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

Abstract

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 Hsp (ℝn). In most of the studies on this subject, the initial data U0(cursive Greek chi) belongs to Lebesgue spaces Lp(ℝn) or to supercritical fractional Sobolev spaces Hsp(ℝn) (s > n/p). Our purpose is to study the intermediate cases (namely for 0 < s