The Behavior of the Free Boundary for Reaction-Diffusion Equations with Convection in an Exterior Domain with Neumann or Dirichlet Boundary Condition

Abstract

Let \begin{equation*} L=\sum_{i,j=1}^da_{i,j}\frac{\partial^2}{\partial x_i\partial x_j}-\sum_{i=1}^db_i\frac{\partial}{\partial x_i} \end{equation*} be a second order elliptic operator and consider the reaction-diffusion equation with Neumann boundary condition, \begin{equation*} \begin{aligned} &Lu=Λu^p\ \text{in}\ \mathbb{R}^d-D;\\ &\nabla u\cdot \bar n=-h\ \text{on}\ \partial D;\\ &u\ge0 \ \text{is minimal}, \end{aligned} \end{equation*} where $p\in (0,1)$, $d\ge 2$, $h$ and $\Lambda$ are continuous positive functions, $D\subset R^d$ is bounded, and $\bar{n}$ is the unit inward normal to the domain ${\mathbb{R}}^d-\bar{D}$. Consider also the same equations with the Neumann boundary condition replaced by the Dirichlet boundary condition; namely, $u=h$ on $\partial D$. The solutions to the above equations may possess a free boundary. When $D=\{|x|<R\}$ and $L$ and $\Lambda$ are radially symmetric, we write the solution as $u(r)$ with $r=|x|$ and define the radius of the free boundary by $r^{\ast }(h)=\inf \{r>R:u(r)=0\}$. We normalize the diffusion coefficient to be on unit order, consider the convection vector field to be on order $r^m$, $m\in R$, pointing either inward $(-)$ or outward $(+)$, and consider the reaction coefficient $\Lambda$ to be on order $r^{-j}$, $j\in R$. For both the Neumann boundary case and the Dirichlet boundary case, we show for which choices of $m$, $(\pm )$ and $j$ a free boundary exists, and when it exists, we obtain its growth rate in $h$ as a function of $m$, $(\pm )$ and $j$. These results are then used to study the free boundary in the non-radially symmetric case.

The title of this new version of the paper is a little different. This version has some additional results. Also, some errors in the previous version were corrected