带梯度项的反应扩散系统解的存在性及动力学性态

众所周知，反应扩散系统在物理、化学和生物等领域中被广泛应用，引起了许多学者的极大兴趣。从变分的观点看，反应扩散系统形式上具有强不定的变分结构。由于强不定问题一直是非线性变分问题中极具有挑战性的问题，从而使得反应扩散系统的研究变得十分困难，因此对其研究是具有重要意义的课题。本项目将通过变分方法重点研究反应扩散系统的核心问题：稳态解和非稳态解的存在性及动力学性态，如基态解的存在性，半经典解的存在性、集中性、收敛性、衰减性和正则性等。发展和开拓非线性分析方法、技巧，深化数学工具，对所研究的问题获得若干全新的、本质性的结果，推进反应扩散系统定性理论的发展。并且，这些研究将会有助于解决其他非线性变分问题。

It is well known that reaction-diffusion systems have been widely employed in many fields such as physics, chemistry and biology, and it has been recently paid great attention for many scholars. Formally, reaction-diffusion systems have strongly indefinite variational structure from a variational point of view. Since strong indefinite variational provlems are always very challenging problems in nonlinear variational problems, which makes the study on reaction-diffusion systems very difficult, hence it is a significant project to study such problems. In this project, we will use variational methods to explore some key issues for reaction-diffusion systems. Among these issues are the existence of stationary solution and non-stationary solution, as well as some dynamical properties of the semi-classical ground state solutions such as the existence, concentration phenomena, convergence, exponential decay and regularity etc. These are very important yet very challenging mathematical problems, as such, we also expect to develop some novel and more effective techniques which will enable us to obtain some essentially new results and significantly contribute to the theory of reaction-diffusion systems. Moreover, these studies are beneficial to investigate the other nonlinear variational problems.