Abstract
This paper studies the longtime asymptotic behavior of solutions for the nonautonomous pLaplacian equations with dynamic flux boundary conditions in ndimensional bounded smooth domains. We have proved the existence of the uniform attractor in for the nonautonomous pLaplacian evolution equations subject to dynamic nonlinear boundary conditions by using the Sobolev compactness embedding theory, and the existence of the uniform attractor in by asymptotic a priori estimate.
1 Introduction
We are concerned with the existence of uniform attractors for the process associated with the solutions of the following nonautonomous pLaplacian equation:
Equation (1) is subject to the dynamic flux boundary condition
and the initial condition
where () is a bounded domain with smooth boundary Γ, ν denotes the outer unit normal on Γ, , the nonlinearity f and the external force g satisfy some conditions specified later.
Nonautonomous equations appear in many applications in the natural sciences, so they are of great importance and interest. The longtime behavior of solutions of such equations has been studied extensively in recent years (e.g., see [14]). The first attempt was to extend the notion of a global attractor to the nonautonomous case, leading to the concept of the socalled uniform attractor (see [5]). It is remarkable that the conditions ensuring the existence of a uniform attractor are parallel with those for the autonomous case. A uniform attractor need not be ‘invariant’, unlike a global attractor for autonomous systems. Moreover, it is well known that the trajectories may be unbounded for many nonautonomous systems when the time tends to infinity, and there does not exist a uniform attractor for these systems.
Dynamic boundary conditions are very natural in many mathematical models such as heat transfer in a solid in contact with a moving fluid, thermoelasticity, diffusion phenomena, heat transfer in two mediums, problems in fluid dynamics (see [14,611]).
In recent years, many authors have studied pLaplacian equations (see [1217]) and the problem (1)(3) for (see [3,7,9,10]) by discussing the existence and uniqueness of local solutions, the blowup of solutions, the global existence of solutions, the global attractors of solutions and the eigenvalue problems, etc. In [18], the authors have proved the global existence of solutions for quasilinear elliptic equations with dynamic boundary conditions. Due to the complications inherent to nonlinear dynamic boundary conditions, these problems (1)(3) still need to be investigated. In [1517,19], the authors have considered the eigenvalue problem
and obtained some results, and some pLaplacian elliptic equations with nonlinear boundary condition have been studied by using these results mentioned in [1517,19]. In [14,20], the authors have proved the existence of uniform attractors for the nonautonomous pLaplacian equations with Dirichlet boundary conditions in a bounded and an unbounded domain in . The authors have proved the existence of global attractors for the autonomous pLaplacian equations with dynamic flux boundary conditions in [21]. In [11], the authors have used a new type of uniformly Gronwall inequality and proved the existence of a pullback attractor in of the following equation:
under the assumptions that f, g satisfy the polynomial growth condition with order , and satisfies some weak assumption
for all , where θ is some positive constant.
Moreover, the existence of uniform attractors for the nonautonomous pLaplacian equations with dynamical boundary conditions remains unsolvable.
To study problem (1)(3), we assume the following conditions.
(H_{1}) The functions and satisfy
(H_{2}) The external force is locally Lipschitz continuous, , and satisfies
(H_{3}) Furthermore, is uniformly bounded in with respect to , i.e., there exists a positive constant K such that
The main purpose of this paper is to study the longtime dynamical behavior for the nonautonomous pLaplacian evolutionary equations (1)(3) under quite general assumptions (4)(6). We first prove the existence and the uniqueness of solutions for (1)(5), and then the existence of uniformly (w.r.t. ) absorbing sets for the process corresponding to (1)(5) in and , respectively, is obtained. Finally, the existence of the uniform (w.r.t. ) attractor for the process corresponding to (1)(5) in is obtained by the Sobolev compactness embedding theory and the existence of the uniform (w.r.t. ) attractor for the process corresponding to (1)(5) in is obtained by asymptotic a priori estimate.
This paper is organized as follows. In Section 2, we give some notations and lemmas used in the sequel. The existence and the uniqueness of solutions for the problem (1)(5) have been proved in Section 3. Section 4 is devoted to proving the existence of the uniformly (w.r.t. ) absorbing sets in , and , respectively, for the process corresponding to (1)(5) and the existence of the uniform (w.r.t. ) attractors in , and , respectively, for the process corresponding to (1)(5).
