The behavior of the free boundary for reaction–diffusion equations with convection in an exterior domain with Neumann or Dirichlet boundary condition

Abstract

LetL=A(r)d2dr2−B(r)ddr be a second order elliptic operator and consider the reaction–diffusion equation with Neumann boundary condition,Lu=Λupforr∈(R,∞);u′(R)=−h;u≥0is minimal, where p∈(0,1), R>0, h>0 and Λ=Λ(r)>0. This equation is the radially symmetric case of an equation of the formLu=ΛupinRd−D¯;∇u⋅n¯=−hon∂D;u≥0is minimal, whereL=∑i,j=1dai,j∂2∂xi∂xj−∑i=1dbi∂∂xi is a second order elliptic operator, and where d≥2, h>0 is continuous, D⊂Rd is bounded, and n¯ is the unit inward normal to the domain Rd−D¯. Consider also the same equations with the Neumann boundary condition replaced by the Dirichlet boundary condition; namely, u(R)=h in the radial case and u=h on ∂D in the general case. The solutions to the above equations may possess a free boundary. In the radially symmetric case, if r⁎(h)=inf⁡{r>R:u(r)=0}<∞, we call this the radius of the free boundary; otherwise there is no free boundary. We normalize the diffusion coefficient A to be on unit order, consider the convection vector field B to be on order rm, m∈R, pointing either inward (−) or outward (+), and consider the reaction coefficient Λ to be on order r−j, j∈R. For both the Neumann boundary case and the Dirichlet boundary case, we show for which choices of m, (±) and j a free boundary exists, and when it exists, we obtain its growth rate in h as a function of m, (±) and j. These results are then used to study the free boundary in the non-radially symmetric case.