Long-Time Dynamics and Regularity Properties of Strongly Coupled Parabolic Systems

反应扩散系统一直是应用数学中活跃研究的一个重要来源。物种/颗粒在不同地点的分布影响着它们与其他物种/颗粒的相互作用以及它们的运动。因此，应考虑交叉扩散。然而，直到二十多年前，交叉扩散系统才被研究过，几乎没有发现结果；我们对其解的定性性质知之甚少。交叉扩散项的存在使得这些系统强耦合抛物型系统，因为耦合也以高阶项(扩散项)存在。这种强耦合不仅给分析处理带来了巨大的困难，而且重新提出了许多基本问题，揭示了抛物型系统理论中许多有趣的现象。我们的研究集中在两个问题上：解的正则性和长时间动态性。研究强耦合抛物方程组解的正则性在整体存在理论中占有重要的地位。我们不会在最一般的情况下考虑这些系统，在这种情况下，我们只会关注在应用中出现并满足某些结构条件的系统，这些系统可以保证对正则性问题的完整答案。第二个目标是研究某些抛物型系统的长时间动力学和共存，其中强耦合被假定为不是最大的。特别是，我们将考虑一类描述生态学、生物学、粒子物理等领域中许多重要过程的三角交叉扩散系统。我们建议通过扩展我们在先前对反应扩散的研究中的发现来研究这个问题。人、物种和粒子在它们的栖息地移动、扩散和相互作用。为了理解这些现象，反应-扩散系统的数学模型已经被引入到应用科学的许多领域。很好地理解他们的解决方案的动态可以帮助回答重要的生活问题。在通常的扩散中，物种(或粒子)的运动完全由其自身的特性决定，而不取决于所讨论的其他物种的存在。也就是说，未知成分之间的相互作用只存在于反应项中。交叉扩散使用从环境中存在的其他物种/粒子收集的信息来研究物种/粒子的运动。虽然人们自然认为，生物在栖息地内不同地点的分布会影响它们与其他生物的相互作用以及它们的移动或扩散，但确实会发生交叉扩散。交叉扩散项的引入使问题在数学上更具挑战性，扩大了反应扩散方程的应用范围。交叉扩散系统最近引起了特别的兴趣并受到了高度的科学关注，但很少有关于解的长期动力学的结果。广义地说，这项建议的目的是研究一类在某些化学、生态和生物应用中出现的具有趋化反应的交叉扩散系统。在暂态效应消失后，我们主要研究了解的整体存在性、正则性和大时间解的渐近行为。这一领域的进步可以推动新的数学工具的发展，也有助于理解生命问题，如相互作用的种群是否以及如何能够持续存在(生存和避免灭绝)。最近和部分结果的类似系统的趋化反应的引入鼓励我们在这个新的方向走得更远。我们建议继续和推广我们在具有趋化作用的模型上的结果，该模型模拟扩散的微生物有机体的相互作用，并研究趋化作用在生物体动力学中的作用。该项目的成功完成将代表着在理解扩散策略(细胞运动、趋化性等)的作用方面向前迈出了重要的一步。以及在许多生态和生物应用方面的竞争能力。

Reaction diffusion systems have been a great source for active research in appliedmathematics. The distribution of species/particles among different locations affects their interaction with other species/particles as well as their movement. Thus, cross diffusion should be taken into account. However, cross diffusion systems have only been studied, and few results were discovered, no more than two decades ago; and very little that we know about the qualitative properties of their solutions. The presence of the cross diffusion terms makes these systems strongly coupled parabolic systems since the couplings are also present in higher order terms (diffusion terms). This strong coupling has introduced not only enormous difficulties in analytical treatments but also reopened many fundamental questions as well as unveiled many interesting phenomena in the theory of parabolic systems. Our proposed research focuses on two problems: regularity and long time dynamics of solutions. The study of regularity properties of solutions of strongly coupled parabolic systems, as we shall explain in details, plays an essential and fundamental role in global existence theory. Instead of considering these systems in their most general settings, where it is known that only partial answers could be expected, our focus will be on systems that arise in applications, and satisfy certain structure conditions, which can guarantee a complete answer to the regularity question. The second goal is to investigate long time dynamics and coexistence for certain parabolic systems where strong couplings are assumed to be not in their full force. In particular, we will consider a class of triangular cross diffusion systems that describe many important processes in ecology, biology, particle physics, etc. We propose to study this issue by extending our findings in our previous research on reaction diffusion counterparts. People, species and particles move, or diffuse, and interact with each other in their habitats. In order to understand these phenomena, mathematical models of reaction-diffusion systems have been introduced in many areas in applicable sciences. A good understanding of the dynamics of their solutions can help to answer important life questions. In ordinary diffusion, motility of the species (or particles) is determined solely by its own characteristics but not on the presence of other species in question. That is, the interaction among the unknown components is present only in the reaction terms. Cross diffusion studies the motion of species/particles using the information gathered from others present in the environment. While it is naturally believed that the distribution of organisms among different locations within a habitat affects their interaction with others as well as their movement or dispersal, cross diffusion does occur. The introduction of cross-diffusion terms into the systems makes the problem much more mathematically challenging and extends the application range of reaction-diffusion equations. Cross diffusion systems have recently drawn special interests and received heightened scientific attention, but few are results concerning the long time dynamics of solutions. Broadly speaking, the aim of this proposal is to study a class of cross diffusion systems arising in certain chemical, ecological and biological applications with chemotactic response. Our main focus is on the global existence, regularity property and the asymptotic behavior of solutions for large times, after transient effects have disappeared. Progress in this area can force the development of new mathematical tools, and also help to understand life questions such as whether and how a community of interacting populations can persist (survive and avoid extinction). Recent and partial results for similar systems with chemotactic response introduced have encouraged us to go further in this new direction. We propose to continue and extend our results on models with chemotaxis, which simulate the interaction of diffused microbial organisms, and investigate the role of chemotactic effects on the dynamics of organisms. The successful completion of this project will represent a significant step forward in the understanding of the roles of dispersal strategies (cell motilities, chemotaxis, etc.) and competitive abilities in many ecology and biology applications.