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The Stability of Travelling Waves for Generalized Fisher Equation(Systems) and Viscous Balance Law

Author: XingXiuXia
Tutor: WuYaPing
School: Capital Normal University
Course: Basic mathematics
Keywords: Traveling wave solutions Algebraic decay Spectral analysis Evans function Semigroup Comparison principle Asymptotically stable Global stability
CLC: O175.1
Type: PhD thesis
Year: 2005
Downloads: 117
Quote: 6
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Abstract


This article consists of three parts: the first part of the study before the generalized Fisher equation wave asymptotic stability of. The classic spectral analysis method and semigroup methods combined, we have been traveling wave solutions with critical and non-critical wave speed in the weighted space the locally asymptotically index stability. On this basis, the use of the comparison principle, prove the critical wave speed traveling wave solutions exponentially weighted space of global asymptotic stability. Further, the Evans function method, the appropriate space decomposition semigroup method and the classic combination of spectral analysis method, we were obtained the locally asymptotically algebra with critical and non-critical wave speed traveling wave solutions in polynomial weighted space stability. Stability of traveling wave solutions with non-critical wave speed in the study, due to the traveling wave attenuation ∞ algebra at a slower rate, which leads to research linearization operator eigenvalue problem, often not directly based on the classic asymptotic theory of differential equations to determine the solution of the eigenvalue problem ∞ the asymptotic behavior, making it impossible to define the Evans function. By using more general asymptotic theory of ordinary differential equations, we get a detailed portrait ∞ the asymptotic behavior of the solution of the eigenvalue problem. Based on this, we can define the Evans function to construct the Evans function here and prove that the same has to resolve, and still corresponding linear operator eigenvalues ??zero, which indicates that the Evans function definition is extended to more general situation. Stability of traveling wave solutions in the critical wave speed, we verify that the Evans function D (λ) has some important properties of linear operator of the MS is not zero, especially at the origin: D (0 ) = 0, but D_λ (0) ≠ 0, this to the appropriate spatial decomposition and semigroup estimate is very useful. The second part of a class of viscous balance law equation asymptotic stability of traveling wave solutions. First, the nature of the use of the method of spectral analysis, comparison principle, and ω-limit sets, we prove asymptotic exponential stability of traveling wave solutions for the global general viscous balance law equation connecting the saddle point, this results generalize the classic reaction-diffusion saddle saddle wave equation global stability results. Secondly, we study the existence and stability of viscous balance law of traveling wave solutions degraded. Phase plane analysis, we obtain the existence of solutions in its wavefront. On this basis, the use of the Evans function method, semigroup method and the method of spectral analysis, similar to the first part, we obtain the local asymptotic stability of traveling wave solutions exponentially weighted space and polynomial weighted space. Stability of traveling wave solutions of the third part of the study of a class of self-catalytic chemical reaction equations. Clever spectral analysis and semigroup theory, we get the diffusion coefficient d = 1 has local asymptotic critical wave speed and non-critical Traveling Waves in the appropriate weighted space exponential stability; addition, we preliminarily diffusion The coefficient d ≠ the spectral properties of the linearity of the operator.

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CLC: > Mathematical sciences and chemical > Mathematics > Mathematical Analysis > Differential equations, integral equations > Ordinary Differential Equations
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