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

Boundary Value Problems20132013:128

DOI: 10.1186/1687-2770-2013-128

Received: 27 December 2012

Accepted: 1 May 2013

Published: 17 May 2013

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 L 2 ( Ω ¯ , d μ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq1_HTML.gif 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 ( W 1 , p ( Ω ) L q ( Ω ) ) × L q ( Γ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq2_HTML.gif 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:
u t Δ p u + | u | p 2 u + f ( u ) = g ( x , t ) , ( x , t ) Ω × R . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equ1_HTML.gif
(1)
Equation (1) is subject to the dynamic flux boundary condition
u t + | u | p 2 u ν + f ( u ) = 0 , ( x , t ) Γ × R , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equ2_HTML.gif
(2)
and the initial condition
u ( x , τ ) = u 0 ( x ) , x Ω ¯ , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equ3_HTML.gif
(3)

where Ω R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq3_HTML.gif ( n 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq4_HTML.gif) is a bounded domain with smooth boundary Γ, ν denotes the outer unit normal on Γ, p 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq5_HTML.gif, 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 [14]). 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 [14, 611]).

In recent years, many authors have studied p-Laplacian equations (see [1217]) and the problem (1)-(3) for p = 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq6_HTML.gif (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 [1517, 19], the authors have considered the eigenvalue problem
{ Δ p u + | u | p 2 u = 0 , x Ω , | u | p 2 u ν = λ | u | p 2 u , x Γ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equa_HTML.gif
and obtained some results, and some p-Laplacian 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 non-autonomous p-Laplacian equations with Dirichlet boundary conditions in a bounded and an unbounded domain in R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq7_HTML.gif. 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 L r 1 ( Ω ) × L r 2 ( Γ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq8_HTML.gif of the following equation:
{ u t Δ p u + | u | p 2 u + f ( u ) = h ( t ) , ( x , t ) Ω × R , u t + | u | p 2 u ν + g ( u ) = 0 , ( x , t ) Γ × R , u ( x , τ ) = u 0 ( x ) , x Ω ¯ , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equb_HTML.gif
under the assumptions that f, g satisfy the polynomial growth condition with order r 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq9_HTML.gif, r 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq10_HTML.gif and h ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq11_HTML.gif satisfies some weak assumption
t e θ s h ( s ) L 2 ( Ω ) 2 d s < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equc_HTML.gif

for all t R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq12_HTML.gif, 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 f C 1 ( R , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq13_HTML.gif and satisfy
f ( u ) l http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equ4_HTML.gif
(4)
for some l 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq14_HTML.gif, and
c 1 | u | q k f ( u ) u c 2 | u | q + k , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equ5_HTML.gif
(5)

where c i > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq15_HTML.gif ( i = 1 , 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq16_HTML.gif), q > 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq17_HTML.gif, k > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq18_HTML.gif.

(H2) The external force g : Ω × R R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq19_HTML.gif is locally Lipschitz continuous, d g d t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq20_HTML.gif, g L loc 2 ( R , L 2 ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq21_HTML.gif and satisfies
sup r R r r + 1 g ( s ) L 2 ( Ω ) 2 d s < . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equ6_HTML.gif
(6)
(H3) Furthermore, g ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq22_HTML.gif is uniformly bounded in L 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq23_HTML.gif with respect to t R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq12_HTML.gif, i.e., there exists a positive constant K such that
sup t R g ( t ) L 2 ( Ω ) K . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equd_HTML.gif

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. σ H w ( g ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq24_HTML.gif) absorbing sets for the process { U σ ( t , τ ) } σ H w ( g ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq25_HTML.gif corresponding to (1)-(5) in L 2 ( Ω ¯ , d μ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq1_HTML.gif and ( W 1 , p ( Ω ) L q ( Ω ) ) × L q ( Γ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq2_HTML.gif, respectively, is obtained. Finally, the existence of the uniform (w.r.t. σ H w ( g ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq24_HTML.gif) attractor for the process { U σ ( t , τ ) } σ H w ( g ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq25_HTML.gif corresponding to (1)-(5) in L 2 ( Ω ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq26_HTML.gif is obtained by the Sobolev compactness embedding theory and the existence of the uniform (w.r.t. σ H w ( g ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq24_HTML.gif) attractor for the process { U σ ( t , τ ) } σ H w ( g ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq25_HTML.gif corresponding to (1)-(5) in ( W 1 , p ( Ω ) L q ( Ω ) ) × L q ( Γ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq27_HTML.gif 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. σ H w ( g ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq24_HTML.gif) absorbing sets in L 2 ( Ω ¯ , d μ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq1_HTML.gif, L q ( Ω ¯ , d μ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq28_HTML.gif and ( L q ( Ω ) W 1 , p ( Ω ) ) × L q ( Γ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq29_HTML.gif, respectively, for the process { U σ ( t , τ ) } σ H w ( g ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq25_HTML.gif corresponding to (1)-(5) and the existence of the uniform (w.r.t. σ H w ( g ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq24_HTML.gif) attractors in L 2 ( Ω ¯ , d μ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq1_HTML.gif, L q ( Ω ¯ , d μ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq28_HTML.gif and ( L q ( Ω ) W 1 , p ( Ω ) ) × L q ( Γ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq29_HTML.gif, respectively, for the process { U σ ( t , τ ) } σ H w ( g ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq25_HTML.gif corresponding to (1)-(5).

