Finite-time blowup for some nonlinear complex Ginzburg-Landau equations

TL;DR

This paper reviews conditions under which solutions to complex Ginzburg-Landau equations blow up in finite time, focusing on how parameters influence blowup behavior and timing.

Contribution

It provides a comprehensive review of finite-time blowup criteria for complex Ginzburg-Landau equations, analyzing parameter effects on blowup conditions and timing.

Findings

Blowup criteria depend on parameters θ and γ.

Blowup time varies with parameter values.

Certain parameter regimes guarantee finite-time blowup.

Abstract

In this article, we review finite-time blowup criteria for the family of complex Ginzburg-Landau equations $u_t = e^{ i\theta } [\Delta u + |u|^\alpha u] + \gamma u$ on ${\mathbb R}^N $, where $0 \le \theta \le \frac {\pi } {2}$, $\alpha >0$ and $\gamma \in {\mathbb R} $. We study in particular the effect of the parameters $\theta $ and $\gamma $, and the dependence of the blowup time on these parameters.