We study the initial value problem with unbounded nonnegative functions or
measures for the equation \prt_tu-\Gd_p u+f(u)=0 in \BBR^N\ti(0,\infty)
where p>1, \Gd_p u = \text{div}(\abs {\nabla u}^{p-2} \nabla u) and f is
a continuous, nondecreasing nonnegative function such that f(0)=0. In the
case p>N+12N, we provide a sufficient condition on f for existence
and uniqueness of the solutions satisfying the initial data k\gd_0 and we
study their limit when k→∞ according f−1 and F−1/p are
integrable or not at infinity, where F(s)=\int_0^s f(\gs)d\gs. We also give
new results dealing with non uniqueness for the initial value problem with
unbounded initial data. If p>2, we prove that, for a large class of
nonlinearities f, any positive solution admits an initial trace in the class
of positive Borel measures. As a model case we consider the case f(u)=u^\ga
\ln^\gb(u+1), where \ga>0 and \gb\geq 0