Global Existence and Exponential Decay for a Dynamic Contact Problem of Thermoelastic Timoshenko Beam with Second Sound

In this paper, we study the global existence and exponential decay for a dynamic contact problem between a Timoshenko beam with second sound and two rigid obstacles, of which the heat flux is given by Cattaneo’s law instead of the usual Fourier’s law. The main difficulties arise from the irregular boundary terms, from the low regularity of the weak solution and from the weaker dissipative effects of heat conduction induced by Cattaneo’s law. By considering related penalized problems, proving some a priori estimates and passing to the limit, we prove the global existence of the solutions. By considering the approximate framework, constructing some new functionals and applying the perturbed energy method, we obtain the exponential decay result for the approximate solution, and then prove the exponential decay rate to the original problem by utilizing the weak lower semicontinuity arguments.

Figure 1: A thermoelastic Timoshenko beam and the tip at x = l with clearance g = g 1 + g 2 .
The tip at x = l is modeled with the Signorini non-penetration condition, see [21,28].In particular, the tip with gap g is the asymmetrical so that g = g 1 + g 2 , where g 1 > 0 and g 2 > 0 are, respectively, the upper and lower clearances, when the system is at rest (see Figure 1).Then, the right end of the beam is assumed to move vertically only between two stops, namely −g 2 ≤ ϕ(l, t) ≤ g 1 , t ∈ (0, T ). (1.5) We denote by σ(t) the shear stress at x = l, i.e., σ(t) := k[ϕ x (l, t) + ψ(l, t)].
Accordingly, we prescribe −σ(t) ∈ ∂X (ϕ(l, t)), t ∈ [0, T ], (1.6) where ∂X denotes the subdifferential of the indicator function X +∞, otherwise, namely Many researchers got interested in studying the dynamics contact problems involving only a single displacement and/or a single variation of temperature, see for example [2,3,21,22,28,38,44]. Carlson [12] and Day [18] found that two or more materials may come in contact as a result of thermoelastic expansion or contraction in industrial processes.Copetti [15], Kuttler and Shillor [27] proposed the dynamic evolution of a thermoviscoelastic rod which may contact or impact a rigid or reactive obstacle, whereas the exponential energy decay rate for weak solutions of a thermoelastic rod, contacting a rigid obstacle, has been analyzed in [36].Copetti [16] proved existence and uniqueness results and proposed finite element approximations in space with backward Euler discretization in time for a contact problem in generalized thermoelasticity under the theory of thermoelasticity proposed by Green and Lindsay [24].Berti and Naso [10] considered the existence and longtime behavior of solutions for a dynamic contact problem between a nonlinear viscoelastic beam and two rigid obstacles.Afterward, thermal effects have been also taken into account in [7,9], where Berti et al. proved the existence and uniqueness of the solution as well as the exponential decay of the related energy.
Timoshenko beam with thermal contribution have been investigated by many authors and some results related to global existence and decay properties have been obtained, see for example [13,14,19,20,23,25,29,32,35,43,46].For the case of nonlinear internal frictional damping and without thermal effects, we refer the readers to Boussouira [1], Rivera and Racke [37], Raposo et al. [42] and Soufyane [45].The boundary stabilization and boundary control have been studied in [26,48] (see also references therein).Arantes and Rivera in [5] proved that the energy associated with the thermoelastic Timoshenko beam system decays exponentially as time goes to infinity.Meanwhile, a great number of researchers have devoted considerable amount time studying Timoshenko beam with contact problems.For instances, in [6], Araruna et al. showed the existence of solutions and the exponential stability of the energy for a contact problem associated with an elastic Timoshenko beam and a rigid obstacle under the assumption of a dissipative boundary feedback.Berti et al. [8] proved global existence in time of solutions and exponential decay for a dynamic contact problem between a Timoshenko beam and two rigid obstacles.In [17], well-posedness and fully discrete approximations for a dynamic contact problem between a viscoelastic Timoshenko beam and a deformable obstacle was analyzed.
In the above-mentioned result of Berti et al. [8], the heat dissipation is given through Fourier's law.As it is well known, by using the Fouriers law for the heat conduction, the thermal effect is propagated in an infinite speed in thermoelasticity.To overcome this physical paradox, many theories have been developed.Lord and Shulman [31] suggested that Fouriers law was replaced by Cattaneos law to describe the heat conduction, which transforms the classical thermoelastic system into the thermoelastic system with second sound, in which the thermal disturbance is propagated in a finite speed.Over the past decade, sevaral asymptotic behavior results have been obtained for the thermoelasticity system with second sound ([4, 11, 30, 33, 34, 39, 40, 41, 47]).
Berti et al. [9] investigated a dynamic contact problem describing the mechanical and thermal evolution of a damped extensible thermoviscoelastic beam under the Cattaneo law.
Motivated by these results, the aim of the present paper is to establish a global in time existence result to problem (1.1)-(1.6)and analyze its longtime behavior.In particular, we prove that the system possesses an energy decaying exponentially as time goes to infinity.Problem (1.1)- (1.6) can be regarded as an extension and improvement of Berti et al. [8] to the thermoelastic Timoshenko beam with second sound.It has been shown in [23]

Main results
To give a variational formulation of the problem, we introduce the following spaces: The initial data We define as the energy associated with system (1.1)-(1.5).
Definition 2.1 Let ϕ 0 , ψ 0 , θ 0 , q 0 , ϕ 1 , ψ 1 be given as in (2.1) and 0 < T ≤ ∞.We say that with initial data satisfying (1.2), the inequality ) ) Here are the main results of the paper.By a regularization, a priori estimates, and passage to the limit procedure, the proof of this result will be carried out in Section 3. In Section 4, we shall prove the following exponential decay result.
Theorem 2.3 (Exponential decay) Let ϕ be a weak solution to problem (1.1)-(1.6)provided by Theorem 2.2.Then there exist two positive constants R and ω, independent of t, such that (2.7)

Global existence
In this section, we show that the solution for problem (1.1)-(1.6) is global.Firstly, in Section 3.1, we approximate problem (1.1)-(1.6)by a penalization procedure and we prove well-posedness for the regularized problem (Proposition 3.1 below).Then, in Section 3.2, we show that a sequence of approximate solutions converges to a solution to the original problem.

