Attractor and traveling wave solutions for 3D Ginzburg-Landau type equation

In this chapter, we consider the initial value problem of a Ginzburg- Landau equation with spatial periodicity condition in three dimension. Firstly, In Section 2 we prove the existence of local solution (see Theorem 2.1). Furthermore, the existence and uniqueness of global solution are proved by making a prior estimate for solution u(t) (see Theorem 2.2 and 2.3). In Section 3, the existence of global attractor and upper bound estimates for its Hausdorff dimensions and fractal dimensions are obtained (see Theorem 3.2 and 3.3). Then, in Section 4 the existence of finite dimensional exponential attractor is proved (see Theorem 4.2). In Section 5, we construct a fully discrete Fourier spectral approximation scheme for problem (1.4)-(1.6) and then prove the existence and convergence of approximate attractors (see Theorem 5.2.1 and 5.3.2) are presented. Moreover, the long-time