  Using the O(4) linear $\sigma$ model, we address the topic of non-equilibrium
relaxation of an inhomogeneous initial configuration due to quantum and thermal
fluctuations. The space-time evolution of an inhomogeneous fluctuation of the
condensate in the isoscalar channel decaying via the emission of pions in the
medium is studied within the context of disoriented chiral condensates. We use
out of equilibrium closed time path methods in field theory combined with the
amplitude expansion. We give explicit expressions for the asymptotic space-time
evolution of an initial inhomogeneous configuration including the contribution
of thresholds at zero and non-zero temperature. At non-zero temperature we find
new relaxational processes due to thermal cuts that have no counterpart in the
homogeneous case. Within the one-loop approximation, we find that the space
time evolution of such inhomogeneous configuration out of equilibrium is
effectively described in terms of a rapidity dependent temperature
$T(\vartheta)=T/\cosh[\vartheta]$ as well as a rapidity dependent decay rate
$\Gamma(\vartheta, T(\vartheta))$. This rate is to be interpreted as the
production minus absorption rate of pions in the medium and approaches the zero
temperature value at large rapidities. An initial configuration localized on a
bounded region spreads and decays in spherical waves with slower relaxational
dynamics at large rapidity.
