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

We consider the system of partial differential equations {eta(t) - alpha u(xxx) - beta eta(xx) = 0 u(t) + eta(x) + beta u(xx) = 0 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 x = alpha - beta(2). In particular, existence and uniqueness occur if and only if x > 0. In which case, an explicit representation for the solutions is given. Nonetheless, for the case x <= 0 we have uniqueness in the class of strong solutions, and sufficient conditions to guarantee exponential instability are provided.