Nonlinearity in Reaction-Diffusion and Kinetic Equations

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Diverse phenomena such as the spread of an invasive species, turbulent combustion, and the evolution of a cloud of particles in a gas can be modeled using partial differential equations. These examples necessarily involve nonlinearity, which is when the rate of change of a quantity depends on the quantity itself. A simple illustration of nonlinearity is the so-called Allee effect, where the reproduction rate of certain species is negative below a minimum population size, perhaps due to factors such as cooperative defense or mate limitation. Typically, nonlinearities present serious difficulties in the analysis of the model. In the context of various scientifically relevant classes of partial differential equations, this project aims to develop an understanding of when a model can be approximated by a (simpler) linear one and, more generally, which essential features are required by a minimal model to faithfully represent the essential behavior of the original phenomenon. The intent is to aid scientists to identify and implement the most tractable model for their investigations. The project will provide training opportunities for undergraduate students.A focus of the project is to develop technical tools for understanding the fundamental nature of the long-time behavior of several reaction-diffusion equations. These model systems exhibiting growth (reaction) and spreading (diffusion) in which an interface (front) forms and propagates with a constant speed. Classically, these systems are divided into two categories based on the underlying mechanism driving the movement of the front: linear behavior far beyond the front ('pulled' fronts) or nonlinear behavior at the front ('pushed' fronts). Often, the shape and speed of these fronts can depend strongly on intrinsic properties of the system, generally represented by a parameter. As the parameter changes, the character of the fronts may change from 'pulled' to 'pushed' or vice versa. New advances stemming from the recent introduction of ideas such as relative entropy and quantitative steepness as well as a careful understanding of the regularity of equations have opened the door to a high level of precision in characterizing this pulled-pushed transition. A second focus of the project is the development of the well-posedness theory of various collisional kinetic equations.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.