Asymptotics for the generalized two-dimensional Ginzburg-Landau equation (Q1576976)

This paper deals with the study of the generalized Ginzburg-Landau equation \(u_t=\alpha_0u+\alpha_1\Delta u+\alpha_2|u|^2u_x+\alpha_3|u|^2u_y+ \alpha_4u^2\overline{u}_x+\alpha_5u^2\overline{u}_y-\alpha_6|u|^{2\sigma}u\) in \(\Omega =(0,L_1)\times (0,L_2)\), where \(\alpha_0>0\), \(\alpha_j=a_j+ib_j\), \(j=1,\cdots ,6\), \(a_1>0\), \(a_6>0\), \(\sigma >0\), \(t>0\), under periodic boundary conditions and usual initial condition. The main result of the paper establishes the existence of a maximal, compact, connected attractor with finite Hausdorff and fractal dimensions. The proof is based on refined elliptic estimates and on the existence of a unique global solution in \(H^2(\Omega)\) of the considered initial-boundary value problem.