On the linearized Whitham–Broer–Kaup system on bounded domains

We consider the system of partial differential equations on bounded domains, known in the literature as the Whitham-Broer-Kaup system. The well-posedness of the problem, under suitable boundary conditions, is addressed, and it is shown to depend on the sign of the number In particular, existence and uniqueness occur if and only if 0$]]>. In which case, an explicit representation for the solutions is given. Nonetheless, for the case we have uniqueness in the class of strong solutions, and sufficient conditions to guarantee exponential instability are provided.