Electron. J. Differential Equations, Vol. 2020 (2020), No. 40, pp. 1-18.

Complex Ginzburg-Landau equations with a delayed nonlocal perturbation

Jesus Ildefonso Diaz, Juan Francisco Padial, Jose Ignacio Tello, Lourdes Tello

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 $\Omega\subset\mathbb{R}^{N}$ ($N\leq3$),

for t>0, with

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.

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J. Ildefonso Díaz
Instituto de Matemática Interdisciplinar
Universidad Complutense de Madrid
28040 Madrid, Spain
email: jidiaz@ucm.es
J. Francisco Padial
Departamento de Matemática Aplicada
E.T.S. de arquitectura
Universidad Politécnica de Madrid
28040 Madrid, Spain
email: jf.padial@upm.es
J. Ignacio Tello
Departamento de Matemáticas Fundametales
Facultad de Ciencias
Universidad Nacional de Educación a Distancia
28040 Madrid, Spain
email: jtello@mat.uned.es
Lourdes Tello
Departamento de Matemática Aplicada
E.T.S. de arquitectura
Universidad Politécnica de Madrid
28040 Madrid, Spain
email: l.tello@upm.es

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