### 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
*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
*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:

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

and the initial condition

where
*ν* denotes the outer unit normal on Γ,
*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
*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
*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

under the assumptions that *f*, *g* satisfy the polynomial growth condition with order

for all
*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.

(H_{1}) The functions

for some

where

(H_{2}) The external force

(H_{3}) Furthermore,
*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.
*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.

Throughout this paper, we denote the inner product in
*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

and denote by
*X*. We also define the Lebesgue spaces as follows:

where

for

and

for any

Denote

and let the operator

Next, we recall briefly some lemmas used to prove the well-posedness of the solutions
and the existence of the uniform (w.r.t.
*f*.

**Lemma 2.1**[22]

*Let*
*be a bounded domain in*
*and*
*let*
*be given*. *Assume that*
*where**C**is independent of**n*,
*as*
*almost everywhere in*
*and*
*Then*
*as*
*weakly in*

**Lemma 2.2**[13]

*Let*
*and*
*be the standard scalar product in*
*Then*, *for any*
*there exist two positive constants*
*which depend on**p*, *such that*

**Lemma 2.3**[23]

*Let*
*and* Ω *be a bounded subset of*
*with smooth boundary* Γ. *Then the inclusion*

*is compact for any*
*where*

**Lemma 2.4**[24]

*Let**A**be defined in* (7) *and*
*Then*, *for any*
*one has*

*Furthermore*,
*if and only if*
*a*.*e*. *in*

**Lemma 2.5**[25]

*Let**X**be a given Banach space with dual*
*and let**u**and**g**be two functions belonging to*
*Then the following three conditions are equivalent*:

(i) *u**is almost everywhere equal to a primitive function of**g*, *i*.*e*.,

*for almost every*

(ii) *For each test function*

(iii) *For each*

*in the scalar distribution sense on*

*If* (i)-(iii) *are satisfied*, *u**is almost everywhere equal to a continuous function from*
*into**X*.

### 3 The well-posedness of solutions

In what follows, we assume that

**Definition 3.1** A function

and

for all test functions

**Theorem 3.1***Let* Ω *be a bounded domain in*
*Assume that**f**satisfies* (H_{1}),
*is locally Lipschitz continuous and*
*Then*, *for any*
*any initial data*
*and any*
*there exists a unique weak solution*
*of* (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

where

Since *f* is continuous and *g* is locally Lipschitz continuous, using the Peano theorem, we get the local existence
of
*a priori* estimates for

Thanks to (5), we obtain

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

Let

Integrating (13) over

for any

Due to (14), we get

Therefore,
*n* in the
*n* in the

Let

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

Using the boundedness of

Since

Therefore we can extract a subsequence such that

By virtue of the Aubin compactness theorem, we can extract a further subsequence
(still denoted by

Due to the boundedness of

Therefore, we have

for any

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

it follows from the formulation of

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

Therefore, from Lemma 2.2, the Hölder inequality and the Young inequality, we deduce
that for any

which implies that

Finally, we prove the uniqueness and continuous dependence of the initial data of
the solutions. Let
*w*, we deduce that

By virtue of (4) and Lemma 2.2, we obtain

which implies that

Therefore,

Since

by use of Lemma 2.5, we know that

Therefore,

By Theorem 3.1, we can define a family of continuous processes

where

### 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,
*X*. Denote by
*X* and

A set

for any

A set

for an arbitrary fixed

**Definition 4.2**[5]

A closed set

**Definition 4.3**[5]

Define the uniform (w.r.t.
*ω*-limit set of *B* by

**Definition 4.4**[5]

A family of processes
*i.e.*, for any bounded subset
*Y*.

**Lemma 4.1**[20]

*If a family of processes*
*is*
*uniformly asymptotically compact*, *then for any*

(i) *for any sequences*
*as*
*there is a convergent subsequence of*
*in**Y*,

(ii)
*is nonempty and compact in**Y*,

(iii)

(iv)

(v) *if**A**is a closed set and*
*uniformly* (*w*.*r*.*t*.
*attracting**B*, *then*

**Assumption 1** Let

(i)

(ii) translation identity:

**Definition 4.5**[5]

The kernel
*X* consists of all bounded complete trajectories of the process

The set

**Definition 4.6**[5]

A family of processes
*Y*.

**Assumption 2** Let Σ be a weakly compact set and

**Lemma 4.2**[20]

*Under Assumptions* 1 *and* 2 *with*
*which is a weakly continuous semigroup*, *if*
*acting on**X**is*
*uniformly* (*w*.*r*.*t*.
*asymptotically compact*, *then it possesses an*
*uniform* (*w*.*r*.*t*.
*attractor*
*which is compact in**Y**and attracts all the bounded subsets of**X**in the topology of**Y*.

*Moreover*,

*where*
*is a bounded neighborhood of the compact*
*uniformly attracting set in**Y*; *i*.*e*.,
*is a bounded*
*uniformly* (*w*.*r*.*t*.
*absorbing set of*
*is the section at*
*of kernel*
*of the process*
*with 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]

*Let*
*be a family of processes on*
*and suppose*
*has a bounded*
*uniformly* (*w*.*r*.*t*.
*absorbing set in*
*Then*, *for any*
*and any bounded subset*
*there exist two positive constants*
*and*
*such that*

*for any*

*Let a family of processes*
*be*
*uniformly* (*w*.*r*.*t*.
*asymptotically compact*, *then*
*is*
*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 constants*
*and*
*such that*

From Theorem 3.1, we know that the problem (1)-(5) generates a process
*g* in
*i.e.*, the closure of the set

**Lemma 4.5**[5]

*If* ℰ *is reflective separable and*
*then*

(i) *for all*

(ii) *the translation group*
*is weakly continuous on*

(iii)
*for*

(iv)
*is weakly compact*.

Due to Lemma 4.5,

**Theorem 4.1***The family of processes*
*corresponding to problem* (1)-(5) *is*
*weakly continuous and*
*weakly continuous*.

*Proof* For any fixed
*τ*,

for any

By the same method as the proof of Theorem 3.1, we know that
*V* is the weak solution of (1)-(5) with the initial condition

**Lemma 4.6**[25]

(*The uniform Gronwall lemma*) *Let*
*be three positive locally integrable functions on*
*and for some*
*and all*
*satisfy the following inequalities*:

*and*

*where**R*, *A*, *B**are 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.

**Theorem 4.2***Assume that**f**and**g**satisfy* (H_{1})-(H_{2}). *Then the family of processes*
*corresponding to problem* (1)-(5) *has a bounded*
*and*
*uniformly* (*w*.*r*.*t*.
*absorbing set*. *That is*, *for any bounded subset**B**of*
*and any*
*there exist*
*and two positive constants*
*such that*

*for any*
*and*

*for any*
*where*
*and*
*are 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

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

Let
*β* such that

and

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