Localization, singular perturbations and reaction-diffusion systems

应用数学的渐近和奇异摄动方法将被开发和应用于分析各种各样的线性和非线性偏微分方程模型的扩散和反应扩散型与局部解行为在多维空间域。 该提案的主要部分是建立一个详细的理论框架，以分析多维空间域中各种反应扩散模型的局部化斑点和条纹形式的局部化粒子状溶液的稳定性和动力学。目前有一个相当有限的理论理解这样的本地化结构在多维的背景下。局部化模式出现在反应扩散模型的各种应用中，包括理论化学中的化学模式、干旱环境中的植被模式和生物形态发生中的色素沉着模式。我们关于局部模式的理论结果将应用于其中一些特定的科学应用。 本建议的第二部分是关于多维域中的局部化效应的研究，这些研究涉及共享某些关键数学特征的各种问题的集合。这些问题包括细胞信号在生物物理学从本地化的细胞内室，平均第一通道时间的测定与陷阱域中的扩散，和浓度现象的非线性四阶模型的微机电系统。 虽然所考虑的具体问题出现在非常不同的应用中，但它们都需要在解决方案中解决小尺度空间特征。 本研究的预期影响范围从偏微分方程的理论问题到生物物理建模的具体问题。

Asymptotic and singular perturbation methods of applied mathematics will be developed and applied to analyze a wide variety of linear and nonlinear partial differential equation models of diffusion and reaction-diffusion type with localized solution behavior in multi-dimensional spatial domains. The main part of the proposal is to develop a detailed theoretical framework to analyze the stability and dynamics of localized particle-like solutions in the form of localized spots and stripes to various reaction-diffusion models in multi-dimensional spatial domains. There is currently a rather limited theoretical understanding of such localized structures in a multi-dimensional context. Localized patterns occur in diverse applications of reaction-diffusion modeling, including chemical patterns in theoretical chemistry, vegetation patterns in arid environments, and pigmentation patterns in biological morphogenesis. Our theoretical results for localized patterns will be applied to some of these specific scientific applications. The second part of this proposal is related to the study of localization effects in multi-dimensional domains for a collection of diverse problems that share certain key mathematical features. These problems include cell-signaling in biophysics from localized intracellular compartments, the determination of the mean first passage time for diffusion in domains with traps, and concentration phenomena for nonlinear fourth-order models of a Micro-Electrical-Mechanical System. Although the specific problems considered arise in very diverse applications, they all share the need to resolve small scale spatial features in the solution. The intended impact of this study ranges from theoretical questions in partial differential equations to specific issues in biophysical modeling.