| 分数阶Laplacian反应扩散方程和Belousov-Zhabotinskii系统的行波解 | |
| Alternative Title | Traveling wave Fronts in Reaction Diffusion Equation with Fractional Laplacian and Belousov--Zhabotinskii System |
| 马陆一 | |
| Subtype | 博士 |
| Thesis Advisor | 王智诚 |
| 2021-05-21 | |
| Degree Grantor | 兰州大学 |
| Place of Conferral | 兰州 |
| Degree Name | 理学博士 |
| Degree Discipline | 应用数学 |
| Keyword | 反应扩散方程 行波解 分数阶Laplacian Belousov-Zhabotinskii系统 稳定性. |
| Abstract | ~：反应扩散方程在物理及生物等领域有极其丰富的应用背景, 对于其行波解以及解的渐近行为的研究在应用数学领域具有重要的地位. 近年来带分数阶 Laplacian 的反应扩散方程得到了众多学者的关注, 双稳和点火情形时平面行波解的存在性以及Fisher-KPP情形时解的加速传播现象等得到了系统的研究. 另外, Belousov-Zhabotinskii反应扩散系统在化学领域有重要应用, 其平面行波解也得到了充分研究. 而在高维空间中, 研究表明有更复杂的非平面行波解. 目前, 针对分数阶Laplacian方程和Belousov-Zhabotinskii系统的非平面波研究也有了初步进展, 取得的主要成果是分数阶Laplacian方程三维棱锥波的存在性以及Belousov-Zhabotinskii系统的二维V形波的存在唯一性及稳定性. 本文将继续研究以上两类反应扩散方程的行波解以及解的渐近行为, 主要包括分数阶Laplacian反应扩散方程一维平面行波解的稳定性, 二维 V 形波的存在唯一性和稳定性, 三维圆锥波的存在性, 以及 Belousov-Zhabotinskii系统三维棱锥波的存在稳定性. 本文主要内容分为以下四章: 本文第二章首先研究了带分数阶Laplacian的双稳反应扩散方程一维行波解的稳定性. 我们证明了适度解意义下的比较原理, 并克服了分数阶Laplacian在0处产生的积分奇异性,构造了所需要的上下解, 进而我们用挤压方法得到了行波解的全局渐近稳定性. 本文第三章研究了带分数阶Laplacian双稳反应扩散方程的二维V形波. 由于分数阶Laplacian算子作用到函数上会产生代数衰减, 这使得以往使用双曲函数来构造上解的方法不再适用. 因此, 我们使用重新定义的磨光曲线来构造合适的函数作为问题的上解, 进而我们利用分数阶 Laplacian 反应扩散方程的正则性理论证明了V形波的存在性. 在V形波存在的基础上, 我们借助初值函数的性质以及强连续半群理论得到了这个非平面波的稳定性. 本文的第四部分主要研究带分数阶Laplacian双稳反应扩散方程的圆柱对称形行波解. 我们首先分析了分数阶Laplacian核的对称性, 并利用这个结论克服了强最大值原理使用时的局限性, 同时获知了三维棱锥波的一些对称和单调性质. 随后, 我们构造了一个棱锥波序列并对这个序列取极限, 最终得到了圆柱对称形行波解. 最后一章, 我们研究Belousov-Zhabotinskii反应扩散系统棱锥波的存在性和稳定性. 我们将Belousov-Zhabotinskii系统二维V型波的研究过程中构造的辅助函数推广到三维情形,随后构造了合适的上解, 并用上下解方法得到了棱锥波的存在性. 在此基础上, 我们运用比较原理等工具证明了这个棱锥波的稳定性. |
| Other Abstract | ~The reaction diffusion equation has many applications in physics and biology, and thus the study of its traveling wave fronts and asymptotic behavior of solutions plays an important role in the field of applied Mathematics. In recent years, the reaction diffusion equation with fractional Laplacian has attracted a great deal of attention. The planar traveling wave fronts for the bistable and combustion case have been well studied. For the Fisher-KPP case, the results include the accelerated propagation of solutions. In addition, Belousov-Zhabotinskii reaction diffusion system has important applications in chemistry and its planar traveling wave fronts have been fully studied. But in high dimensional space, it is shown that there exist more complex nonplanar traveling wave fronts. Nowadays, the research on nonplanar traveling wave fronts of reaction diffusion equations with fractional Laplacian and Belousov-Zhabotinskii system has made initial progress. The main results include the three-dimensional pyramidal traveling waves of reaction diffusion equations with fractional Laplacian and the existence and stability of V-shaped traveling wave fronts of Belousov-Zhabotinskii system. In this thesis, we are going to study the traveling wave fronts for the above two kinds of equations. For bistable reaction-diffusion equation with fractional Laplacian, we study the stability of one-dimensional traveling wave fronts, the existence, uniqueness and stability of two-dimensional V-shaped traveling wave fronts and the existence of cylindrical symmetric traveling wave fronts. For the Belousov-Zhabotinskii system, we study the existence and stability of the pyramidal traveling wave fronts. This paper will be divided into four parts. In this thesis, we first study the stability of one-dimensional traveling wave fronts of bistable reaction-diffusion equation with fractional Laplacian. We prove the comparison principle in the sense of mild solutions. We also overcome the singularity of integral generated by fractional Laplacian at $0$ and construct proper super and subsolutions. Then we obtain the global asymptotic stability of traveling wave fronts by using the classical squeezing method. The third chapter studies the V-shaped traveling wave fronts of the bistable reaction-diffusion equation with fractional Laplacian. Since the fractional Laplacian will produce the algebraic decay on functions, the previous method of using hyperbolic function to construct the supersolutions is no longer applicable. Thus we use a redefined smoothing curve to construct the suitable supersolutions. After that, we use the regularity theory of reaction-diffusion equation with fractional Laplacian to prove the existence of V-shaped traveling wave fronts. Based on the existence of V-shaped traveling wave fronts, we obtain the stability of this nonplanar traveling wave fronts by using the properties of the initial value and the theory of strongly continuous semigroups. In the fourth part of this thesis, we mainly study the cylindrical symmetric traveling wave fronts of the bistable reaction-diffusion equation with fractional Laplacian. We first analyze the symmetry of the kernel of fractional Laplacian, and use this conclusion to overcome the limitations of the strong maximum principle. At the same time, we obtain the symmetric and monotone properties of three-dimensional pyramidal traveling wave fronts. Then we construct a sequence of pyramidal traveling wave fronts and take the limit for this sequence to obtain the cylindrical symmetric traveling wave fronts. In the last chapter, we study the existence and stability of pyramidal traveling wave fronts of the Belousov-Zhabotinskii reaction-diffusion system. We extend the auxiliary function constructed in the two-dimensional V-shape traveling wave fronts of the Belousov-Zhabotinskii system to the three-dimensional case. Then we construct a suitable supersolution, and obtain the existence of pyramidal traveling wave fronts by using the super and subsolutions. After that, we get the stability of the pyramidal traveling wave fronts by using the comparsion principle. |
| Pages | 135 |
| URL | 查看原文 |
| Language | 中文 |
| Document Type | 学位论文 |
| Identifier | https://ir.lzu.edu.cn/handle/262010/461957 |
| Collection | 数学与统计学院 |
| Affiliation | 数学与统计学院 |
| First Author Affilication | School of Mathematics and Statistics |
| Recommended Citation GB/T 7714 | 马陆一. 分数阶Laplacian反应扩散方程和Belousov-Zhabotinskii系统的行波解[D]. 兰州. 兰州大学,2021. |
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