Global attractors and convergence to equilibrium for degenerate Ginzburg–Landau and parabolic equations

Abstract

We study the dynamics of a degenerate complex Ginzburg–Landau (CGL), and a real parabolic equation with a variable, generally nonsmooth diffusion coefficient, which may vanish at some points or be unbounded. For the CGL equation, there exists a global attractor in L 2 ( Ω ) . For the parabolic equation, we show the existence of a global branch of nonnegative stationary states. The global bifurcation result, is used in order to establish—in conjunction with the definition of a gradient dynamical system in the natural phase space—that any solution with nonnegative initial data, tends to the trivial or the nonnegative equilibrium.