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)up-1u - γu = 0, where λ > 0, κ > 0, α, β, γ ε ℝ, p ≥ 1, and κ-1β ≤ 2 √p(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 →up-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 + ∥up-1u = 0 for all p ≥ 1. © 2002 Elsevier Science (USA).