Cauchy problem for semilinear parabolic equations with initial data in $H_p^s({\mathbb{R}}^n)$

Abstract

We study local and global Cauchy problems for the Semilinear Parabolic Equations ${\partial }_{t}U–\Delta U=P(D)F(U)$ with initial data in fractional Sobolev spaces $H_p^s({\mathbb{R}}^n)$. In most of the studies on this subject the initial data $U_0(x)$ belongs to Lebesgue spaces $L^p({\mathbb{R}}^n)$ or to supercritical fractional Sobolev spaces $H_p^s({\mathbb{R}}^n)(s>n/p)$. 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 $H_p^s({\mathbb{R}}^n)$ 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.