Long time existence results for bore-type initial data for BBM-Boussinesq systems

In this paper we deal with the long time existence for the Cauchy problem associated to BBM-type Boussinesq systems of equations which are asymptotic models for long wave, small amplitude gravity surface water waves. As opposed to previous papers devoted to the long time existence issue, we consider initial data with nontrivial behaviour at innity which may be used to model bore propagation.