Hyperbolic equations with growing coefficients in unbounded domains

The author considers the weighted wave equation u_{tt}-|x|^sDelta u=0, quad t>0, xinOmega, where sge0, in the unknown u=u(t,x) on an unbounded domain OmegasubsetBbb{R}^N, Nge2, with smooth boundary Gamma. Additionally, it is assumed that u(0,x)=f(x), quad u_t(0,x)=g(x), quad xin Omega, u|_{Gamma}=0, quad t>0. par Motivation for this work comes from examining the propagation of waves in non-homogeneous isotropic media. The difficulty with such mixed problems is treating the unbounded coefficient. In this presentation, the author only considers domains Omega that are star-shaped with respect to the (deleted) origin. The main theorem (Theorem 1) builds on results in [A.~V. Filinovskii, in {it Proceedings of the International Conference Dedicated to the 85th Anniversary of L.~D. Kudryavtsev}, 344--346, RUDN, Moscow, 2008; per bibl.] to prove a scattering result.