  We classify the finite time blow-up profiles for the following
reaction-diffusion equation with unbounded weight: $$ \partial_tu=\Delta
u^m+|x|^{\sigma}u^p, $$ posed in any space dimension $x\in\mathbf{R}^N$,
$t\geq0$ and with exponents $m>1$, $p\in(0,1)$ and $\sigma>2(1-p)/(m-1)$. We
prove that blow-up profiles in backward self-similar form exist for the
indicated range of parameters, showing thus that the unbounded weight has a
strong influence on the dynamics of the equation, merging with the nonlinear
reaction in order to produce finite time blow-up. We also prove that all the
blow-up profiles are \emph{compactly supported} and might present two different
types of interface behavior and three different possible \emph{good behaviors}
near the origin, with direct influence on the blow-up behavior of the
solutions. We classify all these profiles with respect to these different local
behaviors depending on the magnitude of $\sigma$. This paper generalizes in
dimension $N>1$ previous results by the authors in dimension $N=1$ and also
includes some finer classification of the profiles for $\sigma$ large that is
new even in dimension $N=1$.
