Polynomial stability of an abstract system with local damping

In this paper, we consider the polynomial stability for an abstract system of the type u_tt+Lu+Bu_t=0, where L is a self-adjoint operator on a Hilbert space and operator B represents the local damping. By establishing precise estimates on the resolvent, we prove polynomial decay of the corresponding semigroup. The results reveal that the rate of decay depends strongly on the concentration of eigenvalues of operator L and non-degeneration of operator B. Finally, several examples are given as an application of our abstract results.