Approximating problem
For any ε > 0, we introduce the families of initial data (ϕ ε 0 , ψ ε 0 , θ ε 0 , q ε 0 ) ε>0 , satisfying We introduce a penalized version of problem (1.1)-(1.6)by regularizing the Signorini contact condition with a normal compliance condition.We consider the following system: together with The boundary conditions at x = 0 are At the tip x = l, for t ∈ [0, T ], we set where Here and in the sequel, [f ] + := max{f, 0} denotes the positive part of a function f .
Henceforth, we will also use the following functionals: where with initial data satisfying (3.1), (3.3) and compatible with the boundary conditions (3.4)- (3.6) for t = 0.
Proof.(Construction of Faedo-Galerkin approximations) Let {w j } ∞ j=1 , be a basis of V and We construct the approximate solutions of the form verifying, for j = 1, ..., n, where and initial data Accordingly, the standard theory of ordinary differential equations guarantees, under Lipschitz conditions, system (3.10)-(3.13)appended by initial conditions (3.14) admits a local solution.We now need the a priori estimates that permit us to extend the solution to the whole interval [0, T ], for any T > 0.
(A priori estimates) We multiply (3.10) by h n jt , (3.11) by p n jt , (3.12) by u n j , (3.13) by v n j , respectively, summing over j and adding the resulting equations, we infer where E n,ε (t) = E(t, ϕ n,ε , ψ n,ε , θ n,ε , q n,ε ).Note that if we denote by f − = max{−f, 0} the negative part of a function f , we have f Thus, the previous equality becomes An integration over (0, t) and initial conditions (3.14) ensure that where K is a positive constant independent of n.Note that for any ϕ ε 0 ∈ K we get that σn,ε (0) = −εϕ n,ε t (l, 0).After a differentiation of Eqs.(3.10)-(3.13)with respect to t, we have where E n,ε t (t) = E(t, ϕ n,ε t , ψ n,ε t , θ n,ε t , q n,ε t ) and E n,ε t (t) , B n (t) are defined as follows In addition, by applying the Young's and Sobolev's inequalities and noting that |(f we can estimate the last term in (3.20) as follows (see [8]) where C ε is a positive constant depending on ε but independent of n, which is allowed to vary even in the same formula.From (3.20), we have An integration over (0, t) implies We can show that the second order energy is initially bounded, independently of n, namely is bounded independently of n.To this aim, we multiply (3.10) by h n jtt , we sum up over j = 1, ..., n and we let t → 0. By (3.14), we have After an integration by parts and owing to the compatibility conditions (3.4)-(3.6)for t = 0, we In the light of Hölder's inequality and Young's inequality, we deduce that there exists a constant Similarly, multiplying (3.11) by p n jtt , (3.12) by u n jt , summing up over j = 1, ..., n and letting t → 0, we get which leads to the inequality Finally, multiplying (3.13) by v n jt , summing up over j = 1, ..., n and letting t → 0, we obtain By (3.1), we infer that Thus, from (3.22) and applying Gronwall's inequality, we find that E n,ε t (t) is bounded in [0, T ]. (Passage to the limit) Inequalities (3.15) and (3.22) guarantee that Therefore we deduce, up to a subsequence, the convergence 2) 4 , we deduce that θ ε x , q ε x ∈ L ∞ (0, T ; L 2 (0, l)) and hence the regularity (ϕ ε , ψ ε , θ ε , q ε ) verifies the regularity specified in (3.9).Proposition 3.2 (Uniqueness) For any T > 0, the solution (ϕ ε , ψ ε , θ ε , q ε ) to problem (3.2), with initial data satisfying (3.3) and compatible with the boundary conditions (3.4)-(3.6), is unique.

Exponential decay
In this section, we prove an exponentially stability result of system (1.1)-(1.6).We introduce the following Lyapunov functional: where for δ 1 , δ 2 , δ 3 are positive constants which will be fixed later.
It is easy to check that, by using Young's inequality, Poincaré's inequality and Sobolev embedding theorem, there exist two constants β 1 and β 2 such that Next, we estimate the derivative of L ε (t) according to the following lemmas.
Lemma 4.1 Let (ϕ ε , ψ ε , θ ε , q ε ) be the solution provided by Proposition 3.1.Then there holds where C p is a Poincaré constant, J ε (t) is defined in (3.7) and η 1 is a positive constant to be chosen later.
Proof.By differentiating (4.2) with respect to t and by means of equation (3.2), we have By Hölder's inequality and Young's inequality, we get For the last term on the right-hand side of (4.7), we can obtain Substituting into the previous inequality, we reach the conclusion.

r − η 3 l
holds.By passing to lim ε→0 inf and on account of (3.27) and (3.31), we reach the conclusion.