  We consider the low-temperature coarsening dynamics of a one-dimensional
Ising ferromagnet with conserved Kawasaki-like dynamics in the domain
representation. Domains diffuse with size-dependent diffusion constant, $D(l)
\propto l^\gamma$ with $\gamma = -1$. We generalize this model to arbitrary
$\gamma$, and derive an expression for the domain density, $N(t) \sim
t^{-\phi}$ with $\phi=1/(2-\gamma)$, using a scaling argument. We also
investigate numerically the persistence exponent $\theta$ characterizing the
power-law decay of the number, $N_p(t)$, of persistent (unflipped) spins at
time $t$, and find $N_{p}(t)\sim t^{-\theta}$ where $\theta$ depends on
$\gamma$. We show how the results for $\phi$ and $\theta$ are related to
similar calculations in diffusion-limited cluster-cluster aggregation (DLCA)
where clusters with size-dependent diffusion constant diffuse through an
immobile `empty' phase and aggregate irreversibly on impact. Simulations show
that, while $\phi$ is the same in both models, $\theta$ is different except for
$\gamma=0$. We also investigate models that interpolate between symmetric
domain diffusion and DLCA.
