Blow-up of solutions for weakly coupled systems of complex Ginzburg-Landau equations

TL;DR

This paper investigates the blow-up behavior of solutions in weakly coupled systems of complex Ginzburg-Landau equations using an ODE approach, providing insights into blow-up rates and lifespan estimates.

Contribution

It introduces a straightforward ODE method to analyze blow-up phenomena in coupled complex Ginzburg-Landau systems, avoiding traditional test-function techniques.

Findings

Established natural blow-up rates for the systems.

Derived lower estimates of solution lifespan.

Extended the approach to heat systems for lifespan analysis.

Abstract

Blow-up phenomena ofvweakly coupled systems of several evolution equations, especially complex Ginzburg-Landau equationsvis shown by a straightforward ODE approach not so-called test-function method, which gives the natural blow-up rate. The difficulty of the proof is that, unlike the single case, terms which come from the fact that the Laplacian cannot be absorbed into the weakly coupled nonlinearities. A similar ODE approach is applied to heat systems by Mochizuki to obtain the lower estimate of lifespan.