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

Ross G. Pinsky

Published 2013-12-12, updated 2014-02-19Version 2

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=\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 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 $r^*(h)=\inf\{r>R:u(r)=0\}$. 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.