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On the linearized Whitham–Broer–Kaup system on bounded domains
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics , First View
- Published online by Cambridge University Press:
- 07 September 2023, pp. 1-20
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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.
