Polynomial Stabilization of Solutions to a Class of Damped Wave Equations

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.