| Summary: | 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.
|
|---|