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 $1<\ga<2$. 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 $\gf(0)=\infty$.