Complex Ginzburg-Landau equations with a delayed nonlocal perturbation

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. For more information see https://ejde.math.txstate.edu/Volumes/2020/40/abstr.html