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

Abstract EN

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+sinu, 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 sinu, 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.