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 \begin{equation*} \partial_t\rho=\Delta\rho^m+\rho g(\rho),\quad (x,t)\in \mathbb{R}^n\times \mathbb{R}_+,\quad n\geq2,\quad m>1, \end{equation*} where $g$ is the given monotonic decreasing function with the density critical threshold $\rho_M>0$ satisfying $g(\rho_M)=0$. We prove that the pressure $P:=\frac{m}{m-1}\rho^{m-1}$ in $L_{loc}^{\infty}(\mathbb{R}^n)$ tends to the pressure critical threshold $P_M:=\frac{m}{m-1}(\rho_M)^{m-1}$ 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 $\rho$ 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 $\partial \{(x,t): \rho(x,t)>0\}\cap \{t>T_0\}$ is locally Lipschitz continuous. In addition, under the same initial assumptions of compact support, we verify that the free boundary $\partial \{(x,t): \rho(x,t)>0\}\cap \{t>T_0\}$ is a local $C^{1,\alpha}$ surface.