Throughout this paper, we denote the inner product in L 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq23_HTML.gif (or L 2 ( Γ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq30_HTML.gif) by ( , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq31_HTML.gif, 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 W 1 , p ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq32_HTML.gif defined as the closure of C ( Ω ) W 1 , p ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq33_HTML.gif in the norm
u 1 , p = ( Ω | u | p + | u | p d x ) 1 p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Eque_HTML.gif
and denote by X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq34_HTML.gif the dual space of X. We also define the Lebesgue spaces as follows:
L r ( Γ ) = { v : v L r ( Γ ) < } , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equf_HTML.gif
where
v L r ( Γ ) = ( Γ | v | r d S ) 1 r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equg_HTML.gif
for r [ 1 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq35_HTML.gif. Moreover, we have
L s ( Ω ) L s ( Γ ) = L s ( Ω ¯ , d μ ) , s [ 1 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equh_HTML.gif
and
U L s ( Ω ¯ , d μ ) = ( Ω | u | s d x ) 1 s + ( Γ | v | s d S ) 1 s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equi_HTML.gif

for any U = ( u v ) L s ( Ω ¯ , d μ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq36_HTML.gif, where the measure d μ = d x | Ω d S | Γ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq37_HTML.gif on Ω ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq38_HTML.gif is defined for any measurable set A Ω ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq39_HTML.gif by μ ( A ) = | A Ω | + S ( A Γ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq40_HTML.gif. In general, any vector θ L s ( Ω ¯ , d μ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq41_HTML.gif will be of the form ( θ 1 θ 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq42_HTML.gif with θ 1 L s ( Ω , d x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq43_HTML.gif and θ 2 L s ( Γ , d S ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq44_HTML.gif, and there need not be any connection between θ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq45_HTML.gif and θ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq46_HTML.gif.

Denote
v = γ u , p = p p 1 , Ω T = Ω × ( τ , T ) , Γ T = Γ × ( τ , T ) , V = ( L p ( τ , T ; W 1 , p ( Ω ) ) L 2 ( Ω T ) L q ( Ω T ) ) V = × ( L p ( τ , T ; W 1 1 p , p ( Γ ) ) × L 2 ( Γ T ) L q ( Γ T ) ) , V = ( L p ( τ , T ; ( W 1 , p ( Ω ) ) ) + L 2 ( Ω T ) + L q ( Ω T ) ) V = × ( L p ( τ , T ; ( W 1 1 p , p ( Γ ) ) ) + L 2 ( Γ T ) + L q ( Γ T ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equj_HTML.gif
and let the operator A : L p ( τ , T ; W 1 , p ( Ω ) ) ( L p ( τ , T ; W 1 , p ( Ω ) ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq47_HTML.gif be defined as follows:
A ( u ) , v = Ω T | u | p 2 u v + | u | p 2 u v . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equ7_HTML.gif
(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. σ Σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq48_HTML.gif) attractors for (1)-(3) under some assumptions on f.

Lemma 2.1 [22]

Let O http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq49_HTML.gif be a bounded domain in R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq7_HTML.gif and { g n } n = 1 L q ( O ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq50_HTML.gif, let 1 < q < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq51_HTML.gif be given. Assume that g n L q ( O ) C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq52_HTML.gif, where C is independent of n, g n g http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq53_HTML.gif, as n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq54_HTML.gif, almost everywhere in O http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq49_HTML.gif, and g L q ( O ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq55_HTML.gif. Then g n g http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq53_HTML.gif, as n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq54_HTML.gif weakly in L q ( O ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq56_HTML.gif.

Lemma 2.2 [13]

Let x , y R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq57_HTML.gif and , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq58_HTML.gif be the standard scalar product in R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq7_HTML.gif. Then, for any p 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq5_HTML.gif, there exist two positive constants C 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq59_HTML.gif, C 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq60_HTML.gif, which depend on p, such that
| x | p 2 x | y | p 2 y , x y C 1 | x y | p , | | x | p 2 x | y | p 2 y | C 2 ( | x | + | y | ) p 2 | x y | . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equk_HTML.gif

Lemma 2.3 [23]

Let 1 p < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq61_HTML.gif and Ω be a bounded subset of R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq7_HTML.gif with smooth boundary Γ. Then the inclusion
W 1 , p ( Ω ) L r ( Γ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equl_HTML.gif
is compact for any r [ 1 , p ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq62_HTML.gif, where
p = { ( n 1 ) p n p , p < n ; , p = n . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equm_HTML.gif

Lemma 2.4 [24]

Let A be defined in (7) and X = L p ( τ , T ; W 1 , p ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq63_HTML.gif. Then, for any u , v X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq64_HTML.gif, one has
A ( u ) A ( v ) , u v ( u X p 1 v X p 1 ) ( u X v X ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equn_HTML.gif

Furthermore, A ( u ) A ( v ) , u v = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq65_HTML.gif if and only if u = v http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq66_HTML.gif a.e. in Ω T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq67_HTML.gif.

