Existence and Boundary Stabilization of the Semilinear Mindlin-Timoshenko System

We consider dynamics of the one-dimensional Mindlin-Timoshenko model for beams with a nonlinear external forces and a boundary damping mechanism. We investigate existence and uniqueness of strong and weak solution. We also study the boundary stabilization of the solution, i.e., we prove that the energy of every solution decays exponentially as t ! 1.


Introduction
A widely accepted dynamical model describing the transverse vibrations of beams is the Mindlin-Timoshenko system of equations.This system is chosen because it is a more accurate model than the Euler-Bernoulli beam one and because it also takes into account transverse shear effects.The Mindlin-Timoshenko system is used, for example, to model aircraft wings.For a beam of length L > 0 this one-dimensional system reads as ρh 3  12 where Q = (0, L) × (0, T ) and T > 0 is a given time.In (1.1) subscripts mean partial derivatives.Here the function u = u (x, t) is the angle of deflection of a filament (it is measure of transverse shear effects) and v = v (x, t) is the transverse displacement of the beam at time t.The constant h > 0 represents the thickness of the beam that, for this model, is considered to be small and uniform, independent of x.The constant ρ is the mass density per unit volume of the beam and the parameter k is the so called modulus of elasticity in shear.It is given by the formula k = kEh/2 (1 + µ) , where k is a shear correction coefficient, E is the Young's modulus and µ is the Poisson's ratio, 0 < µ < 1/2.The functions f and g represent nonlinear external forces.For details concerning the Mindlin-Timoshenko hypotheses and governing equations see, for example, Lagnese [7] and Lagnese-Lions [8].
We impose the following boundary conditions: u (0, •) = v (0, •) = 0 on (0, T ) , u x (L, •) + u t (L, •) = 0 on (0, T ) , u (L, •) + v x (L, •) + v t (L, •) = 0 on (0, T ) . (1. 2) The conditions (1.2) assure that the beam stays clamped in the end x = 0 and in the end x = L it is supported and suffering action of a dissipative force.To complete the system, let us include the initial conditions: Several authors analyzed different aspects of the Mindlin-Timosheko system.In the linear case (f ≡ g ≡ 0) we can cite Lagnese-Lions [8], Medeiros [12], which studied the exact controllability property using the Hilbert Uniqueness Method (HUM) introduced by Lions (see [11]) and Lagnese [7] which analyzed the asymptotic behavior (as t → ∞) of the system.In Araruna-Zuazua [2] was made a spectral analysis of the system allowing to obtain a controllability using HUM combined with arguments of non-harmonic analysis.In the semilinear case, we can mention Parente et.al. [16], which treat about existence and uniqueness for the problem (1.1) − (1.3) , with the functions f and g being Lipschitz continuous, applying the same method used in Milla Miranda-Medeiros [15].The existence of a compact global attractor, in the 2-dimensional case, was studied in Chueshov-Lasiecka [4] with the nonlinearities f and g being locally Lipschitz.All the mentioned papers are treated with different boundary conditions involving several situations that appear in the engineering.
In this work we state a result of existence of solutions for the system (1.1) − (1.3) , when the nonlinearities f and g satisfy the following conditions: f, g are continuous function, such that f (s) s ≥ 0 and g(s)s ≥ 0, ∀s ∈ R.
(1.4) Furthermore, we analyze the asymptotic behavior (as t → ∞) of the solutions with the nonlinearities satisfy the additional growth condition: Precisely, we show the existence of positive constants C > 0 and κ > 0 such that the energy of the system (1.1) defined by verifies the estimate E(t) ≤ CE (0) e −κt , ∀t ≥ 0. (1.8) The uniqueness for the semilinear Mindlin-Timoshenko system (1.1) − (1.3) with the general nonlinearities considered here is a open problem.
To obtain existence of solution of the semilinear Mindlin-Timoshenko problem (1.1) − (1.3) , we found difficulties to show that the solution verifies the boundary conditions (1.2) and to overcome them, we use the same techniques applied in [1], that consists essentially in to combine results involving non-homogeneous boundary value problem with hidden regularity arguments.Boundary stability is also analyzed, that is, we show that the energy (1.7) associated to weak solution of the problem (1.1)−(1.3)tends to zero exponentially as t → ∞.In order, the exponential decay was obtained by constructing perturbed energy functional for which differential inequality leads to this rate decay.We apply this method motivated by work of Komornik-Zuazua [6], whose authors treated this issue for semilinear wave equation.
