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

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:260

Article

Pinsky, Ross G.¹ 

[1] Technion Israel Inst Technol, Dept Math, IL-32000 Haifa, Israel

关键词: Reaction-diffusion equation;    Free boundary;    Exterior domain;    Neumann boundary condition;    Boundary flux;    Dirichlet boundary condition;   

DOI  :  10.1016/j.jde.2015.11.035

来源: Elsevier

【 摘 要 】

Let L = A(r) d(2)/dr(2) - B(r) d/dr be a second order elliptic operator and consider the reaction-diffusion equation with Neumann boundary condition, Lu = Lambda u(p) for r is an element of (R, infinity); u'(R) = -h; u >= 0 is minimal, where p is an element of (0, 1), R > 0, h > 0 and Lambda = Lambda (r) > 0. This equation is the radially symmetric case of an equation of the form Lu = Lambda u(p) in R-d - (D) over bar; del u . (n) over bar = -h on partial derivative D; u >= 0 is minimal, where L = Sigma(d)(i,j=1) a(i,j) partial derivative(2)/partial derivative x(i)partial derivative x(j) - Sigma(d)(i=1) b(i) partial derivative/partial derivative x(i) is a second order elliptic operator, and where d >= 2, h > 0 is continuous, D subset of R-d is bounded, and (n) over bar is the unit inward normal to the domain R-d - (D) over bar. 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 partial derivative 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} < infinity, 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 r(m), m is an element of R, pointing either inward (-) or outward (+), and consider the reaction coefficient A to be on order r(-j), j is an element of 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. (C) 2015 Elsevier Inc. All rights reserved.

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