Stability of a stationary solution for the Lugiato-Lefever equation

We study the stability of a stationary solution for the Lugiato-Lefever equation with the periodic boundary condition in one space dimension, which is a damped and driven nonlinear Schrodinger equation introduced to model the optical cavity. In this paper, we prove the Strichartz estimates for the linear damped Schrodinger equation with potential and external forcing and investigate the stability of certain stationary solutions under the initial perturbation within the framework of $L^2$