Blow-up profile for the complex Ginzburg–Landau equation

Arch Rational Mech Anal

2017

Self Cite

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.

“…Unlike in subcritical case [MZ08] and [Zaa01]. the criticality of the problem induces substantial changes in the blow-up profile as pointed out in the comments following Theorem 1.…”

Section: Remark 14mentioning

confidence: 92%

“…Hence, we only prove the existence result (Theorem 1) and kindly refer the reader to [MZ97] and [MZ08] for the proof of the stability.…”

Section: Remark 14mentioning

confidence: 99%

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

Nouaili

Arch Rational Mech Anal

2017

Self Cite

“…In the following, we would like to find its equivalent as x → 0 and show that it is in 444 singular at the origin. We argue as in Masmoudi and Zaag [MZ08]. Consider K 0 > 0 to be fixed large enough later.…”

Section: Proof Of Proposition 38mentioning

confidence: 92%

“…In [EZ11], Ebde and Zaag use the same ideas to show the persistence of the profile (4) under perturbations of equation (1) in the real case by lower order terms involving u and ∇u. In [MZ08], Masmoudi and Zaag adapted that method to the case of the following complex Ginzburg-Landau equation, where no gradient structure exists:…”

Section: Introductionmentioning

confidence: 99%

Profile for a Simultaneously Blowing up Solution to a Complex Valued Semilinear Heat Equation

Nouaili

Communications in Partial Differential Equations

2015

Self Cite

We construct a solution to a complex nonlinear heat equation which blows up in finite time T only at one blow-up point. We also give a sharp description of its blow-up profile. The proof relies on the reduction of the problem to a finite dimensional one and the use of index theory to conclude. We note that the real and imaginary parts of the constructed solution blow up simultaneously, with the imaginary part dominated by the real.Mathematical Subject classification: 35K57, 35K40, 35B44.

Construction of a Multisoliton Blowup Solution to the Semilinear Wave Equation in One Space Dimension

Côte

2013

Comm Pure Appl Math

126

We consider the semilinear wave equation with power nonlinearity in one space dimension. Given a blowup solution with a characteristic point, we refine the blowup behavior first derived by Merle and Zaag. We also refine the geometry of the blowup set near a characteristic point and show that, except for perhaps one exceptional situation, it is never symmetric with respect to the characteristic point. Then, we show that all blowup modalities predicted by those authors do occur. More precisely, given any integer k ≥ 2 and $\zeta _0 \in {\cal R}$, we construct a blowup solution with a characteristic point a such that the asymptotic behavior of the solution near (a,T(a)) shows a decoupled sum of k solitons with alternate signs whose centers (in the hyperbolic geometry) have ζ0 as a center of mass for all times. © 2013 Wiley Periodicals, Inc.