Some infinite-dimensional Morse-Smale systems defined by parabolic partial differential equations (Q1062329)

Under suitable assumptions on \(f: {\mathbb{R}}\to {\mathbb{R}}\) the Chafee-Infante problem \(u_ t=u_{xx}+\lambda f(u)\) on \(0<x<\pi\), \(u=0\) at \(x=0\), \(x=\pi\) for \(\lambda\geq 0\) is considered. The map \(F_{\lambda}: H^ 1_ 0(0,\pi)\to H^ 1_ 0(0,\pi)\), \(u|_{t=0}\to u|_{t=1}\) is shown to be a \(C^ 2\) Morse-Smale map, except for an exceptional set of \(\lambda\). The main point is the proof of transversality for the stable and unstable manifolds of equilibrium points.