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 & 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 & 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 & Continuous Dynamical Systems - B, 2004, 4 (4) : 1033-1064. doi: 10.3934/dcdsb.2004.4.1033
[3]

Shin-Ichiro Ei, Kota Ikeda, Yasuhito Miyamoto. Dynamics of a boundary spike for the shadow Gierer-Meinhardt system. Communications on Pure & Applied Analysis, 2012, 11 (1) : 115-145. doi: 10.3934/cpaa.2012.11.115
[4]

Siu-Long Lei. Adaptive method for spike solutions of Gierer-Meinhardt system on irregular domain. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 651-668. doi: 10.3934/dcdsb.2011.15.651
[5]

Georgia Karali, Takashi Suzuki, Yoshio Yamada. Global-in-time behavior of the solution to a Gierer-Meinhardt system. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2885-2900. doi: 10.3934/dcds.2013.33.2885
[6]

Juncheng Wei, Matthias Winter. On the Gierer-Meinhardt system with precursors. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 363-398. doi: 10.3934/dcds.2009.25.363
[7]

Manuel del Pino, Patricio Felmer, Michal Kowalczyk. Boundary spikes in the Gierer-Meinhardt system. Communications on Pure & Applied Analysis, 2002, 1 (4) : 437-456. doi: 10.3934/cpaa.2002.1.437
[8]

Henghui Zou. On global existence for the Gierer-Meinhardt system. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 583-591. doi: 10.3934/dcds.2015.35.583
[9]

Kota Ikeda. The existence and uniqueness of unstable eigenvalues for stripe patterns in the Gierer-Meinhardt system. Networks & Heterogeneous Media, 2013, 8 (1) : 291-325. doi: 10.3934/nhm.2013.8.291
[10]

Rui Peng, Xianfa Song, Lei Wei. Existence, nonexistence and uniqueness of positive stationary solutions of a singular Gierer-Meinhardt system. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4489-4505. doi: 10.3934/dcds.2017192
[11]

Kazuhiro Kurata, Kotaro Morimoto. Construction and asymptotic behavior of multi-peak solutions to the Gierer-Meinhardt system with saturation. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1443-1482. doi: 10.3934/cpaa.2008.7.1443
[12]

Mi-Young Kim. Uniqueness and stability of positive periodic numerical solution of an epidemic model. Discrete & Continuous Dynamical Systems - B, 2007, 7 (2) : 365-375. doi: 10.3934/dcdsb.2007.7.365
[13]

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
[14]

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
[15]

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
[16]

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
[17]

Yajing Zhang, Xinfu Chen, Jianghao Hao, Xin Lai, Cong Qin. Dynamics of spike in a Keller-Segel's minimal chemotaxis model. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 1109-1127. doi: 10.3934/dcds.2017046
[18]

Kazuhiro Kurata, Kotaro Morimoto. Existence of multiple spike stationary patterns in a chemotaxis model with weak saturation. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 139-164. doi: 10.3934/dcds.2011.31.139
[19]

Haifeng Hu, Kaijun Zhang. Stability of the stationary solution of the cauchy problem to a semiconductor full hydrodynamic model with recombination-generation rate. Kinetic & Related Models, 2015, 8 (1) : 117-151. doi: 10.3934/krm.2015.8.117
[20]

Chunhua Jin. Global classical solution and stability to a coupled chemotaxis-fluid model with logistic source. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3547-3566. doi: 10.3934/dcds.2018150

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (25)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences