讲座提要：

We study the wave equation on geometric perturbations of Euclidean space, where the support of the perturbation is a compact set $K$. Local energy (i.e. energy over a bounded spatial region $U$) decays in time, in a way depending on the dynamics of the geodesic flow over $K$ and on the geometric relationship between $U$ and $K$. Simple examples show that $K$ can influence decay rates over $U$ even when the distance between $U$ and $K$ is large. For many radial problems it is possible to compute precisely the critical distance at which the influence stops and interpret this distance geometrically. In the general case the picture is less clear but some partial sharp results are known. Our approach to this problem is based on semiclassical resolvent estimates, proven in part using Olver's WKB approximations and in part using Carleman estimates. This talk is based on joint works with Long Jin and with Jeffrey Galkowski and Jacob Shapiro.

主讲人介绍：

Associate Professor of Mathematics. Research interests: Analysis, partial differential equations, mathematical physics.

Personal homepage：

https://www.math.purdue.edu/people/profile/kdatchev.html