本学位论文建立了一类边界快速扩散问题正解的长时间存在性并证明了正解关于分离变量解的稳定性, 得到多项式速率的渐近行为. 进一步, 在一定的条件下获得指数式的衰减率并讨论出最佳速率.

      根据 Steklov 第一特征根的符号, 问题分为三种情形, 对应三种分离变量解.

      第一特征根为正时, 通过积分估计得到正解一定在有限时刻消失. 得到消失时刻的渐近性并讨论出最佳速率. 创新点是 引入了伪平均曲率, 并利用伪平均曲率的发展方程建立了一个椭圆型的 Harnack 不等式.
      第一特征根为负时, 正解会在时间趋于无穷时爆破. 得到时间趋于无穷时的渐近性并讨论出最佳收敛速率. 创新之处是采用上下解方法得到解的上下界估计.
      第一特征根为零时, 正解既不消失也不爆破.
      本文采用能量方法, 利用 Lojasiewicz-Simon 不等式, 在全部三种情形下, 建立了多项式速率的渐近性.

This thesis establishes existence of smooth positive solutions to a boundary fast diffusion problem and proves stability about separable solutions. Then we obtain asymptotic behavior in polynomial rate. Furthermore, under some condition, we acquire exponential rate and discuss the sharp rate.

According to the sign of Steklov first characteristic root, the problem is divided into three cases which are corresponding to three types of separable solutions.

When the first characteristic root is positive, by integral estimates we know positive solutions must extinct at a finite extinction time. We get asymptotic behavior near extinction time and discuss the sharp rate. The innovation point is that we introduce a pseudo average curvature and use its evolution equation to establish an elliptic
Harnack inequality.

When the first characteristic root is negative, positive solutions must blow up as time goes to infinity. We derive asymptotic behavior as time goes to infinity and discuss the sharp rate. The innovation point is adapting the method of super and sub solutions to get the upper and lower bounds of solutions.

When the first characteristic root is zero,neither positive solutions extinct nor they blow up.

In the thesis, we use energy method and a Lojasiewicz-Simon inequality to establish asymptotic behavior in polynomial rate in all three cases.