Vibrations of elastic string with nonhomogeneous material

Abstract

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 u 0 and u 1 , uniqueness of solutions and a rate decay estimate for the energy.

Study of a nonlinear Kirchhoff equation with non-homogeneous material

“…The problem (1.2) was proposed by J. L. Lions [17] (see also [16] and [24]). The elliptic version of (1.2) was studied in [22] and [26] for bounded and unbounded domains, respectively.…”

On a nonlinear thermoelastic system with nonlocal coefficients

“…This operator B was also considered in Fernandez-Cara et al [10]. In Límaco et al [17] was derived a model for small transverse vibrations of an elastic membrane assuming that the density of the material is not constant. More precisely, when u = u(x, t) is the deformation of the membrane an approximate model for this physical phenomenon can be written for a dissipative equation of membrane type with variable coefficients like this…”

“…The motivations significants to our article are contained in Chipot-Lovar [6] and Límaco et al [17]. In what follows we comment in few words these two works, in order to establish the problem to be studied here.…”

Asymptiotic Behavior for the Viscoelastic Kirchhoff Type Equation With an Internal Time-Varying Delay Term

“…Recently, problems with the extended Kirchhoff type equation which is concerning axially moving heterogeneous or non heterogeneous materials (nonlinear vibrations of beams, strings, plates, and membranes) have been considered by many authors (See [14,15,16]). …”