Georgia Institute of Technology College of Sciences

In this talk, we generalize the classical Kupka-Smale theorem for ordinary differential equations on R^n to the case of scalar parabolic equations. More precisely, we show that, generically with respect to the non-linearity, the semi-flow of a reaction-diffusion equation defined on a bounded domain in R^n or on the torus T^n has the "Kupka-Smale" property, that is, all the critical elements (i.e. the equilibrium points and periodic orbits) are hyperbolic and the stable and unstable manifolds of the critical elements intersect transversally. In the particular case of T1, the semi-flow is generically Morse-Smale, that is, it has the Kupka-Smale property and, moreover, the non-wandering set is finite and is only composed of critical elements. This is an important property, since Morse-Smale semi-flows are structurally stable. (Joint work with P. Brunovsky and R. Joly).