Local and global properties of solutions of heat equation with superlinear absorption

We study the limit, when $k\to \mathrm{\infty }$ of solutions of $u_t-\mathrm{\Delta }u+f(u)=0$ in $R^N\times (0,\mathrm{\infty })$ with initial data $k\text{gd}$, when $f$ is a positive increasing function. We prove that there exist essentially three types of possible behaviour according $f^{-1}$ and $F^{-1\mathrm{/}2}$ belong or not to $L^1(1,\mathrm{\infty })$, 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.