Flat Blow-up Solutions for the Complex Ginzburg Landau Equation

Abstract

In this paper, we consider the complex Ginzburg-Landau equation ∂_t u = (1 + iβ)Δu + (1 + iδ)|u|^(p-1)u - αu, where β, δ, α ∈ ℝ.The study focuses on investigating the finite-time blow-up phenomenon, which remains an open question for a broad range of parameters, particularly for β and δ. Specifically, for a fixed β ∈ ℝ, the existence of finite-time blow-up solutions for arbitrarily large values of |δ| is still unknown. According to a conjecture made by Popp et al. (Physica D Nonlinear Phenom 114:81–107 1998), when β = 0 and δ is large, blow-up does not occur for generic initial data. In this paper, we show that their conjecture is not valid for all types of initial data, by presenting the existence of blow-up solutions for β = 0 and any δ ∈ ℝ with different types of blowup.