The paper is organized as follows.Section 2 contains some notations and essential results which we apply in this work.In Section 3 we prove existence and uniqueness of strong solution for (1.1) − (1.3) employing the Faedo-Galerkin's method with a special basis like in [15] with f and g being Lipschitz continuous functions satisfying a sign condition.Section 4 is devoted to get existence of weak solution of (1.1) − (1.3) , with f and g satisfying (1.4).For this, we approached the functions f and g by Lipschitz functions, as in Strauss [17], and we obtain the weak solution as limit of sequence of strong solutions acquired in the Section 3. We still analyze the uniqueness only for some particular cases of f and g which permit the application of the energy method as in Lions [9].Finally, in Section 5 we prove the exponential decay for the energy associated to weak solution of the problem (1.1) − (1.3) making use of the perturbed energy method as in [6].

Some Notations and Results
Let us represent by D(0, T ) the space of the test functions defined in (0, T ) and H 1 (0, L) the usual Sobolev space.We define the Hilbert space equipped with the inner product and norm given by Let us represent by E the Banach space with the norm The trace application γ : In what follows, we will use C to denote a generic positive constant which may vary from line to line (unless otherwise stated).
We will now establish some results of elliptic regularity essential for the development of this work.
Let w be the unique solution of the following boundary value problem: Since f ∈ L 2 (0, L), we have by classical elliptic result (see for instance [3]) that w ∈ D and the existence of a constant C > 0 such that We would like to prove existence and uniqueness of solution for the problem (2.5) Formally, we obtain from (2.5) that Taking in (2.6) v ∈ D, we obtain We adopt (2.7) as definition of solution of (2.5) in the sense of transposition (see [10]).To guarantee the existence and uniqueness of (2.5) we consider the follow result: , then there exists a unique function u ∈ E satisfying (2.7).The application T : Proof.Let g ∈ L 2 (0, L) and v be a solution of the problem (2.8) We have v ∈ D.
Let us consider the application S : L 2 (0, L) → C 0 ([0, L]) such that Sg = v, where v is the solution of (2.8).Then S is linear and continuous.Let S * be the transpose of S, that is, where •, • represents different pairs of duality.Let us prove that the function u = S * f satisfies (2.7).In fact, we have S * f, g = f, Sg , which means For the uniqueness, we consider u 1 , u 2 ∈ L 2 (0, L) satisfying (2.7).Then Considering g ∈ L 2 (0, L) and v be a solution of (2.8), we get Therefore u 1 = u 2 and the uniqueness is proved.Since T = S * and S * is linear and continuous, it follows that T has the same properties.
For the non-homogeneous boundary value problem we consider the following result: Then there exists a unique solution u ∈ E for the problem (2.10).
Proof.Let us consider the function ξ : [0, L] → R, defined by ξ(x) = βx.Let w be the solution of the problem Since f ∈ L 1 (0, L) , by Proposition 2.2, it follows that w ∈ E. Taking u = w + ξ, we have u ∈ E is a solution of (2.10).
For the uniqueness, let u 1 and u 2 two solutions of (2.10).Then Hence, by Proposition 2.2, we have v = 0, which implies u 1 = u 2 .
Let us consider w (1) to be a solution of the problem According to Proposition 2.1, it follows that w (1) ∈ V ∩ H 2 (0, L) and x Analogously, let us consider w (2) to be a solution of the problem −w (2) By Proposition 2.1, we have that w (2) ∈ V ∩ H 2 (0, L) and

Strong Solution
Our goal in this section is to prove existence and uniqueness of solutions for the problem (1.1) − (1.3) , when u 0 , v 0 , u 1 and v 1 are smooth.EJQTDE, 2008 No. 34, p. 7 Let be f, g functions defined in R and u 0 , v 0 , u 1 , v 1 functions defined in (0, L) satisfying f, g : R → R are Lipschitz function with constant c f , c g , respectively, and sf (s) ≥ 0, sg (s) ≥ 0, ∀s ∈ R, (3.1) ) ) Proof.We employ the Faedo-Galerkin's method with the special basis in V ∩ H 2 (0, L) .
