交叉单稳型Nicholson模型行波解的稳定性 Alternative Title Stability of Traveling Waves in Nicholson's Model With Crossing-monostability 张海波 Thesis Advisor 李万同 2012-05-27 Degree Grantor 兰州大学 Place of Conferral 兰州 Degree Name 硕士 Keyword Nicholson模型 行波解 指数稳定性 交叉单稳 加权能量方法 Abstract Nicholson苍蝇模型是在Nicholson所做的澳大利亚铜绿蝇的实验研究中所导出的一类重要的反应扩散方程，已被许多学者广泛研究。 而在反应扩散方程的研究中，行波解是一个重要课题， 行波解可以很好地刻画以有限速度传播问题和振荡现象。 因此，本文主要研究如下带有非局部时滞的Nicholson模型行波解的稳定性： \frac{\partial u(x,t)}{\partial t}=D\triangle u(x,t)-\gamma u(x,t)+\int_{-\infty}^{\infty}f_{\alpha}(y)b(u(x-y,t-\tau))dy,x\in \Bbb{R} 其中$b(u)=pue^{-au}$。当非线性项为单稳（$1<\frac{p}{\gamma}\leq e$）时，行波解的稳定性可以用加权能量方法结合比较原理得到。但当非线性项为交叉单稳（$\frac{p}{\gamma}>e$）时， 此时比较原理不再适用，我们将用加权能量方法结合连续性方法来考虑行波解的稳定性。本文在假设$e<\frac{p}{\gamma}< e^{\frac{9}{5}}$下，首先给出该方程相应的Cauchy问题解的局部存在性，利用加权能量方法建立解的先验估计。然后在解的局部存在性和先验估计的基础上，用连续性方法证明了该方程在小初始扰动（即在行波解附近的初始扰动在一个加权范数意义下是适当小的）下，大波速行波解的指数稳定性。 Other Abstract In the study of reaction-diffusion equations, traveling wave solution is an important topic, which can well describe the propagation with finite speed and the oscillations. But when the nonlinear term is crossing-monostable (~$\frac{p}{\gamma}>e$~), the comparison principle is no longer applicable. So we'll consider the stability of the traveling waves by means of the weighted energy method combining continuation method. In this paper under the assumption of $e<\frac{p}{\gamma}< e^{\frac{9}{5}}$, we first state the local existence of solutions of the Cauchy problem for the equation, and establish a priori estimate with the weighted energy method. Then based on the local existence and the priori estimate of solutions, we apply the continuation method to prove the exponential stability of the traveling waves with large speed under the so-called small initial perturbation . URL 查看原文 Language 中文 Document Type 学位论文 Identifier https://ir.lzu.edu.cn/handle/262010/225020 Collection 数学与统计学院 Recommended CitationGB/T 7714 张海波. 交叉单稳型Nicholson模型行波解的稳定性[D]. 兰州. 兰州大学,2012.
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