The Lugiato-Lefever equation is a nonlinear Schrodinger equation with damping and forcing terms, which models the optical cavity or optical resonator. I will talk about the asymptotic stability of certain stationary solutions. In the case of nonlinear parabolic equations, the smoothing property and the fractional power of infinitesimal generator for the holomorphic semigroup are useful. But in the case of nonlinear Schrodinger equations, the Strichartz estimate plays a crucial role, which enables us to treat the rougher perturbation than the $H^1$ perturbation in the previous papers by Ghidaglia (1988) and X.-M. Wang (1995).
This is a joint work with Tomoyuki Miyaji and Isamu Ohnishi