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

We consider the system of partial differential equations \[ \begin{cases} \eta_t - \alpha u_{xxx} - \beta \eta_{xx} = 0 \\ u_t + \eta_x + \beta u_{xx} = 0 \end{cases} \] 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 \[ \varkappa=\alpha-\beta^2. \] In particular, existence and uniqueness occur if and only if $\varkappa >0$ . In which case, an explicit representation for the solutions is given. Nonetheless, for the case $\varkappa \leq 0$ we have uniqueness in the class of strong solutions, and sufficient conditions to guarantee exponential instability are provided.