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.

反应扩散方程组是应用数学研究的一个重要来源。物质/颗粒在不同位置之间的分布影响它们与其他物质/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，交叉扩散系统只被研究过，发现的结果很少，不超过二十年前;我们对它们的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系统，因为耦合也存在于高阶项（扩散项）中。这种强耦合不仅给解析处理带来了巨大的困难，而且也重新开启了许多基本问题，并揭示了抛物系统理论中许多有趣的现象。我们的研究主要集中在两个问题：规律性和长时间动态的解决方案。强耦合抛物型方程组解的正则性研究在整体存在性理论中起着重要的基础性作用，我们将对此进行详细的解释。而不是考虑这些系统在其最一般的设置，它是已知的，只有部分答案可以预期，我们的重点将是在应用中出现的系统，并满足一定的结构条件，这可以保证一个完整的答案的规律性问题。第二个目标是调查长时间的动力学和共存的某些抛物型系统的强耦合被假定为不是在他们的全部力量。特别是，我们将考虑一类三角交叉扩散系统，描述了许多重要的过程，在生态学，生物学，粒子物理学等，我们建议研究这个问题，通过扩展我们的研究结果在我们以前的研究反应扩散对应。人、物种和粒子在其栖息地中移动或扩散，并相互作用。为了理解这些现象，反应扩散系统的数学模型被引入到应用科学的许多领域。很好地理解他们解决方案的动态可以帮助回答重要的生活问题。在普通扩散中，物质（或粒子）的运动性仅由其自身的特性决定，而不是由其他物质的存在决定。也就是说，未知组分之间的相互作用仅存在于反应项中。交叉扩散利用从环境中其他物种收集的信息来研究物种/粒子的运动。虽然人们自然地认为，生物体在栖息地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，但交叉扩散确实会发生。交叉扩散项的引入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。 交叉扩散系统近年来引起了人们的特别兴趣和高度的科学关注，但很少有关于长时间动力学的结果。 从广义上讲，这个建议的目的是研究一类交叉扩散系统在某些化学，生态和生物应用中产生的趋化反应。我们主要研究了在瞬态效应消失后解的整体存在性、正则性和大时间渐近性。这一领域的进展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持续存在（生存和避免灭绝）。最近引入趋化反应的类似系统的部分结果鼓励我们在这个新方向上走得更远。我们建议继续和扩展我们的结果模型与趋化性，模拟扩散的微生物的相互作用，并探讨趋化作用的生物体的动力学的作用。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）的作用方面迈出了重要的一步。在许多生态学和生物学应用中具有竞争力。