Construction of a blow-up solution for the Complex Ginzburg-Landau equation in some critical case

TL;DR

This paper constructs a finite-time blow-up solution for the Complex Ginzburg-Landau equation in a critical case, providing a detailed profile and stability analysis using finite-dimensional reduction and index theory.

Contribution

It introduces a novel method to explicitly construct and analyze stable blow-up solutions for the Complex Ginzburg-Landau equation in critical regimes.

Findings

Existence of a finite-time blow-up solution at a single point.

Precise description of the blow-up profile.

Proof of stability of the constructed solution.

Abstract

We construct a solution for the Complex Ginzburg-Landau equation in some critical case, which blows up in finite time $T$ only at one blow-up point. We also give a sharp description of its profile. The proof relies on the reduction of the problem to a finite dimensional one, and the use of index theory to conclude. The interpretation of the parameters of the finite dimension problem in terms of the blow-up point and time allows to prove the stability of the constructed solution.