Lemma 2.5 [25]

Let X be a given Banach space with dual X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq68_HTML.gif, and let u and g be two functions belonging to L 1 ( a , b ; X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq69_HTML.gif. Then the following three conditions are equivalent:
  1. (i)
    u is almost everywhere equal to a primitive function of g, i.e.,
    u ( t ) = ζ + a t g ( s ) d s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equo_HTML.gif
     
for almost every t [ a , b ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq70_HTML.gif;
  1. (ii)
    For each test function ϕ D ( a , b ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq71_HTML.gif,
    a b u ( t ) ϕ ( t ) d t = a b g ( t ) ϕ ( t ) d t ( ϕ ( t ) = d ϕ d t ) ; http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equp_HTML.gif
     
  2. (iii)
    For each η X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq72_HTML.gif,
    d d t u , η = g , η http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equq_HTML.gif
     

in the scalar distribution sense on ( a , b ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq73_HTML.gif.

If (i)-(iii) are satisfied, u is almost everywhere equal to a continuous function from [ a , b ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq74_HTML.gif into X.

3 The well-posedness of solutions

In what follows, we assume that u 0 L 2 ( Ω ¯ , d μ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq75_HTML.gif is given.

Definition 3.1 A function u ( x , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq76_HTML.gif is called a weak solution of (1)-(3) on ( τ , T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq77_HTML.gif if
( u , v ) V , ( u t , v t ) V , u | t = τ = u 0 a.e. in Ω ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equr_HTML.gif
and
Ω T ( u t ξ + | u | p 2 u ξ + | u | p 2 u ξ + f ( u ) ξ ) + Γ T ( v t ξ + f ( v ) ξ ) = Ω T g ( x , t ) ξ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equs_HTML.gif

for all test functions ξ V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq78_HTML.gif.

Theorem 3.1 Let Ω be a bounded domain in R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq7_HTML.gif ( n 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq4_HTML.gif). Assume that f satisfies (H1), g : Ω × R R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq19_HTML.gif is locally Lipschitz continuous and g L loc 2 ( R , L 2 ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq21_HTML.gif. Then, for any τ R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq79_HTML.gif, any initial data u 0 L 2 ( Ω ¯ , d μ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq75_HTML.gif and any T > τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq80_HTML.gif, there exists a unique weak solution u ( x , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq76_HTML.gif of (1)-(3), and the mapping
( u 0 , γ u 0 ) ( u ( t ) , v ( t ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equt_HTML.gif

is continuous on L 2 ( Ω ¯ , d μ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq1_HTML.gif.

Proof We first prove the existence of solutions for (1)-(5) by the Faedo-Galerkin method (see [25]).

Consider the approximating solution u n ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq81_HTML.gif in the form
u n ( t ) = i = 1 n u n i ( t ) e i , v n ( t ) = i = 1 n u n i ( t ) γ e i , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equu_HTML.gif
where { ( e j , γ e j ) } j = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq82_HTML.gif is an orthogonal basis of L 2 ( Ω ¯ , d μ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq1_HTML.gif, which is included in ( W 1 , p ( Ω ) L q ( Ω ) ) × L q ( Γ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq27_HTML.gif. We get u n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq83_HTML.gif from solving the following problem:
d u n d t , e k + d v n d t , e k + A ( u n ) + | u n | p 2 u n , e k + f ( u n ) , e k + f ( v n ) , e k = g ( x , t ) , e k , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equ8_HTML.gif
(8)
( u n ( τ ) , e k ) = ( u 0 , e k ) , k = 1 , , n . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equ9_HTML.gif
(9)
Since f is continuous and g is locally Lipschitz continuous, using the Peano theorem, we get the local existence of ( u n , v n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq84_HTML.gif. Next, we establish some a priori estimates for ( u n , v n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq84_HTML.gif. We have
1 2 d d t u n ( t ) L 2 ( Ω ) 2 + 1 2 d d t v n ( t ) L 2 ( Γ ) 2 + u n 1 , p p + Ω f ( u n ) u n d x + Γ f ( v n ) v n d S = Ω g ( x , t ) u n d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equv_HTML.gif
Thanks to (5), we obtain
1 2 d d t u n ( t ) L 2 ( Ω ) 2 + 1 2 d d t v n ( t ) L 2 ( Γ ) 2 + u n 1 , p p + c 1 u n L q ( Ω ) q + c 1 v n L q ( Γ ) q http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equ10_HTML.gif
(10)
1 2 g ( t ) L 2 ( Ω ) 2 + 1 2 u n L 2 ( Ω ) 2 + k | Ω | + k | Γ | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equ11_HTML.gif
(11)
by virtue of the following inequality (see Theorem 2.3.1 in [26]):
μ z q + λ z 2 C μ 2 q 2 λ q q 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equ12_HTML.gif
(12)
Let μ = c 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq85_HTML.gif and λ = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq86_HTML.gif, we deduce from (10) and (12) that
d d t u n ( t ) L 2 ( Ω ) 2 + d d t v n ( t ) L 2 ( Γ ) 2 + 2 u n 1 , p p + c 1 u n L q ( Ω ) q + 2 c 1 v n L q ( Γ ) q g ( t ) L 2 ( Ω ) 2 + C . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equ13_HTML.gif
(13)
Integrating (13) over [ τ , t ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq87_HTML.gif, we obtain
u n ( t ) L 2 ( Ω ) 2 + v n ( t ) L 2 ( Γ ) 2 + 2 τ t u n 1 , p p d s + c 1 τ t u n L q ( Ω ) q d s + 2 c 1 τ t v n L q ( Γ ) q d s C ( T τ ) + τ t g ( s ) L 2 ( Ω ) 2 d s + u 0 L 2 ( Ω ¯ , d μ ) 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equ14_HTML.gif
(14)

for any t ( τ , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq88_HTML.gif.

