Classification and evolution of bifurcation curves for a one-dimensional prescribed mean curvature problem. (English) Zbl 1389.34060

Cheng, Yan-Hsiou; Hung, Kuo-Chih; Wang, Shin-Hwa

Differ. Integral Equ. 29, No. 7-8, 631-664 (2016).

This paper deals with the classification and evolution of the bifurcation curves of positive solutions for the one-dimensional prescribed mean curvature problem \[ \begin{aligned} &-\left( \frac{u'(x)}{\sqrt{1+(u'(x))^2}}\right)'=\lambda \left[\exp\left(\frac{au(x)}{a+u(x)}\right)-1\right],\quad x\in (-L,L),\\ &u(-L)=u(L)=0,\end{aligned} \] where \(\lambda >0\) is a bifurcation parameter, and \(L\), \(a>0\) are evolution parameters. The authors classify the shape of the bifurcation curve depending on the value of the parameter \(a\) and consider three cases: \(0<a<2\), \(a=2\) and \(a>2\). Note that for \(0<a\leq 2\), the map \(\exp\left(\frac{au(x)}{a+u(x)}\right)-1\) is concave on \((0,\infty)\); whereas for \(a>2\) this is convex-concave on \((0,\infty)\). Moreover, the authors state conjectures, supported by numerical simulations, on the complete classification and evolution of the bifurcation curves for \(a>2\).

Reviewer: Daniel Franco Leis (Madrid)

MSC:

34B09	Boundary eigenvalue problems for ordinary differential equations
34B18	Positive solutions to nonlinear boundary value problems for ordinary differential equations
34C23	Bifurcation theory for ordinary differential equations

Keywords:

global evolution; classical solution; non-classical solution; numerical simulation

Software:

Mathematica