On the existence of a free-boundary for a class of reaction-diffusion systems

Abstract

Let Ω⊂R N be a bounded smooth domain, f,g∈C 1 (R) , φ 1 ,φ 2 ∈C 2 (∂Ω) , b,c∈C 2 (R) nondecreasing, α,β:R→2 R maximal monotone such that 0∈α(0)∩β(0) and consider the weakly coupled elliptic system (∗) Δu∈α(u)f(v) , −Δv∈β(u)g(v) on Ω with Dirichlet-boundary conditions u=φ 1 , v=φ 2 on ∂Ω , or with the nonlinear boundary conditions ∂u/∂n+b(u)=φ 1 , ∂v/∂n+c(v)=φ 2 on ∂Ω . Systems of this type arise in several applications, in particular as models for certain chemical reactions; here one typically has α(u)=β(u)=|u| q sgnu , where q≥0 is the order of the reaction. Assuming 0≤m 1 ≤f(s) , 0≤g(s)≤m 2 and α(s)≠∅≠β(s) on R , the authors construct pairs of bounded sub/supersolutions of (∗) and then prove, by a standard application of Schauder's theorem, existence of a solution (u,v) of (∗) with either boundary condition satisfying u,v∈W 1,p (Ω) for each p∈[1,∞) . The second part of the paper is devoted to existence of a so-called "dead core'' for u , i.e., the set Ω 0 =u −1 (0) is of positive Lebesgue measure. For α(u)=μ 2 |u| q sgnu it is shown that a dead core only exists if 0≤q<1 holds; in this case the dead core is proved to exist for μ large and estimates for the measure of Ω 0 are derived by means of comparison techniques.