Porous medium equations is a class of mathematical models, primarily partial differential equations, developed to describe and analyze physical processes occurring within materials characterized by interconnected pores. This research area investigates phenomena such as fluid flow, heat transfer, and mass transport within porous structures, often incorporating constitutive relations like Darcy's Law or its extensions to represent macroscopic behavior. Their significance lies in providing the theoretical and computational framework essential for understanding and predicting behavior in diverse scientific and engineering applications, including hydrology, petroleum engineering, and materials science.