Exponential Stability of Traveling Wavefronts for ReactionDiffusion Equations With Delayed Nonlocal Responses

LI Panxiao

doi:10.21656/1000-0887.380336

Volume 39 Issue 11

Turn off MathJax

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2018 > 39(11): 1300-1312

LI Panxiao. Exponential Stability of Traveling Wavefronts for ReactionDiffusion Equations With Delayed Nonlocal Responses[J]. Applied Mathematics and Mechanics, 2018, 39(11): 1300-1312. doi: 10.21656/1000-0887.380336

Citation:

PDF( 410 KB)

Exponential Stability of Traveling Wavefronts for ReactionDiffusion Equations With Delayed Nonlocal Responses

doi: 10.21656/1000-0887.380336

LI Panxiao

School of Mathematics and Statistics, Xidian University, Xi’an 710071, P.R.China

Funds: The National Natural Science Foundation of China(11671315)

Received Date: 2017-12-28
Rev Recd Date: 2018-02-02
Publish Date: 2018-11-01

Abstract

Abstract

A spatially nonlocal diffusion model with a class of delayed nonlocal responses was considered. The asymptotic stability and the convergence rate of the traveling wavefronts were mainly studied. Through construction of weighted functions and establishment of a comparison principle for the related linear equations, the conclusion that if the initial function is within a bounded distance from a certain traveling wavefront with respect to a weighted maximum norm, the solution satisfying the initial value will converge to the traveling wavefront exponentially in time, was proved, and the exponential convergence rate was also obtained.
- traveling wave solution,
- nonlocal time delay,
- stability

FullText(HTML)

References(26)

References

[1]	FANG J, WEI J, ZHAO X Q. Spatial dynamics of a nonlocal and time-delayed reaction-diffusion system[J]. Journal of Differential Equations,2008,245(10): 2749-2770.
[2]	GOURLEY S A, WU J. Delayed non-local diffusive systems in biological invasion and disease spread[J]. Nonlinear Dynamics & Evolution Equations,2006,48: 137-200.
[3]	OU C, WU J. Persistence of wavefronts in delayed nonlocal reaction-diffusion equations[J]. Journal of Differential Equations,2007,235(1): 219-261
[4]	JOSEPH W H S, WU J, ZOU X. A reaction-diffusion model for a single species with age structure I: travelling wavefronts on unbounded domains[C]//Proceedings: Mathematical, Physical and Engineering Sciences,2012,457: 1841-1853.
[5]	THIEME H R, ZHAO X Q. Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models[J]. Journal of Differential Equations,2003,195(2): 430-470.
[6]	WANG Z C, LI W T, RUAN S. Traveling fronts in monostable equations with nonlocal delayed effects[J]. Journal of Dynamics & Differential Equations,2008,20(3): 573-607.
[7]	BRITTON N F. Spatial structures and periodic travelling waves in an integro-differential reaction-diffusion population model[J]. SIAM Journal on Applied Mathematics,1990,50(6): 1663-1688.
[8]	BRITTON N F. Aggregation and the competitive exclusion principle[J]. Journal of Theoretical Biology,1989,136(1): 57-66.
[9]	BATES P W, FIFE P C, REN X, et al. Traveling waves in a convolution model for phase transitions[J]. Archive for Rational Mechanics & Analysis,1997,138(2): 105-136.
[10]	BATES P W, CHMAJ A. A discrete convolution model for phase transitions[J]. Archive for Rational Mechanics & Analysis,1999,150(4): 281-368.
[11]	COVILLE J. On uniqueness and monotonicity of solutions of non-local reaction diffusion equation[J]. Annali di Matematica Pura ed Applicata,2006,185(3): 461-485.
[12]	COVILLE J, DUPAIGNE L. Propagation speed of travelling fronts in non local reaction-diffusion equations[J]. Nonlinear Analysis: Theory, Methods & Applications,2005,60(5): 797-819.
[13]	COVILLE J, DUPAIGNE L. On a non-local eqution arising in population dynamics[J]. Proceedings of the Royal Society of Edinburgh Section A: Mathematics,2007,137(4): 727-755.
[14]	YU Z, YUAN R. Existence, asymptotic and uniqueness of traveling waves for nonlocal diffusion systems with delayed nonlocal response[J]. Taiwanese Journal of Mathematics,2013,17(6): 2163-2190.
[15]	GUO S, ZIMMER J. Travelling wavefronts in nonlocal diffusion equations with nonlocal delay effects[J]. Bulletin of the Malaysian Mathematical Sciences Society,2018,41(2): 919-943.
[16]	CHENG H, YUAN R. Stability of traveling wave fronts for nonlocal diffusion equation with delayed nonlocal response[J]. Taiwanese Journal of Mathematics,2016,20(4): 801-822.
[17]	GUO S, ZIMMER J. Stability of travelling wavefronts in discrete reaction-diffusion equations with nonlocal delay effects[J]. Nonlinearity,2015,28(2): 463-492.
[18]	MEI M, OU C, ZHAO X Q. Global stability of monostable traveling waves for nonlocal time-delayed reaction-diffusion equations[J]. SIAM Journal on Mathematical Analysis,2010,42(6): 2762-2790.
[19]	MA S, ZHAO X Q. Global asymptotic stability of minimal fronts in monostable lattice equations[J]. Discrete and Continuous Dynamical Systems,2008,21(1): 259-275.
[20]	OUYANG Z, OU C. Global stability and convergence rate of traveling waves for a nonlocal model in periodic media[J]. Discrete and Continuous Dynamical Systems (Series B),2012,17(3): 993-1007.
[21]	邢伟, 颜七笙, 杨志辉, 等. 一类具有非线性传染率的SEIS传染病模型的稳定性分析[J]. 应用数学和力学, 2016, 37(11): 1247-1254.(XING Wei, YAN Qisheng, YANG Zhihui. Stability analysis of an SEIS epidemic model with a nonlinear incidence rate[J]. Applied Mathematics and Mechanics,2016,37(11): 1247-1254.(in Chinese))
[22]	谢英超, 程燕, 贺天宇. 一类具有非线性发生率的时滞传染病模型的全局稳定性[J]. 应用数学和力学, 2015,36(10): 1107-1116.(XIE Yingchao, CHENG Yan, HE Tianyu. Global stability of a class of delayed epidemic models with nonlinear incidence rates[J]. Applied Mathematics and Mechanics,2015,36(10): 1107-1116.(in Chinese))
[23]	WU S L, CHEN G. Uniqueness and exponential stability of traveling wave fronts for a multi-type SIS nonlocal epidemic model[J]. Nonlinear Analysis: Real World Applications,2017,36: 267-277.
[24]	CHEN X, GUO J S. Existence and asymptotic stability of traveling waves of discrete quasilinear monostable equations[J]. Journal of Differential Equations,2002,184(2): 549-569.
[25]	WANG X S, ZHAO X Q. Pulsating waves of a partially degenerate reaction-diffusion system in a periodic habitat[J]. Journal of Differential Equations,2015,259(12): 7238-7259.
[26]	WU Y, XING X. Stability of traveling waves with critical speeds for p-degree Fisher-type equations[J]. Discrete and Continuous Dynamical Systems,2008,20(4): 1123-1139.