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We study the initial value problem with unbounded nonnegative functions ormeasures 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 $f$ isa continuous, nondecreasing nonnegative function such that $f(0)=0$. In thecase $p>\frac{2N}{N+1}$, we provide a sufficient condition on $f$ for existenceand uniqueness of the solutions satisfying the initial data $k\gd_0$ and westudy their limit when $k\to\infty$ according $f^{-1}$ and $F^{-1/p}$ areintegrable or not at infinity, where $F(s)=\int_0^s f(\gs)d\gs$. We also givenew results dealing with non uniqueness for the initial value problem withunbounded initial data. If $p>2$, we prove that, for a large class ofnonlinearities $f$, any positive solution admits an initial trace in the classof positive Borel measures. As a model case we consider the case $f(u)=u^\ga\ln^\gb(u+1)$, where $\ga>0$ and $\gb\geq 0$.

Initial trace of positive solutions of a class of degenerate heat equation with absorption