Due to (14), we get
{ u n } is uniformly bounded in L ( τ , T ; L 2 ( Ω ) ) , { v n } is uniformly bounded in L ( τ , T ; L 2 ( Γ ) ) , { u n } is uniformly bounded in L p ( τ , T ; W 1 , p ( Ω ) ) , { u n } is uniformly bounded in L q ( Ω T ) , { v n } is uniformly bounded in L q ( Γ T ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equw_HTML.gif
Therefore, { u n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq89_HTML.gif is uniformly bounded in n in the L p ( τ , T ; W 1 , p ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq90_HTML.gif, L q ( Ω T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq91_HTML.gif, respectively, and { v n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq92_HTML.gif is uniformly bounded in n in the L q ( Γ T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq93_HTML.gif, and one can extract a subsequence { u n j } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq94_HTML.gif of { u n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq89_HTML.gif such that
{ u n j } u weakly in L p ( τ , T ; W 1 , p ( Ω ) ) , { u n j } u weakly in L q ( Ω T ) , { v n j } v weakly in L q ( Γ T ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equx_HTML.gif
Let P n : V span { ( e j , γ e j ) } j = 1 n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq95_HTML.gif be a projection. For any ϕ V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq96_HTML.gif, set ϕ n = P n ϕ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq97_HTML.gif, we have
d u n d t , ϕ n + d v n d t , ϕ n + A ( u n ) , ϕ n + f ( u n ) , ϕ n + f ( v n ) , ϕ n = g ( x , t ) , ϕ n . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equ15_HTML.gif
(15)
We perform the following estimate deduced from the Hölder inequality and the Young inequality:
| A ( u n ) , ϕ n | = | Ω T | u n | p 2 u n ϕ n + | u n | p 2 u n ϕ n d x d s | u n L p ( Ω T ) p 1 ϕ n L p ( Ω T ) + u n L p ( Ω T ) p 1 ϕ n L p ( Ω T ) u n L p ( τ , T ; W 1 , p ( Ω ) ) p 1 ϕ n L p ( τ , T ; W 1 , p ( Ω ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equy_HTML.gif
Using the boundedness of { u n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq89_HTML.gif in L p ( τ , T ; W 1 , p ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq90_HTML.gif again, we infer that
{ A ( u n ) } is uniformly bounded in L p ( τ , T ; ( W 1 , p ( Ω ) ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equz_HTML.gif
Since g L loc 2 ( R , L 2 ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq21_HTML.gif, f ( u n ) L q ( Ω T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq98_HTML.gif, f ( v n ) L q ( Γ T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq99_HTML.gif, we find
( u n , v n ) V . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equaa_HTML.gif
Therefore we can extract a subsequence such that
( u n , v n ) ( u , v ) in V , A ( u n ) ξ in L p ( τ , T ; ( W 1 , p ( Ω ) ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equab_HTML.gif
By virtue of the Aubin compactness theorem, we can extract a further subsequence (still denoted by { u n j } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq94_HTML.gif) such that additionally
u n j u in L p ( Ω T ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equ16_HTML.gif
(16)
v n j v in L p ( Γ T ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equ17_HTML.gif
(17)
Due to the boundedness of { u n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq89_HTML.gif in L q ( Ω T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq91_HTML.gif and (5), we obtain that { f ( u n ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq100_HTML.gif is uniformly bounded in L q ( Ω T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq101_HTML.gif and hence f ( u n ) χ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq102_HTML.gif in L q ( Ω T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq101_HTML.gif, similarly, f ( v n ) η http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq103_HTML.gif in L q ( Γ T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq104_HTML.gif. By virtue of (16)-(17), we see that u n j u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq105_HTML.gif a.e. in Ω T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq67_HTML.gif and v n j v http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq106_HTML.gif a.e. in Γ T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq107_HTML.gif, then f ( u n j ) f ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq108_HTML.gif a.e. in Ω T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq67_HTML.gif and f ( v n j ) f ( v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq109_HTML.gif a.e. in Γ T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq107_HTML.gif. Thanks to Lemma 2.1, we know that
χ = f ( u ) , η = f ( v ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equac_HTML.gif
Therefore, we have
u , ϕ + v , ϕ + ξ , ϕ + f ( u ) , ϕ + f ( v ) , ϕ = g ( x , t ) , ϕ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equ18_HTML.gif
(18)

for any ϕ V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq96_HTML.gif.

