Construction of a Blow-Up Solution for the Complex Ginzburg-Landau Equation in a Critical Case

We construct a solution for the Complex Ginzburg-Landau equation in a 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 us to prove the stability of the constructed solution.