October  2008, 20(4): 1123-1139. doi: 10.3934/dcds.2008.20.1123

Stability of traveling waves with critical speeds for $P$-degree Fisher-type equations

1. 

Department of Mathematics, Capital Normal University, Beijing 100037

2. 

College of Applied Science, Beijing University of Technology, Beijing 100022

Received  January 2007 Revised  October 2007 Published  January 2008

This paper is concerned with the stability of the traveling front solutions with critical speeds for a class of $p$-degree Fisher-type equations. By detailed spectral analysis and sub-supper solution method, we first show that the traveling front solutions with critical speeds are globally exponentially stable in some exponentially weighted spaces. Furthermore by Evans's function method, appropriate space decomposition and detailed semigroup decaying estimates, we can prove that the waves with critical speeds are locally asymptotically stable in some polynomially weighted spaces, which verifies some asymptotic phenomena obtained by numerical simulations.
Citation: Yaping Wu, Xiuxia Xing. Stability of traveling waves with critical speeds for $P$-degree Fisher-type equations. Discrete and Continuous Dynamical Systems, 2008, 20 (4) : 1123-1139. doi: 10.3934/dcds.2008.20.1123
[1]

Ramon Plaza, K. Zumbrun. An Evans function approach to spectral stability of small-amplitude shock profiles. Discrete and Continuous Dynamical Systems, 2004, 10 (4) : 885-924. doi: 10.3934/dcds.2004.10.885

[2]

Hongmei Cheng, Rong Yuan. Existence and asymptotic stability of traveling fronts for nonlocal monostable evolution equations. Discrete and Continuous Dynamical Systems - B, 2017, 22 (7) : 3007-3022. doi: 10.3934/dcdsb.2017160

[3]

Mohammad Ghani, Jingyu Li, Kaijun Zhang. Asymptotic stability of traveling fronts to a chemotaxis model with nonlinear diffusion. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6253-6265. doi: 10.3934/dcdsb.2021017

[4]

Peter Howard, K. Zumbrun. The Evans function and stability criteria for degenerate viscous shock waves. Discrete and Continuous Dynamical Systems, 2004, 10 (4) : 837-855. doi: 10.3934/dcds.2004.10.837

[5]

Shi-Liang Wu, Tong-Chang Niu, Cheng-Hsiung Hsu. Global asymptotic stability of pushed traveling fronts for monostable delayed reaction-diffusion equations. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 3467-3486. doi: 10.3934/dcds.2017147

[6]

Todd Kapitula, Björn Sandstede. Eigenvalues and resonances using the Evans function. Discrete and Continuous Dynamical Systems, 2004, 10 (4) : 857-869. doi: 10.3934/dcds.2004.10.857

[7]

Yuri Latushkin, Alim Sukhtayev. The Evans function and the Weyl-Titchmarsh function. Discrete and Continuous Dynamical Systems - S, 2012, 5 (5) : 939-970. doi: 10.3934/dcdss.2012.5.939

[8]

Cheng-Hsiung Hsu, Jian-Jhong Lin. Stability analysis of traveling wave solutions for lattice reaction-diffusion equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (5) : 1757-1774. doi: 10.3934/dcdsb.2020001

[9]

Hongmei Cheng, Rong Yuan. Multidimensional stability of disturbed pyramidal traveling fronts in the Allen-Cahn equation. Discrete and Continuous Dynamical Systems - B, 2015, 20 (4) : 1015-1029. doi: 10.3934/dcdsb.2015.20.1015

[10]

Lina Wang, Xueli Bai, Yang Cao. Exponential stability of the traveling fronts for a viscous Fisher-KPP equation. Discrete and Continuous Dynamical Systems - B, 2014, 19 (3) : 801-815. doi: 10.3934/dcdsb.2014.19.801

[11]

Zhen-Hui Bu, Zhi-Cheng Wang. Stability of pyramidal traveling fronts in the degenerate monostable and combustion equations Ⅰ. Discrete and Continuous Dynamical Systems, 2017, 37 (5) : 2395-2430. doi: 10.3934/dcds.2017104

[12]

Hirokazu Ninomiya, Masaharu Taniguchi. Global stability of traveling curved fronts in the Allen-Cahn equations. Discrete and Continuous Dynamical Systems, 2006, 15 (3) : 819-832. doi: 10.3934/dcds.2006.15.819

[13]

Dmitry Jakobson and Iosif Polterovich. Lower bounds for the spectral function and for the remainder in local Weyl's law on manifolds. Electronic Research Announcements, 2005, 11: 71-77.

[14]

Shiwang Ma, Xiao-Qiang Zhao. Global asymptotic stability of minimal fronts in monostable lattice equations. Discrete and Continuous Dynamical Systems, 2008, 21 (1) : 259-275. doi: 10.3934/dcds.2008.21.259

[15]

Kun Li, Jianhua Huang, Xiong Li. Asymptotic behavior and uniqueness of traveling wave fronts in a delayed nonlocal dispersal competitive system. Communications on Pure and Applied Analysis, 2017, 16 (1) : 131-150. doi: 10.3934/cpaa.2017006

[16]

Björn Sandstede, Arnd Scheel. Evans function and blow-up methods in critical eigenvalue problems. Discrete and Continuous Dynamical Systems, 2004, 10 (4) : 941-964. doi: 10.3934/dcds.2004.10.941

[17]

Linghai Zhang. Wave speed analysis of traveling wave fronts in delayed synaptically coupled neuronal networks. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 2405-2450. doi: 10.3934/dcds.2014.34.2405

[18]

Wei-Jie Sheng, Wan-Tong Li. Multidimensional stability of time-periodic planar traveling fronts in bistable reaction-diffusion equations. Discrete and Continuous Dynamical Systems, 2017, 37 (5) : 2681-2704. doi: 10.3934/dcds.2017115

[19]

Zhen-Hui Bu, Zhi-Cheng Wang. Global stability of V-shaped traveling fronts in combustion and degenerate monostable equations. Discrete and Continuous Dynamical Systems, 2018, 38 (5) : 2251-2286. doi: 10.3934/dcds.2018093

[20]

Shi-Liang Wu, Wan-Tong Li, San-Yang Liu. Exponential stability of traveling fronts in monostable reaction-advection-diffusion equations with non-local delay. Discrete and Continuous Dynamical Systems - B, 2012, 17 (1) : 347-366. doi: 10.3934/dcdsb.2012.17.347

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (138)
  • HTML views (0)
  • Cited by (32)

Other articles
by authors

[Back to Top]