Throughout this paper, we denote the inner product in (or ) by , and let C be a positive constant, which may be different from line to line (and even in the same line); we denote the trace operator by γ.
2 Preliminaries
In order to study the problem (1)(5), we recall the Sobolev space defined as the closure of in the norm
and denote by the dual space of X. We also define the Lebesgue spaces as follows:
where
and
for any , where the measure on is defined for any measurable set by . In general, any vector will be of the form with and , and there need not be any connection between and .
Denote
and let the operator be defined as follows:
Next, we recall briefly some lemmas used to prove the wellposedness of the solutions and the existence of the uniform (w.r.t. ) attractors for (1)(3) under some assumptions on f.
Lemma 2.1[22]
Letbe a bounded domain inand, letbe given. Assume that, whereCis independent ofn, , as, almost everywhere in, and. Then, asweakly in.
Lemma 2.2[13]
Letandbe the standard scalar product in. Then, for any, there exist two positive constants, , which depend onp, such that
Lemma 2.3[23]
Letand Ω be a bounded subset ofwith smooth boundary Γ. Then the inclusion
Lemma 2.4[24]
LetAbe defined in (7) and. Then, for any, one has
Furthermore, if and only ifa.e. in.
Lemma 2.5[25]
LetXbe a given Banach space with dual, and letuandgbe two functions belonging to. Then the following three conditions are equivalent:
(i) uis almost everywhere equal to a primitive function ofg, i.e.,
in the scalar distribution sense on.
If (i)(iii) are satisfied, uis almost everywhere equal to a continuous function fromintoX.
3 The wellposedness of solutions
In what follows, we assume that is given.
Definition 3.1 A function is called a weak solution of (1)(3) on if
and
Theorem 3.1Let Ω be a bounded domain in (). Assume thatfsatisfies (H_{1}), is locally Lipschitz continuous and. Then, for any, any initial dataand any, there exists a unique weak solutionof (1)(3), and the mapping
Proof We first prove the existence of solutions for (1)(5) by the FaedoGalerkin method (see [25]).
Consider the approximating solution in the form
where is an orthogonal basis of , which is included in . We get from solving the following problem:
Since f is continuous and g is locally Lipschitz continuous, using the Peano theorem, we get the local existence of . Next, we establish some a priori estimates for . We have
Thanks to (5), we obtain
by virtue of the following inequality (see Theorem 2.3.1 in [26]):
Let and , we deduce from (10) and (12) that
Integrating (13) over , we obtain
Due to (14), we get
Therefore, is uniformly bounded in n in the , , respectively, and is uniformly bounded in n in the , and one can extract a subsequence of such that
Let be a projection. For any , set , we have
We perform the following estimate deduced from the Hölder inequality and the Young inequality:
Using the boundedness of in again, we infer that
Therefore we can extract a subsequence such that
By virtue of the Aubin compactness theorem, we can extract a further subsequence (still denoted by ) such that additionally
Due to the boundedness of in and (5), we obtain that is uniformly bounded in and hence in , similarly, in . By virtue of (16)(17), we see that a.e. in and a.e. in , then a.e. in and a.e. in . Thanks to Lemma 2.1, we know that
Therefore, we have
In order to prove that u is a weak solution of (1)(3), it remains to show that . Noticing that
it follows from the formulation of and that in and in . Moreover, by the lower semicontinuity of and , we obtain
Meanwhile, by the Lebesgue dominated theorem, one can check that
This fact and (20)(21) imply
In view of (18), we have
This and (22) deduce
To this end, we first observe that
On the other hand, it follows from Lemma 2.4 that
Hence
Combining (24) with in , we obtain
Therefore, from Lemma 2.2, the Hölder inequality and the Young inequality, we deduce that for any ,
which implies that in , hence .
Finally, we prove the uniqueness and continuous dependence of the initial data of the solutions. Let , be two solutions of (1)(5) with the initial data , , respectively. Let . Taking the inner product of the equation with w, we deduce that
By virtue of (4) and Lemma 2.2, we obtain
which implies that
Therefore, a.e. in if in , and is continuously dependent on the initial data.
Since
by use of Lemma 2.5, we know that
By Theorem 3.1, we can define a family of continuous processes in as follows: For all ,
where is the solution of (1)(5) with initial data . That is, a family of mappings satisfies
4 Existence of uniform attractors
In this section, we prove the existence of uniform attractors for (1)(3).
4.1 Abstract results
