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在这篇论文中，我们考虑二维中的Gray-Scott 模型核心问题的解: / [ 附圖]. / 对于足够小的ε，我们会构造一个“多个凸 的解。这些解的“凸 会分布在一个正多边形的顶点上。在这个解的U 方向上，经过一个合适的放缩之后，它会看起来像下列方程的唯一对称解： / [ 附圖]. / 此外，我们同时也会构造单个“凸 和两个“凸 的解。 / In this thesis, we consider solutions to the core problem for Gray-Scott model in R²: / [With mathematic formula]. / We construct multi-bump solutions for this problem for all sufficiently small ε. The centers of these bumps are located at the vertices of a regular polygon, andthey resemble, after a suitable scaling in their U-coordinate, the unique radial solution of / [With mathematic formula]. / The solutions with one single bump and two bumps are also constructed. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Yip, Chit Ming. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 45-46). / Abstracts also in Chinese. / Chapter 1 --- Introduction: Derivation of the Core Problem --- p.6 / Chapter 2 --- One-dimensional core problem --- p.12 / Chapter 3 --- Main results on two-dimensional core problem --- p.19 / Chapter 4 --- Proof of Theorem 3.1 --- p.22 / Chapter 4.1 --- Estimate for S₁({U+03A6}) --- p.24 / Chapter 4.2 --- Estimate for N₁({U+03A6}) --- p.25 / Chapter 4.3 --- Estimate for S₁({U+03A6}₁) - S₁({U+03A6}₂) --- p.26 / Chapter 4.4 --- Estimate for N₁({U+03A6}₁) - N₁({U+03A6}₂) --- p.26 / Chapter 5 --- Proof of Theorem 3.2 --- p.29 / Chapter 5.1 --- Estimate for S₂({U+03A6}) --- p.33 / Chapter 5.2 --- Estimate for N₂({U+03A6}) --- p.34 / Chapter 5.3 --- Estimate for S₂({U+03A6}₁) - S₂({U+03A6}₂) --- p.34 / Chapter 5.4 --- Estimate for N₂({U+03A6}₁) - N₂({U+03A6}₂) --- p.35 / Chapter 5.5 --- The reduced problem --- p.35 / Chapter 6 --- Proof of Theorem 3.3 --- p.40 / Chapter 6.1 --- Invariance under permutations --- p.41 / Chapter 6.2 --- Reducing number of equations for regular polygons --- p.42 / Bibliography --- p.45

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published_or_final_version / Mechanical Engineering / Master / Master of Philosophy

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Fluctuations are known to radically alter the behaviour of reaction-diffusion systems. Below a certain upper critical dimension d_c , this effect results in the breakdown of traditional approaches, such as mean field rate equations. In this thesis we tackle this fluctuation problem by employing systematic field theoretic/renormalisation group methods, which enable perturbative calculations to be made below d_c. We first consider a steady state reaction front formed in the two species irreversible reaction A + B → Ø. In one dimension we demonstrate that there are two components to the front - one an intrinsic width, and one caused by the ability of the centre of the front to wander. We make theoretical predictions for the shapes of these components, which are found to be in good agreement with our one dimensional simulations. In higher dimensions, where the intrinsic component dominates, we also make calculations for its asymptotic profile. Furthermore, fluctuation effects lead to a prediction of asymptotic power law tails in the intrinsic front in all dimensions. This effect causes high enough order spatial moments of a time dependent reaction front to exhibit multiscaling. The second system we consider is a time dependent multispecies reaction-diffusion system with three competing reactions A+A → Ø, B + B → Ø, and A + B → Ø, starting with homogeneous initial conditions. Using our field theoretic formalism we calculate the asymptotic density decay rates for the two species for d ≤ d_c. These calculations are compared with other approximate methods, such as the Smoluchowski approach, and also with previous simulations and exact results.

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Thesis (Ph. D.)--University of Texas at Austin, 2006. / Vita. Includes bibliographical references.

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In this thesis, we establish the asymptotic analysis of the singularly perturbed reaction diffusion equation [cataloger unable to transcribe mathematical equations].... Our results establish the specific dependency on the coefficients of this equation and the size of the parameter [delta] with respect to [epsilon]. The analyses include equation subject to Dirichlet and Neumann boundary conditions. In both cases, the solutions u[superscript epsilon] converge locally uniformally to the equilibria of the reaction term f. We characterize the limiting behavior of the solutions through the viscosity solution of a variational inequality. To construct the coefficients defining the variational inequality, we apply concepts developed for the homogenization of elliptic operators. In chapter two, we derive the convergence results in the Neumann case. The third chapter is dedicated to the analysis of the Dirichlet case. / text