Since the data u 0 , v 0 , u 1 and v 1 verify (3.2)−(3.5), it follows by Proposition 2.5 the existence of four sequences (u 0ν ) ν∈N , (u 1ν ) ν∈N , (v 0ν ) ν∈N and (v 1ν ) } is a linearly independent set, we take as being the first four vectors of the basis.By Gram-Schmidt's orthonormalization process, we construct, for each ν ∈ N, a basis in V ∩H 2 (Ω) represented by {w ν 1 , w ν 2 , w ν 3 , w ν 4 , . . ., w ν n , . ..}.Otherwise, if A is a linearly dependent set, we can extract a linearly independent subset of A and continue the above process.For each m ∈ N, we consider V ν m = [w ν 1 , w ν 2 , w ν 3 , w ν 4 , . . ., w ν m ] the subspace of V ∩ H 2 (Ω) generated by the first m vectors of basis.Let us find an "approximate solution" (u νm , v νm ) ∈ V ν m × V ν m of the type where µ jνm (t) and h jνm (t) are solutions of the initial value problem for all ψ, ϕ ∈ V ν m , where (3.21) The system (3.20) has solution on an interval [0, t νm ] , with t νm < T .This solution can be extended to the whole interval [0, T ] as a consequence of a priori estimates that shall be proved in the next step.Adding the equations in (3.20) results for all ψ, ϕ ∈ V ν m .Estimates I. Making ψ = 2u νm t (t) , ϕ = 2v νm t (t) in (3.22) , integrating from 0 to t ≤ t νm and using (3.21), we get where ds and the constant C > 0 is independent of m, ν and t.We must obtain estimates for the terms 2 L 0 F (u 0νm )dx and 2 L 0 G(v 0νm )dx.Since EJQTDE, 2008 No. 34, p. 9 f (s)s ≥ 0 and g(s)s ≥ 0, it follows that F (t) ≥ 0 and G(t) ≥ 0, for all t ∈ [0, T ] and f (0) = g(0) = 0. So, by (3.1), we have (3.24) From (3.21) and (3.24), the inequality (3.23) becomes ) where C > 0 is a constant which is independent of m, ν and t.In this way, we can prolong the solution to the whole interval [0, T ] .Estimates II.Considering the temporal derivative of the approximate equation (3.22), setting ψ = u νm tt (t) and ϕ = v νm tt (t) in the resulting equation and integrating from 0 to t ≤ T we get (3.26)We need estimates for the terms involving u νm tt (0) , v νm tt (0) and for last two integrals in (3.26) .For this, we consider in (3.22) t = 0, ψ = u νm tt (0) and ϕ = v νm tt (0).So, using (3.1) and (3.21) we obtain ρh 3  12 where C > 0 is a constant independent of m, ν and t.We also have by ( From (3.30) − (3.35), we can obtain subsequences of (u νm ) and (v νm ), which will be also denoted by (u νm ) and (v νm ), such that According to (3.1), (3.30), (3.33) and the compact injection of H 1 (Q) in L 2 (Q), there exists a subsequence of (u νm ) and (v νm ), which will be also denoted by (u νm ) and (v νm ), such that We can see that the estimates (3.25) and (3.29) are also independent of ν.So, using the same arguments to obtain u ν and v ν , we can pass to the limit, as ν → ∞, to obtain functions u and v such that To complete the proof of the theorem, we need to show that u, v ∈ L 2 (0, T, H 2 (0, L)).For this, we consider the following boundary value problem: (3.59) , it follows by Proposition 2.1 that u, v ∈ L 2 (0, T, H 2 (0, L)).Using a standard argument, we can verify the initial conditions.The uniqueness of solution is proved by energy method.

Weak Solution
The purpose of this section is to obtain existence of solutions for the problem (1.1) − (1.3), with less regularity on the initial data and now f, g being continuous functions and sf (s) ≥ 0, sg (s) ≥ 0, ∀s ∈ R. Owing to few regularity of the initial data, the corresponding solutions shall be called weak.
Then there exist at least two functions u, v : ) )
EJQTDE, 2008 No. 34, p. 13 Since the initial data u 0 and v 0 are not necessarily bounded, we approximate u 0 and v 0 by bounded functions of V .We consider the functions ξ j : R → R defined by Considering ξ j (u 0 ) = u 0j and ξ j (v 0 ) = v 0j , we have by Kinderlehrer-Stampacchia [5] that the sequences (u 0j ) j∈N and (v 0j ) j∈N in V are bounded in [0, L] and Let us take the sequences (u 0jp ) p∈N , (v 0jp ) p∈N in V ∩ H 2 (0, L) and (u 1p ) p∈N , (v 1p ) p∈N in V such that u 0jp → u 0j strongly in V, (4.13) We fix (j, p, ν) ∈ N.For the initial data (u 0jp , u 1p , v 0jp , v 1p ) ∈ {[V ∩ H 2 (0, L)] × V } 2 , there exist unique functions u jpν , v jpν : Q → R in the conditions of the Theorem 3.1.By the same argument employed in the Estimates I (see (3.23)), we obtain We need estimates for the terms From (4.11) and (4.12), there exist subsequences of (u 0j ) j∈N and (v 0j ) j∈N , which still be also denoted by (u 0j ) j∈N and (v 0j ) j∈N , such that u 0j → u 0 a. e. in (0, L) , v 0j → v 0 a. e. in (0, L) .
