A Double S-Shaped Bifurcation Curve for a Reaction-Diffusion Model with Nonlinear Boundary Conditions

Jerome Goddard,Eunkyoung Lee,R Shivaji

doi:10.1155/2010/357542

Jerome Goddard, Eunkyoung Lee + Show 1 more

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https://doi.org/10.1155/2010/357542

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Abstract

We study the positive solutions to boundary value problems of the form ; , ; , where is a bounded domain in with , is the Laplace operator, is a positive parameter, is a continuous function which is sublinear at , is the outward normal derivative, and is a smooth function nondecreasing in . In particular, we discuss the existence of at least two positive radial solutions for when is an annulus in . Further, we discuss the existence of a double S-shaped bifurcation curve when , , and with .

Highlights

We consider the reaction-diffusion model with nonlinear boundary condition given by ut dΔu λf u ; Ω, 1.1 dα x, u
1 − α x, u u 0; ∂Ω, 1.2 where Ω is a bounded domain in Rn with n ≥ 1, Δ is the Laplace operator, λ is a positive parameter, d is the diffusion coefficient, ∂u/∂η is the outward normal derivative, f : 0, ∞ → 0, ∞ is a smooth function, and α x, u : Ω × R → 0, 1 is a smooth function nondecreasing
The motivating example for this study comes from combustion theory see 5–15 where f u takes the form: f u eβu/ β u

Summary

Introduction

We consider the reaction-diffusion model with nonlinear boundary condition given by ut dΔu λf u ; Ω, 1.1 dα x, u. The boundary condition 1.2 arises naturally in applications and has been studied by the authors of 1–4 , among others, in particular in the context of population dynamics. We will be interested in the study of positive steady state solutions of 1.1 - 1.2 when d 1, namely,. U ≡ 0 Dirichlet boundary condition case there is already a very rich history in the literature about positive solutions of 1.4 - 1.5. When f u eβu/ β u and β 1 the bifurcation diagram of positive solutions is known to be S-shaped see 16, 17. The main purpose of this paper is to extend this study to the nonlinear boundary condition 1.5 , namely, when α x, u u u1. Studying 1.10 is equivalent to analyzing the two boundary value problems. We show that for β large enough, 1.10 has a double S-shaped bifurcation curve with exactly 6 positive solutions for a certain range of λ see Figure 1

Here we consider the existence of positive radial solutions for

Fu u

This follows from observing that