  This is the first of a two-parts work on the qualitative properties and large
time behavior for the following quasilinear equation involving a spatially
inhomogeneous absorption $$ \partial_tu=\Delta u^m-|x|^{\sigma}u^p, $$ posed
for $(x,t)\in\mathbf{R}^N\times(0,\infty)$, $N\geq1$, and in the range of
exponents $1<m<p<\infty$, $\sigma>0$. We give a complete classification of
(singular) self-similar solutions of the form $$
u(x,t)=t^{-\alpha}f(|x|t^{-\beta}), \
\alpha=\frac{\sigma+2}{\sigma(m-1)+2(p-1)}, \
\beta=\frac{p-m}{\sigma(m-1)+2(p-1)}, $$ showing that their form and behavior
strongly depends on the critical exponent $$ p_F(\sigma)=m+\frac{\sigma+2}{N}.
$$ For $p\geq p_F(\sigma)$, we prove that all self-similar solutions have a
tail as $\xi\to\infty$ of one of the forms $$ u(x,t)\sim
C|x|^{-(\sigma+2)/(p-m)} \quad {\rm or} \quad u(x,t)\sim
\left(\frac{1}{p-1}\right)^{1/(p-1)}|x|^{-\sigma/(p-1)}, $$ while for
$m<p<p_F(\sigma)$ we add to the previous the \emph{existence and uniqueness} of
a \emph{compactly supported very singular solution}. These solutions will be
employed in describing the large time behavior of general solutions in a
forthcoming paper.
