Vibrations of elastic string with nonhomogeneous material

In this paper we analyze from the mathematical point of view a model for small vertical vibrations of an elastic string with fixed ends and the density of the material being not constant. We employ techniques of functional analysis, mainly a theorem of compactness for the analysis of the approximation of Faedo–Galerkin method. We obtain strong global solutions with restrictions on the initial data u0 and u1, uniqueness of solutions and a rate decay estimate for the energy.