Refined asymptotics for the blow-up solution of the complex Ginzburg–Landau equation in the subcritical case

Duong, Giao Ky; Nouaili, Nejla; Zaag, Hatem

doi:10.4171/aihpc/2

Cited by 5 publications

(1 citation statement)

References 48 publications

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“…Regarding the construction of Type I blowup solutions to (1.3), the authors in [1] (also in [57]) used an important method consisting in two main steps: first, a reduction to a finite-dimensional problem, then a topological argument based on index theory to solve the finite-dimensional problem. We also mention that the method has been proven robust in a lot of situations such as [56] for the reconnection of vortex; in [71,16,61] for perturbated nonlinear source terms; in [14,15,59,63] for blowup solutions to complex Ginzburg-Landau equations; in [60,23,22] for complex-valued heat equations which has no variational structure; in parabolic systems as in [27]; and also in [24] for MEMS models.…”

Section: Introductionmentioning

confidence: 99%

Blowup solutions for the shadow limit model of singular Gierer-Meinhardt system with critical parameters

Duong,

Ghoul,

Kavallaris

et al. 2021

Preprint

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We consider a nonlocal parabolic PDE, which may be regarded as the standard semilinear heat equation with power nonlinearity, where the nonlinear term is divided by some Sobolev norm of the solution. Unlike the earlier work in [13] where we consider a subcritical regime of parameters, we focus here on the critical regime, which is much more complicated. Our main result concerns the construction of a blow-up solution with the description of its asymptotic behavior. Our method relies on a formal approach, where we find an approximate solution. Then, adopting a rigorous approach, we linearize the equation around that approximate solution, and reduce the question to a finite dimensional problem. Using an argument based on index theory, we solve that finite-dimensional problem, and derive an exact solution to the full problem. We would like to point out that our constructed solution has a new blowup speed with a log correction term, which makes it different from the speed in the subcritical range of parameters and the standard heat equation.

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Section: Introductionmentioning

confidence: 99%

Blowup solutions for the shadow limit model of singular Gierer-Meinhardt system with critical parameters

Duong,

Ghoul,

Kavallaris

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

Gradient profile for the reconnection of vortex lines with the boundary in type-II superconductors

Huang,

Zaag

2023

J. Evol. Equ.

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B-methods for the numerical solution of evolution problems with blow-up solutions part II: Splitting methods

Beck,

Gander,

Kwok

2023

Computers & Mathematics with Applications

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Refined asymptotics for the blow-up solution of the complex Ginzburg–Landau equation in the subcritical case

Abstract: In this paper, we aim to refine the blow-up behavior for the complex Ginzburg-Landau (CGL) equation in some subcritical case. More precisely, we construct blow-up solutions and refine their blow-up profile to the next order.

Cited by 5 publications

References 48 publications

Blowup solutions for the shadow limit model of singular Gierer-Meinhardt system with critical parameters

Blowup solutions for the shadow limit model of singular Gierer-Meinhardt system with critical parameters

Gradient profile for the reconnection of vortex lines with the boundary in type-II superconductors

B-methods for the numerical solution of evolution problems with blow-up solutions part II: Splitting methods

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