Flat blow-up solutions for the complex Ginzburg Landau equation

In this paper, we consider the complex Ginzburg-Landau equation $${\mathrm{\partial }}_{t}u=(1+i\beta )\mathrm{\Delta }u+(1+i\delta )\mathrm{\mid }u{\mathrm{\mid }}^{p-1}u-\alpha u,\text{where }\beta ,\delta ,\alpha \in \mathbb{R}\mathrm{.}$$ 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 $\mathrm{\mid }\delta \mathrm{\mid }$ 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 $\mathit{\text{generic}}\mathit{\text{initial}}\mathit{\text{data}}$. 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.