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On the Dynamics of Zero-Speed Solutions for Camassa-Holm-Type Equations

Title
On the Dynamics of Zero-Speed Solutions for Camassa-Holm-Type Equations
Authors
Alejo, Miguel A.Cortez, Manuel FernandoKwak, ChulkwangMunoz, Claudio
Ewha Authors
곽철광
SCOPUS Author ID
곽철광scopus
Issue Date
2021
Journal Title
INTERNATIONAL MATHEMATICS RESEARCH NOTICES
ISSN
1073-7928JCR Link

1687-0247JCR Link
Citation
INTERNATIONAL MATHEMATICS RESEARCH NOTICES vol. 2021, no. 9, pp. 6543 - 6585
Publisher
OXFORD UNIV PRESS
Indexed
SCIE; SCOPUS WOS
Document Type
Article
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 ->+infinity 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 vertical bar x vertical bar less than or similar to t(1/2-) as t ->+infinity. 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 a 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 L-x(1) and the 2nd one in the energy space H-x(1). Both functionals combined lead to local-in-space decay to zero in vertical bar x vertical bar less than or similar to t(1/2-) as t -> +infinity. Our methods do not rely on the integrable character of the equation, applying to other nonintegrable families of CH-type equations as well.
DOI
10.1093/imrn/rnz038
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자연과학대학 > 수학전공 > Journal papers
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