On global attractors for 2D damped driven nonlinear Schrödinger equations

Abstract

The well-posedness and global attraction are proved for nonautonomous 2D nonlinear damped Schrödinger equations with almost periodic external source in a bounded region. The proofs rely on a development of the method of energy equation. The research is inspired by the problems of laser and maser coherent radiation.

On absorbing set for 3D Maxwell--Schrödinger damped driven equations in bounded region

“…• Dissipative autonomous evolutionary PDEs. The theory of attractors and long-time behaviour of solutions of such equations originated in the works of Ball, Foias, Hale, Henry, Temam, and was developed further by Babin and Vishik, Chepyzhov, Haraux, Ilyin, Miranville, Pata, Zelik, and others for the Navier-Stokes, reactiondiffusion, Ginzburg-Landau, damped wave and nonlinear Schrödinger, and sine-Gordon equations [22]- [40].…”

“…A compact global attractor in the energy space for 2D weakly damped driven nonlinear Schrödinger equation with general nonlinearity was constructed in joint paper of the PI with E. Kopylova [40] for general bounded region V ⊂ R 2 and almost periodic driving.…”

“…Then (3.7) together with div A(y,t) ≡ 0 for y ∈ Ω implies that ∂ 3 A 3 (y,t) = 0 (A. 40) if A ∈ C 1 (Ω) ⊗ R 3 . Hence, the boundary conditions (3.6), (A.40) can be written as…”