Abstract
We consider the diffusive logistic equation with nonlinear boundary conditions δu + λu(1- u) = 0 x ∈Ω, α(u)(∂u/ ∂v) + (1-α(u))u = 0 x ∈ Ω, u 0 for i∈Ω, where Ω is a bounded domain in R N with v its outer unit normal and A is a positive parameter. The boundary conditions of the equation considered are nonlinear, with the function α satisfying α([0,1]) C [0,1] and increasing. In this work we will study the case α(0) = 0, α′(0) > 0, which implies that, as A varies, the above equation has two continua of solutions, one having Dirichlet boundary conditions, and another one in which each solution is positive at the boundary. We show that the second continuum of solutions may contain infinitely many solutions for a fixed value of λ.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 445-455 |
| Number of pages | 11 |
| Journal | Rocky Mountain Journal of Mathematics |
| Volume | 41 |
| Issue number | 2 |
| DOIs | |
| State | Published - 2011 |
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ASJC Scopus subject areas
- Mathematics(all)
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Steady state solutions of a logistic equation with nonlinear boundary conditions. / Cantrell, Robert; Cosner, George; Martínez, Salomé.
In: Rocky Mountain Journal of Mathematics, Vol. 41, No. 2, 2011, p. 445-455.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Steady state solutions of a logistic equation with nonlinear boundary conditions
AU - Cantrell, Robert
AU - Cosner, George
AU - Martínez, Salomé
PY - 2011
Y1 - 2011
N2 - We consider the diffusive logistic equation with nonlinear boundary conditions δu + λu(1- u) = 0 x ∈Ω, α(u)(∂u/ ∂v) + (1-α(u))u = 0 x ∈ Ω, u 0 for i∈Ω, where Ω is a bounded domain in R N with v its outer unit normal and A is a positive parameter. The boundary conditions of the equation considered are nonlinear, with the function α satisfying α([0,1]) C [0,1] and increasing. In this work we will study the case α(0) = 0, α′(0) > 0, which implies that, as A varies, the above equation has two continua of solutions, one having Dirichlet boundary conditions, and another one in which each solution is positive at the boundary. We show that the second continuum of solutions may contain infinitely many solutions for a fixed value of λ.
AB - We consider the diffusive logistic equation with nonlinear boundary conditions δu + λu(1- u) = 0 x ∈Ω, α(u)(∂u/ ∂v) + (1-α(u))u = 0 x ∈ Ω, u 0 for i∈Ω, where Ω is a bounded domain in R N with v its outer unit normal and A is a positive parameter. The boundary conditions of the equation considered are nonlinear, with the function α satisfying α([0,1]) C [0,1] and increasing. In this work we will study the case α(0) = 0, α′(0) > 0, which implies that, as A varies, the above equation has two continua of solutions, one having Dirichlet boundary conditions, and another one in which each solution is positive at the boundary. We show that the second continuum of solutions may contain infinitely many solutions for a fixed value of λ.
UR - http://www.scopus.com/inward/record.url?scp=79958726881&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=79958726881&partnerID=8YFLogxK
U2 - 10.1216/RMJ-2011-41-2-445
DO - 10.1216/RMJ-2011-41-2-445
M3 - Article
VL - 41
SP - 445
EP - 455
JO - Rocky Mountain Journal of Mathematics
JF - Rocky Mountain Journal of Mathematics
SN - 0035-7596
IS - 2
ER -