In this subsection, let Σ be a parameter set, let X, Y be two Banach spaces, continuously. is a family of processes in a Banach space X. Denote by the set of all bounded subsets of X and . In the following, we give some basic definitions and some abstract results about the existence of bispace uniform (with respect to (w.r.t.) ) attractors.
A set is called to be uniformly (w.r.t. ) absorbing for if for any and any bounded subset , there exists a positive constant such that
A set is said to be uniformly (w.r.t. ) attracting for the family of processes , if
for an arbitrary fixed and any bounded set .
Definition 4.2[5]
A closed set is said to be an uniform (w.r.t. ) attractor for the family of processes if it is uniformly (w.r.t. ) attracting and it is contained in any closed uniformly (w.r.t. ) attracting set for the family of processes : .
Definition 4.3[5]
Define the uniform (w.r.t. ) ωlimit set of B by . This can be characterized by the following: if and only if there are sequences , , , such that ().
Definition 4.4[5]
A family of processes possessing a compact uniformly (w.r.t. ) absorbing set is called uniformly compact. A family of processes is called uniformly asymptotically compact if it possesses a compact uniformly (w.r.t. ) attracting set, i.e., for any bounded subset and any sequences , as and , is precompact in Y.
Lemma 4.1[20]
If a family of processesisuniformly asymptotically compact, then for any, ,
(i) for any sequences, , , as, there is a convergent subsequence ofinY,
(ii) is nonempty and compact inY,
(v) ifAis a closed set anduniformly (w.r.t. ) attractingB, then.
Assumption 1 Let be a family of operators acting on Σ and satisfying:
(ii) translation identity:
Definition 4.5[5]
The kernel of the process acting on X consists of all bounded complete trajectories of the process :
The set is said to be kernel section at time , .
Definition 4.6[5]
A family of processes is said to be weakly continuous if for any fixed , , the mapping is weakly continuous from to Y.
Assumption 2 Let Σ be a weakly compact set and be weakly continuous.
Lemma 4.2[20]
Under Assumptions 1 and 2 with, which is a weakly continuous semigroup, ifacting onXisuniformly (w.r.t. ) asymptotically compact, then it possesses anuniform (w.r.t. ) attractor, which is compact inYand attracts all the bounded subsets ofXin the topology ofY.
Moreover,
whereis a bounded neighborhood of the compactuniformly attracting set inY; i.e., is a boundeduniformly (w.r.t. ) absorbing set of. is the section atof kernelof the processwith symbol. Furthermore, is nonempty for all.
From the ideas of [4,20,28], we give the following results, which are very useful for the existence of a uniform attractor in .
Lemma 4.3[20]
Letbe a family of processes on () and supposehas a boundeduniformly (w.r.t. ) absorbing set in. Then, for any, and any bounded subset, there exist two positive constantsandsuch that
Let a family of processesbeuniformly (w.r.t. ) asymptotically compact, thenisuniformly asymptotically compact for, if
(i) has a boundeduniformly (w.r.t. ) absorbing set ,
(ii) for any, and any bounded subset, there exist two positive constantsandsuch that
From Theorem 3.1, we know that the problem (1)(5) generates a process acting in and the time symbol is . We denote by the space endowed with a locally weak convergence topology. Let be the hull of g in , i.e., the closure of the set in and .
Lemma 4.5[5]
If ℰ is reflective separable and, then
(ii) the translation groupis weakly continuous on,
Due to Lemma 4.5, is weakly compact and the translation semigroup satisfies that and is weakly continuous on . Because of the uniqueness of solution, the following translation identity holds:
Theorem 4.1The family of processescorresponding to problem (1)(5) isweakly continuous andweakly continuous.
Proof For any fixed and τ, , , let () weakly in and weakly in as , denote by . The same estimates for given in the Galerkin approximations (in Section 3) are valid for the here. Therefore, for some subsequence and such that for any , , weakly in and . And the sequence , is bounded in . Denote by , and the weak limits of , and in , and , respectively. So, we get the following equation for :
By the same method as the proof of Theorem 3.1, we know that , and , which means that in V is the weak solution of (1)(5) with the initial condition . Due to the uniqueness of the solution, we state that weakly in and . For any other subsequence, and satisfy weakly in and , by the same process, we obtain the analogous relation weakly in and holds. Then it can be easily seen that for any weakly convergent initial sequence and weakly convergent sequence , we have weakly in and . □
Lemma 4.6[25]
(The uniform Gronwall lemma) Let, , be three positive locally integrable functions on, and for someand all, , , satisfy the following inequalities:
and
whereR, A, Bare three positive constants. Then
4.2 The existence of uniformly absorbing sets
In this subsection, we prove the existence of uniformly (w.r.t. ) absorbing sets for the process corresponding to (1)(5).
