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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共 99 条
  • [61] Mielke A., Reduction of PDEs on domains with several unbounded directions: a first step towards modulation equations, Z. Angew. Math. Phys., 43, pp. 449-470, (1992)
  • [62] Mielke A., The complex Ginzburg-Landau equation on large and unbounded domains: sharper bounds and attractors, Nonlinearity, 10, (1997)
  • [63] Mielke A., Instability and stability of rolls in the Swift–Hohenberg equation, Commun. Math. Phys., 189, pp. 829-853, (1997)
  • [64] Mielke A., Bounds for the solutions of the complex Ginzburg-Landau equation in terms of the dispersion parameters, Phys. D: Nonlinear Phenom., 117, pp. 106-116, (1998)
  • [65] Mielke A., Schneider G., Attractors for modulation equations on unbounded domains-existence and comparison, Nonlinearity, 8, (1995)
  • [66] Murray J.D., Mathematical biology, Bioinformatics, (1989)
  • [67] Nefedov N.N., Schneider K.R., Delay of exchange of stabilities in singularly perturbed parabolic problems, Tr. Inst. Mat. Mekh. UrO RAN, 9, pp. 121-130, (2003)
  • [68] Neishtadt A., Persistence of stability loss for dynamical bifurcations I, Differ. Equ., 23, pp. 1385-1391, (1987)
  • [69] Neishtadt A., Persistence of stability loss for dynamical bifurcations II, Differ. Equ., 24, pp. 171-176, (1988)
  • [70] Newell A.C., Whitehead J.A., Finite bandwidth, finite amplitude convection, J. Fluid Mech., 38, pp. 279-303, (1969)
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