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 parametrized by Cartesian coordinates x and y. We assume the damping a to be strictly positive and constant for $$x<\sigma $$ and zero for $$x>\sigma $$ . We prove the exact $$t^{-4/3}$$ -decay rate for the energy of classical solutions. Our main result (Theorem 1) answers question (1) of Anantharaman and Léautaud (Anal PDE 7(1):159–214, 2014, Section 2C).