Analysis of dynamic singularities in nonlinear evolution equations

本文研究了一类非线性抛物型偏微分方程具有运动奇异点解的定性性质。首先，我们研究了具有奇异性解整体存在的条件。其次，在一定条件下，解收敛于一个奇异稳态。进一步，我们证明了收敛速度取决于初始值的衰减速率。这意味着存在两种以收敛速率为特征的奇异稳态。

In this project, we studied the qualitative properties of solutions with moving singularities for various nonlinear parabolic partial differential equations. First, we investigated conditions for the global existence of solutions with singularities. Next, under certain conditions, the solution converges to a singular steady state. Furthermore, we showed that the convergence rate depends on the decay rate of initial values. This implies that there are two types of singular steady states that arecharacterized by the convergence rate.