Existence of homoclinic solutions to periodic orbits in a center manifold

We study the Cauchy problem for a class of p-evolution operators P(t,x,Dt,Dx) in [0,T]×Rn, p>1, with less than C1 coefficients with respect to the time variable. According to Lipschitz, log-lipschitz or Holder regularity we find well-posedness in Sobolev spaces or in Gevrey classes.