In order to prove that u is a weak solution of (1)-(3), it remains to show that ξ = A ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq110_HTML.gif. Noticing that
A ( u n ) , u n = τ T u n 1 , p p d s = 1 2 u n ( τ ) L 2 ( Ω ) 2 1 2 u n ( T ) L 2 ( Ω ) 2 + 1 2 v n ( τ ) L 2 ( Γ ) 2 1 2 v n ( T ) L 2 ( Γ ) 2 τ T Ω f ( u n ) u n d x d s τ T Γ f ( v n ) v n d S d s + τ T Ω g ( x , t ) u n d x d s , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equ19_HTML.gif
(19)
it follows from the formulation of u n ( τ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq111_HTML.gif and v n ( τ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq112_HTML.gif that u n ( τ ) u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq113_HTML.gif in L 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq23_HTML.gif and v n ( τ ) θ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq114_HTML.gif in L 2 ( Γ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq30_HTML.gif. Moreover, by the lower semi-continuity of L 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq115_HTML.gif and L 2 ( Γ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq116_HTML.gif, we obtain
u ( T ) L 2 ( Ω ) 2 lim inf n u n ( T ) L 2 ( Ω ) 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equ20_HTML.gif
(20)
v ( T ) L 2 ( Ω ) 2 lim inf n v n ( T ) L 2 ( Ω ) 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equ21_HTML.gif
(21)
Meanwhile, by the Lebesgue dominated theorem, one can check that
τ T Ω f ( u ) u d x d s + τ T Γ f ( v ) v d S d s = lim n τ T Ω f ( u n ) u n d x d s + lim n τ T Γ f ( v n ) v n d S d s , lim n τ T Ω g ( x , t ) u n d x d s = τ T Ω g ( x , t ) u d x d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equad_HTML.gif
This fact and (20)-(21) imply
lim sup n A ( u n ) , u n 1 2 u ( τ ) L 2 ( Ω ) 2 1 2 u ( T ) L 2 ( Ω ) 2 + 1 2 v ( τ ) L 2 ( Γ ) 2 1 2 v ( T ) L 2 ( Γ ) 2 τ T Ω f ( u ) u d x d s τ T Γ f ( v ) v d S d s + τ T Ω g ( x , t ) u d x d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equ22_HTML.gif
(22)
In view of (18), we have
ξ , u = 1 2 u ( τ ) L 2 ( Ω ) 2 1 2 u ( T ) L 2 ( Ω ) 2 + 1 2 v ( τ ) L 2 ( Γ ) 2 1 2 v ( T ) L 2 ( Γ ) 2 τ T Ω f ( u ) u d x d s τ T Γ f ( v ) v d S d s + τ T Ω g ( x , t ) u d x d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equae_HTML.gif
This and (22) deduce
lim sup n A ( u n ) , u n ξ , u . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equ23_HTML.gif
(23)
To this end, we first observe that
lim n A ( u n ) A ( u ) , u n u = lim n ( A ( u n ) , u n A ( u n ) , u A ( u ) , u n u ) ξ , u ξ , u = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equaf_HTML.gif
On the other hand, it follows from Lemma 2.4 that
A ( u n ) A ( u ) , u n u ( u n L p ( τ , T ; W 1 , p ( Ω ) ) p 1 u L p ( τ , T ; W 1 , p ( Ω ) ) p 1 ) ( u n L p ( τ , T ; W 1 , p ( Ω ) ) u L p ( τ , T ; W 1 , p ( Ω ) ) ) 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equag_HTML.gif
Hence
u n L p ( τ , T ; W 1 , p ( Ω ) ) u L p ( τ , T ; W 1 , p ( Ω ) ) , as n . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equ24_HTML.gif
(24)
Combining (24) with u n u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq117_HTML.gif in L p ( τ , T ; W 1 , p ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq90_HTML.gif, we obtain
u n u in L p ( τ , T ; W 1 , p ( Ω ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equah_HTML.gif
Therefore, from Lemma 2.2, the Hölder inequality and the Young inequality, we deduce that for any ϕ L p ( τ , T ; W 1 , p ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq118_HTML.gif,
| A ( u n ) A ( u ) , ϕ | = | Ω T ( | u n | p 2 u n | u | p 2 u ) ϕ + ( | u n | p 2 u n | u | p 2 u ) ϕ d x d s | C 2 Ω T ( | u n | + | u | ) p 2 | u n u | | ϕ | d x d s + C 2 Ω T ( | u n | + | u | ) p 2 | u n u | | ϕ | d x d s C ( u n L p ( τ , T ; W 1 , p ( Ω ) ) p 2 + u L p ( τ , T ; W 1 , p ( Ω ) ) p 2 ) × u n u L p ( τ , T ; W 1 , p ( Ω ) ) ϕ L p ( τ , T ; W 1 , p ( Ω ) ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equai_HTML.gif

which implies that A ( u n ) A ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq119_HTML.gif in ( L p ( τ , T ; W 1 , p ( Ω ) ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq120_HTML.gif, hence ξ = A ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq110_HTML.gif.

