论文标题
具有双向培养基方程的不可压缩极限
Incompressible limit of porous medium equation with bistable and monostable reaction terms
论文作者
论文摘要
我们研究了与压力变量相对于无单酮的反应项的多孔培养基方程的不可压缩极限。更具体地说，我们认为反应术语是可动的或可单位的。我们表明，这种类型的反应术语在解决方案的定性行为上产生了许多有趣的差异，与单调反应术语的问题相反，在最近的文献中已经进行了广泛研究。在表征了极限问题之后，我们开始对一个空间维度中的问题进行全面研究，以说明问题的微妙性质，包括非唯一性和不稳定性的通用性质。对于紧凑的初始数据，我们表明，即使密度从相同的初始数据开始，具体取决于其初始压力配置，即使它从相同的初始数据开始。当初始压力是一种特征函数时，我们确定了将两种行为分开的尖锐阈值的存在。最后，我们详细介绍了在不可压缩的极限下行驶波的行为。我们研究了限制模型的行进波的存在，并证明收敛导致不可压缩极限（取决于反应项）。
We study the incompressible limit of the porous medium equation with a reaction term that is non-monotone with respect to the pressure variable. More specifically we consider reaction terms that are either bistable or monostable. We show that this type of reaction term generates many interesting differences in the qualitative behavior of solutions, in contrast to the problem with monotone reaction terms that have been extensively studied in recent literature. After characterizing the limit problem, we embark on a comprehensive study of the problem in one space dimension, to illustrate the delicate nature of the problem, including the generic nature of non-uniqueness and instability. For compactly supported initial data, we show that the density can either perish or thrive, even if it starts from the same initial data, depending on its initial pressure configuration. When the initial pressure is a characteristic function, we establish the existence of the sharp threshold separating the two behaviors. Lastly we present a detailed analysis of the behavior of traveling waves in this incompressible limit. We study the existence of traveling waves for the limiting model and prove convergence results in the incompressible limit (depending on the reaction term).
