Global and Local Structures of Bifurcation Curves of ODE with Nonlinear Diffusion

Tetsutaro Shibata

doi:10.1155/2018/5053415

Abstract

We consider the nonlinear eigenvalue problem Duu′′+λfu=0, u(t)>0, t∈I≔(0,1), u(0)=u(1)=0, where D(u)=uk, f(u)=u2n-k-1+sin⁡u, and λ>0 is a bifurcation parameter. Here, n∈N and k (0≤k<2n-1) are constants. This equation is related to the mathematical model of animal dispersal and invasion, and λ is parameterized by the maximum norm α=uλ∞ of the solution uλ associated with λ and is written as λ=λ(α). Since f(u) contains both power nonlinear term u2n-k-1 and oscillatory term sin⁡u, it seems interesting to investigate how the shape of λ(α) is affected by f(u). The purpose of this paper is to characterize the total shape of λ(α) by n and k. Precisely, we establish three types of shape of λ(α), which seem to be new.

Highlights

This paper is concerned with the following nonlinear eigenvalue problems:
Bifurcation problems have a long history and there are so many results concerning the asymptotic properties of bifurcation diagrams
Bifurcation problems with nonlinear diffusion have been proposed in the field of population biology, and several model equations of logistic type have been considered

Summary

Introduction

This paper is concerned with the following nonlinear eigenvalue problems:. where D(u) = uk, f(u) = u2n−k−1 + sin u, and λ > 0 is a bifurcation parameter. Bifurcation problems with nonlinear diffusion have been proposed in the field of population biology, and several model equations of logistic type have been considered. The case D(u) = uk (k > 0) has been derived from a model equation of animal dispersal and invasion in [10, 11] In this situation, λ is a parameter which represents the habitat size and diffusion rate. There are several papers which treat the asymptotic behavior of oscillatory bifurcation curves. The purpose of this paper is to find the difference between the structures of bifurcation curves of the equations with only oscillatory term and those with both nonlinear diffusion term and the oscillatory term in (1). It should be mentioned that if we apply Lemma 4 to our situation, careful consideration about the regularity of the functions is necessary

Proof of Theorem 2

Proof of Theorem 3