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

Abstract

We consider the nonlinear eigenvalue problem ${\left[D\left(u\right)u^{\mathrm{\prime }}\right]}^{\mathrm{\prime }}+\lambda f\left(u\right)=\mathrm{0}$, $u(t)>\mathrm{0}$, $t\in I≔(\mathrm{0,1})$, $u(\mathrm{0})=u(\mathrm{1})=\mathrm{0}$, where $D(u)=u^k$, $f(u)=u^{\mathrm{2}n-k-\mathrm{1}}+\sin u$, and $\lambda >\mathrm{0}$ is a bifurcation parameter. Here, $n\in \mathbb{N}$ and $k$ () are constants. This equation is related to the mathematical model of animal dispersal and invasion, and $\lambda$ is parameterized by the maximum norm $\alpha ={‖u_{\lambda }‖}_{\mathrm{\infty }}$ of the solution $u_{\lambda }$ associated with $\lambda$ and is written as $\lambda =\lambda (\alpha )$. Since $f(u)$ contains both power nonlinear term $u^{\mathrm{2}n-k-\mathrm{1}}$ and oscillatory term $\sin u$, it seems interesting to investigate how the shape of $\lambda (\alpha )$ is affected by $f(u)$. The purpose of this paper is to characterize the total shape of $\lambda (\alpha )$ by $n$ and $k$. Precisely, we establish three types of shape of $\lambda (\alpha )$, which seem to be new.