American Institute of Mathematical Sciences

Advanced
May  2008, 20(2): 259-274. doi: 10.3934/dcds.2008.20.259

Stability from the point of view of diffusion, relaxation and spatial inhomogeneity

Fang Li 1, , Kimie Nakashima 2, and  Wei-Ming Ni 3,

1. 

School of Mathematics, University of Minnesota, Minneapolis, MN55455, United States
2. 

Department of Ocean Science, Tokyo University of Marine Sciences and Technology, 4-5-7, Konan, Minato-ku, Tokyo 108-8477, Japan
3. 

School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455

Received  January 2007 Revised  August 2007 Published  November 2007

In this paper, we study how stability properties of solutions to general reaction-diffusion systems are related to the diffusion coefficients, response rate, and spatial inhomogeneity. In particular, all eigenvalues of steady states to general shadow systems are completely determined, and some consequences are discussed.
Keywords: Reaction-diffusion equations, response rate, instability, spatial inhomogeneity., shadow systems.
Mathematics Subject Classification: 35K57, 35B3.
Citation: Fang Li, Kimie Nakashima, Wei-Ming Ni. Stability from the point of view of diffusion, relaxation and spatial inhomogeneity. Discrete & Continuous Dynamical Systems, 2008, 20 (2) : 259-274. doi: 10.3934/dcds.2008.20.259
[1]

Shin-Ichiro Ei, Kota Ikeda, Eiji Yanagida. Instability of multi-spot patterns in shadow systems of reaction-diffusion equations. Communications on Pure & Applied Analysis, 2015, 14 (2) : 717-736. doi: 10.3934/cpaa.2015.14.717
[2]

Masaharu Taniguchi. Instability of planar traveling waves in bistable reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2003, 3 (1) : 21-44. doi: 10.3934/dcdsb.2003.3.21
[3]

Toshi Ogawa. Degenerate Hopf instability in oscillatory reaction-diffusion equations. Conference Publications, 2007, 2007 (Special) : 784-793. doi: 10.3934/proc.2007.2007.784
[4]

Shuangquan Xie, Theodore Kolokolnikov. Oscillations of many interfaces in the near-shadow regime of two-component reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 959-975. doi: 10.3934/dcdsb.2016.21.959
[5]

Rebecca McKay, Theodore Kolokolnikov. Stability transitions and dynamics of mesa patterns near the shadow limit of reaction-diffusion systems in one space dimension. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 191-220. doi: 10.3934/dcdsb.2012.17.191
[6]

Weihua Jiang, Xun Cao, Chuncheng Wang. Turing instability and pattern formations for reaction-diffusion systems on 2D bounded domain. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021085
[7]

Xiaoyan Zhang, Yuxiang Zhang. Spatial dynamics of a reaction-diffusion cholera model with spatial heterogeneity. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2625-2640. doi: 10.3934/dcdsb.2018124
[8]

Hideki Murakawa. Fast reaction limit of reaction-diffusion systems. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1047-1062. doi: 10.3934/dcdss.2020405
[9]

Ching-Shan Chou, Yong-Tao Zhang, Rui Zhao, Qing Nie. Numerical methods for stiff reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2007, 7 (3) : 515-525. doi: 10.3934/dcdsb.2007.7.515
[10]

Laurent Desvillettes, Klemens Fellner. Entropy methods for reaction-diffusion systems. Conference Publications, 2007, 2007 (Special) : 304-312. doi: 10.3934/proc.2007.2007.304
[11]

A. Dall'Acqua. Positive solutions for a class of reaction-diffusion systems. Communications on Pure & Applied Analysis, 2003, 2 (1) : 65-76. doi: 10.3934/cpaa.2003.2.65
[12]

Piermarco Cannarsa, Giuseppe Da Prato. Invariance for stochastic reaction-diffusion equations. Evolution Equations & Control Theory, 2012, 1 (1) : 43-56. doi: 10.3934/eect.2012.1.43
[13]

Martino Prizzi. A remark on reaction-diffusion equations in unbounded domains. Discrete & Continuous Dynamical Systems, 2003, 9 (2) : 281-286. doi: 10.3934/dcds.2003.9.281
[14]

Wilhelm Stannat, Lukas Wessels. Deterministic control of stochastic reaction-diffusion equations. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020087
[15]

Angelo Favini, Atsushi Yagi. Global existence for Laplace reaction-diffusion equations. Discrete & Continuous Dynamical Systems - S, 2020, 13 (5) : 1473-1493. doi: 10.3934/dcdss.2020083
[16]

Guangying Lv, Jinlong Wei, Guang-an Zou. Noise and stability in reaction-diffusion equations. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021005
[17]

Dieter Bothe, Michel Pierre. The instantaneous limit for reaction-diffusion systems with a fast irreversible reaction. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 49-59. doi: 10.3934/dcdss.2012.5.49
[18]

Marek Fila, Hirokazu Ninomiya, Juan-Luis Vázquez. Dirichlet boundary conditions can prevent blow-up in reaction-diffusion equations and systems. Discrete & Continuous Dynamical Systems, 2006, 14 (1) : 63-74. doi: 10.3934/dcds.2006.14.63
[19]

Peter E. Kloeden, Thomas Lorenz, Meihua Yang. Reaction-diffusion equations with a switched--off reaction zone. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1907-1933. doi: 10.3934/cpaa.2014.13.1907
[20]

Jacson Simsen, Mariza Stefanello Simsen, Marcos Roberto Teixeira Primo. Reaction-Diffusion equations with spatially variable exponents and large diffusion. Communications on Pure & Applied Analysis, 2016, 15 (2) : 495-506. doi: 10.3934/cpaa.2016.15.495

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (107)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences