On the solutions of non-planar oscillations for a nonlinear coupled Kirchhoff beam equations with moving boundary

J. Ferreira1, E. Pişkin2, S. M. S. Cordeiro3 and C. A. Raposo4,∗ 1 Department of Exact Sciences, Federal Fluminense University 27213-145, Volta Redonda, Brazil. 2 Department of Mathematics, Dicle University 21280, Diyarbakir, Turkey. 3 Faculty of Exact Sciences and Technology, Federal University of Pará, 68440-000, Abaetetuba, PA, Brazil. 4 Department of Mathematics, Federal University of São João del-Rei, 36307-352, São João del-Rey, Brazil. * Correspondence: raposo@ufsj.edu.br


Introduction
I n this work, we focus on the existence and uniqueness of strong solution for a nonlinear coupled Kirchhoff beam equations with moving boundary given by where a and b are real constants, Q t is the non-cylindrical domain of R 2 defined by being α( · ) and β( · ) functions of the class C 3 such that We denote by Σ t the lateral boundary of Q t given by The coupled system (1) describes non-planar oscillations of beams with a small thickness and variable limit length. Steady non-planar motions are characterized by each point on the beam centerline tracing an elliptical path perpendicular to the beam axis, then the functions u(x, t) and v(x, t) are the components of the displacement on a point of the beam in the direction of symmetry. The method used here consists in transform an initial boundary-value problem defined in a noncylindrical domain into another defined over a cylindrical domain whose sections are not time-dependent (see Dal Passo and Ughi [1]).
In the last decades, several types of equations have been used as some mathematical models that describe physical, chemical, biological, and engineering systems. Among them, mathematical models of vibrating and flexible structures have been considerably stimulated by an increasing number of questions of practical interest. In this sense, we stick to the study of a strong solution, for the proposed nonlinear model.
A mathematical model for transverse deflection of an extensible beam of length L, with ends attached at a certain fixed distance is given by the equation: For cylindrical domain, the stability of nonlinear oscillations of an elastic rod was studied by Haight and King [2]. Later, non-planar and nonlinear oscillations of a beam were considered by Ho at al., [3]. Following the authors, the equations of motion describing the non-planar response of a beam of length L in (0, L) × (0, ∞) are given by where p(x, t), q(x, t) are external forces. For non-planar oscillation of beams under periodic forcing, (see Fix and Kannan [4] and references therein).
Moving boundary problems occur in many physical applications like in heat transfer where a phase transition occurs, in moisture transport, such as swelling grains or polymers, and in deformable porous media problems where solid displacement is governed by diffusion. The noncylindrical domain is created by smooth perturbations of a fixed open set which are defined by diffeomorphism and was initially investigated by Lions [5] applying a method, called by himself the penalty method. From then on, problems in noncylindrical domain have been extensively studied by several authors.
To close this section, we mention some references on the partial differential equations in non-cylindrical domains in several contexts. Following these references, the reader will have an overview of the subject.
For the nonlinear vibrations of an elastic string, see for instance: Narasimha [6], Lions [7], Ho et al., [8], Ebihara et al., [9], Feireisl [10], Matsuyama and Ikehata [11], Cavalcanti [12]. Hyperbolic-parabolic equations with the nonlinearity of Kirchhoff-Carrier type in domain with moving boundary was studied Benabidallah and Ferreira [13]. A variational approach to evolution problems with variable domains was considered by Bonaccorsi and Guatteri [14]. Later, Límaco et al., [15] proved the existence, uniqueness, and controllability for parabolic equations in non-cylindrical domains. Asymptotic behaviour for the nonlinear beam equation in the noncylindrical domain was analyzed by Benabidallah and Ferreira [16]. For asymptotic behaviour for wave equations with memory in a non-cylindrical domain see Ferreira and Santos [17]. The stability for a Kirchhoff beam equation with memory in noncylindrical domain was given by Ferreira et al., [18]. The exponential decay for a Kirchhoff wave equations with a nonlocal condition in a non-cylindrical domain was analyzed by Ferreira et al., [19]. Santos et al.,in [20] studied the existence and uniform decay for a nonlinear beam equation with nonlinearity of Kirchhoff type in domains with moving boundary. Later, Robalo et al., in [21] considered a reaction diffusion model for a class of nonlinear parabolic equations with moving boundaries. We cite the work of Lopez et al., [22] for remarks on a nonlinear wave equation in a noncylindrical domain. The dynamics of stochastic Boissonade system on the time-varying domain was considered by Zhang and Huang [23]. For Pullback attractors for non-autonomous reaction-diffusion equation in non-cylindrical domains, we mention Xiao [24]. Finally, we refer the recent manuscript of Almeida et al., [25] where was applied the finite element scheme for a class of nonlocal parabolic systems with moving boundaries.
As far as we know, the strong solutions of non-planar oscillations for a nonlinear coupled Kirchhoff beam equations with moving boundary for system (1) were not considered previously. This paper consists of two sections in addition to the introduction. In Section 2, we recall some definitions of Sobolev spaces with their properties and present the technique to deal with our mobile boundary domain. In Section 3, we provide the existence and uniqueness of strong solutions.

