This article is a continuation of the investigations on
nonlinear evolution
equations of parabolic type with singular initial data. Most of the results
available in the literature are for initial data measures, e.g. for
semilinear
parabolic equations with nonlinearity of the type f(u) we cite
the pioneering work by Brezis and Friedman while
for the Navier-Stokes
equation we mention the recent papers by Hato and Kato and Ponce.
The present paper proposes a link between the growth at
infinity
of the nonlinearity g(u), the maximal order k of the singularity
of the initial data, the index p and weighted Holder spaces of possible
solutions which implies existence-uniqueness
results for local and global solutions to the Cauchy problem.
In the case of Burgers' equation we give examples
which show that the corresponding link is optimal in order to have uniqueness. We study also
the regularity
down to t=0 and we
estimate the lifespan of the local solutions.