On the dynamics of zero-speed solutions for camassa-holm-type equations
Author
dc.contributor.author
Alejo, Miguel A.
Author
dc.contributor.author
Cortez, Manuel Fernando
Author
dc.contributor.author
Kwak, Chulkwang
Author
dc.contributor.author
Muñoz, Claudio
Admission date
dc.date.accessioned
2021-11-26T18:46:50Z
Available date
dc.date.available
2021-11-26T18:46:50Z
Publication date
dc.date.issued
2021
Cita de ítem
dc.identifier.citation
International Mathematics Research Notices, Volume 2021, Issue 9, May 2021, Pages 6543–6585
es_ES
Identifier
dc.identifier.other
10.1093/imrn/rnz038
Identifier
dc.identifier.uri
https://repositorio.uchile.cl/handle/2250/182902
Abstract
dc.description.abstract
In this paper, we consider globally defined solutions of Camassa–Holm (CH)-type equations outside the well-known nonzero-speed, peakon region. These equations include the standard CH and Degasperis–Procesi (DP) equations, as well as nonintegrable generalizations such as the b-family, elastic rod, and Benjamin-Bona-Mahony (BBM) equations. Having globally defined solutions for these models, we introduce the notion of zero-speed and breather solutions, i.e., solutions that do not decay to zero as t→+∞ on compact intervals of space. We prove that, under suitable decay assumptions, such solutions do not exist because the identically zero solution is the global attractor of the dynamics, at least in a spatial interval of size |x|≲t1/2− as t→+∞. As a consequence, we also show scattering and decay in CH-type equations with long-range nonlinearities. Our proof relies in the introduction of suitable virial functionals à la Martel–Merle in the spirit of the works of [74, 75] and [50] adapted to CH-, DP-, and BBM-type dynamics, one of them placed in L1x and the 2nd one in the energy space H1x. Both functionals combined lead to local-in-space decay to zero in |x|≲t1/2− as t→+∞. Our methods do not rely on the integrable character of the equation, applying to other nonintegrable families of CH-type equations as well.
es_ES
Patrocinador
dc.description.sponsorship
Conselho Nacional de Desenvolvimento Cientifico e Tecnologico (CNPQ) 305205/2016-1
Comision Nacional de Investigacion Cientifica y Tecnologica (CONICYT)
CONICYT FONDECYT 3170067
1150202
Centro de Modelamiento Matematico (CMM) Comision Nacional de Investigacion Cientifica y Tecnologica (Conicyt) PIA AFB170001
es_ES
Lenguage
dc.language.iso
en
es_ES
Publisher
dc.publisher
Oxford
es_ES
Type of license
dc.rights
Attribution-NonCommercial-NoDerivs 3.0 United States