Ill-Posedness Issues on (abcd)-Boussinesq System

In this paper, we consider the Cauchy problem for (abcd)-Boussinesq system posed on one- and two-dimensional Euclidean spaces. This model, initially introduced by Bona et al. (J Nonlinear Sci 12:283–318, 2002, Nonlinearity 17:925–952, 2004), describes a small-amplitude waves surface of an inviscid fluid, is derived as first order approximation incompressible, irrotational Euler equations. We mainly establish ill-posedness under various parameter regimes, which generalize result one-dimensional BBM–BBM case Chen Liu (Anal Math 121:299–316, 2013). Among results established here, emphasize that optimal. The proof follows from observation high to low frequency cascade present in nonlinearity, motivated Bejenaru Tao Funct Anal 233:228–259, 2006).