Award Abstract #0707229
Higher dimension cross diffusion systems

NSF Org:	DMS Division of Mathematical Sciences
Initial Amendment Date:	May 11, 2007
Latest Amendment Date:	May 11, 2007
Award Number:	0707229
Award Instrument:	Standard Grant
Program Manager:	Joe W. Jenkins DMS Division of Mathematical Sciences MPS Directorate for Mathematical & Physical Sciences
Start Date:	June 1, 2007
Expires:	May 31, 2011 (Estimated)
Awarded Amount to Date:	$107528
Investigator(s):	Dung Le dle@math.utsa.edu (Principal Investigator)
Sponsor:	University of Texas at San Antonio One UTSA Circle San Antonio, TX 78249 210/458-4340
NSF Program(s):	ANALYSIS PROGRAM
Field Application(s):	0000099 Other Applications NEC
Program Reference Code(s):	OTHR, 0000
Program Element Code(s):	1281

ABSTRACT

This project is to study the properties of solutions of boundary value problems for strongly coupled quasilinear parabolic systems of equations. The proofs of many of the well-known properties of solutions of a single parabolic equation do not extend to coupled systems and some interesting new phenomena have been observed in numerical simulations of these equations. A particular interest is in proving the regularity of solutions of these systems. Some results have recently been attained for systems of two equations and we will investigate systems with more than two components. Another goal is to investigate long time dynamics and coexistence for strongly coupled parabolic systems with certain degeneracies. Such systems arise in fluid flow in porous media, and material mixing problems.

Species and particles move, or diffuse, and interact with each other in their habitats. Cross diffusion studies the motion of species/particles using information about the immediate environment. We will study some classes of cross diffusion systems with a large number of variables that arise in modeling chemical, ecological and mechanical applications. Progress in this objective will require the development of new mathematical tools and methods, and also help to understand life questions such as whether and how a community of interacting populations can persist. That is survive and avoid extinction. 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 certain ecological and biological applications.

PUBLICATIONS PRODUCED AS A RESULT OF THIS RESEARCH

Kuiper, Hank; Le, Dung. "Global Attractors for Cross Diffusion Systems on Domains of Arbitrary Dimension," Rocky Mountain J. Math, v.37, 2007, p. 1645.

Le, Dung. "L^p estimates for gradients of solutions to degenerate parabolic systems.," Discrete and Continuous Dynamical Systems. AIMS conference., v.26, 2010.

Le, Dung; Nguyen, Linh; Nguyen, Toan. "Coexistence in Cross Diffusion systems," Indiana Univ. J. Math., v.56, 2007, p. 1749.

Le, Dung; Nguyen, Toan. "Global attractors and uniform persistence for cross diffusion systems," Dynamical Systems and Applications, v.16, 2007, p. 361.

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