Existence of Periodic Solutions for 3-D Complex Ginzberg-Landau Equation

Abstract

In this paper, the authors consider complex Ginzburg-Landau equation (CGL) in three spatial dimensions u_t = ρu + (1 + iϒ)Δu - (1 + iμ) |u|^{2σ u + f, where u is an unknown complex-value function defined in 3+1 dimensional space-time R^{3+1}, Δ is a Laplacian in R^3, Δ > 0, ϒ, μ are real parameters, Ω ∈ R^3 is a bounded domain. By using the method of Galërkin and Faedo-Schauder fix point theorem we prove the existence of approximate solution uN of the problem. By establishing the uniform boundedness of the norm ||uN|| and the standard compactness arguments, the convergence of the approximate solutions is considered. Moreover, the existence of the periodic solution is obtained .