  We study the admissible growth at infinity of initial data of positive
solutions of $\prt\_t u-\Gd u+f(u)=0$ in $\BBR\_+\ti\BBR^N$ when $f(u)$ is a
continuous function, {\it mildly} superlinear at infinity, the model case being
$f(u)=u\ln^\ga (1+u)$ with $1\textless{}\ga\textless{}2$. We prove in
particular that if the growth of the initial data at infinity is too strong,
there is no more diffusion and the corresponding solution satisfies the ODE
problem $\prt\_t \gf+f(\gf)=0$ on $\BBR\_+$ with $\gf(0)=\infty$.
