Energy Decay Estimate For A Nonhomogeneous Damped String

This article is mainly concerned about the rate at which energy decays in a nonhomogeneous string which can be described by the equation,??x?y_tt-???x?y_x?_x+2a?x?y_t=0.We estimate the decay rate and prove that the rate achieves its negative minimum over those dampings whose total variation does not exceed a prescribed value.First we use ”shooting method” to identify the eigenvalues of the equation with the zeros of some special complex valued function.Then,by analyzing the spectrum of this equation in some detail,we prove that the rate of decay can be estimated by the supremum of the real part of the spectrum,and thus decompose the spectrum into two parts one of which is a finite system of eigenvalues.Finally,by the continuity of the system under the perturbation of the operator in the sense of generalized convergence,we conclude that the best rate of decay can be achieved.