  We present a simulation study of the one-dimensional $\varphi^4$ lattice
theory with long-range interactions decaying as an inverse power
$r^{-(1+\sigma)}$ of the intersite distance $r$, $\sigma>0$. We consider the
cases of single and double-well local potentials with both attractive and
repulsive couplings. The double-well, attractive case displays a phase
transition for $0<\sigma \le 1$ analogous to the Ising model with long-range
ferromagnetic interactions. A dynamical scaling analysis of both energy
structure factors and excess energy correlations shows that the effective
hydrodynamics is diffusive for $\sigma>1$ and anomalous for $0<\sigma<1$ where
fluctuations propagate superdiffusively. We argue that this is accounted for by
a fractional diffusion process and we compare the results with an effective
model of energy transport based on L\'evy flights. Remarkably, this result is
fairly insensitive on the phase transition. Nonequilibrium simulations with an
applied thermal gradient are in quantitative agreement with the above scenario.
