  The time evolution of the Wigner function for Gaussian states generated by
Lindblad quantum dynamics is investigated in the semiclassical limit. A new
type of phase-space dynamics is obtained for the centre of a Gaussian Wigner
function, where the Lindblad terms generally introduce a non-Hamiltonian flow.
In addition to this, the Gaussian approximation yields dynamical equations for
the covariances. The approximation becomes exact for linear Lindblad operators
and a quadratic Hamiltonian. By viewing the Wigner function as a wave function
on a coordinate space of doubled dimension, and the phase-space Lindblad
equation as a Schr\"{o}dinger equation with a non-Hermitian Hamiltonian, a
further set of semiclassical equations are derived. These are capable of
describing the interference terms in Wigner functions arising in superpositions
of Gaussian states, as demonstrated for a cat state in an anharmonic oscillator
subject to damping.
