Porous medium type reaction-diffusion equation: large time behaviors and regularity of free boundary

We consider the Cauchy problem of the porous medium type reaction-diffusion equation {equation*} ∂_tρ=Δρ^m+ρ g(ρ), (x,t)∈ R^n× R_+, n≥2, m>1, {equation*} 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:=m/m-1ρᵐ⁻¹ in Lloc∞(Rⁿ) tends to the pressure critical threshold PM:=m/m-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 : 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\∩ >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\∩ >T₀\ is a local C1,α surface.