Abstract
We study the long-time behavior of the Kirchhoff type equation with linear damping. We prove the existence of strong solution and the semigroup associated with the solution possesses a global attractor in the higher phase space.
1. Introduction
We consider the following nonlinear Kirchhoff type equation with the initial-boundary conditions:where is a bounded domain in with smooth boundary, denotes the Laplace operator, is a given function lying in , independent of time, and with fulfills the dissipation inequalityandwhere is a real number and is the first eigenvalue of on with Dirichlet boundary conditions
When , this problem describes, for instance, the motion of a vibrating string with fixed boundary in a viscous medium. In particular, the function represents the displacement from equilibrium, is the velocity, and the term may correspond to a (nonlinear) elastic force. For more details on the model of Kirchhoff, one can refer to [1–3] and the reference therein.
When the coefficient of is a positive function , which depends on , then the term is a resistance force and the model impresses that the viscous medium embedding the string is stratified. In this case, the existence of the global and exponential attractors has been proven by S. Kolbasin in [4]. The attractor is in the phase space and it is bounded in , where is a bounded domain in with smooth boundary.
Similar models have been considered in [5–9], such as plate equationwhen , the existence, regularity, and finite dimensionality of a global attractor in with a localized damping and a critical exponent were proven in [5]. For a three-dimensional unbounded domain and under suitable conditions on and , A.Kh. Khanmamedov in [6] showed that this equation possesses a global attractor in . About Kirchhoff models, long-time dynamics properties were studied by Yang and I. Chueshov in [7–9] and their references. For example, Yang et al. discussed the long-time behavior of solutions to the Cauchy problem of some Kirchhoff type equations with a strong dissipation in [7, 8] and proved that the dynamical system possesses a global attractor under suitable conditions in the phase space , where .
Thus, to the best of our knowledge, the research about global attractors of the weak solutions for problem (1) with respect to the norm of is much more; however, the results about the existence of the strong solutions and strong global attractors for (1) are relatively fewer. The purpose of this paper is to supplement some conclusions for the above problem. In particular, we do not demand the function to be bounded by polynomials and present here a method different from [9–14]. Furthermore, the global attractor is established in the higher energy space in .
The paper is organized as follows. In Section 2, we recall some notations and some general facts about the dynamical systems theory. In Section 3, we prove well-posedness and the existence of a bounded absorbing set for problem (1). Sections 4 and 5 contain our main results, and we prove the existence of strong solution and a global attractor in the space of higher order.
2. Preliminaries
Throughout the paper we will denote
The norm and the scalar product in is denoted by and , respectively, the norm in is denoted by , and the Hilbert spaces , are
For any given function , we will write for short and endow space with the standard inner product and norm , .
For convenience, the letters and present different positive constants and different positive increasing functions, respectively.
We collect some basic concepts and general theorems, which are important for getting our main results. We refer to [14–18] and the references therein for more details.
Definition 1 (see [16, 18]). Let be a Banach space and be a family operator on . We say that is a norm-to-weak continuous semigroup on , if satisfies(i) (the identity);(ii), ;(iii), if in .
Definition 2 (see [15]). A semigroup in a Banach space is said to satisfy the condition () if, for any and for any bounded set of , there exists and a finite dimensional subspace of , such that is bounded andwhere is a bounded projector.
Theorem 3 (see [14]). Let X be a Banach space and be a norm-to-weak continuous semigroup on X. Then has a global attractor in X provided that the following conditions hold true:
(i) has a bounded absorbing set in X.
(ii) satisfies the condition (C).
Theorem 4 (see [16, 18]). Let X and Y be two Banach spaces such that with a continuous injection. If a function belongs to and is weakly continuous with values in Y, then is weakly continuous with values in X.
Theorem 5 (see [16, 18]). Let X, Y be two Banach spaces and be their dual spaces, respectively, such thatwhere the injection is continuous and its adjoint, , is a densely injective. Let be a semigroup on X and Y, respectively, and be a continuous semigroup or a weak continuous semigroup on Y. Then for any bounded subset B of X, is norm-to-weak continuous on .
Theorem 6 (see [16–18]). Assume that is a semigroup on Banach space X and satisfies that
(i) has a bounded absorbing set in X;
(ii) satisfies condition (C) or is -limit compact in X.
And assume furthermore that is norm-to-weak continuous on . Then has a global attractor in X; i.e., is nonempty, invariant, compact in X and attracts every bounded subset of X in the norm topology of X.
3. A Priori Estimate
Next we iterate some main results in [4], which are important for getting a priori estimate.
