Complex Ginzburg-Landau equations with a delayed nonlocal perturbation
| dc.contributor.author | Diaz, Jesus Ildefonso | |
| dc.contributor.author | Padial, J. Francisco | |
| dc.contributor.author | Tello, J. Ignacio | |
| dc.contributor.author | Tello, Lourdes | |
| dc.date.accessioned | 2021-09-22T20:06:34Z | |
| dc.date.available | 2021-09-22T20:06:34Z | |
| dc.date.issued | 2020-04-30 | |
| dc.description.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. | |
| dc.description.department | Mathematics | |
| dc.format | Text | |
| dc.format.extent | 18 pages | |
| dc.format.medium | 1 file (.pdf) | |
| dc.identifier.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. | |
| dc.identifier.issn | 1072-6691 | |
| dc.identifier.uri | https://hdl.handle.net/10877/14546 | |
| dc.language.iso | en | |
| dc.publisher | Texas State University, Department of Mathematics | |
| dc.rights | Attribution 4.0 International | |
| dc.rights.holder | This work is licensed under a Creative Commons Attribution 4.0 International License. | |
| dc.rights.uri | https://creativecommons.org/licenses/by/4.0/ | |
| dc.source | Electronic Journal of Differential Equations, 2020, San Marcos, Texas: Texas State University and University of North Texas. | |
| dc.subject | Complex Ginzburg-Landau equation | |
| dc.subject | Nonlocal delayed perturbation | |
| dc.subject | Existence of weak solutions | |
| dc.subject | Uniqueness | |
| dc.subject | Qualitative properties | |
| dc.title | Complex Ginzburg-Landau equations with a delayed nonlocal perturbation | |
| dc.type | Article |