Traveling Sharp Waves of Porous Media Equations in Spatially Periodic Environment

We consider one dimensional porous media equations in spatially periodic environment. We will construct a periodic traveling sharp wave whose profile tends to a positive steady state at left infinity and takes zero on the right half line, with a free boundary satisfying the Darcy's law. Our method is to take the limit for a sequence of normalized solutions starting at Heaviside type of initial data. The crucial step is to give a uniform positive lower bound for the instantaneous speed of the free boundary.