Global and Local Structures of Bifurcation Curves of ODE with Nonlinear Diffusion - Tetsutaro Shibata (International Journal of Differential Equations)
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.