Attractor of Beam Equation with Structural Damping under Nonlinear Boundary Conditions

0 (u x ) 2 dx)u xxt = q(x), in [0, L] × R+ with the structural damping and the rotational inertia term. Little attention is paid to the longtime behavior of the beam equation under nonlinear boundary conditions. In this paper, under nonlinear boundary conditions, we prove not only the existence and uniqueness of global solutions by prior estimates combined with some inequality skills, but also the existence of a global attractor by the existence of an absorbing set and asymptotic compactness of corresponding solution semigroup. In addition, the same results also can be proved under the other nonlinear boundary conditions.


Introduction
In this paper, we will consider a kind of more general Kirchhoff-type beam equation.The physical origin of the problem lies in the theory of vibrations of an extensible beam of length ; moreover, during vibration, the elements of a beam not only perform a translatory motion but also rotate.
In (1), when the structural damping term and the rotational inertia term are absent, (1) is a model for vibrations of tensible beam.This was proposed by Woinowsky-Krieger [1] in the form was done by Ball [2] which was later extended to an abstract setting by defining a linear operator  by Medeiros [3].In [4], Patcheu obtained the decay of the energy for above equation when a nonlinear damping (  ) was effective in Ω.In addition, the attractor on extensible beams with null boundary conditions was considered by several authors.We quote, for instance, [5][6][7], and so on.But the longtime behavior of the beam equation with nonlinear boundary conditions was paid little attention.We also refer the reader to a few works.One of the first studies in this direction was done by Pazoto and Menzala [8], where stabilization of a thermoelastic extensible beam was considered.Motivated by the result, Ma proved the existence of global solutions and the existence of a global attractor in [9] and [10], respectively, for the Kirchhoff-type beam equation with the absence of the structural damping and the rotational inertia, subjected to the nonlinear boundary conditions In the following, we mentioned some results on longtime behavior of beam equation with the rotational inertia term.Under null boundary conditions, Geredeli and Lasiecka [11] considered the existence of a compact attractor of beam with a rotational inertia term.Under nonlinear boundary conditions Ji and Lasiecka [12] considered the semilinear Kirchhoff equation with rotational inertia, and they showed that the above problem is uniformly stabilized with uniform energy decay rates.
In addition, we also mentioned some results on longtime behavior of the equation with the structural damping term.Chueshov [13] studied the global attractor with a structural damping of the form (‖∇‖ 2 )(−Δ)    with 1/2 ≤  ≤ 1.
Chueshov [14] and Yang et al. [15] considered the global attractor for the Kirchhoff-type equation (12) with structural damping under null boundary conditions, respectively.
Under the above assumptions, we prove the existence of global solutions and the existence of a global attractor of extensible beam equation system (1)- (4).And the paper is organized as follows.In Section 2, we introduce some Sobolev spaces.In Section 3, we discuss the existence of global strong and weak solutions.In Section 4, we establish the result of the existence of a global attractor in  × .

The Existence of Global Solutions
Firstly, using the classical Galerkin method, we can establish the existence and uniqueness of regular solution to problem (1)-( 4).We state it as follows.
Estimate 2. In approximate equation of (28), integrating by parts with  =    (0) and  = 0 and considering the compatibility condition (22) and then using Schwarz inequality and the mean value inequality, we see that there exists for all  ∈ [0, ] and for all  ∈ .
Estimate 3. Let us fix ,  > 0 such that  <  − .Taking the difference of approximate equation of (28) with  =  +  and  =  and replacing  by    ( + ) −    (), we can find a constant  3 > 0, depending only on , such that Uniqueness.Let , V be two solutions of ( 1)-( 4) with the same initial data.Then writing  =  − V and taking the difference (28) with  =  and  = V and replacing  by   and then using mean value theorem and the Young inequalities combined with Estimates 1 and 3, we deduce that, for some constant  > 0, Then from Gronwall's lemma we see that  = V.
Theorem 2. Assume the assumptions of Theorem 1 and ( 24) hold; if the initial data ( 0 ,  1 ) ∈  0 , then there exists a unique weak solution () of problem ( 1)-( 4) which depends continuously on initial data with respect to the norm of  × .
Proof.Let us consider { 0 ,  1 } ∈  ×  =  ×  2 , and since We observe that, for each  ∈ , there exists   , smooth solution of the initial boundary value problem (1)-( 4) which satisfies Considering the arguments used in the estimate of the existence of solution, we obtain where  0 is a positive constant independent of  ∈ .
Defining  , =   −   : ,  ∈  where   and   are regular solutions of (35), following the steps already used in the uniqueness of regular solution for (1)-( 4) and considering the convergence given in (34) ( 0  →  0 in  and  1  →  1 in ), we deduce that there exists  such that   →  strongly in  ([0, ) ; ) ,   →   strongly in  ([0, ) ; ) . (37) From the above convergence, we can pass to the limit using standard arguments in order to obtain Theorem 2 is proved.

The Existence of Global Attractor
In this section, we give the existence of a global attractor.
A global attractor for a  0 -semigroup () defined on a complete metric space  is a bounded closed subset A ⊂  which is positive fully invariant, that is, ()A = A, for all  ≥ 0, and uniformly attracting, that is, for any bounded set  ⊂ .
A bounded set B ⊂  is an absorbing set for () if, for any bounded set  ⊂ , there exists   = () ≥ 0 such that which defines (, ()) as a dissipative dynamical system.
A semigroup () is asymptotically smooth in  if for any bounded positive invariant set  ⊂ , there exists a compact set  ⊂  such that dist ( () , ) → 0 as  → ∞. ( Then the following lemma is well known. Lemma 5 (see [16], Theorem 2.3).Let () be a dissipative  0semigroup defined on a metric space ; then () has a compact global attractor in  if and only if it is asymptotically smooth in .

limRemark 8 .Theorem 10 .
() −   ()     2 () is asymptotically smooth in  0 .That is, Lemma 6 holds.Thus Theorem 7 is proved.The novelty and difficulty of Theorems 4 lie in the appropriate definition on the functions () = ∫  0 (  −   ) , () = ∫  0 (  −   )  and the relationship between Ẽ () and the energy function Ẽ() and the relationship between   () and the function ().Lemma 9([18], Theorem 1.1 in Chapter I).One assumes that  is a metric space and that the operators {()} ≥0 are given and satisfy the usual group properties and () is continuous operator from  into itself and the semigroup {()} ≥0 is asymptotically compact.One also assumes that there exist an open set  and a bounded set B of  such that  is absorbing in B. Then (B) is a compact attractor.By view of Lemma 9, with Theorems 4 and 7, the main result of a global attractor reads as follows.The corresponding semigroup () of problem (1)-(4) has a compact global attractor in the phase space  0 .