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We prove existence and uniqueness of self-similar solutions with exponentialform $$ u(x,t)=e^{\alpha t}f(|x|e^{-\beta t}), \qquad \alpha, \ \beta>0 $$ tothe following quasilinear reaction-diffusion equation $$ \partial_tu=\Deltau^m+|x|^{\sigma}u^p, $$ 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 resultsholding true in dimension $N=1$ under the extra assumption $1<p<(m+1)/2$. Suchself-similar solutions are usually known in literature as \emph{eternalsolutions} since they exist for any $t\in(-\infty,\infty)$. As an applicationof the existence of these eternal solutions, we show existence of \emph{globalin time weak solutions} with any initial condition $u_0\inL^{\infty}(\real^N)$, and in particular that these weak solutions remaincompactly supported at any time $t>0$ if $u_0$ is compactly supported.

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