Electronic Journal of Differential Equations, 
Vol. 2020 (2020), No. 40, pp. 1-18.

Title: Complex Ginzburg-Landau equations with a delayed nonlocal perturbation

Authors: Jesus Ildefonso Diaz (Univ. Complutense de Madrid, Spain)
         Juan  Francisco Padial (Univ. Politecnica de Madrid, Spain)
         Jose Ignacio Tello (Univ. Nacional de Educacion a Distancia, Madrid, Spain)
         Lourdes Tello (Univ. Politecnica de Madrid, Spain)

Abstract:
  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 $\Omega\subset\mathbb{R}^{N}$ ($N\leq3$),
 $$
 \frac{\partial u}{\partial t}-(1+i\epsilon)\Delta u
  +(1+i\beta)  | u| ^2u-(1-i\omega) u=F(u(x,t-\tau))
 $$
 for t>0, with
 $$
 F(u(x,t-\tau))  =e^{i\chi_0}\big\{  \frac{\mu}{|
 \Omega| }\int_{\Omega}u(x,t-\tau)  dx+\nu u(x,t-\tau)  \big\}  ,
 $$
 where $\mu$, $\nu\geq0$, $\tau>0$ but the rest of real parameters $\epsilon$,
 $\beta$, $\omega$ and $\chi_0$ 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 $\nu=0$ and $\mu>0$ we prove that only the initial
 history of the integral on $\Omega$  of the unknown on $(-\tau,0)$ 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 $|u| ^2u$ is
 replaced by $|u| ^{m-1}u$, for some $m\in(0,1)$. We extend
 to the delayed case some previous results in the literature of complex
 equations without any delay.
 
Submitted March 31, 2020. Published April 30, 2020.
Math Subject Classifications: 35K15, 35B40, 35Q35.
Key Words: Complex Ginzburg-Landau equation; nonlocal delayed perturbation;
           existence of weak solutions; uniqueness; qualitative properties.