Study On The Spatial Behavior Of Solutions For A Class Of Strongly Interacting Dynamical Systems

The study of the asymptotic behavior of singular perturbed equations and system of elliptic or parabolic type is a very active and difficult subject of research.In the last three decades,people show a lot of interests in strongly competing systems arising in Bose-Einstein condensates and in competing models in population dynamics.In the limiting configuration as the competition rate grows infinity,different components of the solutions are spatially segregated,therefore induing a free boundary problem.These problem have been investigated extensively by many famous mathematicians,including Caffarelli,Fang Hua Lin,Dancer,S.Terracini.Some interesting results are obtained,such as the uniform bounds with respected to the competition rates,the regularity of the free boundary and the uniqueness of the limiting configuration.The study of these issues involves many branches of mathematics,including partial differential equations,geometric measure theory,functional analysis,and so on.Bose on these works,in this dissertation we continue the study of some unsolved and interesting problems.Firstly,we study a competition-diffusion-advection parabolic system for the competitive species inhabiting a spatially heterogeneous environment.Using a blow-up method,we prove the optimal uniform Lipschitz estimates for solutions of the system.This result improve the existing results.We also study the asymptotic of the solutions to above system with strong advection interactions.We prove that,if one of the competing species has strong tendency to move upward along the environment gradients,then if concentrates at places of locally most favorable environments;if both species have such strong tendency,it can lead to competing of the whole population at places of locally most favorable environments,which cause the extinction of the species with strong competition.At last,we study a strongly coupled elliptic system.Based on the maximum principle and some iterative techniques,an alternative proof to show uniqueness is given.Our proof is more simple and doesn't require results from regularity of the free boundary and procedures of integration by parts to avoid singularity of free boundary.