Abstract
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.
Publisher
Cambridge University Press (CUP)