Stabilization rates for the damped wave equation with polynomial and oscillatory damping

Abstract: In this talk I will discuss energy decay of solutions of the Damped wave equation on the torus when the geometric control condition is not satisfied. In this setup properties of the damping at the boundary of its support determine the decay rate, however a general sharp rate is not known.

I will discuss damping 0 on a strip and vanishing either like a polynomial x^b or an oscillating exponential e^{-1/x} sin^2(1/x). Polynomial damping produces decay of the semigroup at exactly t^{-(b+2)/(b+3)}, while oscillating damping produces decay at least as fast as t^{-4/5+\delta} for any \delta>0. I will explain how these model cases are proved and how they direct study of the general sharp rate. 

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