  We study inhomogeneous heat equation with inverse square potential, namely,
\[\partial_tu + \mathcal{L}_a u= \pm |\cdot|^{-b} |u|^{\alpha}u,\] where
$\mathcal{L}_a=-\Delta + a |x|^{-2}.$ We establish some fixed-time decay
estimate for $e^{-t\mathcal{L}_a}$ associated with inhomogeneous nonlinearity
$|\cdot|^{-b}$ in Lebesgue spaces. We then develop local theory in $L^q-$
scaling critical and super-critical regime and small data global well-posedness
in critical Lebegue spaces. We further study asymptotic behaviour of global
solutions by using self-similar solutions, provided the initial data satisfies
certain bounds. Our method of proof is inspired from the work of
Slimene-Tayachi-Weissler (2017) where they considered the classical case, i.e.
$a=0$.
