We study the admissible growth 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 weakly superlinear function mildly at infinity, the model being $f(u)=u\ln^\ga (u)$ with $1<\ga<2$. We prove that if the growth of the initial data 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 $\gf(0)=\infty$.