Localization, singular perturbations and reaction-diffusion systems

应用数学中的渐近摄动和奇异摄动方法将被用来分析在多维空间域中具有局部化解行为的各种线性和非线性扩散和反应扩散偏微分方程模型。 该建议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域化斑点和条带形式的局域类粒子解的稳定性和动力学。目前，在多维背景下对这种局域化结构的理论理解相当有限。局部化模式存在于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以及生物形态发生中的色素模式。我们关于局域模式的理论结果将应用于其中一些具体的科学应用。 这一建议的第二部分涉及对一系列具有某些关键数学特征的不同问题在多维领域中的本地化效应的研究。这些问题包括来自局部细胞内隔间的生物物理学中的细胞信号，具有陷阱的区域中扩散的平均第一通过时间的确定，以及微电子机械系统的非线性四阶模型的浓度现象。 虽然所考虑的具体问题在非常不同的应用中出现，但它们都需要在解决方案中解决小比例尺空间要素。这项研究的预期影响范围从偏微分方程式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.