非局部时滞反应扩散方程行波解的存在性

数学物理学报 ›› 2011, Vol. 31 ›› Issue (5): 1273-1281.

非局部时滞反应扩散方程行波解的存在性

梁飞^1,2, 高洪俊¹

1.南京师范大学数学科学学院南京 210046|2.安徽科技学院理学院数学系安徽凤阳 233100

收稿日期:2009-10-20 修回日期:2011-08-20 出版日期:2011-10-25 发布日期:2011-10-25
基金资助:
国家自然科学基金(10871097)和安徽省高等学校省级优秀人才基金(2011SQRL115)资助

Existence of Travelling Wave Solutions for a Reaction-diffusion Equation with Nonlocal Delay

LIANG Fei^1,2, GAO Hong-Jun¹

1.Institute of Mathematics, School of Mathematical Science, Nanjing Normal University, Nanjing 210046;
2.Department of Mathematics, An Hui Science &|Technology University, Anhui Fengyang |233100

Received:2009-10-20 Revised:2011-08-20 Online:2011-10-25 Published:2011-10-25
Supported by:
国家自然科学基金(10871097)和安徽省高等学校省级优秀人才基金(2011SQRL115)资助

摘要/Abstract

摘要：

该文主要考虑非局部时滞反应扩散方程行波解的存在性. 对于特殊的核函数, 通过线性链技巧和几何奇异扰动理论有机结合, 建立了带有非局部时滞反应扩散方程和对应的不带时滞反应扩散方程行波解存在性之间的自然联系. 得到如果不带时滞反应扩散方程行波解存在, 则在时滞充分小的条件下对应的带时滞反应扩散方程行波解也存在.

关键词: 行波解, 非局部时滞, 几何奇异扰动理论

Abstract:

This paper is concerned with the existence of a reaction-diffusion equation with nonlocal delay. For special kernels, by linear chain trick and geometric singular perturbation theory, the authors consider a natural connection between the existence of travelling wave solutions for the reaction-diffusion equation with nonlocal delay and the existence of travelling wave solutions for the corresponding undelayed reaction-diffusion equation. It is showed that if the corresponding undelayed reaction-diffusion equation has a travelling wave solution, then the reaction-diffusion with nonlocal delay also has a travelling wave solution for any sufficiently small delay.

Key words: Travelling wave solution, Nonlocal delay, Geometric singular perturbation

中图分类号:

35K57

梁飞, 高洪俊. 非局部时滞反应扩散方程行波解的存在性[J]. 数学物理学报, 2011, 31(5): 1273-1281.

LIANG Fei, GAO Hong-Jun. Existence of Travelling Wave Solutions for a Reaction-diffusion Equation with Nonlocal Delay[J]. Acta mathematica scientia,Series A, 2011, 31(5): 1273-1281.

参考文献

[1] Ai S. Traveling wavefronts for generalized Fisher equation with spatio-temporal delays. J Differential Equations, 2007, 232: 104--133

[2] Al-omari J, Gourley A S. Monotone traveling fronts in age-structed reaction-diffusion of a single species. J Math Boil, 2002, 45: 294--312
[3] Al-omari J, Gourley A S. Monotone wave-fronts in a structed population model with distributed maturation delay. IMA J Appl Mtah, 2005, 70: 858--879

[4] Britton F N. Aggregation and competitive exclusion principle. J Theoret Biol, 1989, 136: 57--66

[5] Britton F N. Spatial structures and periodic travelling waves in an intego-differential reaction-diffusion population model. SIAM J Appl Math, 1990, 50: 1663--1688

[6] Fenichel N. Geometric singular perturbation theory for ordinary differential equations. J Differential Equations, 1979, 31: 53--98

[7] Fisher A P. The advance of advantageous genes. Ann Eugenics, 1937, 7: 355--369

[8] Feng W, Lu X. On diffusive population models with toxicants and time delays. J Math Anal Appl, 1999, 233: 373--386

[9] Gourley A S. Travelling fronts in the diffusive Nicholson's blowflied equation with distributed delays. Math Comput Modelling, 2000, 32: 843--853

[10] Jones T R K C. Geometric Singular Perturbation Theory. Lecture Notes in Math. Berlin: Springer, 1995

[11] Lin G, Li W T. Bistable wavefronts in a diffusive and competitive Lotka-Volterra type system with delays. J Differential Equations, 2008, 244: 487--513

[12] Ma S. Travelling wavefronts for delayed reaction-diffusion systems via a fixed point theorem. J Differential Equations, 2001, 171: 294--314

[13] Ou C, Wu H J. Persistence of wavefronts in delayed nonlocal reaction-diffusion equations. J Differential Equations, 2007, 238: 219--261

[14] Yahong Peng, Yongli Song. Existence of travelling wave solutions for a reaction-diffusion equation with distributed delays. Nonlinear Anal, 2007, 67: 2415--2423

[15] Zhicheng Wang, Wantong Li, Shigui Ruan. Existence and stability of travelling wave fronts in reaction advection diffusion equations with nonlocal delay. J Differential Equations, 2007, 238: 153--200

[16] Wu J, Zou X. Travelling wave fronts of reaction diffusion systems with delay. J Dynam Diff Eqns, 2008, 20: 531--533

[17] Jianming Zhang, Yahong Peng. Travelling waves of the diffusive Nicholson's blowflies equation with strong generic delay kernel and non-local effect. Nonlinear Anal, 2008, 68: 1263--1270