Exponential attractor for the 3D Ginzburg-Landau type equation (Q2381491)

The authors consider the following initial value problem for 3D Ginzburgh-Landau type equation to \(\Omega\)-periodic function \(u\), \(\Omega=[0,L]\times [0,L]\times [0,L]\) \[ u_t-(1+i\nu)\Delta u+(1+i\mu)| u| ^{2\sigma}u-\gamma u=0,\quad u(x,0)=u_0(x) \] Under some additional assumptions on parameters \(\sigma,\mu,\nu\) and \(\gamma>0\) the existence and uniqueness of a global solution are proved. Also the existence of the global attractor with upper estimates for its Hausdorff and fractal dimensions and the existence of the exponential attractor are proved.