Complex Ginzburg-Landau equations with a delayed nonlocal perturbation




Diaz, Jesus Ildefonso
Padial, J. Francisco
Tello, J. Ignacio
Tello, Lourdes

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Texas State University, Department of Mathematics


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 Ω ⊂ ℝN (N ≤ 3), ∂u/∂t - (1 + iε)∆u + (1 + iβ)|u|2u - (1 - iω)u = F(u(x, t - τ)) for t > 0, with F(u(x, t - τ)) = eix0 {u/|Ω| ∫Ω u(x, t - τ)}, where μ, v ≥ 0, τ > 0 but the rest of real parameters ε, β, ω and X0 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 μ > 0 we prove that only the initial history of the integral on Ω of the unknown on (-τ, 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-1u, for some m ∈ (0, 1). We extend to the delayed case some previous results in the literature of complex equations without any delay.



Complex Ginzburg-Landau equation, Nonlocal delayed perturbation, Existence of weak solutions, Uniqueness, Qualitative properties


Díaz, J. I., Padial, J. F., Tello, J. I., & Tello, L. (2020). Complex Ginzburg-Landau equations with a delayed nonlocal perturbation. <i>Electronic Journal of Differential Equations, 2020</i>(40), pp. 1-18.


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