A logarithmic convexity approach to quantitative unique continuation for the complex Ginzburg-Landau operator

We establish quantitative unique continuation estimates for solutions of the complex Ginzburg-Landau equation in the framework of two-sphere and one-cylinder inequalities. While prior work [Dou et al. SIAM J. Control Optim. (2023)] relied on Carleman estimates, the present study develops a novel logarithmic convexity approach to prove the quantitative unique continuation property. We derive an explicit quantitative unique continuation constant and fully characterize its dependence on the parameters.