Monotonicity Method Applied to the Complex Ginzburg–Landau and Related Equations

Global existence of unique strong solutions is established for the complex Ginzburg–Landau equation ∂tu − λ + iα u + κ + iβ u p−1u − γu = 0 where λ > 0 κ > 0 α β γ ∈ p ≥ 1, and κ−1 β ≤ 2√p/ p − 1 .The key is a new inequality in monotonicity methods.It is based on the sectorial estimates of − in Lp+1 and the nonlinear operator u → u p−1u appearing in the equation.The key inequality also yields the global existence of unique strong solutions of the nonlinear Schr¨odinger type equation with monotone nonlinearity ∂tu − i u + u p−1u = 0 for all p ≥ 1.