Abstract
We prove in this paper the existence of a global attractor for the plate equations of Kirchhoff type with nonlinear damping and memory using the contraction function method.
1. Introduction
Let us consider the long-time behavior for the following Kirchhoff plate equations with fading memory and nonlinear damping:where is a bounded domain with smooth boundary ; , , and are positive constants, , ; is the unit outer normal on ; is the prescribed past history of .
Problem (1) arises from the isothermal viscoelastic theory; it describes a fourth-order viscoelastic plate with a lower order perturbation and also models transversal vibrations of a thin extensible elastic plate in a history space, which is established based on the framework of elastic vibration by Woinowsky-Krieger [1] and Berger [2], and can be seen as an elastoplastic flow equation with some kind of memory effects (see [3, 4] for details). The convolution term means that the stress at any instant depends on the whole history of strains which the material has undergone and produced a weak damping mechanism (see [5, 6]).
In the case where , , (1) becomes the normal plate equation which has been treated in many papers such as [7–14]. For instance, the authors investigated the existence of the compact attractor for the plate equation on both the bounded domain [8, 10, 13] and the unbounded domain in [7, 11, 12], respectively. Yue and Zhong [9] proved the existence of global attractors to the plate equations when the nonlinear function satisfies the critical exponent in a locally uniform space. In [14], the authors studied the existence of the random attractor for the stochastic strongly damping plate equations with additive noise and critical nonlinearity.
The case of in problem (1) has been studied by several authors (see [2, 5, 15–23] and references therein). For instance, Wu [15] scrutinized the existence of global attractors for the nonlinear plate equation with thermal memory effects due to non-Fourier heat flux laws when . Recently, Conti and Geredeli [5] paid attention to the existence of a smooth global attractor for the nonlinear viscoelastic equations with memory, in which they required the nonlinear damping to be the polynomial growth. Shen and Ma [21] studied the existence of the random attractor for plate equations with memory and additive white noise. On the other hand, the asymptotic behavior of solutions for the extensible plate equations without memory affection was studied by several authors in [24–27].
We focus on the existence of the extensible plate equations with nonlinear damping and history memory in the present paper. To prove the existence of a compact global attractor, the key goal is to establish the compact property of the semigroup associated with the dynamical system. Regarding problem (1), we need to overcome the following difficulties. One difficulty is caused by the critical nonlinearity and nonlinear damping. In order to overcome these obstacles, we apply the contraction function method into our problem. Another difficulty is brought about by the memory kernel, because there is no compact embedding in the history space; besides, we cannot use the finite rank method. We solve this term by introducing a new variable and defining an extending phase space (see [20] for details). In addition, the terms make the estimates more complex, so we have to deal with them through accurate computation. Our main result is Theorem 12.
As in [18], we define
By assuming that and taking , the original memory term can be rewritten as and then problem (1) can be transformed into the following system: Here, (5) is obtained by differentiating (2). Initial-boundary value conditions are given as follows: where
This paper is organized as follows: in Sections 2 and 3, we make some preparations for our consideration; in Section 4, we will show the existence of bounded absorbing set and compact global attractors.
2. Assumptions
The following conditions are necessary for our main result.
Concerning the nonlinear term , there exists a constant such that where Condition (9) implies that . Also, we say that is a critical exponent for the growth of when . In addition, we assume that where is the principal eigenvalue of in .
The nonlinear damping function satisfies with if and if .
With respect to the memory kernel , we assume that and that there exists a constant such that
Now, we consider the Hilbert spaces that will be used in our analysis. Let equipped with the respective inner products and norms, where is -inner product and we use to denote -norms.
We define the following weighted -space: which is a Hilbert space endowed with inner product and norm
Finally, we introduce the phase space equipped with the norm
In order to obtain the global attractor of problems (4)–(6), we need the following theorem. Under our hypotheses, we can derive an existence result by standard Faedo-Galerkin method (see [12, 28, 29]). For arguments involving the memory term, we follow Giorgi et al. [19, 20].
Theorem 1. Let (8)–(15) hold. If , then the following results hold. (i)Provided that the initial data , then problems (4)–(6) have a weak solution satisfying (ii)Assume that are weak solutions of problems (4)–(6) corresponding to initial data , . Then, for some constant .
Remark 2. Let be the unique weak solution of problems (4)–(6); by Theorem 1, we can define a semigroup as follows: And is continuous on .
3. Some Abstract Results
In this section, we will recall some basic theories of infinite dimensional dynamical systems; we refer to [30] for more details.
Definition 3. The global attractor is the maximal compact invariant set and the minimal set that attracts all bounded sets: for any bounded set , where is the Hausdorff semidistance in .
