American Institute of Mathematical Sciences

Advanced
  • Previous Article
    The uniform boundedness and threshold for the global existence of the radial solution to a drift-diffusion system
  • CPAA Home
  • This Issue
  • Next Article
    On the uniqueness of ground state solutions of a semilinear equation containing a weighted Laplacian
March  2006, 5(1): 85-95. doi: 10.3934/cpaa.2006.5.85

Spike solutions to a nonlocal differential equation

Changfeng Gui 1, and  Zhenbu Zhang 2,

1. 

Department of Mathematics, University of Connecticut, 196 Auditorium Road, U-3009, Storrs, CT 06269-3009, United States
2. 

Department of Mathematics, Jackson State University, P.O. Box 17610, Jackson, MS 39217, United States

Received  March 2005 Revised  October 2005 Published  December 2005

In this paper we consider a nonlocal differential equation, which is a limiting equation of one dimensional Gierer-Meinhardt model. We study the existence of spike steady states and their stability. We also construct a single-spike quasi-equilibrium solution and investigate the dynamics of spike-like solutions.
Keywords: stability., spike solution, Gierer-Meinhardt model.
Mathematics Subject Classification: 35B25, 35B35, 35B40, 35K5.
Citation: Changfeng Gui, Zhenbu Zhang. Spike solutions to a nonlocal differential equation. Communications on Pure and Applied Analysis, 2006, 5 (1) : 85-95. doi: 10.3934/cpaa.2006.5.85
[1]

Nabil T. Fadai, Michael J. Ward, Juncheng Wei. A time-delay in the activator kinetics enhances the stability of a spike solution to the gierer-meinhardt model. Discrete and Continuous Dynamical Systems - B, 2018, 23 (4) : 1431-1458. doi: 10.3934/dcdsb.2018158
[2]

Theodore Kolokolnikov, Michael J. Ward. Bifurcation of spike equilibria in the near-shadow Gierer-Meinhardt model. Discrete and Continuous Dynamical Systems - B, 2004, 4 (4) : 1033-1064. doi: 10.3934/dcdsb.2004.4.1033
[3]

Nancy Khalil, David Iron, Theodore Kolokolnikov. Stability and dynamics of spike-type solutions to delayed Gierer-Meinhardt equations. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022117
[4]

Shin-Ichiro Ei, Kota Ikeda, Yasuhito Miyamoto. Dynamics of a boundary spike for the shadow Gierer-Meinhardt system. Communications on Pure and Applied Analysis, 2012, 11 (1) : 115-145. doi: 10.3934/cpaa.2012.11.115
[5]

Siu-Long Lei. Adaptive method for spike solutions of Gierer-Meinhardt system on irregular domain. Discrete and Continuous Dynamical Systems - B, 2011, 15 (3) : 651-668. doi: 10.3934/dcdsb.2011.15.651
[6]

Georgia Karali, Takashi Suzuki, Yoshio Yamada. Global-in-time behavior of the solution to a Gierer-Meinhardt system. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 2885-2900. doi: 10.3934/dcds.2013.33.2885
[7]

Juncheng Wei, Matthias Winter. On the Gierer-Meinhardt system with precursors. Discrete and Continuous Dynamical Systems, 2009, 25 (1) : 363-398. doi: 10.3934/dcds.2009.25.363
[8]

Manuel del Pino, Patricio Felmer, Michal Kowalczyk. Boundary spikes in the Gierer-Meinhardt system. Communications on Pure and Applied Analysis, 2002, 1 (4) : 437-456. doi: 10.3934/cpaa.2002.1.437
[9]

Henghui Zou. On global existence for the Gierer-Meinhardt system. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 583-591. doi: 10.3934/dcds.2015.35.583
[10]

Mengxin Chen, Ranchao Wu, Yancong Xu. Dynamics of a depletion-type Gierer-Meinhardt model with Langmuir-Hinshelwood reaction scheme. Discrete and Continuous Dynamical Systems - B, 2022, 27 (4) : 2275-2312. doi: 10.3934/dcdsb.2021132
[11]

Jan-Phillip Bäcker, Matthias Röger. Analysis and asymptotic reduction of a bulk-surface reaction-diffusion model of Gierer-Meinhardt type. Communications on Pure and Applied Analysis, 2022, 21 (4) : 1139-1155. doi: 10.3934/cpaa.2022013
[12]

Kota Ikeda. The existence and uniqueness of unstable eigenvalues for stripe patterns in the Gierer-Meinhardt system. Networks and Heterogeneous Media, 2013, 8 (1) : 291-325. doi: 10.3934/nhm.2013.8.291
[13]

Rui Peng, Xianfa Song, Lei Wei. Existence, nonexistence and uniqueness of positive stationary solutions of a singular Gierer-Meinhardt system. Discrete and Continuous Dynamical Systems, 2017, 37 (8) : 4489-4505. doi: 10.3934/dcds.2017192
[14]

Kazuhiro Kurata, Kotaro Morimoto. Construction and asymptotic behavior of multi-peak solutions to the Gierer-Meinhardt system with saturation. Communications on Pure and Applied Analysis, 2008, 7 (6) : 1443-1482. doi: 10.3934/cpaa.2008.7.1443
[15]

Chunqing Lu. Existence and uniqueness of single spike solution of the carrier-pearson problem. Conference Publications, 2001, 2001 (Special) : 259-264. doi: 10.3934/proc.2001.2001.259
[16]

Antonio Di Crescenzo, Maria Longobardi, Barbara Martinucci. On a spike train probability model with interacting neural units. Mathematical Biosciences & Engineering, 2014, 11 (2) : 217-231. doi: 10.3934/mbe.2014.11.217
[17]

Mi-Young Kim. Uniqueness and stability of positive periodic numerical solution of an epidemic model. Discrete and Continuous Dynamical Systems - B, 2007, 7 (2) : 365-375. doi: 10.3934/dcdsb.2007.7.365
[18]

Aniello Buonocore, Luigia Caputo, Enrica Pirozzi, Maria Francesca Carfora. A leaky integrate-and-fire model with adaptation for the generation of a spike train. Mathematical Biosciences & Engineering, 2016, 13 (3) : 483-493. doi: 10.3934/mbe.2016002
[19]

Giuseppe D'Onofrio, Enrica Pirozzi. Successive spike times predicted by a stochastic neuronal model with a variable input signal. Mathematical Biosciences & Engineering, 2016, 13 (3) : 495-507. doi: 10.3934/mbe.2016003
[20]

Yajing Zhang, Xinfu Chen, Jianghao Hao, Xin Lai, Cong Qin. Dynamics of spike in a Keller-Segel's minimal chemotaxis model. Discrete and Continuous Dynamical Systems, 2017, 37 (2) : 1109-1127. doi: 10.3934/dcds.2017046

2021 Impact Factor: 1.273

Tools

Metrics

  • PDF downloads (69)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy