Research

# Uniform attractors for the non-autonomous p-Laplacian equations with dynamic flux boundary conditions

Kun Li1 and Bo You2*

Author Affiliations

1 Department of Basic, Henan Mechanical and Electrical Engineering College, Xinxiang, 453003, P.R. China

2 School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, P.R. China

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Boundary Value Problems 2013, 2013:128  doi:10.1186/1687-2770-2013-128

 Received: 27 December 2012 Accepted: 1 May 2013 Published: 17 May 2013

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

This paper studies the long-time asymptotic behavior of solutions for the non-autonomous p-Laplacian equations with dynamic flux boundary conditions in n-dimensional bounded smooth domains. We have proved the existence of the uniform attractor in for the non-autonomous p-Laplacian 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 non-autonomous p-Laplacian equation:

(1)

Equation (1) is subject to the dynamic flux boundary condition

(2)

and the initial condition

(3)

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.

Non-autonomous equations appear in many applications in the natural sciences, so they are of great importance and interest. The long-time behavior of solutions of such equations has been studied extensively in recent years (e.g., see [1-4]). The first attempt was to extend the notion of a global attractor to the non-autonomous case, leading to the concept of the so-called 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 non-autonomous 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 [1-4,6-11]).

In recent years, many authors have studied p-Laplacian equations (see [12-17]) and the problem (1)-(3) for (see [3,7,9,10]) by discussing the existence and uniqueness of local solutions, the blow-up 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 quasi-linear 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 [15-17,19], the authors have considered the eigenvalue problem

and obtained some results, and some p-Laplacian elliptic equations with nonlinear boundary condition have been studied by using these results mentioned in [15-17,19]. In [14,20], the authors have proved the existence of uniform attractors for the non-autonomous p-Laplacian 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 p-Laplacian 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 non-autonomous p-Laplacian equations with dynamical boundary conditions remains unsolvable.

To study problem (1)-(3), we assume the following conditions.

(H1) The functions and satisfy

(4)

for some , and

(5)

where (), , .

(H2) The external force is locally Lipschitz continuous, , and satisfies

(6)

(H3) 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 long-time dynamical behavior for the non-autonomous p-Laplacian 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

for . Moreover, we have

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:

(7)

Next, we recall briefly some lemmas used to prove the well-posedness 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

is compact for any, where

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.,

for almost every;

(ii) For each test function,

(iii) For each,

in the scalar distribution sense on.

If (i)-(iii) are satisfied, uis almost everywhere equal to a continuous function fromintoX.

### 3 The well-posedness 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

for all test functions .

Theorem 3.1Let Ω be a bounded domain in (). Assume thatfsatisfies (H1), is locally Lipschitz continuous and. Then, for any, any initial dataand any, there exists a unique weak solutionof (1)-(3), and the mapping

is continuous on.

Proof We first prove the existence of solutions for (1)-(5) by the Faedo-Galerkin 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:

(8)

(9)

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

(10)

(11)

by virtue of the following inequality (see Theorem 2.3.1 in [26]):

(12)

Let and , we deduce from (10) and (12) that

(13)

Integrating (13) over , we obtain

(14)

for any .

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

(15)

We perform the following estimate deduced from the Hölder inequality and the Young inequality:

Using the boundedness of in again, we infer that

Since , , , we find

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

(16)

(17)

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

(18)

for any .

In order to prove that u is a weak solution of (1)-(3), it remains to show that . Noticing that

(19)

it follows from the formulation of and that in and in . Moreover, by the lower semi-continuity of and , we obtain

(20)

(21)

Meanwhile, by the Lebesgue dominated theorem, one can check that

This fact and (20)-(21) imply

(22)

In view of (18), we have

This and (22) deduce

(23)

To this end, we first observe that

On the other hand, it follows from Lemma 2.4 that

Hence

(24)

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

(25)

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

Therefore, is meaningful. □

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 bi-space uniform (with respect to (w.r.t.) ) attractors.

Definition 4.1[5,27]

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

for any .

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. ) ω-limit 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 processesis-uniformly asymptotically compact, then for any, ,

(i) for any sequences, , , as, there is a convergent subsequence ofinY,

(ii) is nonempty and compact inY,

(iii) ,

(iv) ,

(v) ifAis a closed set and-uniformly (w.r.t. ) attractingB, then.

Assumption 1 Let be a family of operators acting on Σ and satisfying:

(i) , ,

(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 onXis-uniformly (w.r.t. ) asymptotically compact, then it possesses an-uniform (w.r.t. ) attractor, which is compact inYand attracts all the bounded subsets ofXin the topology ofY.

Moreover,

whereis a bounded neighborhood of the compact-uniformly attracting set inY; i.e., is a bounded-uniformly (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 bounded-uniformly (w.r.t. ) absorbing set in. Then, for any, and any bounded subset, there exist two positive constantsandsuch that

for any, , .

Lemma 4.4[4,28]

Let a family of processesbe-uniformly (w.r.t. ) asymptotically compact, thenis-uniformly asymptotically compact for, if

(i) has a bounded-uniformly (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]

Ifis reflective separable and, then

(i) for all, ,

(ii) the translation groupis weakly continuous on,

(iii) for,

(iv) is weakly compact.

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) is-weakly continuous and-weakly 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 :

for any .

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

for all.

#### 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 (H1)-(H2). Then the family of processescorresponding to problem (1)-(5) has a bounded- and-uniformly (w.r.t. ) absorbing set. That is, for any bounded subsetBofand any, there exist, and two positive constants, such that

(26)

for anyand

(27)

for any, where, , , andare specified in (33), (41), (32) and (40), respectively.

Proof Taking the inner product of (1) with u, we deduce that

(28)

By virtue of (5), the Hölder inequality and the Young inequality, we obtain

(29)

Let and , we deduce from (12) and (29) that

(30)

It follows from the classical Gronwall inequality and Lemma 4.5 that

(31)

where we have used the following inequality:

From (31), we deduce that

where

(32)

(33)

Integrating (30) over , we obtain

(34)

Let , we deduce from (5) that there exist three positive constants , , β such that

and

(35)

(36)

Thanks to (34), we deduce from (35)-(36) that

(37)

On the other hand, taking the inner product of (1) with , we obtain

which implies

(38)

Combining (37) with (38), by virtue of the uniform Gronwall Lemma 4.6, we get

(39)

which implies that for any and , there exists a positive constant such that

where

(40)

(41)

□

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 an-uniform (w.r.t. ) attractor, which is compact inand attracts every bounded subset ofin the topology of. Moreover,

whereis the-uniformly (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 an-uniform (w.r.t. ) attractor, which is compact inand attracts every bounded subsetBofin the topology of. Moreover,

whereis the-uniformly (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 (H3), 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,

Set and , we have

Due to (5), we can choose large enough such that

for some positive constant c. Therefore,

(42)

Since

(43)

(44)

and

(45)

From (42)-(45), we deduce that

Since for all , we obtain

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

Setting , we have

for all and .

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 (H1)-(H3), 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

(46)

(47)

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

for any , and , where

□

Next, we prove the process is uniformly (w.r.t. ) asymptotically compact in .

Theorem 4.5Assume thatfandgsatisfy (H1)-(H3). Then the family of processescorresponding to problem (1)-(5) with initial datais-uniformly (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 pre-compact in .

Thanks to Lemma 4.2, it is sufficient to verify that for any , and , is pre-compact in .

In fact, from Corollary 4.1 and Theorem 4.3, we know that is pre-compact 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

(48)

and

(49)

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.

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