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研究生: 林穎志
Ying-Chih Lin
論文名稱: 摺積型非線性項之網格動態系統研究
On a lattice dynamical system with convolution type nonlinearity
指導教授: 郭忠勝
Guo, Jong-Shenq
學位類別: 博士
Doctor
系所名稱: 數學系
Department of Mathematics
論文出版年: 2012
畢業學年度: 100
語文別: 英文
論文頁數: 59
中文關鍵詞: 網格動態系統摺積型行進波全域解
英文關鍵詞: lattice dynamical system, convolution type, traveling wave, entire solution
論文種類: 學術論文
相關次數: 點閱:126下載:1
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  • 在本篇論文中,我們研究非線性項為摺積型的網格動態系統。本論文分成兩個部份。在第一部分,我們考慮在非線性項為單穩定摺積型的網格動態系統中的行進波,我們討論行進波的漸近行為、單調性及唯一性。在第二部份中,我們考慮在非線性項為雙穩定摺積型的網格動態系統下之雙波峰全域解。我們建構三種不同類型之雙波峰全域解。在當時間趨近負無窮大時,每一種雙波峰全域解的行為近似於連接三個平衡態中的其中兩個平衡態的兩個行進波。

    In this thesis, we study a lattice dynamical system with convolution type nonlinearity. This thesis is divided into two parts. In the first part, we consider traveling front solutions of a lattice dynamical system with monostable convolution type nonlinearity. We discuss the asymptotic behaviors, monotonicity and uniqueness of traveling waves. Then, in the second part, we consider two-front entire solutions of a lattice dynamical system with bistable convolution type nonlinearity. We construct three different types of two-front entire solutions. Each of these entire solutions behaves as two traveling fronts connecting two of those three equilibria as time approaches minus infinity.

    1 Introduction …………………………………………………………1 1.1 Traveling wave solution in the monostable case ……1 1.2 Two-front entire solutions …………………………………2 2 Traveling wave solution in the monostable case ……… 5 2.1 Introduction ………………………………………………………5 2.2 Preliminaries ………………………………………………………9 2.3 Asymptotic behavior ……………………………………………23 2.4 Monotonicity and the proof of Theorem 2.1.4 ………24 2.5 Uniqueness ……………………………………………………… 27 3 Two-front entire solutions ……………………………………33 3.1 Introduction ………………………………………………………33 3.2 Preliminaries ……………………………………………………38 3.3 Entire solutions ……………………………………………… 41 3.3.1 Proof of Theorem 3.1.1 ……………………………………41 3.3.2 Proof of Theorem 3.1.2 …………………………………… 45 3.3.3 Proof of Theorem 3.1.3 …………………………………… 53 4 References………………………………………………………………55

