Qualitative Properties of Solutions of Differential Equations Modeling Biological Pattern Formation

模式坍缩是一些具有奇异非线性的反应扩散系统所特有的一种新发现的现象，在这些系统中，模式最初形成，但最终收敛到一个不规则的稳态。我们已经给出了模式坍缩和解在有限时间内爆破的充分条件。此外，还详细研究了解的定性性质，如最大点的动力学和/或解的渐近形式。此外，在弯曲能量的驱动下，平面闭合曲线的运动被认为是决定红细胞形状的几何变分问题的低维模拟。在一定的约束条件下，找到了能量泛函的所有临界点，构造了约束最小化问题的梯度流。

Collapse of patterns is a newly found phenomenon characteristic to some reaction-diffusion systems possessing singular nonlinearities, where patterns are formed at first but eventually converge to a nonregular steady state. We have given sufficient conditions for patterns to collapse and also for solutions to blow-up in finite time. In addition, qualitative properties of solutions such as the dynamics of maximum points and/or asymptotic forms of solutions have been studied in detail.Moreover, movement of planar closed curves driven by bending energy is considered as a lower dimensional analogue for the geometric variational problem which determines the shape of red blood cells. All the critical points of the energy functional under some constraints are found and the gradient flow of the constraint minimization problem has been constructed.