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 $${\partial }_{t}u=\Delta u^m+|x{|}^{\sigma }{u}^p,$$ posed in ${\Re }^N$, 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 $m+p\ge 2$. 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.