Local energy decay of solutions to the wave equation for nontrapping metrics
The paper analyzes asymptotic decay of local energy of a solution to a generalized linear wave equation on a Riemannian manifold of an arbitrary dimension, i.e., energy calculated in a bounded part of the manifold, in the case when the metric on the manifold is nontrapping, i.e., the energy initially confined to a bounded area decays because of emission of radiation. As a result, an estimate is obtained for the decay law of the local energy as a negative power of time. The proofs are based on resolvent estimates for high-frequency components of the wave field. The general results are applied to asymptotically Euclidean spaces, and some other cases of particular interest (the model with long-range nonnegative potentials).
scientific article; zbMATH DE number 2103856