Finite-Time Blowup for a Complex Ginzburg--Landau Equation

Abstract

We prove that negative energy solutions of the complex Ginzburg-Landau equation e −iθ ut = ∆u + |u| α u blow up in finite time, where α > 0 and −π/2 < θ < π/2. For a fixed initial value u(0), we obtain estimates of the blow-up time T θ max as θ → ±π/2. It turns out that T θ max stays bounded (respectively, goes to infinity) as θ → ±π/2 in the case where the solution of the limiting nonlinear Schrödinger equation blows up in finite time (respectively, is global).2010 Mathematics Subject Classification. 35Q56, 35B44.

Some stability results for the complex Ginzburg–Landau equation

“…On the other hand, there are not many results concerning the blow-up of the solutions of (CGL): we refer e.g. [2] and [12]. Furthermore, concerning the existence of standing wave solutions, some partial results were obtained in the case of a bounded connected domain; cf.…”

Local well-posedness of the complex Ginzburg–Landau equation in bounded domains

“…On the other hand, there are few treatment for the case where κ > 0 especially on the well-posedness: Cazenave et al studied blow-up of solutions( [4,3]) and the existence of standing wave solutions ( [5]). In their papers [4,3] they made restriction on parameters λ, κ, α, β to be α λ = β κ . They proved the existence of the unique local solution based on the semi-group theory for sufficiently smooth initial data in the whole space R N for any q > 2.…”

Finite-time Blowup for some Nonlinear Complex Ginzburg–Landau Equations

“…On the other hand, this strategy does not give any information on how the blowup occurs, nor on the mechanism that leads to blowup. Concerning the family (1.1), this is the type of approach used in [15,19,2] for the nonlinear heat equation; in [45,12,17,41,30,31] for the nonlinear Schrödinger equation; in [38,6,5] for the intermediate case of (1.1).…”