
Autor(a) principal: | |
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Data de Publicação: | 2011 |
Outros Autores: | , |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
Texto Completo: | http://hdl.handle.net/1822/16905 |
Resumo: | We investigate the initial value problem (IVP) associated to the equation \begin{equation*} u_{t}+\partial_x^3u+g(\omega t)\partial_x(u^5) =0, \end{equation*} where $g$ is a periodic function. 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^5) =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. |
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Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
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7160 |
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On the critical KdV equation with time-oscillating nonlinearityKorteweg-de vries equationCauchy problemLocal & global well-posednessScience & TechnologyWe investigate the initial value problem (IVP) associated to the equation \begin{equation*} u_{t}+\partial_x^3u+g(\omega t)\partial_x(u^5) =0, \end{equation*} where $g$ is a periodic function. 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^5) =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.Fundação para a Ciência e a Tecnologia (FCT)Khayyam PublishingUniversidade do MinhoCarvajal, XavierPanthee, Mahendra PrasadScialom, Marcia20112011-01-01T00:00:00Zinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articleapplication/pdfhttp://hdl.handle.net/1822/16905eng0893-4983http://www.aftabi.com/die-24a.htmlinfo:eu-repo/semantics/openAccessreponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãoinstacron:RCAAP2023-07-21T12:04:28Zoai:repositorium.sdum.uminho.pt:1822/16905Portal AgregadorONGhttps://www.rcaap.pt/oai/openaireopendoar:71602024-03-19T18:54:46.904569Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãofalse |
dc.title.none.fl_str_mv |
On the critical KdV equation with time-oscillating nonlinearity |
title |
On the critical KdV equation with time-oscillating nonlinearity |
spellingShingle |
On the critical KdV equation with time-oscillating nonlinearity Carvajal, Xavier Korteweg-de vries equation Cauchy problem Local & global well-posedness Science & Technology |
title_short |
On the critical KdV equation with time-oscillating nonlinearity |
title_full |
On the critical KdV equation with time-oscillating nonlinearity |
title_fullStr |
On the critical KdV equation with time-oscillating nonlinearity |
title_full_unstemmed |
On the critical KdV equation with time-oscillating nonlinearity |
title_sort |
On the critical KdV equation with time-oscillating nonlinearity |
author |
Carvajal, Xavier |
author_facet |
Carvajal, Xavier Panthee, Mahendra Prasad Scialom, Marcia |
author_role |
author |
author2 |
Panthee, Mahendra Prasad Scialom, Marcia |
author2_role |
author author |
dc.contributor.none.fl_str_mv |
Universidade do Minho |
dc.contributor.author.fl_str_mv |
Carvajal, Xavier Panthee, Mahendra Prasad Scialom, Marcia |
dc.subject.por.fl_str_mv |
Korteweg-de vries equation Cauchy problem Local & global well-posedness Science & Technology |
topic |
Korteweg-de vries equation Cauchy problem Local & global well-posedness Science & Technology |
description |
We investigate the initial value problem (IVP) associated to the equation \begin{equation*} u_{t}+\partial_x^3u+g(\omega t)\partial_x(u^5) =0, \end{equation*} where $g$ is a periodic function. 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^5) =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. |
publishDate |
2011 |
dc.date.none.fl_str_mv |
2011 2011-01-01T00:00:00Z |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/article |
format |
article |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
http://hdl.handle.net/1822/16905 |
url |
http://hdl.handle.net/1822/16905 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
0893-4983 http://www.aftabi.com/die-24a.html |
dc.rights.driver.fl_str_mv |
info:eu-repo/semantics/openAccess |
eu_rights_str_mv |
openAccess |
dc.format.none.fl_str_mv |
application/pdf |
dc.publisher.none.fl_str_mv |
Khayyam Publishing |
publisher.none.fl_str_mv |
Khayyam Publishing |
dc.source.none.fl_str_mv |
reponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação instacron:RCAAP |
instname_str |
Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
instacron_str |
RCAAP |
institution |
RCAAP |
reponame_str |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
collection |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
repository.name.fl_str_mv |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
repository.mail.fl_str_mv |
|
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1799132330787340288 |