Flat blow-up solutions for the complex Ginzburg Landau equation

Abstract

In this paper, we consider the complex Ginzburg-Landau equation $$ \partial_t u = (1 + i \beta) \Delta u + (1 + i \delta) |u|^{p-1}u - \alpha u, \quad \text{where } \beta, \delta, \alpha \in \mathbb{R}. $$ The study focuses on investigating the finite-time blow-up phenomenon, which remains an open question for a broad range of parameters, particularly for \(\beta\) 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. \cite{POPphd98}, when \(\beta = 0\) and \(\delta\) 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 \(\beta = 0\) and any \(\delta \in \mathbb{R}\) with different types of blowup.