Flat Blow-up Solutions for the Complex Ginzburg Landau Equation

Abstract

In this paper, we consider the complex Ginzburg-Landau equation

$$\begin{aligned} \partial _t u = (1 + i \beta ) \Delta u + (1 + i \delta ) |u|^{p-1}u - \alpha u, \quad \text {where } \beta , \delta , \alpha \in {\mathbb {R}}. \end{aligned}$$

The study focuses on investigating the finite-time blow-up phenomenon, which remains an open question for a broad range of parameters, particularly for \(\beta \) and \(\delta \). Specifically, for a fixed \(\beta \in {\mathbb {R}}\), the existence of finite-time blow-up solutions for arbitrarily large values of \( |\delta | \) is still unknown. According to a conjecture made by Popp et al. (Physica D Nonlinear Phenom 114:81–107 1998), when \(\beta = 0\) and \(\delta \) 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 \(\beta = 0\) and any \(\delta \in {\mathbb {R}}\) with different types of blowup.