On the Hardy-H\'enon heat equation with an inverse square potential

We study Cauchy problem for the Hardy-H\'enon parabolic equation with an inverse square potential, namely, $$\partial_tu -\Delta u+a|x|^{-2} u= |x|^{\gamma} F_{\alpha}(u),$$ where $a\ge-(\frac{d-2}{2})^2,$ $\gamma\in \mathbb R$, $\alpha>1$ and $F_{\alpha}(u)=\mu |u|^{\alpha-1}u, \mu|u|^\alpha$ or $\mu u^\alpha$, $\mu\in \{-1,0,1\}$. We establish sharp fixed time-time decay estimates for heat semigroups $e^{-t (-\Delta + a|x|^{-2})}$ in weighted Lebesgue spaces. This may be of independent interest. As an application, we establish local well-posedness in scale subcritical and critical weighted Lebesgue spaces and small data global existence in critical weighted Lebesgue spaces. Further, under certain conditions on $\gamma$ and $\alpha,$ we show that local solution cannot be extended to global one for certain initial data in the subcritical regime. Thus, finite time blow-up in the subcritical Lebesgue space norm is exhibited. We also demonstrate nonexistence of local positive weak solution (and hence failure of local well-posedness) in supercritical case for $\alpha>1+\frac{2+\gamma}{d}$ the Fujita exponent.