We consider the Cauchy problem for semilinear parabolic equations with strongly singular initial data and nonlinear terms with superlinear or sublinear growth at infinity. 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 Hölder type spaces, following T. Kato in. We examine the regularity up to the initial plane of these solutions.
"Semilinear parabolic equations with singular initial data in anisotropic weighted spaces." Differential Integral Equations 12 (5) 613 - 636, 1999.