Sharp polynomial decay rates for the damped wave equation with Hölder-like damping

Datchev, Kiril; Kleinhenz, Perry

doi:10.1090/proc/15018

Sharp polynomial decay rates for the damped wave equation with Hölder-like damping
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by Kiril Datchev and Perry Kleinhenz PDF

Proc. Amer. Math. Soc. 148 (2020), 3417-3425 Request permission

Abstract:

We study decay rates for the energy of solutions of the damped wave equation on the torus. We consider dampings invariant in one direction and bounded above and below by multiples of $x^{\beta }$ near the boundary of the support and show decay at rate $1/t^{\frac {\beta +2}{\beta +3}}$. In the case where $W$ vanishes exactly like $x^{\beta }$ this result is optimal by [Comm. Math. Phys. 369 (2019), pp. 1187–1205]. The proof uses a version of the Morawetz multiplier method.

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