Incompressible Limit of a Porous Media Equation with Bistable and Monostable Reaction Term

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).