Polynomial Stabilization of Solutions to a Class of Damped Wave Equations

TL;DR

This paper establishes polynomial decay rates for solutions to damped wave equations with various damping types, linking decay rates to eigenvalue and eigenfunction properties, with applications to rectangular domains.

Contribution

It provides new resolvent estimates for the associated operator, leading to precise polynomial decay results depending on eigenvalue distribution and domain geometry.

Findings

Decay rate varies with eigenvalue concentration.

Rational vs. irrational domain ratios affect energy decay.

Abstract results apply to specific geometric configurations.

Abstract

We consider a class of wave equations of the type $\partial_{tt} u + Lu + B\partial_{t} u = 0$, with a self-adjoint operator $L$, and various types of local damping represented by $B$. By establishing appropriate and raher precise estimates on the resolvent of an associated operator $A$ on the imaginary axis of ${{\Bbb C}}$, we prove polynomial decay of the semigroup $\exp(-tA)$ generated by that operator. We point out that the rate of decay depends strongly on the concentration of eigenvalues and that of the eigenfunctions of the operator $L$. We give several examples of application of our abstract result, showing in particular that for a rectangle $\Omega := (0,L_{1})\times (0,L_{2})$ the decay rate of the energy is different depending on whether the ratio $L_{1}^2/L_{2}^2$ is rational, or irrational but algebraic.