Periodic solutions of an infinite dimensional hamiltonian system

We establish existence and multiplicity of periodic solutions to the infinite dimensional Hamiltonian system (∂_tu - Δ_xu = H_v(t, x, u, v) -∂_tu - Δ_xu = H_v(t, x, u, v) for (t, x) ∈R × Ω, where Ω ⊂ R^N is a bounded domain or Ω = R^N. When Ω is bounded, we treat the situations where H(t, x, z) is, with respect to z = (u, v), sub- or superquadratic, or concave and convex, and discuss also the convergence to homoclinics of sequences of subharmonic orbits. If Ω = R^N, we handle the case of superquadratic nonlinearities.