    [1] P. W. Bates, X. Chen, A. Chmaj, Traveling waves of bistable dynamics on a lattice, SIAM J. Math. Anal. 35(2003), 520--546.
    [2] P. W. Bates, P. C. Fife, X. Ren, X. Wang, Traveling waves in a nonlocal model of phase transitions, Arch. Ration. Mech. Anal. 138(1997), 105--136.
    [3] J. Carr, A. Chmaj, Uniqueness of traveling waves for nonlocal monostable equations, Proc. Amer. Math. Soc. 132(2004), 2433--2439.
    [4] X. Chen, Existence, uniqueness, and asymptotic stability of traveling waves
    in nonlocal evolution equations , Advances in Differential Equation. 2(1997), 125--160.
    [5] X. Chen, S.-C. Fu, J.-S. Guo, Uniqueness and asymptotics of traveling waves of monostable dynamics on lattices, SIAM J. Math. Anal. 38(2006), 233--258.
    [6] X. Chen, J.-S. Guo, Existence and asymptotic stability of traveling waves of a
    discrete monostable equation, J. Differential Equations. 184(2002), 549--569.
    [7] X. Chen, J.-S. Guo, Uniqueness and existence of traveling waves for discrete
    quasilinear monostable dynamics, Math. Ann. 326(2003), 123--146.
    [8] X. Chen, J.-S. Guo, Existence and uniqueness of entire solutions for a reaction-diffusion equation, J. Differential Equations. 212(2005), 62--84.
    [9] X. Chen, J.-S. Guo, C.-C. Wu, Traveling waves in discrete periodic media for bistable dynamics, Arch. Rational Mech. Anal. 189(2008), 189--236.
    [10] J. Coville, L. Dupaigne, Travelling fronts in integrodifferential equations, C. R.
    Math. Acad. Sci. Paris. 337(2003), 25--30.
    [11] J. Coville, L. Dupaigne, Propagation speed of travelling fronts in non local
    reaction-diffusion equations, Nonlinear Anal. 60(2005), 797--819.
    [12] J. Coville, L. Dupaigne, On a non-local equation arising in population dynamics, Proc. Roy. Soc. Edinburgh Sect. A. 137 (2007), 727--755.
    [13] S.-C. Fu, J.-S. Guo, S.-Y. Shieh,
    Traveling wave solutions for some discrete quasilinear parabolic equations , Nonlinear Analysis. 48(2002), 1137--1149.
    [14] Y. Fukao, Y. Morita, H. Ninomiya, Some entire solutions of the Allen-Cahn equation, Taiwanese J. Math. 8(2004), 15--32.
    [15] J.-S. Guo, Y.-C. Lin, Traveling wave solution for a lattice dynamical system with
    convolution type nonlinearity, Discrete Contin. Dyn. Syst. 32(2012), 101--124.
    [16] J.-S. Guo, Y.-C. Lin, Entire solutions for a discrete diffusive equation with bistable convolution type nonlinearity, Osaka J. Math. (to appear).
    [17] J.-S. Guo, Y. Morita, Entire solutions of reaction-diffusion equations and an application to discrete diffusive equations, Discrete Contin. Dyn. Syst. 12(2005), 193--212.
    [18] Y.-J.~Lin~Guo, Entire solutions for a discrete diffusive equation, J. Math.
    Anal. Appl. 347(2008), 450--458.
    [19] F. Hamel, N. Nadirashvili, Entire solutions of the KPP equation, Comm. Pure Appl. Math. 52(1999), 1255--1276.
    [20] F. Hamel, N. Nadirashvili, Travelling fronts and entire solutions of the Fisher-KPP equation in , Arch. Ration. Mech. Anal. 157(2001), 91--163.
    [21] W. Hudson, B. Zinner, Existence of traveling waves for a generalized discrete Fisher's equation, Comm. Appl. Nonlinear Analysis. 1(1994), 23--46
    [22] G. Lv, Asymptotic behavior of traveling fronts and entire solutions for a
    nonlocal monostable equation, Nonlinear Analysis. 72(2010), 3659--3668.
    [23] S. Ma, P. Weng, X. Zou, Asymptotic speed of propagation and traveling wavefronts in a non-local delayed lattice differential equation, Nonlinear Analysis. 65(2006), 1858--1890.
    [24] S. Ma, X. Zou, Existence, uniqueness and stability of traveling waves in a
    discrete reaction-diffusion monostable equation with delay, J. Differential Equations. 217(2005), 54--87.
    [25] S. Ma, X. Zou, Propagation and its failure in a lattice delayed differential
    equation with global interaction, J. Differential Equations. 212(2005), 129--190.
    [26] Y. Morita, H. Ninomiya, Entire solutions with merging fronts to reaction-diffusion equations, J. Dynam. Differential Equations. 18(2006), 841--861.
    [27] K. Schumacher, Traveling fronts solutions for integro-differential equations I,
    J. Reine Angew. Math. 316(1980), 54--70.
    [28] Z.-C. Wang, W.-T. Li, J. Wu, Entire solutions in delayed lattice differential equations with monostable nonlinearity, SIAM J. Math. Anal. 40(2009), 2392--2420.
    [29] P. X. Weng, H. X. Huang, J.H. Wu, Asymptotic speed of propagation of wave fronts in a lattice delay differential equation with gloval interaction, IMA J. Appl.
    Math. 68(2003), 409--439.
    [30] B. Zinner, Stability of traveling wavefronts for the discrete Nagumo equation, SIAM J. Math. Anal. 22(1991), 1016--1020.
    [31] B. Zinner, Existence of traveling wavefronts for the discrete Nagumo equation, J. Differential Equations. 96(1992), 1--27.
    [32] B.Zinner, G. Harris, W. Hudson, Traveling wavefronts for the discrete Fisher's equation, J. Differential Equations.105(1993), 46--62.

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