Finally, we prove the uniqueness and continuous dependence of the initial data of the solutions. Let u 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq121_HTML.gif, u 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq122_HTML.gif be two solutions of (1)-(5) with the initial data u 0 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq123_HTML.gif, u 0 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq124_HTML.gif, respectively. Let w = u 1 u 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq125_HTML.gif. Taking the inner product of the equation with w, we deduce that
1 2 d d t w ( t ) L 2 ( Ω ) 2 + 1 2 d d t w ( t ) L 2 ( Γ ) 2 + Ω ( | u 1 | p 2 u 1 | u 2 | p 2 u 2 , u 1 u 2 ) d x + Ω ( | u 1 | p 2 u 1 | u 2 | p 2 u 2 , u 1 u 2 ) d x + Ω ( f ( u 1 ) f ( u 2 ) , u 1 u 2 ) d x + Γ ( f ( v 1 ) f ( v 2 ) , v 1 v 2 ) d S = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equ25_HTML.gif
(25)
By virtue of (4) and Lemma 2.2, we obtain
1 2 d d t w ( t ) L 2 ( Ω ) 2 + 1 2 d d t w ( t ) L 2 ( Γ ) 2 l w ( t ) L 2 ( Ω ) 2 + l w ( t ) L 2 ( Γ ) 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equaj_HTML.gif
which implies that
w ( t ) L 2 ( Ω ) 2 + w ( t ) L 2 ( Γ ) 2 exp ( 2 l ( t τ ) ) ( w ( τ ) L 2 ( Ω ) 2 + w ( τ ) L 2 ( Γ ) 2 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equak_HTML.gif

Therefore, u 1 ( x , t ) = u 2 ( x , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq126_HTML.gif a.e. in Ω T ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq127_HTML.gif if u 0 1 ( x ) = u 0 2 ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq128_HTML.gif in Ω ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq38_HTML.gif, and u ( x , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq76_HTML.gif is continuously dependent on the initial data.

Since
( u ( t ) , v ( t ) ) V , ( u t ( t ) , v t ( t ) ) V , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equal_HTML.gif
by use of Lemma 2.5, we know that
( u ( t ) , v ( t ) ) C ( [ τ , T ] ; L 2 ( Ω ¯ , d μ ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equam_HTML.gif

Therefore, ( u ( τ ) , v ( τ ) ) L 2 ( Ω ¯ , d μ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq129_HTML.gif is meaningful. □

By Theorem 3.1, we can define a family of continuous processes { U ( t , τ ) : < τ t < } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq130_HTML.gif in L 2 ( Ω ¯ , d μ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq1_HTML.gif as follows: For all t τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq131_HTML.gif,
U ( t , τ ) ( u 0 , γ u 0 ) = ( u ( t ) , v ( t ) ) : = ( u ( t ; τ , ( u 0 , γ u 0 ) ) , v ( t ; τ , ( u 0 , γ u 0 ) ) ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equan_HTML.gif
where u ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq132_HTML.gif is the solution of (1)-(5) with initial data ( u ( τ ) , v ( τ ) ) = ( u 0 , γ u 0 ) L 2 ( Ω ¯ , d μ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq133_HTML.gif. That is, a family of mappings U ( t , τ ) : L 2 ( Ω ¯ , d μ ) L 2 ( Ω ¯ , d μ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq134_HTML.gif satisfies
U ( τ , τ ) = i d (identity) , U ( t , τ ) = U ( t , r ) U ( r , τ ) for all τ r t . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equao_HTML.gif

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, Y X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq135_HTML.gif continuously. { U σ ( t , τ ) } σ Σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq136_HTML.gif is a family of processes in a Banach space X. Denote by B ( X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq137_HTML.gif the set of all bounded subsets of X and R τ = [ τ , + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq138_HTML.gif. 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.) σ Σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq48_HTML.gif) attractors.

Definition 4.1 [5, 27]

A set B 0 B ( Y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq139_HTML.gif is called to be ( X , Y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq140_HTML.gif-uniformly (w.r.t. σ Σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq48_HTML.gif) absorbing for { U σ ( t , τ ) } σ Σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq141_HTML.gif if for any τ R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq79_HTML.gif and any bounded subset B X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq142_HTML.gif, there exists a positive constant t 0 = t 0 ( τ , B ) τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq143_HTML.gif such that
σ Σ U σ ( t , τ ) B B 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equap_HTML.gif

for any t t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq144_HTML.gif.

A set P Y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq145_HTML.gif is said to be ( X , Y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq140_HTML.gif-uniformly (w.r.t. σ Σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq48_HTML.gif) attracting for the family of processes { U σ ( t , τ ) } σ Σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq136_HTML.gif, if
sup σ Σ dist Y ( U σ ( t + τ , τ ) B , P ) 0 ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equaq_HTML.gif

for an arbitrary fixed τ R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq79_HTML.gif and any bounded set B X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq142_HTML.gif.

Definition 4.2 [5]

A closed set A Σ Y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq146_HTML.gif is said to be an ( X , Y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq140_HTML.gif-uniform (w.r.t. σ Σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq48_HTML.gif) attractor for the family of processes { U σ ( t , τ ) } σ Σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq136_HTML.gif if it is ( X , Y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq140_HTML.gif-uniformly (w.r.t. σ Σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq48_HTML.gif) attracting and it is contained in any closed ( X , Y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq140_HTML.gif-uniformly (w.r.t. σ Σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq48_HTML.gif) attracting set A http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq147_HTML.gif for the family of processes { U σ ( t , τ ) } σ Σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq136_HTML.gif: A Σ A http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq148_HTML.gif.

