Eternal solutions for a reaction-diffusion equation with weighted reaction

<p style='text-indent:20px;'>We prove existence and uniqueness of <i>eternal solutions</i> in self-similar form growing up in time with exponential rate for the weighted reaction-diffusion equation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \partial_tu = \Delta u^m+|x|^{\sigma}u^p, $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>posed in <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^N $\end{document}</tex-math></inline-formula>, with <inline-formula><tex-math id="M2">\begin{document}$ m&gt;1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ 0&lt;p&lt;1 $\end{document}</tex-math></inline-formula> and the critical value for the weight</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \sigma = \frac{2(1-p)}{m-1}. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>Existence and uniqueness of some specific solution holds true when <inline-formula><tex-math id="M4">\begin{document}$ m+p\geq2 $\end{document}</tex-math></inline-formula>. On the contrary, no eternal solution exists if <inline-formula><tex-math id="M5">\begin{document}$ m+p&lt;2 $\end{document}</tex-math></inline-formula>. We also classify exponential self-similar solutions with a different interface behavior when <inline-formula><tex-math id="M6">\begin{document}$ m+p&gt;2 $\end{document}</tex-math></inline-formula>. Some transformations to reaction-convection-diffusion equations and traveling wave solutions are also introduced.</p>