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Hysteresis-driven pattern formation in reaction-diffusion-ODE systems

<p style='text-indent:20px;'>The paper is devoted to analysis of <i>far-from-equilibrium</i> pattern formation in a system of a reaction-diffusion equation and an ordinary differential equation (ODE). Such systems arise in modeling of interactions between cellular process...

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Main Authors: Köthe, Alexandra (Author) , Marciniak-Czochra, Anna (Author) , Takagi, Izumi (Author)
Format: Article (Journal)
Language:English
Published: [June 2020]
In: Discrete and continuous dynamical systems
Year: 2020, Volume: 40, Issue: 6, Pages: 3595-3627
ISSN:1553-5231
DOI:10.3934/dcds.2020170
Online Access:Verlag, lizenzpflichtig, Volltext: https://doi.org/10.3934/dcds.2020170
Verlag, lizenzpflichtig, Volltext: https://www.aimsciences.org/article/doi/10.3934/dcds.2020170
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Author Notes:Alexandra Köthe, Anna Marciniak-Czochra, Izumi Takagi

MARC

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520 |a <p style='text-indent:20px;'>The paper is devoted to analysis of <i>far-from-equilibrium</i> pattern formation in a system of a reaction-diffusion equation and an ordinary differential equation (ODE). Such systems arise in modeling of interactions between cellular processes and diffusing growth factors. Pattern formation results from hysteresis in the dependence of the quasi-stationary solution of the ODE on the diffusive component. Bistability alone, without hysteresis, does not result in stable patterns. We provide a systematic description of the hysteresis-driven stationary solutions, which may be monotone, periodic or irregular. We prove existence of infinitely many stationary solutions with jump discontinuity and their asymptotic stability for a certain class of reaction-diffusion-ODE systems. Nonlinear stability is proved using direct estimates of the model nonlinearities and properties of the strongly continuous diffusion semigroup.</p> 
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