  We prove existence and uniqueness of \emph{eternal solutions} in self-similar
form growing up in time with exponential rate for the weighted
reaction-diffusion equation $$ \partial_tu=\Delta u^m+|x|^{\sigma}u^p, $$ posed
in $\real^N$, with $m>1$, $0<p<1$ and the critical value for the weight $$
\sigma=\frac{2(1-p)}{m-1}. $$ Existence and uniqueness of some specific
solution holds true when $m+p\geq2$. On the contrary, no eternal solution
exists if $m+p<2$. We also classify exponential self-similar solutions with a
different interface behavior when $m+p>2$. Some transformations to
reaction-convection-diffusion equations and traveling wave solutions are also
introduced.
