Geometric blow-up of a dynamic Turing instability in the Swift-Hohenberg equation

被引：0
作者：
Hummel, F. [1 ]
Jelbart, S. [2 ]
Kuehn, C. [1 ]
机构：
[1] Tech Univ Munich, Sch Computat Informat & Technol, D-85748 Munich, Germany
[2] Univ Adelaide, Sch Comp & Math Sci, Adelaide, SA 5000, Australia
来源：
JOURNAL OF DIFFERENTIAL EQUATIONS | 2025年 / 427卷
基金：
欧盟地平线“2020”;
关键词：
Swift-Hohenberg equation; Geometric blow-up; Modulation theory; Singular perturbation theory; Ginzburg-Landau equation; Delayed stability loss; GINZBURG-LANDAU EQUATION; PERTURBED PARABOLIC EQUATION; MODULATION EQUATIONS; PERIODIC-SOLUTIONS; UNBOUNDED-DOMAINS; GLOBAL EXISTENCE; SLOW MANIFOLDS; STABILITY LOSS; BIFURCATION; VALIDITY;
D O I：
10.1016/j.jde.2025.01.036
中图分类号：
O1 [数学];
学科分类号：
0701 ; 070101 ;
摘要：
We consider the slow passage through a Turing bifurcation in the Swift-Hohenberg equation. We generalise the formally derived multiple scales ansatz from modulation theory for use in the slowly time- dependent setting. The key technique is to reformulate the problem via a geometric blow-up transformation. This leads to non-autonomous modulation equations of Ginzburg-Landau type in the blown-up space. We analyse solutions to the modulation equations in weighted Sobolev spaces in two different cases: (i) A symmetric case featuring delayed stability loss, and (ii) A non-symmetric case with a source term. Rigorous estimates on the error of the dynamic modulation approximation are derived in order to characterise the dynamics of the Swift-Hohenberg equation. This allows for a detailed asymptotic description of solutions to the original Swift-Hohenberg equation in both cases (i)-(ii). We also prove the existence of delayed stability loss in case (i), and provide a lower bound for the delay time.<br /> (c) 2025 Published by Elsevier Inc.
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收藏
页码：219 / 309
页数：91
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