Pullback 𝐷-Attractor of Coupled Rod Equations with Nonlinear Moving Heat Source

Abstract

We consider the pullback -attractor for the nonautonomous nonlinear equations of thermoelastic coupled rod with a nonlinear moving heat source. By Galerkin method, the existence and uniqueness of global solutions are proved under homogeneous boundary conditions and initial conditions. By prior estimates combined with some inequality skills, the existence of the pullback -absorbing set is obtained. By proving the properties of compactness about the nonlinear operator , , and then proving the pullback -condition (C), the existence of the pullback -attractor of the equations previously mentioned is given.

1. Introduction

In this paper, we consider a thermoelastic coupled rod system: with an external force function and a nonlinear moving heat source function. Here is the rod elastic displacement. is the dimensionless temperature. , , are all positive constants, where is the square of wave velocity, is the damping coefficient, and is the thermal diffusivity. is a bounded smooth domain. is the external force and is locally square integrable with respect to time for , ; that is, . is the moving heat source and is locally square integrable in time for , ; that is, . and are all the nonlinear function, and and are continuous on , respectively. We give the pullback -attractor for the nonautonomous nonlinear equations of thermoelastic coupled rod in space , where , .

Recently the research of the nonautonomous infinite dimensional dynamical system has been paid much attention and developed fast as evidence by the references cited in [1–7]. Chepyzhov and Vishik [1] firstly extend the notion of global attractor in the autonomous case to the concept of the uniform attractor for the nonautonomous case. But the uniform attractor is not applicable to the nonautonomous systems in which the trajectories can be unbounded as time increases to infinity. Therefore some new concepts and theories must be brought up for such nonautonomous case, where the concepts and the theorem of existence of the pullback -attractor were advanced in [2–8] and so on.

Caraballo et al. [2] and so forth gave the existence of the pullback -attractor for a nonautonomous N-S equation under the assumptions of asymptotic compactness and existence of a family of absorbing sets. Wang and Zhong [3] advanced the existence of the pullback -attractor for the dissipative Sine-Gordon wave equation in an unbounded domain in which the external force did not need to be bounded. In [4, 5], The author studied the pullback attractor of the reaction-diffusion equation and the generalized Korteweg-de Vries-Burgers equation, respectively. S. H. Park and J. Y. Park [6] considered the nonautonomous modified Swift-Hohenberg equation and proved the existence of the pullback attractor when its external force has exponential growth. The abovementioned systems are all specific systems. For the widespread used nonautonomous structural system in engineering, the study has been paid less attention. Park and Kang [7] studied the existence of the pullback -attractor for nonautonomous suspension bridge equation because of being motivated by Ma et al. [8, 9]:

In this paper, based on Al-Huniti et al. [10] as the relaxation time is not considered and Carlson [11], we study a more general nonlinear thermoelastic coupled system (1)–(4) of a rod due to a nonlinear moving heat source . We give the existence of a pullback -attractor of above system by proving the existence of a pullback -absorbing set and pullback condition (C) for the external force unnecessarily bounded.

In fact, we assume that the external forces and satisfy , , and , respectively, and for any where is a small real number which will be characterized later.

On the assumptions of the nonlinear function , Park and Kang gave the assumption (where ) for nonautonomous suspension bridge equations in [7]. At present, we remove the assumption of [7] and we assume that the nonlinear function satisfies the following assumptions:(H₁)we denote by the primitive of ; that is, , and then (H₂) for some ;(H₃) for some ;(H₄)there exists a constant such that (H₅).

We also assume that the nonlinear function satisfies the following assumptions: there exists a constant such that

Throughout this paper, we introduce the spaces and and endow these spaces with the usual scalar products and norms , , , , where , . Because of defining , with reference to [7] we have the scalar products and norm in the space . By the Poincare inequality, there exists a proper constant , such that

2. Pullback -Attracting Set

By normal Galerkin method (see [1, 12–14]), we have the following theorem of existence and uniqueness of solutions to problems (1)–(4).

Theorem 1. Assume that , , , , and the assumptions of the functions , hold; then, for all and any given , , , problems (1)–(4) have a unique solution such that where .
Moreover , for all , , , ; then

For simplicity, we write , . We denote by the space of vector functions with the norm in and denote by the space of vector functions with the norm in . We can construct the nonautonomous dynamical system generated by problems (1)–(4) in or . We consider , , and then we define The uniqueness of solutions to problems (1)–(4) implies that And, for all , , the mapping (or ) defined by (15) is continuous. Consequently, the mapping defined by (15)  is a continuous cocycle on or .

Let be the set of all functions such that where and and denotes the class of all families such that for some , where denotes the closed ball in centered on with radius .

Theorem 2. Assume that , , and the assumptions of the functions , hold. Suppose that and satisfy (7). Then there exists a pullback -attracting set in for the nonautonomous dynamical system defined by (15).

