Bégout, PascalORCID: https://orcid.org/0000-0002-9172-3057 and Diaz, Jesus IldefonsoORCID: https://orcid.org/0000-0003-1730-9509 (2026) Damped nonlinear Ginzburg-Landau equation with saturation. Part II. Strong Stabilization. Opuscula Mathematica, vol. 46 (n° 2). pp. 185-199.

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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 in
terplay 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.