Control and stabilization problem for a class of fourth-order nonlinear Schr\"odinger equation on boundaryless compact manifold

Abstract

In this paper, we study the control and stabilization problem for a class of fourth-order Schr\"odinger equation on compact manifold without boundary with dimensions $d\in[1,5]$: \begin{align*} i\partial_tu+(\Delta_g^2-\beta\Delta_g)u=|u|^{2k}u, \end{align*} where $k\in\Bbb N$. For $1\leq d\leq4$ and $k\geq1$, we combine the method proposed by Loyola and semiclassical analysis to prove the stabilization result only under the geometric control condition (GCC), which removes the unique continuation assumption in Capistrano-Filho-Pampu [Math. Z. (2022)]. For $d=5$, we focus on a special case, i.e. $\Bbb S^5$. Establishing the propagation of singularity in Bourgain space, we prove the similar control and stabilization result in energy space as lower dimensions, which generalizes the result of Laurent [SIAM J. Math. Anal. (2009)].