Asymptotic dynamics for the small data weakly dispersive one-dimensional hamiltonian abcd system

Abstract

Consider the Hamiltonian abed system in one dimension, with data posed in the energy space H-1 x 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 x 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 > 2/9, a,c < -1/18.

Additionally, decay was obtained inside a proper subset of the light cone (-vertical bar t vertical bar, vertical bar t vertical bar). 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 socalled weakly dispersive regime a, c similar to 0: we prove decay if now

b = d > 1/16, a,c < -1/48.

This result is sharp in the case where a = c, since for a, c bigger, some abed 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 vertical bar x vertical bar similar to vertical bar v vertical bar t for any vertical bar v vertical bar < 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 abed solutions in exterior regions vertical bar x vertical bar >> vertical bar t vertical bar 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.