We consider the Cauchy problem for semilinear
parabolic equations with singular initial data.
We show, under a
certain link between the growth at infinity of the nonlinear term
and the order of the maximal singularity of the initial data, existence
and uniqueness theorems for local and global solutions.
For this we introduce anisotropic weighted Holder type spaces,
following T. Kato.
We examine the regularity up to the initial plane of these solutions.