Cauchy problem for semilinear parabolic equations with initial data in Hps(Rn) spaces

We study local and global Cauchy problems for the Semilinear Parabolic Equations ?tU - ?U = P(D) F(U) with initial data in fractional Sobolev spaces Hps(Rn). In most of the studies on this subject, the initial data U0(x) belongs to Lebesgue spaces Lp(Rn) or to supercritical fractional Sobolev spaces Hps(Rn) (s > n/p). Our purpose is to study the intermediate cases (namely for 0 < s < n/p). We give some mapping properties for functions with polynomial growth on subcritical Hps(Rn) spaces and we show how to use them to solve the local Cauchy problem for data with low regularity. We also give some results about the global Cauchy problem for small initial data