On the linearized Whitham–Broer–Kaup system on bounded domains
- Part of
-
- 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
-
- Article
- Export citation
-
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.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
