Given an (ordinary) super-Brownian motion (SBM) $\varrho$ on
$\mathbf{R}^d$ of dimension $d = 2, 3$, we consider a (catalytic) SBM
$X^{\varrho}$ on $\mathbf{R}^d$ with "local branching rates" $\varrho_s(dx)$.
We show that $X_t^{\varrho}$ is absolutely continuous with a density function
$\xi_t^{\varrho}$, say. Moreover, there exists a version of the map $(t, z)
\mapsto \xi_t^{\varrho}(z)$ which is $\mathscr{C}^{\infty}$ and solves the heat
equation off the catalyst $\varrho$; more precisely, off the (zero set of)
closed support of the time-space measure $ds\varrho_s(dx)$. Using
self-similarity, we apply this result to give the following answer to an open
problem on the long-term behavior of $X^{\varrho}$ in dimension $d = 2$: If
$\varrho$ and $X^{\varrho}$ start with a Lebesgue measure, then does
$X_T^{\varrho}$ converge (persistently) as $T \to \infty$ toward a random
multiple of Lebesgue measure?