  We study the limit, when $k\to\infty$ of solutions of $u_t-\Delta u+f(u)=0$
in $R^N\times(0,\infty)$ with initial data $k\gd$, when $f$ is a positive
increasing function. We prove that there exist essentially three types of
possible behaviour according $f^{-1}$ and $F^{-1/2}$ belong or not to
$L^1(1,\infty)$, where $F(t)=\int_0^t f(s)ds$. We emphasize the case where
f(u)=u((\ln u+1))^{\alpha}. We use these results for giving a general result on
the existence of the initial trace and some non-uniqueness results for regular
solutions with unbounded initial data.
