Eternal solutions for a reaction-diffusion equation with weighted reaction

Abstract: 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.