2013/12/12 by Ross G. Pinsky, Pinsky, Ross G.
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #Differential Equations and Numerical Methods #FOS: Mathematics #Nonlinear Partial Differential Equations #math.AP
paper · pdf · doi:10.48550/arxiv.1312.3431
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
openalex publication_date 2013/12/12 · arxiv created 2014/02/19 · arxiv updated 2014/02/20 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
Let L=∑i,j=1dai,j(∂2)/(∂ xi∂ xj)-∑i=1dbi(∂)/(∂ xi) be a second order elliptic operator and consider the reaction-diffusion equation with Neumann boundary condition, \beginaligned Lu=Λup in ℝd-D;
∇ u⋅ n=-h on ∂ D;
u≥0 is minimal, \endaligned where p∈(0,1), d≥2, h and Λ are continuous positive functions, 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=h on ∂ D. The solutions to the above equations may possess a free boundary. When D=\|x|<R\ and L and Λ 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 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.