Dynamics of the 3-D fractional complex Ginzburg–Landau equation

We study the initial boundary value problem of the fractional complex Ginzburg–Landau equation in three spatial dimensions with the dissipative effect given by a fractional Laplacian. A priori estimates are derived when the nonlinearity satisfies certain growth conditions. Using Galerkin's method, the existence of a global smooth solution is established. Uniqueness is also proved. Furthermore, the existence of a global attractor is proved, and estimates of the Hausdorff and fractal dimensions for the global attractor are obtained.