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We study local and global Cauchy problems for the Semilinear Parabolic Equations ∂U - ΔU = P(D) F(U) with initial data in fractional Sobolev spaces H
(R). In most of the studies on this subject, the initial data U(x) belongs to Lebesgue spaces L(R) or to supercritical fractional Sobolev spaces H
(R) (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...
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