For any initial data , problem (1) possesses a unique weak solution , which satisfies , and, for any ,
Furthermore problem (1) generates a dynamical system of solution within the space . This system possesses a compact global attractor , which is bounded in .
Choosing , taking the scalar product in of the first equation of (1) with , we find
Owing to [4] and inequality, we obtainwhere . Furthermore we see from (11) thatWe denote the energy functions as follows:Combining with (13), (12), and , we get
By the Gronwall inequality, it follows that
Next, we show thatBy inequality, Young’s inequality, Poincaré inequality, and [4], we conclude thatandSelect small enough to verify that
Now if , the ball of , center at of radius , then , provided SoWe denote .
4. Existence of the Strong Solution
Theorem 7. Suppose is a function from R into R satisfying (2)-(3) and . Then given , problem (1) has a unique solution withfor . Moreover is a weakly continuous function from to .
Proof. We prove the existence of strong solutions by using the Faedo-Galerkin schemes. Assume that there exists an orthonormal basis of consisting of eigenvectors of in ; simultaneously they are also orthonormal basis of . The corresponding eigenvalues are , satisfyingFor this purpose, according to the basis theory of ordinary differential equations, we build the sequence of Galerkin approximate solutions. They are smooth functions of the form and it satisfieswhere is the orthogonal projector in , and . Of course for every there is the unique solution to (27).
Choosing , after multiplying (27) by , the similar process of (11) leads toLike the estimates of (12)-(21), we haveIt means that the sequence is bounded in the space for an arbitrary fixed . From (27) we know thatSo is uniformly bounded in ; then we can extract a subsequences, still denoted by , such thatIt is easy to pass the limit in (27) and we obtain that is a solution of (1), such thatFurthermore, from Theorem 4 and [4], we know that is a weakly continuous function from to .
Finally, uniqueness is followed from [4], since any strong solution would be a weak solution.
Thus, the dynamical system generated by (1) can be defined in the phase space , and the corresponding solution semigroup is . By Theorem 7, we have the following results.
Theorem 8. Suppose the conditions of Theorem 7 hold; then there exists a bounded absorbing set in for the semigroup .
5. Global Attractors in
We first prove the following compactness results and the norm-to-weak continuity of semigroup.
Lemma 9. Suppose that (2) and (3) hold; is defined by. Then is continuous compact.
Proof. Suppose that is bounded sequences in ; without lose of generality, we assume that weakly converge to in . By the Sobolev embedding theorem, we know that is bounded and converges to in Denote ; by the results of [4] and the Sobolev embedding theorem, we show that are uniformly bounded in ; that is, there exists a constant , such thatsince there exists constant , such thatBecause converges to in , we haveThe proof is completed.
Similarly, we can prove the following Lemma.
Lemma 10. Let and (2) and (3) hold, . Then and are continuous compact.
Since , from Theorem 5 and [4], we can immediately obtain the following result.
Lemma 11. The semigroup associated with (1) is norm-to weak continuous on , where is abounded absorbing set of in and is the stationary set of defined by
Now we give our main results of the paper.
Theorem 12. Suppose that , and (2) and (3) hold. Then the solution semigroup of problem (1) has global attractor in ; it attracts all bounded subset of in the norm of .
Proof. Applying Theorems 7 and 8, we only need to verify that satisfies condition in .
Let be the eigenvalues of in and be the corresponding eigenvectors such that where , as and forms an orthogonal basis in and .
Let and be the canonical projector on and be the identity. Then, for any , it has a unique decomposition , where Since and are compact continuously verified by Lemma 10, then, for any , there exists , such that, for any , we havewhere is given by (23).
Multiplying (1) by , we can getApplying the Young inequality, inequality, Sobolev embedding theorem, and (40) and (41), the three terms in the right-hand side of (41) can be estimated as follows: Using the above estimates, we transform (41) as follows:Choose small enough such that and . Hence (43) can be rewritten as Denote we arrive at By Gronwall inequalityNext, we show thatIndeed, the right inequality is obtained using inequality, Young inequality, and Lemma 9: and the left one is the same, where .
Thus, combining (46) and (47) and Theorem 8, we deduce Taking , it follows that where, by (29), we denote ,
Together with Theorems 5, 7, and 8, the proof is finished.
Remark 13. If we transform the first equation of (1) to the following form, then the term accounts for dynamical friction, and we only need to assume the damping coefficient is a positive function and by adding some appropriate conditions of continuity, conclusions of Theorem 12 remain valid and the proof’s process has no essential difference.
Data Availability
All data included in this study are available upon request by contact with the corresponding author.
Conflicts of Interest
The author declares that there are no conflicts of interest.
Acknowledgments
This work was supported by the National Science Foundation of China (61703181).