Preliminaries
In this section we present the notation and the technique to transform the system (1) into another initial boundary-value problem defined over a cylindrical domain whose sections are not time-dependent.
For simplicity of notations hereafter we denote by | · | the Lebesgue Space L 2 (Ω) norm and by | · | 2 the Sobolev Space H 2 0 (Ω) norm. Let B be a Banach space and u : [0, T] → B a mensurable function. We denote To prove the existence of solution for system (1), the technic is transformed the non-cylindrical domain Q t into an equivalent problem in a fixed rectangular domain Q = (0, 1) × (0, T), T > 0, using the inverse of the transformation, that is, by using the diffeomorphism The change of variables (x, t) ∈ Q t → (y, t) ∈ Q transforms the equation (1) into the following problem in the cylindrical domain Q: where In order to simplify the calculation, we will take a = 0 and b = 1 in the definition of the function M(u, v) that is, We denote by M the following function To show existence os strong solution we will use the following assumptions sup ess 0≤t≤∞ inf ess 0≤t≤∞ sup ess 0≤t≤∞ Observe that (17) implies that Q t increases, that is, Concerning the functional M and F we assume that and the Equation (1) is satisfied almost everywhere in the corresponding domain.
In the next section we prove the existence of strong solutions for system (1).

Existence of global strong solution
To prove the existence of strong solutions for system (1) in Q t , first we prove the following result in the cylindrical domain Q:  (6) in the sense L ∞ (0, T; L 2 (0, 1)).
Proof. Let A be the operator Aw = w xxxx and D(A) = H 4 (0, 1) ∩ H 2 0 (0, 1). Obviously, A is a positive self adjoint operator in the Hilbert Space L 2 (0, 1) for which there exist sequences {w n } n∈N and {λ n } n∈N of eigenfunctions and eigenvalues of A such that the set of linear combinations of {w n } n∈N is dense in D(A) and Note that for any (u 0 , Let V m be the space generated by w 1 , w 2 , ..., w m . Standard results on ordinary differential equation imply the existence of a local solution u m , v m of the form in [0, t m ), with 0 < t m < T for any arbitrary T > 0. The extension of the solution on the whole interval [0, T] is a consequence of the priori estimates.

Analysis of the nonlinear terms
and M u m (t), v m (t) u m yy , w j −→ M u(x, y), v(x, t) u yy , w j , M u m (t), v m (t) v m yy , w j −→ M u(x, y), v(x, t) v yy , w j .
The convergence (65)-(70), (72) and (75) are sufficient to pass to limit in the approximated system (24)-(27) in order to obtain u m tt + 1 γ 4 u m yyyy − 1 γ 2 M(u, v)u m yy + u m t + F(u m ) + a 1 u m yy + a 2 u m ty + a 3 u m y = 0, v m tt + 1 γ 4 v m yyyy − 1 γ 2 M(u, v)v m yy + v m t + F(v m ) + a 1 v m yy + a 2 v m ty + a 3 v m y = 0, in L 2 (0, T; L 2 (0, 1)). The uniqueness follows by using standard methods. To verify the initial conditions we use the usual argument, as in Lions [5].
Now we present the principal result of this work.

Proof.
To show existence in non-cylindrical domain we return to our original problem by using the change variable given in (3). Let (u, v) the solution obtained from Theorem 1 and (u, v) defined by (5), then (u, v) belongs to the class where I t =]α(t), β(t)[ for any t ≥ 0. Then from (15)- (17), it is easy to see that (u, v) satisfies first and second Equations in (1) in the sense L 2 (0, ∞; L 2 (I t )). The uniqueness follows from the uniqueness of Theorem 1.