Exponential stability of the linear KdV-BBM equation.

In this paper, we consider the following linear Korteweg–de Vries–Benjamin Bona Mahony (KdV-BBM) equation on a finite interval.$ \left\{ \begin{array}{ll} u_t - a^2u_{xxt} + u_x + u_{xxx} = 0, & (x, t)\in(0, L)\times (0, \infty), \\ u(0, t) = u(L, t) = u_x(L, t) = 0, &t\in (0, \infty), \\ u(x, 0) = u_0(x), & x \in (0, L). \end{array} \right. $We show the well-posedness by the semigroup theory. A set of critical length $ \mathcal{L} $ is obtained, for which the system possesses conservative solutions. Then we prove the exponentially stability of the associated semigroup when $ L\not\in \mathcal{L} $ by the frequency domain method. [ABSTRACT FROM AUTHOR]