Opuscula Math. 46, no. 2 (2026), 153-183
https://doi.org/10.7494/OpMath.202603111

 
Opuscula Mathematica

Damped nonlinear Ginzburg-Landau equation with saturation. Part I. Existence of solutions on general domains

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. Under suitable structural conditions on the complex coefficients, we establish the existence and uniqueness of global solutions. The analysis relies on the delicate proofs that the maximal monotone operator theory can be adapted to this framework, even for unbounded domains.

Keywords: damped Ginzburg-Landau equation, saturated nonlinearity, finite time extinction, maximal monotone operators, existence and regularity of weak solutions.

Mathematics Subject Classification: 35Q56, 35A01, 35A02, 35D30, 35D35.

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  1. 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
  2. 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, 2002. https://doi.org/10.1115/1.1483358
  3. I.S. Aranson, L. Kramer, The world of the complex Ginzburg-Landau equation, Rev. Mod. Phys. 74 (2002), no. 1, 99-143. https://doi.org/10.1103/revmodphys.74.99
  4. V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer Monographs in Mathematics, Springer, New York, 2010. https://doi.org/10.1007/978-1-4419-5542-5
  5. D. Battogtokh, A. Mikhailov, Controlling turbulence in the complex Ginzburg-Landau equation, Phys. D 90 (1996), no. 1-2, 84-95. https://doi.org/10.1016/0167-2789(95)00232-4
  6. 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
  7. 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
  8. 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
  9. 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
  10. 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
  11. 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
  12. P. Bégout, J.I. Díaz, On the compactness of the support of solitary waves of the complex saturated nonlinear Schrödinger equation and related problems, Phys. D 472 (2025), 134516. https://doi.org/10.1016/j.physd.2024.134516
  13. P. Bégout, J.I. Díaz, Damped nonlinear Ginzburg-Landau equation with saturation. Part II. Strong stabilization, Opuscula Math. 46 (2026), no. 2, 185-199. https://doi.org/10.7494/OpMath.202603112
  14. J. Bergh, J. Löfström, Interpolation Spaces. An Introduction, Springer, Berlin, 1976. https://doi.org/10.1007/978-3-642-66451-9_5
  15. H. Brezis, Operateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, North-Holland Mathematics Studies, 5, North-Holland, Amsterdam, 1973. https://doi.org/10.1016/s0304-0208(08)x7125-7
  16. 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
  17. 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
  18. 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.
  19. 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
  20. 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), 40. https://doi.org/10.58997/ejde.2020.40
  21. J. Droniou, Intégration et Espaces de Sobolev à Valeurs Vectorielles, HAL preprint, 2001.
  22. R.E. Edwards, Functional Analysis. Theory and Applications, Holt, Rinehart and Winston, New York, 1965. https://doi.org/10.1017/s0008439500028320
  23. J. Ginibre, G. Velo, The Cauchy problem in local spaces for the complex Ginzburg-Landau equation. I. Compactness methods, Phys. D 95 (1996), no. 3-4, 191-228. https://doi.org/10.1016/0167-2789(96)00055-3
  24. J. Ginibre, G. Velo, The Cauchy problem in local spaces for the complex Ginzburg-Landau equation. II. Contraction methods, Comm. Math. Phys. 187 (1997), no. 1, 45-79. https://doi.org/10.1007/s002200050129
  25. V.L. Ginzburg, L.D. Landau, On the theory of superconductivity, Zh. Eksp. Teor. Fiz. 20 (1950), 1064-1082.
  26. Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence, Springer Series in Synergetics, 19, Springer, Berlin, 1984. https://doi.org/10.1007/978-3-642-69689-3_7
  27. L.-P. Lévy, Magnetism and Superconductivity, Springer, Berlin, 2000.
  28. E.H. Lieb, M. Loss, Analysis, Graduate Studies in Mathematics, 14, American Mathematical Society, Providence, RI, 2nd ed., 2001. https://doi.org/10.1090/gsm/014
  29. V.A. Liskevich, M.A. Perelmuter, Analyticity of sub-Markovian semigroups, Proc. Amer. Math. Soc. 123 (1995), no. 4, 1097-1104. https://doi.org/10.1090/s0002-9939-1995-1224619-1
  30. N. Okazawa, Sectorialness of second order elliptic operators in divergence form, Proc. Amer. Math. Soc. 113 (1991), no. 3, 701-706. https://doi.org/10.1090/s0002-9939-1991-1072347-4
  31. N. Okazawa, T. Yokota, Monotonicity method applied to the complex Ginzburg-Landau and related equations, J. Math. Anal. Appl. 267 (2002), no. 1, 247-263. https://doi.org/10.1006/jmaa.2001.7770
  32. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer, New York, 1983.
  33. W.A. Strauss, On continuity of functions with values in various Banach spaces, Pacific J. Math. 19 (1966), 543-551. https://doi.org/10.2140/pjm.1966.19.543
  34. W.A. Strauss, On weak solutions of semi-linear hyperbolic equations, An. Acad. Brasil. Ci. 42 (1970), 645-651.
  35. R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, 68, Springer, New York, 2nd ed., 1997. https://doi.org/10.1007/978-1-4612-0645-3
  36. F. Trèves, Topological Vector Spaces, Distributions and Kernels, Dover, Mineola, NY, 2006.
  37. I.I. Vrabie, Compactness Methods for Nonlinear Evolutions, Pitman Monographs and Surveys in Pure and Applied Mathematics, 75, Longman, Harlow, 1995.
  • Pascal Bégout (corresponding author)
  • ORCID iD 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
  • ORCID iD 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.
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Cite this article as:
Pascal Bégout, Jesús Ildefonso Díaz, Damped nonlinear Ginzburg-Landau equation with saturation. Part I. Existence of solutions on general domains, Opuscula Math. 46, no. 2 (2026), 153-183, https://doi.org/10.7494/OpMath.202603111

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