## Abstract

Consider the Hamiltonian abcd system in one dimension, with data posed in the energy space H^{1} × 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 × 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 2 1 b = d > _{9} , a, c < – _{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 ∼ 0: we prove decay if now 1 b = d > _{16} ^{3} , a, c < – _{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| ∼ |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| |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.

Original language | English |
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Pages (from-to) | 1043-1107 |

Number of pages | 65 |

Journal | Transactions of the American Mathematical Society |

Volume | 373 |

Issue number | 2 |

DOIs | |

State | Published - 2020 |

### Bibliographical note

Funding Information:Received by the editors May 14, 2019. 2010 Mathematics Subject Classification. Primary 35Q35, 35Q51. 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.

Publisher Copyright:

© 2019 American Mathematical Society.