ON THE SHORT WAVE ASYMPTOTIC BEHAVIOUR OF SOLUTIONS OF STATIONARY PROBLEMS AND THE ASYMPTOTIC BEHAVIOUR ASt→ ∞ 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 , we determine the asymptotic behaviour of solutions as . For the corresponding stationary problem we obtain the short-wave asymptotic behaviour of solutions for real and complex frequencies.