## Some infinite-dimensional Morse-Smale systems defined by parabolic partial differential equations.(English)Zbl 0572.58012

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.
Reviewer: G.Warnecke

### MSC:

 37D15 Morse-Smale systems 35K55 Nonlinear parabolic equations
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### References:

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