Bifurcation of a class of reaction-diffusion equations (Q2721543)

The bifurcation of nontrivial solutions to the boundary value problem NEWLINE\[NEWLINEF(u,\mu):= u''+\mu(u-u^k)=0, \qquad u(0)=u(\pi)=0,NEWLINE\]NEWLINE from the trivial ones is studied in this paper (\(k\) is positive integer and \(\mu\) is a real parameter). This paper completes the previous analysis by the author [Appl. Math. Mech., Engl. Ed. 21, No. 3, 265-274 (2000; Zbl 0982.34034)] for all positive integers \(k\) except \(k=1\), which is the trivial case. The dependence of the type of bifurcation (transcritical/pitchfork) on \(k\in \mathbb{Z}^{+}\) is the main result of the paper. Some consequences of this analysis to the evolution equation \(\partial u/\partial t=F(u, \mu)\) are outlined. Main tools are Lyapunov-Schmidt method and singularity theory.