Complex Ginzburg-Landau equations with a delayed nonlocal perturbation
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Date
2020-04-30
Authors
Diaz, Jesus Ildefonso
Padial, J. Francisco
Tello, J. Ignacio
Tello, Lourdes
Journal Title
Journal ISSN
Volume Title
Publisher
Texas State University, Department of Mathematics
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 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.
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Keywords
Complex Ginzburg-Landau equation, Nonlocal delayed perturbation, Existence of weak solutions, Uniqueness, Qualitative properties
Citation
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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Attribution 4.0 International
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This work is licensed under a Creative Commons Attribution 4.0 International License.