Flat blow-up solutions for the complex Ginzburg Landau equation

Abstract

In this paper, we consider the complex Ginzburg-Landau equation $${\partial }_{t}u=(1+i\beta )\Delta u+(1+i\delta )|u{|}^{p-1}u-\alpha u,\text{where }\beta ,\delta ,\alpha \in \mathbb{R}.$$ 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 \(β\in \mathbb{R}\), 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. \cite{POPphd98}, when \(β= 0\) and \(δ\) is large, blow-up does not occur for \textit{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 \(δ\in \mathbb{R}\) with different types of blowup.