Sharp Damping Location for Stability of a Locally Damped Wave Equation and its Galerkin Approximation

Abstract

We study the stability of the variable coefficient wave equation $u_{tt}-u_{xx}-(1+x^{2})u_{yy}+\chi_{\mathbb G}u_{t}=0$ on the rectangle $(-1,1)\times(0,\pi)$ with Dirichlet boundary conditions, where the viscous damping acts on a vertical strip $\mathbb G=(\eta-\delta,\eta+\delta)\times(0,\pi)$. We establish a precise relationship between the location of the damping region and the rate of energy decay. We prove that the associated semigroup is exponentially stable whenever the closure of the damping strip meets the line $x=0$, i.e., $|\eta|\leq\delta$, despite the existence of rays that never enter the damped region. When the damping region is strictly bounded away from $x=0$, the semigroup decays at the sharp rate $(\log(2+t))^{-1}$. The proofs combine separation of variables, one-dimensional resolvent estimates, a reflection identity for the harmonic oscillator, and Gaussian quasimodes. We then consider the spectral Galerkin approximation based on the Dirichlet eigenfunctions of the constant coefficient Laplacian and show that, although the sequence of semigroups resulting from the approximation converges to the continuous one, it fails to be uniformly exponentially stable.