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.Comment: Communications in Contemporary Mathematics, to appea
