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

强耦合抛物线系统的长期动力学和规律性特性

• 批准号: 0305219
• 负责人: Dung Le
• 金额: $ 7.42万
• 依托单位: University of Texas at San Antonio
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-06-01 至 2006-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0305219&HistoricalAwards=false
• 关键词: Long; Time; Dynamics; Regularity; Properties

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.

反应扩散系统已成为应用型载体积极研究的重要来源。不同位置之间物种/颗粒的分布会影响它们与其他物种/颗粒以及运动的相互作用。因此，应考虑交叉扩散。但是，仅研究了交叉扩散系统，很少发现结果，不超过二十年前。我们对解决方案的定性属性的了解很少。交叉扩散项的存在使这些系统具有强烈耦合的抛物线系统，因为耦合也以高阶项（扩散项）存在。这种强大的耦合不仅在分析治疗中引入了巨大的困难，而且还重新开辟了许多基本问题，并在抛物线系统理论中揭示了许多有趣的现象。我们提出的研究重点是两个问题：规律性和解决方案的长期动态。正如我们将详细解释的那样，对强耦合抛物线系统解决方案的规律性性能的研究在全球存在理论中起着重要而基本的作用。我们的重点不是在其最一般的环境中考虑这些系统，而是在其最通用的环境中考虑部分答案，而是在应用程序中出现的系统，并满足某些结构条件，这可以保证对规则性问题的完整答案。第二个目标是研究某些抛物线系统的长时间动态和共存，在某些抛物线系统中，假定强大的耦合并非全力。特别是，我们将考虑一类三角形交叉扩散系统，这些系统描述了生态学，生物学，粒子物理学等许多重要过程。我们建议通过在先前关于反应扩散对应物的研究中扩展我们的发现来研究此问题。人，物种和颗粒在栖息地中移动或扩散，并相互互动。为了理解这些现象，在适用科学的许多领域都引入了反应扩散系统的数学模型。对解决方案的动态有充分的理解可以帮助回答重要的生活问题。在普通的扩散中，该物种（或颗粒）的运动性仅取决于其自身的特征，而不是根据所讨论的其他物种的存在。也就是说，未知组件之间的相互作用仅在反应项中存在。交叉扩散研究物种/颗粒的运动，使用从环境中存在的其他信息中收集的信息。虽然自然而然地认为，栖息地中不同位置之间生物体的分布会影响其与他人的相互作用以及它们的运动或分散，但确实会发生交叉扩散。将交叉扩散术语引入系统使问题在数学上更具挑战性，并扩展了反应扩散方程的应用范围。 交叉扩散系统最近引起了特殊的兴趣，并受到了更高的科学关注，但是关于解决方案的长期动态的结果很少。 从广义上讲，该提案的目的是研究在某些化学，生态和生物学应用中产生的一类交叉扩散系统，并具有趋化性反应。我们的主要重点是在短暂效应消失后，全球存在，规律性特性和解决方案的渐近行为。该领域的进步可以迫使新的数学工具的发展，也有助于了解生活问题，例如互动人群社区是否可以持续存在（生存并避免灭绝）。引入趋化反应的类似系统的最新和部分结果鼓励我们朝这个新方向迈进。我们建议继续使用趋化模型，并扩展我们的结果，这些模型模拟了扩散的微生物生物的相互作用，并研究了趋化作用对生物动力学的作用。该项目的成功完成将代表理解分散策略（细胞运动，趋化性等）和许多生态学和生物学应用中的竞争能力的重要一步。