On the short wave asymptotic behaviour of solutions of stationary problems and the asymptotic behaviour as $t\to\infty$ of solutions of non-stationary problems

In this paper we study the Cauchy problem and boundary-value problem of general form in the exterior of a compact set for hyperbolic operators $L$, whose coefficients depend only on $x$ and are constant near infinity. Assuming that the wave fronts of the Green's matrix for $L$ go off to infinity as $t\to\infty$, we determine the asymptotic behaviour of solutions as $t\to\infty$. For the corresponding stationary problem we obtain the short-wave asymptotic behaviour of solutions for real and complex frequencies.