Proof. Let , , and be fixed. Define Taking the scalar product in of (1) with and taking the scalar product in of (2) with , after a computation of addition, we obtain For simplicity, define . By the assumption (), it is obvious that . By the assumption () of , we have so Considering assumption (11) of , we have By the Young inequality and (12), we have Letting and taking and , we infer from (19) that Also taking , we have Note that and by (25), we have By integrating (27) over the interval , we obtain Since , we have Note that If we take , we infer from (29) that Let be given. For all , and , from the assumption () of , we know that is bounded. So we easily obtain from (31) for all , , and . Set and consider the family of closed balls in defined by . It is easy to check that and is pullback -absorbing for the cocycle by (15).

In order to prove the pullback -attractor, let be the set of all functions which satisfies (17) and denotes the class of all families such that for some , where denotes the closed ball in centered on with radius .

Theorem 3. Assume that , , and the assumptions of , hold. ,, and satisfy (7). Then there exists a pullback -attracting set in for the nonautonomous dynamical system defined by (15).

Proof. Let , and be fixed. Take the scalar product in of (1) with , and take the scalar product in of (2) with ; then make summation to get Since we infer from (34) that Also and consider the assumption () of combined with Sobolev-embed theorem and then we infer from (36) that Let . By the Gronwall lemma we have from (39) Considering that by the assumption () of , we have Set Since , we have from (32) and then we have from (42) Let be given. For all , and , from the assumption () of , we know that is bounded and positive. So we easily obtain from (45) for all , , and . Set The family of closed balls in is pullback -absorbing for the cocycle in .

3. The Pullback -Attractor in

In order to get the existence of the pullback -attractor, we first introduce the following Lemma.

Lemma 4. Let be an infinite dimensional Hilbert space and let the family be an orthonormal basis of . Suppose that , , and for any , Then where is the orthogonal projector.

Proof. Let , , and , so For any and any , we can choose , , large enough so that Then for any and any , we put to get That is, for any , any , and , So

In order to obtain the pullback -attractor in , we also need the following Lemmas of the properties of compactness about the nonlinear operator , .

Lemma 5. Let be a function from into satisfying (); then is continuously compact; that is, is continuous and maps a bounded subset of into a precompact subset of .

Proof. Let be a bounded set in . Assume that is a bounded sequence in . From Sobolev embedding Theorem, the embeddings , and are compact. We assume that is bounded and converges to in and , respectively. By Minkowski inequality, we see that By Holder inequality, we have where and is a constant depending on and the embedding constant. Due to the assumption () of and a classical continuity result, it follows that Also by the Holder inequality The proof is completed.

Lemma 6. Let be a function from into satisfying (11); then is continuously compact.

Proof. Let be a bounded set in and assume to be a bounded sequence in . From Sobolev embedding theorem, the embedding , is compact, so we assume that is bounded and converges to in . Let ; then there exists such that By Holder inequality, we have where is the conjugate of (i.e., ). Combined with the assumption (11), the proof is completed.

Lemma 7. Let be a function from into satisfying ; moreover, . Let be a bounded subset of . Then for any , there exists some such that when where is the orthogonal projection.

Proof. Note that for . By Lemma 5, we see that maps bounded subsets of into precompact subsets of . Let be a bounded subset of and let be given arbitrarily. Since is precompact in , there is a finite number of elements such that We take sufficiently large so that for all , when . Then by , we have where is the orthogonal projection.

Lemma 8. Let be a function from into satisfying (11). Let be a bounded subset of . Then for any , there exists some such that when where is the orthogonal projection.

Theorem 9. Assume that , , and the assumptions of , , hold. , , and satisfy (7); then there exists a pullback -attractor in for the nonautonomous dynamical system defined by (15).

Proof. In order to prove the result of the theorem, we only need to check the pullback -condition (C).
Let be an orthonormal basis of which consists of eigenvectors of . The corresponding eigenvalues are denoted by , and , , as . Then is also an orthonormal basis of   and . We write and is an orthogonal projector. For any , we write
Take the scalar product in of (1) with , and take the scalar product in of (2) with ; then make summation to get Since here we set , and and letting , setting , we have, from (69), By the Gronwall lemma, we obtain So Then given any , we have for any and , . Now we estimate , , , one by one. Given any and any , it is easy to see that so there exists such that for all , .
By Lemmas 7–8, we can choose such that for any , .
By Lemma 4, we can choose large enough such that for .
By (32), there exists such that, for , and the embedding from into is compact combined with , so we can choose large enough such that for , .
By above analysis, if we choose , , then This implies pullback -condition (C).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The project is supported by the National Natural Science Foundation of China (Grant no. 10772131), the Natural Science Foundation of Shanxi Province, China (Grant no. 2010011008), and the Natural Science Foundation for Young Scientists of Shanxi Province, China (Grant no. 2011021002-2).