On inhomogeneous heat equation with inverse square potential

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. $a=0$.