American Institute of Mathematical Sciences

Advanced
December  2006, 5(4): 919-928. doi: 10.3934/cpaa.2006.5.919

Some properties for the solutions of a general activator-inhibitor model

Shaohua Chen 1,

1. 

Department of Mathematics, Physics & Geology, Cape Breton University, Sydney, NS, Canada, B1P 6L2, Canada

Received  February 2006 Revised  July 2006 Published  September 2006

This paper deals with the generalized Activator-Inhibitor model which originally arose in studies of pattern-formation in biology and has interesting and challenging mathematical properties. We study the long time existence of solutions as well as the boundedness and blowup properties for some special cases. We also obtain a priori estimates of stationary solutions followed by some numerical solutions with moving mesh methods.
Keywords: blowup solutions, stationary solutions., boundedness, Activator-Inhibitor model.
Mathematics Subject Classification: Primary: 35K57, 35B40; Secondary: 35J5.
Citation: Shaohua Chen. Some properties for the solutions of a general activator-inhibitor model. Communications on Pure and Applied Analysis, 2006, 5 (4) : 919-928. doi: 10.3934/cpaa.2006.5.919
[1]

Grzegorz Karch, Kanako Suzuki, Jacek Zienkiewicz. Finite-time blowup of solutions to some activator-inhibitor systems. Discrete and Continuous Dynamical Systems, 2016, 36 (9) : 4997-5010. doi: 10.3934/dcds.2016016
[2]

Huiqiang Jiang. Global existence of solutions of an activator-inhibitor system. Discrete and Continuous Dynamical Systems, 2006, 14 (4) : 737-751. doi: 10.3934/dcds.2006.14.737
[3]

Marie Henry. Singular limit of an activator-inhibitor type model. Networks and Heterogeneous Media, 2012, 7 (4) : 781-803. doi: 10.3934/nhm.2012.7.781
[4]

Xiaoli Wang, Guohong Zhang. Bifurcation analysis of a general activator-inhibitor model with nonlocal dispersal. Discrete and Continuous Dynamical Systems - B, 2021, 26 (8) : 4459-4477. doi: 10.3934/dcdsb.2020295
[5]

Victor Ogesa Juma, Leif Dehmelt, Stéphanie Portet, Anotida Madzvamuse. A mathematical analysis of an activator-inhibitor Rho GTPase model. Journal of Computational Dynamics, 2022, 9 (2) : 133-158. doi: 10.3934/jcd.2021024
[6]

Shanshan Chen, Junping Shi, Guohong Zhang. Spatial pattern formation in activator-inhibitor models with nonlocal dispersal. Discrete and Continuous Dynamical Systems - B, 2021, 26 (4) : 1843-1866. doi: 10.3934/dcdsb.2020042
[7]

Zejia Wang, Suzhen Xu, Huijuan Song. Stationary solutions of a free boundary problem modeling growth of angiogenesis tumor with inhibitor. Discrete and Continuous Dynamical Systems - B, 2018, 23 (6) : 2593-2605. doi: 10.3934/dcdsb.2018129
[8]

Shaohua Chen. Boundedness and blowup solutions for quasilinear parabolic systems with lower order terms. Communications on Pure and Applied Analysis, 2009, 8 (2) : 587-600. doi: 10.3934/cpaa.2009.8.587
[9]

Hua Nie, Wenhao Xie, Jianhua Wu. Uniqueness of positive steady state solutions to the unstirred chemostat model with external inhibitor. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1279-1297. doi: 10.3934/cpaa.2013.12.1279
[10]

Yanghong Huang, Andrea Bertozzi. Asymptotics of blowup solutions for the aggregation equation. Discrete and Continuous Dynamical Systems - B, 2012, 17 (4) : 1309-1331. doi: 10.3934/dcdsb.2012.17.1309
[11]

Francesca R. Guarguaglini. Stationary solutions and asymptotic behaviour for a chemotaxis hyperbolic model on a network. Networks and Heterogeneous Media, 2018, 13 (1) : 47-67. doi: 10.3934/nhm.2018003
[12]

Byung-Hoon Hwang, Seok-Bae Yun. Stationary solutions to the boundary value problem for the relativistic BGK model in a slab. Kinetic and Related Models, 2019, 12 (4) : 749-764. doi: 10.3934/krm.2019029
[13]

Yukio Kan-On. Global bifurcation structure of stationary solutions for a Lotka-Volterra competition model. Discrete and Continuous Dynamical Systems, 2002, 8 (1) : 147-162. doi: 10.3934/dcds.2002.8.147
[14]

Walter A. Strauss, Masahiro Suzuki. Large amplitude stationary solutions of the Morrow model of gas ionization. Kinetic and Related Models, 2019, 12 (6) : 1297-1312. doi: 10.3934/krm.2019050
[15]

Chi-Cheung Poon. Blowup rate of solutions of a degenerate nonlinear parabolic equation. Discrete and Continuous Dynamical Systems - B, 2019, 24 (10) : 5317-5336. doi: 10.3934/dcdsb.2019060
[16]

Cemil Tunç. Stability, boundedness and uniform boundedness of solutions of nonlinear delay differential equations. Conference Publications, 2011, 2011 (Special) : 1395-1403. doi: 10.3934/proc.2011.2011.1395
[17]

Haibo Cui, Haiyan Yin. Convergence rate of solutions toward stationary solutions to the isentropic micropolar fluid model in a half line. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 2899-2920. doi: 10.3934/dcdsb.2020210
[18]

Jiashan Zheng. Boundedness of solutions to a quasilinear higher-dimensional chemotaxis-haptotaxis model with nonlinear diffusion. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 627-643. doi: 10.3934/dcds.2017026
[19]

Lu Xu, Chunlai Mu, Qiao Xin. Global boundedness of solutions to the two-dimensional forager-exploiter model with logistic source. Discrete and Continuous Dynamical Systems, 2021, 41 (7) : 3031-3043. doi: 10.3934/dcds.2020396
[20]

Yingshan Chen, Mei Zhang. A new blowup criterion for strong solutions to a viscous liquid-gas two-phase flow model with vacuum in three dimensions. Kinetic and Related Models, 2016, 9 (3) : 429-441. doi: 10.3934/krm.2016001

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (92)
  • HTML views (0)
  • Cited by (15)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy