Steady state solutions of a logistic equation with nonlinear boundary conditions

Robert Cantrell, George Cosner, Salomé Martínez

Research output: Contribution to journalArticle

5 Citations (Scopus)

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 languageEnglish (US)
Pages (from-to)445-455
Number of pages11
JournalRocky Mountain Journal of Mathematics
Volume41
Issue number2
DOIs
StatePublished - 2011

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Logistic Equation
Nonlinear Boundary Conditions
Steady-state Solution
Continuum
Unit normal vector
Infinitely Many Solutions
Dirichlet Boundary Conditions
Bounded Domain
Vary
Boundary conditions
Imply

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

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 journalArticle

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