两类非线性抛物方程解的渐进性质及平衡态的研究

发布时间：2018-11-14 09:30

【摘要】：本文考虑了两类非线性抛物方程解的渐进性质及平衡态.首先我们考虑了一类多孔介质方程解的全局存在与爆破条件.对于该方程在初始能量E(u0)d时的全局存在与爆破条件已有很多研究,其中E(u0)表示初始能量,d是将在正文中给出的一个正常数.本文主要针对初始能量E(u0)=d进行研究并给出了解的全局存在与爆破条件.其次我们研究了具有分数型交错扩散的Lotka-Volterra捕食-食饵模型解的情况.通过分析该模型的线性化问题的特征值问题,并利用分支理论和拓扑度理论我们研究了该模型的正平衡态解的性质,并得到了正平衡态解的多重性条件,此结论推广并完善了已有结果.再次我们研究了交错扩散系数对共存区域的影响,并给出了极限系统的局部分支理论.全文共分为三个部分:·第一章,主要介绍多孔介质方程和具有分数型交错扩散Lotka-Volterra捕食-食饵模型的背景,创新之处.·第二章,主要讨论多孔介质方程的解的全局存在与爆破条件.·第三章,主要讨论具有分数型交错扩散Lotka-Volterra的捕食-食饵模型解的多重性与交错扩散系数对共存区域的影响.
[Abstract]:In this paper, the asymptotic property and equilibrium state of solutions for two classes of nonlinear parabolic equations are considered. First, we consider the global existence and blasting conditions of solutions for a class of porous media equations. Many studies have been done on the global existence and blasting conditions of the equation at the initial energy E (u 0) d, where E (u 0) denotes the initial energy and d is a normal number to be given in the text. In this paper, the initial energy E (u 0) = d is studied and the global existence and blasting conditions of the solution are given. Secondly, we study the solution of Lotka-Volterra predator-prey model with fractional staggered diffusion. By analyzing the eigenvalue problem of the linearization problem of the model, and using the bifurcation theory and topological degree theory, we study the properties of the positive equilibrium solution of the model, and obtain the conditions of multiplicity of the positive equilibrium solution. This conclusion extends and improves the existing results. Thirdly, we study the influence of staggered diffusion coefficient on coexisting region, and give the local bifurcation theory of limit system. The thesis is divided into three parts: chapter 1, mainly introduces the porous medium equation and the background and innovation of the Lotka-Volterra predator-prey model with fractional staggered diffusion. The global existence and blasting conditions of solutions for porous media equations are discussed. In chapter 3, the multiplicity of solutions of predator-prey models with fractional staggered diffusion Lotka-Volterra and the influence of staggered diffusion coefficients on coexisting regions are discussed.
【学位授予单位】：西南大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175.26

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