March  2006, 16(1): 47-66. doi: 10.3934/dcds.2006.16.47

Stability of travelling waves with algebraic decay for $n$-degree Fisher-type equations


Department of Mathematics, Capital Normal University, Beijing 100037, China


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


Department of Mathematics, Beijing Institute of Technology, Beijing 100081, China

Received  September 2005 Revised  February 2006 Published  June 2006

This paper is concerned with the asymptotic stability of travelling wave front solutions with algebraic decay for $n$-degree Fisher-type equations. By detailed spectral analysis, each travelling wave front solution with non-critical speed is proved to be locally exponentially stable to perturbations in some exponentially weighted $L^{\infty}$ spaces. Further by Evans function method and detailed semigroup estimates, the travelling wave fronts with non-critical speed are proved to be locally algebraically stable to perturbations in some polynomially weighted $L^{\infty}$ spaces. It's remarked that due to the slow algebraic decay rate of the wave at $+\infty,$ the Evans function constructed in this paper is an extension of the definitions in [1, 3, 7, 11, 21] to some extent, and the Evans function can be extended analytically in the neighborhood of the origin.
Citation: Yaping Wu, Xiuxia Xing, Qixiao Ye. Stability of travelling waves with algebraic decay for $n$-degree Fisher-type equations. Discrete & Continuous Dynamical Systems, 2006, 16 (1) : 47-66. doi: 10.3934/dcds.2006.16.47

Marcelo M. Cavalcanti, Valéria N. Domingos Cavalcanti, Irena Lasiecka, Flávio A. Falcão Nascimento. Intrinsic decay rate estimates for the wave equation with competing viscoelastic and frictional dissipative effects. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1987-2011. doi: 10.3934/dcdsb.2014.19.1987


Vu Hoang Linh, Volker Mehrmann. Spectral analysis for linear differential-algebraic equations. Conference Publications, 2011, 2011 (Special) : 991-1000. doi: 10.3934/proc.2011.2011.991


Imen Manoubi. Theoretical and numerical analysis of the decay rate of solutions to a water wave model with a nonlocal viscous dispersive term with Riemann-Liouville half derivative. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2837-2863. doi: 10.3934/dcdsb.2014.19.2837


Belkacem Said-Houari, Salim A. Messaoudi. General decay estimates for a Cauchy viscoelastic wave problem. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1541-1551. doi: 10.3934/cpaa.2014.13.1541


Kazuhiro Ishige, Yujiro Tateishi. Decay estimates for Schrödinger heat semigroup with inverse square potential in Lorentz spaces II. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021121


Arnaud Ducrot, Michel Langlais, Pierre Magal. Qualitative analysis and travelling wave solutions for the SI model with vertical transmission. Communications on Pure & Applied Analysis, 2012, 11 (1) : 97-113. doi: 10.3934/cpaa.2012.11.97


Claude-Michel Brauner, Josephus Hulshof, Luca Lorenzi. Stability of the travelling wave in a 2D weakly nonlinear Stefan problem. Kinetic & Related Models, 2009, 2 (1) : 109-134. doi: 10.3934/krm.2009.2.109


Abdelaziz Soufyane, Belkacem Said-Houari. The effect of the wave speeds and the frictional damping terms on the decay rate of the Bresse system. Evolution Equations & Control Theory, 2014, 3 (4) : 713-738. doi: 10.3934/eect.2014.3.713


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


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


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


Roberto Triggiani, Jing Zhang. Heat-viscoelastic plate interaction: Analyticity, spectral analysis, exponential decay. Evolution Equations & Control Theory, 2018, 7 (1) : 153-182. doi: 10.3934/eect.2018008


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


Yan Cui, Zhiqiang Wang. Asymptotic stability of wave equations coupled by velocities. Mathematical Control & Related Fields, 2016, 6 (3) : 429-446. doi: 10.3934/mcrf.2016010


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


Armand Bernou. A semigroup approach to the convergence rate of a collisionless gas. Kinetic & Related Models, 2020, 13 (6) : 1071-1106. doi: 10.3934/krm.2020038


Aslihan Demirkaya, Panayotis G. Kevrekidis, Milena Stanislavova, Atanas Stefanov. Spectral stability analysis for standing waves of a perturbed Klein-Gordon equation. Conference Publications, 2015, 2015 (special) : 359-368. doi: 10.3934/proc.2015.0359


Farah Abdallah, Denis Mercier, Serge Nicaise. Spectral analysis and exponential or polynomial stability of some indefinite sign damped problems. Evolution Equations & Control Theory, 2013, 2 (1) : 1-33. doi: 10.3934/eect.2013.2.1


Yaru Xie, Genqi Xu. The exponential decay rate of generic tree of 1-d wave equations with boundary feedback controls. Networks & Heterogeneous Media, 2016, 11 (3) : 527-543. doi: 10.3934/nhm.2016008


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

2020 Impact Factor: 1.392


  • PDF downloads (72)
  • HTML views (0)
  • Cited by (24)

Other articles
by authors

[Back to Top]