Definition 4.3 [5]

Define the uniform (w.r.t. σ Σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq48_HTML.gif) ω-limit set of B by ω τ , Σ ( B ) = t τ σ Σ s t U σ ( s , τ ) B ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq149_HTML.gif. This can be characterized by the following: y ω τ , Σ ( B ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq150_HTML.gif if and only if there are sequences { x n } B http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq151_HTML.gif, { σ n } Σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq152_HTML.gif, { t n } R τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq153_HTML.gif, t n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq154_HTML.gif such that U σ n ( t n , τ ) x n y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq155_HTML.gif ( n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq54_HTML.gif).

Definition 4.4 [5]

A family of processes { U σ ( t , τ ) } σ Σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq136_HTML.gif possessing a compact ( X , Y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq140_HTML.gif-uniformly (w.r.t. σ Σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq48_HTML.gif) absorbing set is called ( X , Y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq140_HTML.gif-uniformly compact. A family of processes { U σ ( t , τ ) } σ Σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq136_HTML.gif is called ( X , Y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq140_HTML.gif-uniformly asymptotically compact if it possesses a compact ( X , Y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq140_HTML.gif-uniformly (w.r.t. σ Σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq48_HTML.gif) attracting set, i.e., for any bounded subset B X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq142_HTML.gif and any sequences { τ n } R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq156_HTML.gif, t n + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq157_HTML.gif as n + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq158_HTML.gif and { x n } B http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq151_HTML.gif, { U ( t n + τ n , τ n ) x n } n = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq159_HTML.gif is precompact in Y.

Lemma 4.1 [20]

If a family of processes { U σ ( t , τ ) } σ Σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq136_HTML.gif is ( X , Y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq140_HTML.gif-uniformly asymptotically compact, then for any τ R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq79_HTML.gif, B B ( X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq160_HTML.gif,
  1. (i)

    for any sequences { x n } B http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq151_HTML.gif, { σ n } Σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq152_HTML.gif, { t n } R τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq153_HTML.gif, t n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq154_HTML.gif as n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq54_HTML.gif, there is a convergent subsequence of { U σ n ( t n , τ ) x n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq161_HTML.gif in Y,

     
  2. (ii)

    ω τ , Σ ( B ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq162_HTML.gif is nonempty and compact in Y,

     
  3. (iii)

    ω τ , Σ ( B ) = ω 0 , Σ ( B ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq163_HTML.gif,

     
  4. (iv)

    lim t ( sup σ Σ dist Y ( U σ ( t , τ ) B , ω τ , Σ ( B ) ) ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq164_HTML.gif,

     
  5. (v)

    if A is a closed set and ( X , Y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq140_HTML.gif-uniformly (w.r.t. σ Σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq48_HTML.gif) attracting B, then ω τ , Σ ( B ) A http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq165_HTML.gif.

     
Assumption 1 Let { T ( h ) | h 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq166_HTML.gif be a family of operators acting on Σ and satisfying:
  1. (i)

    T ( h ) Σ = Σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq167_HTML.gif, h R + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq168_HTML.gif,

     
  2. (ii)
    translation identity:
    U σ ( t + h , τ + h ) = U T ( h ) σ ( t , τ ) , σ Σ , t τ , τ R , h 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equar_HTML.gif
     

Definition 4.5 [5]

The kernel K http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq169_HTML.gif of the process { U σ ( t , τ ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq170_HTML.gif acting on X consists of all bounded complete trajectories of the process { U σ ( t , τ ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq170_HTML.gif:
K = { u ( ) | U ( t , τ ) u ( τ ) = u ( t ) , dist ( u ( t ) , u ( 0 ) ) C u , t τ , τ R } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equas_HTML.gif

The set K ( s ) = { u ( s ) | u ( ) K } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq171_HTML.gif is said to be kernel section at time t = s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq172_HTML.gif, s R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq173_HTML.gif.

Definition 4.6 [5]

A family of processes { U σ ( t , τ ) } σ Σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq136_HTML.gif is said to be ( X × Σ , Y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq174_HTML.gif-weakly continuous if for any fixed t τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq131_HTML.gif, τ R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq79_HTML.gif, the mapping ( u , σ ) U σ ( t , τ ) u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq175_HTML.gif is weakly continuous from X × Σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq176_HTML.gif to Y.

Assumption 2 Let Σ be a weakly compact set and { U σ ( t , τ ) } σ Σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq136_HTML.gif be ( X × Σ , Y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq174_HTML.gif-weakly continuous.

Lemma 4.2 [20]

Under Assumptions 1 and 2 with { T ( h ) } h 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq177_HTML.gif, which is a weakly continuous semigroup, if { U σ ( t , τ ) } σ Σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq136_HTML.gif acting on X is ( X , Y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq140_HTML.gif-uniformly (w.r.t. σ Σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq48_HTML.gif) asymptotically compact, then it possesses an ( X , Y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq140_HTML.gif-uniform (w.r.t. σ Σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq48_HTML.gif) attractor A Σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq178_HTML.gif, which is compact in Y and attracts all the bounded subsets of X in the topology of Y.

