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

Abstract

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 $\rho =1-\gamma /L$. 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.

19 pages