拟单调中立型反应扩散方程行波解的唯一性
Uniqueness of Traveling Wave Solutions for a Quasi-Monotone Reaction-Diffusion Equation with Neutral Type
DOI: 10.12677/PM.2017.74041, PDF, HTML, XML,  被引量 下载: 1,500  浏览: 4,908  国家自然科学基金支持
作者: 刘玉彬*：惠州学院数学与大数据学院，广东 惠州
关键词: 中立型反应扩散方程；行波解；拟单调反应项；唯一性；Reaction-Diffusion Equation with Neutral Type； Traveling Wave Solution； Quasi-Monotone Reaction； Uniqueness
摘要: 本文考虑了具有拟单调反应项的中立型反应扩散方程行波解的唯一性问题。首先通过线性变换将方程化为具有无限离散时滞的反应扩散方程，并利用Ikehara定理得到了无限时滞方程波速 的单调行波解的指数渐近性态，进而得到方程波速 的单调行波解是唯一的(平移意义下)，最后根据两类方程的解之间的联系得到了原中立型方程波速 的单调行波解的唯一性(平移意义下)。
Abstract: In present paper, we focus on the uniqueness of traveling wave solutions for a quasi-monotone reaction-diffusion equation with neutral type. By using the Ikehara’s Theorem, we firstly establish the asymptotic exponent properties of monotone traveling wave solution with speed for the reaction-diffusion equation with a infinite number of delays, which is transformed from the neutral equation by a linear variable transform, and then the uniqueness (up to translation) of monotone traveling wave solution with speed for the transformed equation. Finally, we obtain the uniqueness (up to translation) of monotone traveling wave solution with speed for the neutral equation by using the relation between solutions of the neutral equation and of the transformed equation.
文章引用：刘玉彬. 拟单调中立型反应扩散方程行波解的唯一性[J]. 理论数学, 2017, 7(4): 310-321. https://doi.org/10.12677/PM.2017.74041

参考文献

[1] Schaaf, K.W. (1987) Asymptotic Behavior and Traveling Wave Solutions for Parabolic Functional Differential Equations. Transactions of the American Mathematical Society, 302, 587-615.
[2] Chow, S.N., Paret, M.J. and Shen, W.X. (1998) Travelling Waves in Lattice Dynamical Systems. Journal of Differential Equations, 149, 248-291.
[3] Ma, S.W. (2001) Traveling Wavefronts for Delayed Reaction-Diffusion Systems via a Fixed Point Theorem. Journal of Differential Equations, 171, 294-314.
[4] Al-Omari, J. and Gourley, S.A. (2002) Monotone Traveling Fronts in Age-Structured Reaction Diffusion Model of a Single Species. Journal of Mathematical Biology, 45, 294-312.
https://doi.org/10.1007/s002850200159
[5] Al-Omari, J. and Gourley, S.A. (2005) Monotone Wavefront in a Structured Population Model with Distributed Maturation Delay. IMA Journal of Applied Mathematics, 70, 858-879.
https://doi.org/10.1093/imamat/hxh073
[6] 黄建华, 黄立宏. 高维格上时滞反应扩散方程的行波解[J]. 应用数学学报, 2005(1): 100-113.
[7] Liang, X. and Zhao, X.Q., (2007) Asymptotic Spreading Speeds of Spread and Traveling Waves for Monotone Semiflows with Application. Communications on Pure Applied Mathematics, 60, 1-40. Erratum: (2008) 61, 137-138.
[8] Wang, H.Y. (2009) On the Existence of Traveling Waves for Delayed Reaction-Diffusion Equations. Journal of Differential Equations, 247, 887-905.
[9] 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(5): 1273-1281.
[10] Fang, J. and Zhao, X.Q. (2014) Traveling Waves for Monotone Semiflows with Weak Compactness. SIAM Journal on Mathematical Analysis, 46, 3678-3704.
[11] 黄业辉, 翁佩萱. 一个具有非局部效应的非线性周期反应扩散方程的渐近形态[J]. 数学学报, 2014(5): 1011- 1030.
[12] 尚德生, 张耀明. 具有分布时滞的一类反应扩散方程的波前解的存在性[J]. 生物数学学报, 2015(3): 415-424.
[13] 边乾坤, 张卫国, 余志先. 具有时滞的时间离散反应扩散系统行波解的存在性[J]. 应用数学, 2016(2): 359-369.
[14] Diekmann, O. and Kapper, H.G. (1978) On the Bounded Solutions of a Nonlinear Convolution Equation, Nonlinear Analysis. Theory, Methods and Applications, 2, 721-737.
[15] Fang, J. and Zhao, X.Q. (2010) Existence and Uniqueness of Traveling Waves for Non-Monotone Integral Equations with Applications. Journal of Differential Equations, 248, 2199-2226.
[16] Aguerrea, M., Gomez, C. and Trofimchuk, S. (2012) On Uniqueness of Semi-Wave Fronts: Diekmann-Kapper Theory of a Nonlinear Convolution Equation Re-Visited. Mathemtische Annalen, 354, 73-109.
https://doi.org/10.1007/s00208-011-0722-8
[17] Xu, Z.Q. and Xiao, D.M. (2016) Spreading Speeds and Uniqueness of Traveling Waves for a Reaction Diffusion Equation with Spatio-Temporal Delays. Journal of Differential Equations, 260, 268-303.
[18] Chen, X.F. and Guo, J.S. (2003) Uniqueness and Existence of Traveling Waves for Discrete Quasilinear Monostable Dynamics. Mathematische Annalen, 326, 123-146.
https://doi.org/10.1007/s00208-003-0414-0
[19] Carr, J. and Chmaj, A. (2004) Uniqueness of Traveling Waves for Nonlocal Monostable Equations. Proceedings of American Mathematical Society, 132, 2433-2439.
https://doi.org/10.1090/S0002-9939-04-07432-5
[20] Ma, S.W. and Zou, X.F. (2005) Existence, Uniqueness and Stability of Travelling Waves in a Discrete Reaction-Diffusion Monostable Equation with Delay. Journal of Differential Equations, 217, 54-87.
[21] Aguerrea, M., Trofimchuk, S. and Valenzuela, G. (2008) Uniqueness of Fast Travelling Fronts in Reaction-Diffusion Equations with Delay. Proceedings of the Royal Society A, 464, 2591-2608.
https://doi.org/10.1098/rspa.2008.0011
[22] Guo, J.S. and Wu, C.H. (2008) Existence and Uniqueness of Traveling Waves for a Monostable 2-D Lattice Dynamical System. Osaka Journal of Mathematics, 45, 327-346.
[23] Wu, S.L. and Liu, S.Y. (2009) Uniqueness of Non-Monotone Traveling Waves for Delayed Reaction-Diffusion Equations. Applied Mathematics Letters, 22, 1056-1061.
[24] Zhao, H.Q., Weng, P.X. and Liu, S.Y. (2014) Uniqueness and Stability of Traveling Wave Fronts for an Age-Structured Population Model in a 2D Lattice Strip. Communications in Nonlinear Science and Numerical Simulation, 19, 507-519.
[25] Yang, Y.R. and Liu, N.W. (2014) Monotonicity and Uniqueness of Traveling Waves in Bistable Systems with Delay. Electronic Journal of Differential Equations, No. 2, 1-12.
[26] Liu, Y.B. and Weng, P.X. (2015) Asymptotic Pattern for a Partial Neutral Functional Differential Equation. Journal of Differential Equations, 258, 3688-3741.

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