Localization, singular perturbations and reaction-diffusion systems

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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.