The global attractor of semilinear parabolic equations on $S^1$

We study the global attractor of semilinear parabolic equations of the form

$u_t=u_{x x}+f(u,u_x),\ x\in\mathbb{R}$/$\mathbb{Z}, \ t>0.$

Under suitable conditions on $f$, the equation generates a global semiflow on a suitable function space. The general theory of inertial manifolds does not apply to this equation due to lack of the so-called spectral gap condition. Using a totally different method, we show that the global attractor is the graph of a continuous mapping of finite dimension. We also show that this dimension is equal to $2[N$/$2]+1$, where $N$ is the maximal value of the generalized Morse index of equilibria and periodic solutions. Note that we do not make any assumption regarding the hyperbolicity of those solutions. We further prove that there exists no homoclinic orbit nor heteroclinic cycle.