Theorem 4.2Assume thatfandgsatisfy (H_{1})(H_{2}). Then the family of processescorresponding to problem (1)(5) has a bounded anduniformly (w.r.t. ) absorbing set. That is, for any bounded subsetBofand any, there exist, and two positive constants, such that
for any, where, , , andare specified in (33), (41), (32) and (40), respectively.
Proof Taking the inner product of (1) with u, we deduce that
By virtue of (5), the Hölder inequality and the Young inequality, we obtain
Let and , we deduce from (12) and (29) that
It follows from the classical Gronwall inequality and Lemma 4.5 that
where we have used the following inequality:
From (31), we deduce that
where
Integrating (30) over , we obtain
Let , we deduce from (5) that there exist three positive constants , , β such that
and
Thanks to (34), we deduce from (35)(36) that
On the other hand, taking the inner product of (1) with , we obtain
which implies
Combining (37) with (38), by virtue of the uniform Gronwall Lemma 4.6, we get
which implies that for any and , there exists a positive constant such that
where
□
From Theorem 4.2, the compactness of the Sobolev embedding , the compactness of the Sobolev trace embedding and Lemma 4.2, we have the following result.
Corollary 4.1The family of processesgenerated by (1)(5) with initial datahas anuniform (w.r.t. ) attractor, which is compact inand attracts every bounded subset ofin the topology of. Moreover,
whereis theuniformly (w.r.t. ) absorbing set inandis the section atof kernelof the processwith symbol.
4.3 The existence of uniform attractor
The main purpose of this subsection is to give an asymptotic a priori estimate for the unbounded part of the modular for the solution of problem (1)(5) in the norm.
Theorem 4.3The family of processescorresponding to problem (1)(5) with initial datahas anuniform (w.r.t. ) attractor, which is compact inand attracts every bounded subsetBofin the topology of. Moreover,
whereis theuniformly (w.r.t. ) absorbing set andis the section atof kernelof the processwith symbol.
Proof We need only prove that the process satisfies the assumption (ii) of Lemma 4.4.
From (H_{3}), we deduce that for any ,
Moreover, from Lemma 4.3 and Theorem 4.2, we know that there exists and such that for any , and ,
Multiplying (1) with and integrating over Ω, we obtain
where denotes the positive part of , that is,
Due to (5), we can choose large enough such that
for some positive constant c. Therefore,
Since
and
From (42)(45), we deduce that
It follows from for any , and the classical Gronwall inequality that
which implies that for any , there exist two positive constants and such that for all and ,
Repeating the same steps as above, just taking instead of , we deduce that there exist two positive constants and such that for all and ,
where
Therefore,
□
4.4 uniform attractor
In this subsection, we prove the existence of an uniform attractor. For this purpose, we first give some a priori estimates about endowed with norm.
Theorem 4.4Under assumptions (H_{1})(H_{3}), for any bounded subset, anyand, there exists a positive constantsuch that
for any, , , whereandis a positive constant which is independent ofBandσ.
Proof First, we differentiate (1) and (2) in time, and denoting , , we get
where ‘⋅’ denotes the dot product in .
Multiplying (46) by ζ and integrating over Ω, and combining (4) with (47), we obtain
On the other hand, for any , integrating (38) from r to and using (39), we find
Therefore, we deduce from the uniformly Gronwall inequality that
which implies that there exist two positive constants and a positive constant such that
□
Next, we prove the process is uniformly (w.r.t. ) asymptotically compact in .
Theorem 4.5Assume thatfandgsatisfy (H_{1})(H_{3}). Then the family of processescorresponding to problem (1)(5) with initial dataisuniformly (w.r.t. ) asymptotically compact, i.e., there exists a compact uniformly attracting set in, which attracts any bounded subsetin the topology of.
Proof Let be an uniformly (w.r.t. ) absorbing set obtained in Theorem 4.2, then we need only to show that for any , and , is precompact in .
Thanks to Lemma 4.2, it is sufficient to verify that for any , and , is precompact in .
In fact, from Corollary 4.1 and Theorem 4.3, we know that is precompact in and .
Without loss of generality, we assume that is a Cauchy sequence in and .
Now, we prove that is a Cauchy sequence in .
Denote by , we deduce from Lemma 2.2 that
We now estimate separately the two terms and . By simple calculations and the Hölder inequality, we deduce that
and
which combining with Corollary 4.1, Theorem 4.3 and Theorem 4.4 yields Theorem 4.5 immediately. □
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors typed, read and approved the final manuscript.
Acknowledgements
The authors would like to thank the referees for their valuable suggestions.
References

