Browsing by Subject "finite time extinction"

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Access status: Open Access ,
Damped nonlinear Ginzburg-Landau equation with saturation. Part I. Existence of solutions on general domains
(Wydawnictwa AGH, 2026) Bégout, Pascal; Díaz, Jesús Ildefonso
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.
Access status: Open Access ,
Damped nonlinear Ginzburg-Landau equation with saturation. Part II. Strong Stabilization
(Wydawnictwa AGH, 2026) Bégout, Pascal; Díaz, Jesús Ildefonso
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.