Decay of small energy solutions in the ABCD Boussinesq model under the influence of an uneven bottom

Abstract

The $abcd$ Boussinesq system, introduced by Bona, Chen, and Saut, describes a four-parameter $(a,b,c,d)$ family of models formulated on the time-space domain $\mathbb{R}_t \times \mathbb{R}_x$. It serves as a first-order two-wave approximation to the two-dimensional incompressible, irrotational water wave equations in shallow water, inspired by Boussinesq's classical derivation. Within the different parameter regimes, the generic regime is described by $b,d>0$ and $a,c<0$ while the system becomes Hamiltonian when $b=d$. Previously, sharp local in space $H^1\times H^1$ decay properties were proved in the case of a large class of $abcd$ model under the small data assumption. In this paper, we generalize [C. Kwak, \emph{et. al.}, \emph{The scattering problem for Hamiltonian ABCD Boussinesq systems in the energy space}. J. Math. Pures Appl. (9) 127 (2019), 121--159] by considering the small data $abcd$ decay problem in the physically relevant \emph{variable bottom regime} described by M. Chen. The nontrivial bathymetry is represented by a smooth space-time dependent function $h=h(t,x)$, which obeys integrability in time and smallness in space. We prove first the existence of small global solutions in $H^1\times H^1$. Then, for a sharp set of dispersive $abcd$ systems (characterized only in terms of parameters $a, b$ and $c$), every $H^1\times H^1$ small solution must converges to zero inside of the light cone $|x|\leq |t|$.