  In this work exact solutions for the equation that describes anomalous heat
propagation in 1D harmonic lattices are obtained. Rectangular, triangular, and
sawtooth initial perturbations of the temperature field are considered. The
solution for an initially rectangular temperature profile is investigated in
detail. It is shown that the decay of the solution near the wavefront is
proportional to $1/ \sqrt{t}$. In the center of the perturbation zone the decay
is proportional to $1/t$. Thus the solution decays slower near the wavefront,
leaving clearly visible peaks that can be detected experimentally.
