Nonlinear Problems From Combustion Theory and Biology

研究者计划研究三类问题：第一类是燃烧理论中的摄动Gelfand问题及其相关问题。重点将放在这些非线性椭圆偏微分方程的精确数值解的完整描述和对更复杂的燃烧模型的理解上；其次，研究者希望更好地理解经典的Lotka-Volterra扩散模型。这些模型虽然看似简单，但涉及到巨大的数学困难。研究者一直在尝试引入新思路，开发新方法来解决这一领域的一些开放性问题；研究者将研究的第三类问题是交叉扩散系统，它是一个强耦合非线性抛物型系统。在与倪伟明的一系列合作中，研究者引入了强大的新方法，可以获得关于这些系统稳态的非常详细的信息。研究者计划研究该交叉扩散系统的全局时间存在性，以及各种稳态的稳定性。研究者希望这种交叉扩散系统的工作可以帮助理解一般的强耦合反应扩散系统。生物学和燃烧理论中的数学问题不仅在数学上具有挑战性，而且在实践中也很重要。为了说明问题，让我们考虑以下生态问题：假设两个不同的物种最初生活在两个不同的地区，它们都能生存。现在，如果我们把这两个地区连接起来，让这两个物种在新的地区混合在一起。除了同一种之间已有的竞争外，还出现了不同种之间新的竞争。一个基本的问题是：在这样的新形势下，这两个物种如何共存？如果假设这两个物种是随机运动的，那么这个问题可以用经典的Lotka-Volterra模型来建模，这个模型被研究者提出研究的第二类问题所覆盖。另一方面，当两个物种为了生存而竞争资源时，仅仅在模型中加入扩散是不太合理的，因为个体并不是随机移动的。相反，它们正在迁移到对自己有利的地方，例如，物种更喜欢竞争对手造成的人口压力较小的地方。基于第二种假设，一些生物学家提出了交叉扩散系统，即研究者将追求的第三类问题。研究者希望这样的数学工作不仅能产生有趣的数学结果，而且还能加强对各种科学分支中现实世界现象的理解。

PI: Yuan LouDMS-9801609The investigator plans to study three classes of problems: first, the perturbed Gelfand problem and its related ones from Combustion theory. The emphasis will be on the complete description of the exact number solutions of these nonlinear elliptic partial differential equations and the understanding of more complex combustion models; second, the investigator wants to pursue a better understanding of the classical Lotka-Volterra models with diffusion. These models, though deceptively simple-looking, involve tremendous mathematical difficulty. The investigator has been trying to introduce new ideas and develop new methods to solve some open problems in this field; The third class of problem that the investigator will study is the cross-diffusion system, which is a strongly-coupled nonlinear parabolic system. The investigator, in a series of joint work with Wei-Ming Ni, has introduced powerful new methods that can yield very detailed information about the steady-states of these systems. The investigator plans to study the global time existence of this cross-diffusion system, along with the stability of various steady-states. The investigator hopes that the work on this cross-diffusionsystem can be instrumental in understanding general strongly-coupled reaction-diffusion systems. Mathematical problems from biology and combustion theory arenot only mathematically challenging, but also practically important. To be illustrative, let us consider the following ecological problem: suppose that two different species originally live in two separate regions and both of them can survive. Now if we connect these tworegions and let these two species be mixed together in the new region. Besides the existing competitions among the same species, there appears the new competition between two different species. A basic questions is: how can these two species manage coexistence under such new situation? If assuming that these two species are moving randomly, then the problem can be modeled by the classical Lotka-Volterra model, and this model is covered by the second class of problems which the investigator proposes to study. On the other hand, when two species are competing for resources to survive, it is not very reasonable to just add diffusion to the model since individuals are not moving around randomly. Instead, they are moving to places to their advantage, e.g., species prefer places with less population pressures created by competitors. Based on this second assumption, some biologists proposed the cross-diffusion system, i.e., the third class of problems which the investigator will pursue. The investigator hopes that such mathematical work not only yields interesting mathematical results, but also strengthens the understanding of realworld phenomena in various branches of science.