By continuity of F and G, it follows that ).Thus, by (4.3) and the Lebesgue's dominated convergence theorem, we get Making the same arguments for F ν and G ν , it follows that where the constant C > 0 is independent of j, p and ν.Using (4.11) − (4.16) and (4.28) in (4.19), we have where C > 0 is independent of j, p, ν and t.From (4.29), we get (u jpν ) is bounded in L ∞ (0, T, V ), (4.30) As the estimates above are hold for all (j, p, ν) ∈ N 3 and, in particular for (ν, ν, ν) ∈ N 3 , we can take subsequences (u ννν ) ν∈N and (v ννν ) ν∈N , which we denote by (u ν ) ν∈N and (v ν ) ν∈N , such that We note that the Theorem 3.1 gives us a. e. in Q.
We also have Making the inner product in L 2 (Q) of (4.46) with u ν (t), we obtain where C > 0 is independent of ν.From (4.52), (4.55) and Strauss' Theorem (see [17]), it follows Analogously, taking the inner product in L 2 (Q) of (4.47) with v ν (t) and after using (4.14) , (4.16) , (4.30), (4.31), (4.33) and (4.35) we get where C > 0 is independent of ν.From (4.53), (4.57) and Strauss' Theorem (see [17]), it follows According to (4.59), we deduce From the uniqueness given by Proposition 2.3, we get ), we can apply the trace theorem in (4.65) to obtain For other side, by (4.46) we have . Thus, as it was done before, we have According to [13] and (4.67), we have From (4.67) − (4.70) and by continuity of the trace, we obtain Taking into account the convergences (4.71) − (4.73), it follows by (4.65) and (4.66) that In this way, comparing (4.42) and (4.74), we can conclude Making the same procedure as before, from (4.60), it follows that with u x , v t ∈ L 2 (Q) and g(v) ∈ L 1 (Q).By Propositions 2.1 and 2.3, there exist functions . So, we can find Applying the trace theorem in (4.78), we obtain We know by (4.47) that which comparing with (4.43), we deduce To verify the initial conditions (4.10) , we use the standard method.The aim of this section is to study the asymptotic behavior of the energy E(t) associated to weak solution of the problem (1.1) − (1.3).As it was mentioned in the introduction, this energy is defined by (5.1) Let us consider the following additional hypotheses: and ∃ δ 2 > 0 such that g(s)s ≥ (2 + δ 2 )G(s), ∀s ∈ R. (5. 3) The functions f and g given in the Remark 4.1 satisfy the conditions (4.1) , (4.3) , (5.2) and (5.3) .
The mean result of this section is: Theorem 5.1 Let L < min {2, 2/k} and f, g, u 0 , v 0 , u 1 , v 1 in the conditions of the Theorem 4.1 plus the hypotheses (5.2) and (5.3).Then there exists a positive constant κ > 0 such that the energy E(t) satisfy E(t) ≤ 4E(0)e −κt , ∀t ≥ 0. (5.4) Proof.Taking the inner product in L 2 (0, L) of (4.46) and (4.47) with u ν t (t) and v ν t (t) , respectively, we obtain where E ν (t) is the energy associated to strong solution (u ν , v ν ), obtained in Section 3, when f and g are replaced by f ν and g ν , respectively.Thus this energy is non-increasing.
It is important to emphasize that, for each ν ∈ N, the functions f ν and g ν of the approximating sequences also satisfy the conditions (5.2) and (5.3), respectively (cf.[17]).
Combining the last inequality with (5.12), we deduce In this way, taking the lim inf ν→∞ in (5.26), we can deduce the inequality (5.4).

Remark 4 . 1
The uniqueness of solution in the conditions of the Theorem 4.1 is a open question.But, for some particular cases of the nonlinearities, for example f (s) = |s| p−1 s and g (s) = |s| q−1 s with p, q ∈ [1, ∞) , we can use the energy method as in Lions[9, p. 15]   to obtain the uniqueness of solution.EJQTDE, 2008 No. 34, p. 20