RT Journal Article
T1 On the existence of a free-boundary for a class of reaction-diffusion systems
A1 Díaz Díaz, Jesús Ildefonso
A1 Hernández, Jesús
AB 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.
PB Society for Industrial and Applied Mathematics
SN 0036-1410
YR 1984
FD 1984
LK https://hdl.handle.net/20.500.14352/64668
UL https://hdl.handle.net/20.500.14352/64668
DS Docta Complutense
RD 3 may 2024