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

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 di ﬀ erential equation problem ∂ 2 u ∂t 2 − ∆ u − ∆ ∂u ∂t = f subject to the contour condition u = 0 over P , and with initial conditions u ( x, 0) = u 0( x ) in Ω , ∂ut ( x, 0) = u 1( 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 ﬁ nite 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 veri ﬁ ed and the uniqueness of the generalized solution is proved.


Introduction
The evolution equations are equations in partial derivatives, which describe processes that develop over time. Likewise, let us consider an nonhomogeneous evolution equation that describes the vibrations of an elastic rod subject to dissipative constraints caused by the medium where the motion occurs in a cylindrical domain; with these physical properties we need to add to the evolution equation certain conditions such as: Dirichlet condition that expresses that the elastic rod is fixed on the lateral boundary of a cylindrical and the Cauchy conditions that fix the state of the elastic rod at the initial instant, also called initial displacement and initial velocity.
The problem encountered when investigating the partial differential equations of non-homogeneous evolution with dissipation, coupled with certain initial and contour values, is reduced to analyzing at first the existence and uniqueness of the solution for the model describing the vibrations of an elastic rod subject to dissipative constraints. But a difficulty presents this non-homogeneous evolution equation with dissipative terms, for example, the data may have initial conditions of functions (solution of this PDE) that are not regular or sufficient to have non-differentiable functions in the classical sense and even be discontinuous, here is the importance of the weak or generalized solution in the study of the wave equations. This new solution concept was introduced by Sobolev and Schwartz (1969). These function spaces are based on the concept of weak solution consisting of L p (Ω) functions whose derivatives in the generalized sense also belong to such a space. Those spaces become the natural environment for the theorems of existence and uniqueness of generalized solutions of partial derivative equations, used by prestigious mathematicians in their research [1,2,3,4,6,7,8,9,10,13,14].

Materials and Methods
In the present article a mathematical model has been considered concerning the vibrations of a bar (acoustic, electromagnetic, etc.) in an elastic medium with dissipation caused by the medium where the motion occurs, subjected to an external force: The method used is from Faedo-Galerkin and Medeiro and consists of approximating the initial evolution problem by equivalent approximate systems, but in finite dimension. Then it is divided into stages: Approximate solution bounding, convergence of the approximate solutions, verification of the initial conditions and finally, uniqueness of the solution.

Results and Discussion
3.1. Existence of the generalized solution of the non-homogeneous hyperbolic differential equation  Let f ∈ L 2 (Q), u 0 ∈ H 1 0 (Ω) and u 1 ∈ L 2 (Ω). Then there exists the function u : [0, T ] → H 1 0 (Ω) fulfilling the conditions: Proof . The demonstration is divided into four stages:

Stage 1 -Approximate problem
The demonstration begins by analyzing H 1 0 (Ω) that the space is a separable Hilbert space, then there exists a succession of vectors (w i ),

Stage 2 -Approximate solution bounding
Two a priori estimates are presented: , be the first a priori estimate and then replacing it in the approximate equation (3.7) we have: The approximate solution u m (t) exists in [0, t m ). Also, integrating equation (3.10) we obtain: Now, applying Gronwall's inequality [7] in (3.11), it is concluded that: Furthermore, by the Poincaré-Friedricks inequality [8], it follows that: If t = T in equality (3.11), then: is a succesion bounded in L 2 (Ω), is a succesion bounded in H 1 0 (Ω).

A priori estimate II
Deriving the approximate equation (3.7) in relation at t, we obtain: Making v = u 00 m (t) ∈ V m , second estimate and replacing in equation (3.18) we obtain: Integrating from 0 to t in (3.19), we have:
Deriving in the sense of the distributions, the first and third integral of equation (3.27) and then, passing to the limit when m → +∞ are obtained: Again, applying the distributional derivative in (3.28), we have: for all v ∈ V m and all θ ∈ D (0, T ) .
Then equation (3.29) can be put in the following form:

Stage 4 -Initial conditions
Once again, we present two tests: Indeed, from (3.22) and (3.23) we have that: and therefore it makes sense to calculate u(0).
Taking in particular z ∈ L 1 ¡ 0, T ; H 1 0 (Ω) ¢ , the convergence becomes: Then, taking z(t) = vθ 0 (t), we have from (3.30) that: . Using the method of integration by parts in the first member of (3.31) and applying the limit when m → +∞, it results: Again, integrating by parts the second member of (3.31) implies that: Then, from (3.32) and (3.33) in (3.31), we obtain: Indeed, from (3.24) and (3.25) we have that: and therefore it makes sense to calculate u 0 (0).

Conclusions
With the theory of distributions, Sobolev spaces and the Faedo-Garleskin -Medeiros method, the existence and uniqueness of weak or generalized solutions u(x, t) defined on Q satisfying the hyperbolic problem (2.1) modeling the vibrations of an elastic bar with dissipation was proved.