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

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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.