Parametrix for a hyperbolic initial value problem with dissipation in some region

We consider the initial value problem for a pseudodifferential equation with first order hyperbolic part, and an order $\gamma >0$ dissipative term. Under an assumption, depending on an integer parameter $L\ge 2$ such that $2 \gamma < L$, we construct for this initial value problem a parametrix that is a Fourier integral operator of type $\rho =1-\gamma \mathrm{/}L$. The assumption implies that where the principal symbol of the dissipative term is zero, the terms of order up to $L-1$ in its Taylor series also vanish.