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

Abstract

Global existence of unique strong solutions is established for the complex Ginzburg–Landau equation ∂tu−(λ+iα)Δu+(κ+iβ)|u|p−1u−γu=0, where λ>0,κ>0,α,β,γ∈R,p≥1, and κ−1|β|≤2p/(p−1). The key is a new inequality in monotonicity methods. It is based on the sectorial estimates of −Δ 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 Schrödinger type equation with monotone nonlinearity ∂tu−iΔu+|u|p−1u=0 for all p≥1.