Definition 4. A semigroup is dissipative if it possesses a bounded absorbing set; that is, for any bounded set , there exists such that
Definition 5. is asymptotically smooth if, for any bounded positively invariant set , there exists a compact set such that
Theorem 6. A dissipative dynamical system has a compact global attractor if and only if it is asymptotically smooth.
Theorem 7. For any bounded positively invariant set , is called asymptotically smooth in , if, for any and any sequence in , there exists such that where satisfies
4. Existence of Attractors
In this section, we will use the abstract results presented in Section 3 to prove our main result.
Lemma 8. Under assumptions (8)–(15), the semigroup corresponding to problems (4)–(6) has a bounded absorbing set in .
Proof. Taking the scalar product in of (4) with , we infer that where .
Thanks to (5), (13), and (15) and Hölder inequality, we get We conclude from (32)–(34) that where Since from (11), we obtain Thus, (35) implies It follows from (10) that there exist () and , such that Using (40) and Poincaré and Young inequalities, we end up with where , .
By (35) and (41), we deduce that Now, we set and rewrite the equation of (4) as follows: We formally take the scalar product in of (43) with ; after a computation, we find where Denote Therefore, together with (33) and (44), we getFrom (40), for any sufficiently small, using Hölder and Young inequalities, we get Thanks to (11), we get Using the Young inequality, we obtain From (11) and (12), we deduce that where is a constant which is independent of .
In line with (52), the Hölder and Young inequalities, and the Sobolev embedding , similar to the progress of [31], we conclude where is a small positive constant and is an embedding constant.
Therefore, we choose and small enough, such that Together with (40), (50), (51), and (53), we have where is a constant which depends on , and , while is a constant which depends on , and .
Notice that where .
Integrating (48) from 0 to , combining with (42), (49), and (55), we arrive at where .
Therefore, for any , there exists , such that Denote by the argument above, we know that is a bounded absorbing set.
Define then is also a bounded absorbing set. This shows that the semigroup corresponding to problems (4)–(6) has a bounded absorbing set in .
Remark 9. If is a solution of (4)–(6) with initial data in a bounded set , then we have where is a constant depending on .
In order to prove that the dynamical system is asymptotically smooth, we need the following lemma.
Lemma 10. Assume that assumptions (8)–(15) hold and ; is a bounded set; let and be two solutions of problems (4)–(6), such that and are in . Then,for any , where and are constants.
Proof. For convenience, we denote and . Then, satisfy the following equations: and initial datum Taking the scalar product in of (63) with , we obtain where .
Noting the similar estimate used in Lemma 8, we obtain Using the Poincaré, Hölder, and Young inequalities and taking small enough, such that then we obtain Combining (69) with (66), we get besides, where is an embedding constant for .
By virtue of the generalized Hölder inequality with , (8), (61), and Young inequality, we have Now, we estimate .
Set where Taking advantage of the Hölder and Young inequalities, (61), it follows that where we have used the fact that .
Plugging the above two inequalities into (73), we obtain Combining with (71), (73), and (76), we deduce from (70) We can choose so small such that Then, we set where By the Gronwall lemma, we get Hence, where .
Namely, It is not difficult to know that (62) holds, where .
Now, we shall show that the dynamical system is asymptotically smooth.
Lemma 11. Assume that assumptions (8)–(15) hold and . Then, the dynamical system is asymptotically smooth.
Proof. Given a bounded positively invariant set of , for , denote and as the solutions of (4)–(6). Then, given , choosing , such that , we can infer from (62) that where is a constant which depends only on the size of .
Now, we estimate the right hand side of (84). Condition (9) implies that if and if . Taking and applying the Galiardo-Nirenberg interpolation inequality, we have Since and are uniformly bounded, there exists a constant such that Using (86), we can rewrite (84) as with We claim that satisfies (31). In fact, let with be given. Denote . It can be shown that is invariant by the property of , and hence are uniformly bounded in .
Particularly, Then, by the fact and the Aubins lemma, it is not hard to show that there exists a subsequence that converges strongly in ; therefore, Without loss of generality (at most by passing subsequences), we assume that following similar argument given in [32, Lemma ] and [31], we get then, the asymptotic smoothness property of follows from Theorem 7.
Finally, we obtain our main results.
Theorem 12. Assume that assumptions (8)–(15) hold; then, the dynamical system corresponding to system (4)–(6) has a compact global attractor .
Proof. We note that Lemmas 8 and 11 imply that is a dissipative dynamical system which is asymptotically smooth. By Theorem 6, it has compact global attractor in the phase space .
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work is partly supported by the NSFC (nos. 11561064 and 11361053) and NWNU-LKQN-14-6.