Optimal decay rate for the wave equation on a square with constant damping on a strip

Abstract

We consider the damped wave equation with Dirichlet boundary conditions on the unit square. We assume the damping to be a characteristic function of a strip. We prove the exact $t^{-4/3}$-decay rate for the energy of classical solutions. This answers a question of Anantharaman and Léautaud (2014).

10 pages. In version 2 we corrected a minor mistake in the formulation of Theorem 1. A slightly extended and revised version of the paper will be published in the "Zeitschrift für angewandte Mathematik und Physik" in April 2017 (online 15th of February)

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