  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, which is of independent interest. As an application, we
establish:
  $\bullet$ Local well-posedness (LWP) in scale subcritical and critical
weighted Lebesgue spaces.
  $\bullet$ Small data global existence in critical weighted Lebesgue spaces.
  $\bullet$ 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.
  $\bullet$ We also demonstrate nonexistence of local positive weak solution
(and hence failure of LWP) in supercritical case for
$\alpha>1+\frac{2+\gamma}{d}$ the Fujita exponent. This indicates that
subcriticality or criticality are necessary in the first point above.
  In summary, we establish a sharp dissipative estimate and addresses short and
long time behaviors of solutions. In particular, we complement several
classical results and shed new light on the dynamics of the considered
equation.
