两类反应扩散方程组的自由边界问题

Applying the free boundary problems to describe the spreading and extending of species not only has important parctical background and theoretical significance, but also have greater difficulty. Firstly, we study Lotka-Volterra competition models with free boundaries in time almost periodic environments. In this part, we shall discuss the existence and uniqueness of solution, spreading vanishing dichotomy, asymptotic behavior. Secondly, we shall research some problem with different free boudaries, establish the existence and uniqueness of solution; obtain the intersection and nonintersectin condition of the two boundaries; give condition and criteria for spreading vanishing.

利用自由边界问题描述物种的蔓延和传播，不仅有具体的应用背景，而且有重要的理论和抽象意义和较大的难度。本项目拟研究反应扩散方程组的自由边界问题。首先研究概周期Lotka-Volterra竞争模型的自由边界问题，探讨解的存在唯一性、蔓延和熄灭的条件，解的渐进性质。其次研究多重自由边界问题，建立解的局部存在唯一性，获得两条自由边界在有限时刻相交和不想交的条件，给出蔓延熄灭的成立条件和判断准则。

本项目主要研究生态学中捕食和竞争模型的自由边界问题。解决了不同自由边界竞争模型的全局解的存在唯一性，给出蔓延和熄灭的判据，并讨论了解的长时间行为。解决了不同自由边界捕食模型的全局解的存在唯一性，给出蔓延和熄灭的判据，并讨论了解的长时间行为。