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${\partial }_t\rho =\Delta {\rho }^m+\rho g\left(\rho \right),(x,t)\in R^n\times R_{+},n\ge 2,m>1,$ where g is the given monotonic decreasing function with the density critical threshold ${\rho }_M>0$ satisfying $g\left({\rho }_M\right)=0$. We prove that the pressure $P:=\frac{m}{m-1}{\rho }^{m-1}$ in $L_{loc}^{\infty }\left(R^n\right)$ tends to the pressure critical threshold $P_M:=\frac{m}{m-1}{\left({\rho }_M\right)}^{m-1}$ at the time decay rate ${(1+t)}^{-1}$. If the initial density $\rho (x,0)$ is compactly supported, we justify that the support $\{x:\rho (x,t)>0\}$ of the density ρ expands exponentially in time. Furthermore, we show that there exists a time $T_0>0$ such that the pressure P is Lipschitz continuous for $t>T_0$, which is the optimal (sharp) regularity of the pressure, and the free surface $\partial \left\{\right(x,t):\rho (x,t)>0\}\cap \{t>T_0\}$ is locally Lipschitz continuous. In addition, under the same initial assumptions of compact support, we verify that the free boundary $\partial \left\{\right(x,t):\rho (x,t)>0\}\cap \{t>T_0\}$ is a local $C^{1,\alpha }$ surface.