Existence and uniqueness of the generalized solution of a non-homogeneous hyperbolic differential equation modeling the vibrations of a dissipating elastic rod

Abstract

The purpose of this mathematical paper is to establish a qualitative research of the existence and uniqueness of the generalized solution to a non-homogeneous hyperbolic partial differential equation problema subject to the contour condition u = 0 over Σ, and with initial conditions u(x, 0) = u0(x) in Ω, ∂ut(x, 0) = u1(x) in Ω. In the development of the research, the deductive method of Faedo-Garleskin and Medeiro is used to demonstrate the existence of the generalized solution that consists in the construction of approximate solutions in a finite dimensional space, obtaining a succession of approximate solutions to the non-homogeneous hyperbolic problem, that is, by means of a priori estimations, these successions of approximate solutions are passed to limit in a suitable topology. Then the initial conditions are verified and the uniqueness of the generalized solution is proved.