## Asymptotic dynamics for the small data weakly dispersive one-dimensional Hamiltonian ABCD system

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- by Chulkwang Kwak and Claudio Muñoz PDF
- Trans. Amer. Math. Soc.
**373**(2020), 1043-1107 Request permission

## Abstract:

Consider the Hamiltonian $abcd$ system in one dimension, with data posed in the energy space $H^1\times H^1$. This model, introduced by Bona, Chen, and Saut, is a well-known physical generalization of the classical Boussinesq equations. The Hamiltonian case corresponds to the regime where $a,c<0$ and $b=d>0$. Under this regime, small solutions in the energy space are globally defined. A first proof of decay for this $2\times 2$ system was given in [J. Math. Pure Appl. (9)**127**(2019), 121–159] in a strongly dispersive regime, i.e., under essentially the conditions \[ b=d > \frac 29, \quad a,c<-\frac 1{18}. \] Additionally, decay was obtained inside a proper subset of the light cone $(-|t|,|t|)$. In this paper, we improve [J. Math. Pure Appl. (9)

**127**(2019), 121–159] in three directions. First, we enlarge the set of parameters $(a,b,c,d)$ for which decay to zero is the only available option, considering now the so-called weakly dispersive regime $a,c\sim 0$: we prove decay if now \[ b=d > \frac 3{16}, \quad a,c<-\frac 1{48}. \] This result is sharp in the case where $a=c$, since for $a,c$ bigger, some $abcd$ linear waves of

*nonzero frequency*do have

*zero group velocity*. Second, we sharply enlarge the interval of decay to consider the whole light cone, that is to say, any interval of the form $|x|\sim |v|t$ for any $|v|<1$. This result rules out, among other things, the existence of nonzero speed solitary waves in the regime where decay is present. Finally, we prove decay to zero of small $abcd$ solutions in exterior regions $|x|\gg |t|$, also discarding super-luminical small solitary waves. These three results are obtained by performing new improved virial estimates for which better decay properties are deduced.

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## Additional Information

**Chulkwang Kwak**- Affiliation: Facultad de Matemáticas, Pontificia Universidad Católica de Chile, 4860 Macul, Chile; and Institute of Pure and Applied Mathematics, Chonbuk National University, Jeonju-si, Jeollabuk-do, 54896 Republic of Korea
- MR Author ID: 1035460
- Email: chkwak@mat.uc.cl
**Claudio Muñoz**- Affiliation: CNRS and Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático (UMI 2807 CNRS), Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile
- MR Author ID: 806855
- Email: cmunoz@dim.uchile.cl
- Received by editor(s): May 14, 2019
- Published electronically: October 18, 2019
- Additional Notes: The first author was supported by FONDECYT Postdoctorado 2017 Proyecto No. 3170067 and project France-Chile ECOS-Sud C18E06.

The second author’s work was partly funded by Chilean research grants FONDECYT 1150202, Fondecyt no. 1191412, project France-Chile ECOS-Sud C18E06 and CMM Conicyt PIA AFB170001. - © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**373**(2020), 1043-1107 - MSC (2010): Primary 35Q35, 35Q51
- DOI: https://doi.org/10.1090/tran/7944
- MathSciNet review: 4068258