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

Abstract

We construct a solution for the Complex Ginzburg-Landau (CGL) 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.Mathematical subject classification: 35K57, 35K40, 35B44.

Profile for the imaginary part of a blowup solution for a complex-valued semilinear heat equation

“…Then, Nouaili and Zaag in [19] has constructed for (1.7) (in case the critical where β = 0 and p = δ 2 ) a blowup solution satisfying…”

Blowup solutions for a reaction–diffusion system with exponential nonlinearities

“…It is worth mentioning that the method of [25] has been also proved to be successful for constructing a solution to some partial differential equation with a prescribed behavior. It was the case of the complex Ginzburg-Landau equation with no gradient structure by Masmoudi and Zaag [23] (see also the earlier work by Zaag [36]) and Nouaili and Zaag [30]; by Nguyen and Zaag [27], [28] for a logarithmically perturbed nonlinear heat equation and for a refined blowup profile for equation (1.13), or by Nouaili and Zaag [29] for a non-variational complexvalued semilinear heat equation. It was also the case of a non-scaling invariant semilinear heat equation with a general nonlinearity treated in [10], and the energy supercritical harmonic heat flow and wave maps by Ghoul, Ibrahim and Nguyen [17,19].…”

Construction and stability of type I blowup solutions for non-variational semilinear parabolic systems

“…This two-step procedure has been successfully applied for various nonlinear evolution equations to construct both Type I and Type II blowup solutions. It was the case of the semilinear heat equation treated in [4], [32], [36] (see also [35], [9] for the case of logarithmic perturbations, [2], [3] and [16] for the exponential source, [37] for the complex-valued case), the Ginzburg-Landau equation in [27], [38] (see also [48] for an earlier work). It was also the nonlinear Schrödinger equation both in the mass critical [28,29,30,31] and mass supercritical [34] cases; the energy critical [10], [22] and supercritical [6] wave equation; the mass critical gKdV equation [24,25,26]; the two dimensional Keller-Segel model [42]; the energy critical and supercritical geometric equations: the wave maps [39] and [18], the Schrödinger maps [33] and the harmonic heat flow [40,41] and [17]; the semilinear heat equation in the energy critical [43] and supercritical [5] cases.…”