Arrieta, JM, Carvalho, AN, Bernal, AR: Attractors of parabolic problems with nonlinear boundary conditions uniform bounds. Partial Differ. Equ.. 25, 1–37 (2000)

Bernal, AR: Attractors for parabolic equations with nonlinear boundary conditions, critical exponents and singular initial data. J. Differ. Equ.. 181, 165–196 (2002). Publisher Full Text

Fan, ZH, Zhong, CK: Attractors for parabolic equations with dynamic boundary conditions. Nonlinear Anal.. 68, 1723–1732 (2008). Publisher Full Text

Yang, L: Uniform attractors for the closed process and applications to the reactiondiffusion with dynamical boundary condition. Nonlinear Anal.. 71, 4012–4025 (2009). Publisher Full Text

Chepyzhov, VV, Vishik, MI: Attractors for Equations of Mathematical Physics, Am. Math. Soc., Providence (2002)

Arrieta, JM, Carvalho, AN, Bernal, AR: Parabolic problems with nonlinear boundary conditions and critical nonlinearities. J. Differ. Equ.. 156, 376–406 (1999). Publisher Full Text

Constantin, A, Escher, J: Global existence for fully parabolic boundary value problems. Nonlinear Differ. Equ. Appl.. 13, 91–118 (2006). Publisher Full Text

Constantin, A, Escher, J, Yin, Z: Global solutions for quasilinear parabolic systems. J. Differ. Equ.. 197, 73–84 (2004). Publisher Full Text

Petersson, J: A note on quenching for parabolic equations with dynamic boundary conditions. Nonlinear Anal.. 58, 417–423 (2004). Publisher Full Text

Popescu, L, Bernal, AR: On a singularly perturbed wave equation with dynamical boundary conditions. Proc. R. Soc. Edinb., Sect. A, Math.. 134, 389–413 (2004). Publisher Full Text

Yang, L, Yang, MH, Kloeden, PE: Pullback attractors for nonautonomous quasilinear parabolic equations with a dynamical boundary condition. Discrete Contin. Dyn. Syst., Ser. B. 17, 2635–2651 (2012)

Anh, CT, Ke, TD: On quasilinear parabolic equations involving weighted pLaplacian operators. Nonlinear Differ. Equ. Appl.. 17, 195–212 (2010). Publisher Full Text

Bartsch, T, Liu, Z: On a superlinear elliptic pLaplacian equation. J. Differ. Equ.. 198, 149–175 (2004). Publisher Full Text

Chen, GX: Uniform attractors for the nonautonomous parabolic equation with nonlinear Laplacian principal part in unbounded domain. Differ. Equ. Appl.. 2(1), 105–121 (2010)

Fernandez, BJ, Rossi, JD: Existence results for the pLaplacian with nonlinear boundary conditions. J. Math. Anal. Appl.. 263, 195–223 (2001). Publisher Full Text

Martinez, S, Rossi, JD: Weak solutions for the pLaplacian with a nonlinear boundary condition at resonance. Electron. J. Differ. Equ.. 27, 1–14 (2003)

Martinez, SR, Rossi, JD: Isolation and simplicity for the first eigenvalue of the pLaplacian with a nonlinear boundary conditions. Abstr. Appl. Anal.. 7, 287–293 (2002). Publisher Full Text

Yin, Z: Global existence for elliptic equations with dynamic boundary conditions. Arch. Math.. 81, 567–574 (2003). Publisher Full Text

Martinez, SR, Rossi, JD: On the Fuc̆ik spectrum and a resonance problem for the pLaplacian with a nonlinear boundary condition. Nonlinear Anal.. 59, 813–848 (2004)

Chen, GX, Zhong, CK: Uniform attractors for nonautonomous pLaplacian equations. Nonlinear Anal.. 68, 3349–3363 (2008). Publisher Full Text

You, B, Zhong, CK: Global attractors for pLaplacian equations with dynamic flux boundary conditions. Adv. Nonlinear Stud.. 13, 391–410 (2013)

Robinson, JC: InfiniteDimensional Dynamical Systems, Cambridge University Press, Cambridge (2001)

Adams, RA, Fournier, JJF: Sobolev Spaces, Academic Press, Amsterdam (2003)

Lê, A: Eigenvalue problems for the pLaplacian. Nonlinear Anal.. 64, 1057–1099 (2006). Publisher Full Text

Temam, R: InfiniteDimensional Systems in Mechanics and Physics, Springer, New York (1997)

Babin, AV, Vishik, MI: Attractors of Evolution Equations, NorthHolland, Amsterdam (1992)

Lu, SS, Wu, HQ, Zhong, CK: Attractors for nonautonomous 2D NavierStokes equations with normal external forces. Discrete Contin. Dyn. Syst.. 13(3), 701–719 (2005)

Li, Y, Zhong, CK: Pullback attractor for the normtoweak continuous process and application to the nonautonomous reactiondiffusion equations. Appl. Math. Comput.. 190, 1020–1029 (2007). Publisher Full Text