COMPLEX GINZBURG-LANDAU EQUATIONS WITH A DELAYED NONLOCAL PERTURBATION

We consider an initial boundary value problem of the complex Ginzburg-Landau equation with some delayed feedback terms proposed for the control of chemical turbulence in reaction diffusion systems. We consider the equation in a bounded domain Q C RN (N < 3), partial derivative u/partial derivative t - (1 + i is an element of)Delta u + (1 _i beta) vertical bar mu vertical bar 2u - 1 omega)u = F(u(x, t -tau)) where it, v > 0, 'T > 0 but the rest of real parameters, /3, w and xo do not have a prescribed sign. We prove the existence and uniqueness of weak solutions of problem for a range of initial data and parameters. When v = 0 and it > 0 we prove that only the initial history of the integral on Q of the unknown on ( T, 0) and a standard initial condition at t = 0 are required to determine univocally the existence of a solution. We prove several qualitative properties of solutions, such as the finite extinction time (or the zero exact controllability) and the finite speed of propagation, when the term lul2u is replaced by lur lu, for some m E (0, 1). We extend to the delayed case some previous results in the literature of complex equations without any delay.