Opuscula Math. 46, no. 2 (2026), 185-199
https://doi.org/10.7494/OpMath.202603112
Opuscula Mathematica
Damped nonlinear Ginzburg-Landau equation with saturation. Part II. Strong Stabilization
Pascal Bégout
Jesús Ildefonso Díaz
Abstract. We study the complex Ginzburg-Landau equation posed on possibly unbounded domains, including some singular and saturated nonlinear damping terms. This model interpolates between the nonlinear Schrödinger equation and dissipative parabolic dynamics through a complex time-derivative prefactor, capturing the interplay between dispersion and dissipation. As a continuation of our previous study on the existence and uniqueness of solutions, we prove here some strong stabilization properties. In particular, we show the finite time extinction of solutions induced by the nonlinear saturation mechanism, which, sometimes, can be understood as a bang-bang control. The analysis relies on refined energy methods. Our results provide a rigorous justification of nonlinear dissipation as an effective stabilization mechanism for this class of complex equations where the maximum principle fails.
Keywords: damped Ginzburg-Landau equation, saturated nonlinearity, finite time extinction.
Mathematics Subject Classification: 35Q56, 35B40, 93D40.
- S. Antontsev, J.-P. Dias, M. Figueira, Complex Ginzburg-Landau equation with absorption: existence, uniqueness and localization properties, J. Math. Fluid Mech. 16 (2014), no. 2, 211-223. https://doi.org/10.1007/s00021-013-0147-0
- S.N. Antontsev, J.I. Díaz, S. Shmarev, Energy Methods for Free Boundary Problems, Progress in Nonlinear Differential Equations and Their Applications, 48, Birkhäuser, Boston, MA, 2002. https://doi.org/10.1115/1.1483358
- P. Bégout, Finite time extinction for a damped nonlinear Schrödinger equation in the whole space, Electron. J. Differential Equations (2020), no. 39, 1-18. https://doi.org/10.58997/ejde.2020.39
- P. Bégout, The dual space of a complex Banach space restricted to the field of real numbers, Adv. Math. Sci. Appl. 31 (2022), no. 2, 241-252. https://doi.org/10.7146/math.scand.a-14305
- P. Bégout, J.I. Díaz, Finite time extinction for the strongly damped nonlinear Schrödinger equation in bounded domains, J. Differential Equations 268 (2020), no. 7, 4029-4058. https://doi.org/10.1016/j.jde.2019.10.016
- P. Bégout, J.I. Díaz, Finite time extinction for a class of damped Schrödinger equations with a singular saturated nonlinearity, J. Differential Equations 308 (2022), 252-285. https://doi.org/10.1016/j.jde.2021.11.010
- P. Bégout, J.I. Díaz, Finite time extinction for a critically damped Schrödinger equation with a sublinear nonlinearity, Adv. Differential Equations 28 (2023), no. 3-4, 311-340. https://doi.org/10.57262/ade028-0304-311
- P. Bégout, J.I. Díaz, Strong stabilization of damped nonlinear Schrödinger equation with saturation on unbounded domains, J. Math. Anal. Appl. 538 (2024), 128329. https://doi.org/10.1016/j.jmaa.2024.128329
- P. Bégout, J.I. Díaz, Damped nonlinear Ginzburg-Landau equation with saturation. Part I. Existence of solutions on general domains, Opuscula Math. 46 (2026), no. 2, 153-183. https://doi.org/10.7494/OpMath.202603111
- H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. https://doi.org/10.1007/978-0-387-70914-7
- A.C. Casal, J.I. Díaz, On the principle of pseudo-linearized stability: applications to some delayed nonlinear parabolic equations, Nonlinear Anal. 63 (2005), e997-e1007. https://doi.org/10.1016/j.na.2005.01.013
- A.C. Casal, J.I. Díaz, On the complex Ginzburg-Landau equation with a delayed feedback, Math. Models Methods Appl. Sci. 16 (2006), no. 1, 1-17. https://doi.org/10.1142/s0218202506001030
- A.C. Casal, J.I. Díaz, M. Stich, On some delayed nonlinear parabolic equations modeling CO oxidation, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 13B (2006), 413-426.
- A.C. Casal, J.I. Díaz, M. Stich, Control of turbulence in oscillatory reaction-diffusion systems through a combination of global and local feedback, Phys. Rev. E 76 (2007), 036209. https://doi.org/10.1103/physreve.76.036209
- J.I. Díaz, J.F. Padial, J.I. Tello, L. Tello, Complex Ginzburg-Landau equations with a delayed nonlocal perturbation, Electron. J. Differential Equations (2020), Paper no. 40. https://doi.org/10.58997/ejde.2020.40
- J. Droniou, Intégration et Espaces de Sobolev à Valeurs Vectorielles, HAL preprint hal-01382368, 2001.
- K. Fenza, M. Labbadi, M. Ouzahra, Finite-time stabilization of a class of nonlinear systems in Hilbert space, [in:] Proc. 2025 IEEE 64th Conference on Decision and Control (CDC), IEEE, 2025, 3003-3008. https://doi.org/10.1109/cdc57313.2025.11312420
- E. Gagliardo, Ulteriori proprietà di alcune classi di funzioni in più variabili, Ricerche Mat. 8 (1959), 24-51.
- L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa (3) 13 (1959), 115-162.
- L. Rosier, B.-Y. Zhang, Null controllability of the complex Ginzburg-Landau equation, Ann. Inst. H. Poincaré C Anal. Non Linéaire 26 (2009), no. 2, 649-673. https://doi.org/10.1016/j.anihpc.2008.03.003
- Pascal Bégout (corresponding author)
https://orcid.org/0000-0002-9172-3057- Toulouse School of Economics, Université Toulouse Capitole, Institut de Mathématiques de Toulouse, 1, Esplanade de l'Université, 31080 Toulouse Cedex 6, France
- Jesús Ildefonso Díaz
- https://orcid.org/0000-0003-1730-9509
- Universidad Complutense de Madrid, Instituto de Matemática Interdisciplinar, Plaza de las Ciencias, 3, 28040 Madrid, Spain
- Communicated by Vicenţiu D. Rădulescu.
- Received: 2026-02-04.
- Revised: 2026-03-10.
- Accepted: 2026-03-11.
- Published online: 2026-04-10.
