On inhomogeneous heat equation with inverse square potential

We study inhomogeneous heat equation with inverse square potential, namely, $${\mathrm{\partial }}_{t}u+{\mathcal{L}}_{a}u=\pm \mathrm{\mid }\cdot {\mathrm{\mid }}^{-b}\mathrm{\mid }u{\mathrm{\mid }}^{\alpha }u,$$ where ${\mathcal{L}}_a=-\mathrm{\Delta }+a\mathrm{\mid }x{\mathrm{\mid }}^{-2}\mathrm{.}$ We establish some fixed-time decay estimate for $e^{-t{\mathcal{L}}_a}$ associated with inhomogeneous nonlinearity $\mathrm{\mid }\cdot {\mathrm{\mid }}^{-b}$ 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. $a=0$.