Eternal solutions for a reaction-diffusion equation with weighted reaction

We prove existence and uniqueness of eternal solutions in self-similar form growing up in time with exponential rate for the weighted reaction-diffusion equation

\begin{document}$ \partial_tu = \Delta u^m+|x|^{\sigma}u^p, $\end{document}

posed in \begin{document}$ \mathbb{R}^N $\end{document}, with \begin{document}$ m>1 $\end{document}, \begin{document}$ 0 and the critical value for the weight

\begin{document}$ \sigma = \frac{2(1-p)}{m-1}. $\end{document}

Existence and uniqueness of some specific solution holds true when \begin{document}$ m+p\geq2 $\end{document}. On the contrary, no eternal solution exists if \begin{document}$ m+p<2 $\end{document}. We also classify exponential self-similar solutions with a different interface behavior when \begin{document}$ m+p>2 $\end{document}. Some transformations to reaction-convection-diffusion equations and traveling wave solutions are also introduced.