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

TL;DR

This paper analyzes the large-time behavior and regularity of solutions to a porous medium reaction-diffusion equation, showing decay rates, exponential support expansion, and regularity of the free boundary.

Contribution

It establishes decay rates of pressure, exponential support growth, and proves the free boundary's Lipschitz and $C^{1,eta}$ regularity for the first time.

Findings

Pressure tends to the critical threshold at rate (1+t)^{-1}.

Support of the density expands exponentially over time.

Free boundary is locally Lipschitz and $C^{1,eta}$ regular after some time.

Abstract

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.