Eternal solutions in exponential self-similar form for a quasilinear reaction-diffusion equation with critical singular potential

We prove existence and uniqueness of self-similar solutions with exponential form $$u(x,t)=e^{\alpha t}f(|x|e^{-\beta t}), \qquad \alpha, \ \beta>0$$ to the following quasilinear reaction-diffusion equation $$\partial_tu=\Delta u^m+|x|^{\sigma}u^p,$$ posed for $(x,t)\in\real^N\times(0,T)$, with $m>1$, $1<p<m$ and $\sigma=-2(p-1)/(m-1)$ and in dimension $N\geq2$, the same results holding true in dimension $N=1$ under the extra assumption $1<p<(m+1)/2$. Such self-similar solutions are usually known in literature as \emph{eternal solutions} since they exist for any $t\in(-\infty,\infty)$. As an application of the existence of these eternal solutions, we show existence of \emph{global in time weak solutions} with any initial condition $u_0\in L^{\infty}(\real^N)$, and in particular that these weak solutions remain compactly supported at any time $t>0$ if $u_0$ is compactly supported.