Generic Morse-Smale property for the parabolic equation on the circle

In this paper, we show that, for scalar reaction-diffusion equations $u_t=u_{xx}+f(x,u,u_x)$ on the circle $S^1$, the Morse-Smale property is generic with respect to the non-linearity $f$. In \cite{CR}, Czaja and Rocha have proved that any connecting orbit, which connects two hyperbolic periodic orbits, is transverse and that there does not exist any homoclinic orbit, connecting a hyperbolic periodic orbit to itself. In \cite{JR}, we have shown that, generically with respect to the non-linearity $f$, all the equilibria and periodic orbits are hyperbolic. Here we complete these results by showing that any connecting orbit between two hyperbolic equilibria with distinct Morse indices or between a hyperbolic equilibrium and a hyperbolic periodic orbit is automatically transverse. We also show that, generically with respect to $f$, there does not exist any connection between equilibria with the same Morse index. The above properties, together with the existence of a compact global attractor and the Poincar\'e-Bendixson property, allow us to deduce that, generically with respect to $f$, the non-wandering set consists in a finite number of hyperbolic equilibria and periodic orbits . The main tools in the proofs include the lap number property, exponential dichotomies and the Sard-Smale theorem. The proofs also require a careful analysis of the asymptotic behavior of solutions of the linearized equations along the connecting orbits.