We consider the Cauchy problem of the porous medium type reaction-diffusion equation∂tρ=Δρm+ρg(ρ),(x,t)∈Rn×R+,n≥2,m>1, where g is the given monotonic decreasing function with the density critical threshold ρM>0 satisfying g(ρM)=0. We prove that the pressure P:=mm−1ρm−1 in Lloc∞(Rn) tends to the pressure critical threshold PM:=mm−1(ρM)m−1 at the time decay rate (1+t)−1. If the initial density ρ(x,0) is compactly supported, we justify that the support {x:ρ(x,t)>0} of the density ρ expands exponentially in time. Furthermore, we show that there exists a time T0>0 such that the pressure P is Lipschitz continuous for t>T0, which is the optimal (sharp) regularity of the pressure, and the free surface ∂{(x,t):ρ(x,t)>0}∩{t>T0} is locally Lipschitz continuous. In addition, under the same initial assumptions of compact support, we verify that the free boundary ∂{(x,t):ρ(x,t)>0}∩{t>T0} is a local C1,α surface.