Some stability results for the complex Ginzburg-Landau equation

Using some classical methods of dynamical systems, stability results and asymptotic decay of strong solutions for the complex Ginzburg-Landau equation (CGL), partial derivative(t)u = (a + i alpha)Delta u - (b + i beta) vertical bar u vertical bar(sigma) u + ku, t > 0, x is an element of Omega, with a > 0, alpha, b, beta, k is an element of R, are obtained. Moreover, we show the existence of bound-states under certain conditions on the parameters and on the domain. We conclude with the proof of asymptotic stability of these bound-states when Omega = R and -k large enough.