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# Jacob Perry (UNC-CH), GMA Seminar

## October 31, 2016 @ 4:00 pm - 5:00 pm

**Title: **Localized energy for wave equations with degene= rate trapping

**Abstract: **Local smoothing estimates for the Schr{\textasciidieresis}odinger equation are well established and show that locally in space and averaged in time, solutions gain one half of a derivative in regularity compared to the initial data. Analogous estimates for solutions to the wave equation, so-called localized energy estimates, have also been studied, and provide a global integrability estimate (in both time and space). When considering such estimates for equations on differentiable manifolds, in either case it is known that geodesic trapping requires a loss. For non-degenerate hyperbolic trapping, the loss is logarithmic. For elliptic trapping, everything is lost except a logarithm. Recently, Christianson and Wunsch demonstrated an algebraic loss for solutions to the Schro{\textasciidieresis}dinger equation on a surface of revolution with degenerate hyperbolic trapping. In this talk, we will review these prior results and consider the analogue for the wave equation on a warped product manifold with degenerate hyperbolic trapping, attaining an algebraic loss of derivative. We will then use a quasimode construction to show that our estimate is sharp. This is a joint work with Robert Booth, Hans Christianson, and Jason Metcalfe