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Let \begin{equation*} L=\sum_{i,j=1}^da_{i,j}\frac{\partial^2}{\partialx_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-diffusionequation with Neumann boundary condition, \begin{equation*} \begin{aligned}&Lu=\Lambda u^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 $p\in(0,1)$, $d\ge2$, $h$ and $\Lambda$ are continuouspositive functions, $D\subset R^d$ is bounded, and $\bar n$ is the unit inwardnormal to the domain $\mathbb{R}^d-\bar D$. Consider also the same equationswith the Neumann boundary condition replaced by the Dirichlet boundarycondition; namely, $u=h$ on $\partial D$. The solutions to the above equationsmay possess a free boundary. When $D=\{|x|<R\}$ and $L$ and $\Lambda$ areradially symmetric, we write the solution as $u(r)$ with $r=|x|$ and define theradius of the free boundary by $r^*(h)=\inf\{r>R:u(r)=0\}$. We normalize thediffusion coefficient to be on unit order, consider the convection vector fieldto be on order $r^m$, $m\in R$, pointing either inward $(-)$ or outward $(+)$,and consider the reaction coefficient $\Lambda$ to be on order $r^{-j}$, $j\inR$. For both the Neumann boundary case and the Dirichlet boundary case, we showfor which choices of $m$, $(\pm)$ and $j$ a free boundary exists, and when itexists, 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-radiallysymmetric case.

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