Moreover,
A Σ = ω τ , Σ ( B 0 ) = σ Σ K σ ( s ) , s R , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equat_HTML.gif

where B 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq179_HTML.gif is a bounded neighborhood of the compact ( X , Y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq140_HTML.gif-uniformly attracting set in Y; i.e., B 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq179_HTML.gif is a bounded ( X , Y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq140_HTML.gif-uniformly (w.r.t. σ Σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq48_HTML.gif) absorbing set of { U σ ( t , τ ) } σ Σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq136_HTML.gif. K σ ( s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq180_HTML.gif is the section at t = s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq172_HTML.gif of kernel K σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq181_HTML.gif of the process { U σ ( t , τ ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq170_HTML.gif with symbol σ Σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq48_HTML.gif. Furthermore, K σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq181_HTML.gif is nonempty for all σ Σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq48_HTML.gif.

From the ideas of [4, 20, 28], we give the following results, which are very useful for the existence of a uniform attractor in L p ( Ω ¯ , d μ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq182_HTML.gif.

Lemma 4.3 [20]

Let { U σ ( t , τ ) } σ Σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq136_HTML.gif be a family of processes on L p ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq183_HTML.gif ( p 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq184_HTML.gif) and suppose { U σ ( t , τ ) } σ Σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq136_HTML.gif has a bounded ( L p ( Ω ) , L p ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq185_HTML.gif-uniformly (w.r.t. σ Σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq48_HTML.gif) absorbing set in L p ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq183_HTML.gif. Then, for any ϵ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq186_HTML.gif, τ R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq79_HTML.gif and any bounded subset B L p ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq187_HTML.gif, there exist two positive constants T = T ( B , τ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq188_HTML.gif and M = M ( ϵ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq189_HTML.gif such that
m ( Ω ( | U σ ( t , τ ) u τ | M ) ) ϵ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equau_HTML.gif

for any u τ B http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq190_HTML.gif, t T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq191_HTML.gif, σ Σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq48_HTML.gif.

Lemma 4.4 [4, 28]

Let a family of processes { U σ ( t , τ ) } σ Σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq136_HTML.gif be ( L p ( Ω ) , L p ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq192_HTML.gif-uniformly (w.r.t. σ Σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq48_HTML.gif) asymptotically compact, then { U σ ( t , τ ) } σ Σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq136_HTML.gif is ( L p ( Ω ) , L q ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq193_HTML.gif-uniformly asymptotically compact for p q < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq194_HTML.gif, if
  1. (i)

    { U σ ( t , τ ) } σ Σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq136_HTML.gif has a bounded ( L p ( Ω ) , L q ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq193_HTML.gif-uniformly (w.r.t. σ Σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq48_HTML.gif) absorbing set  B 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq179_HTML.gif,

     
  2. (ii)
    for any ϵ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq195_HTML.gif, τ R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq79_HTML.gif and any bounded subset B L p ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq196_HTML.gif, there exist two positive constants M = M ( ϵ , B ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq197_HTML.gif and T = T ( ϵ , B , τ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq198_HTML.gif such that
    Ω ( | U σ ( t , τ ) u τ | M ) | U σ ( t , τ ) u τ | q ϵ for all u τ B , t T , σ Σ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equav_HTML.gif
     

From Theorem 3.1, we know that the problem (1)-(5) generates a process { U σ ( t , τ ) } σ Σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq136_HTML.gif acting in L 2 ( Ω ¯ , d μ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq1_HTML.gif and the time symbol is σ ( s ) = g ( x , s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq199_HTML.gif. We denote by L loc 2 , w ( R ; L 2 ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq200_HTML.gif the space L loc 2 ( R ; L 2 ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq201_HTML.gif endowed with a locally weak convergence topology. Let H w ( g ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq202_HTML.gif be the hull of g in L loc 2 , w ( R ; L 2 ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq200_HTML.gif, i.e., the closure of the set { g ( s + h ) | h R } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq203_HTML.gif in L loc 2 , w ( R ; L 2 ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq200_HTML.gif and g ( x , s ) L b 2 ( R ; L 2 ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq204_HTML.gif.

Lemma 4.5 [5]

Ifis reflective separable and ϕ L b 2 ( R ; E ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq205_HTML.gif, then
  1. (i)

    for all ϕ 1 H w ( ϕ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq206_HTML.gif, ϕ 1 L b 2 2 ϕ L b 2 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq207_HTML.gif,

     
  2. (ii)

    the translation group { T ( h ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq208_HTML.gif is weakly continuous on H w ( ϕ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq209_HTML.gif,

     
  3. (iii)

    T ( h ) H w ( ϕ ) = H w ( ϕ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq210_HTML.gif for h 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq211_HTML.gif,

     
  4. (iv)

    H w ( ϕ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq209_HTML.gif is weakly compact.

     
Due to Lemma 4.5, H w ( g ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq202_HTML.gif is weakly compact and the translation semigroup { T ( h ) | h R + } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq212_HTML.gif satisfies that T ( h ) H w ( g ) = H w ( g ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq213_HTML.gif and is weakly continuous on H w ( g ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq202_HTML.gif. Because of the uniqueness of solution, the following translation identity holds:
U σ ( t + h , τ + h ) = U T ( h ) σ ( t , τ ) σ H w ( g ) , t τ , τ R , h 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equaw_HTML.gif

Theorem 4.1 The family of processes { U σ ( t , τ ) } σ H w ( g ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq25_HTML.gif corresponding to problem (1)-(5) is ( L 2 ( Ω ¯ , d μ ) × H w ( g ) , L 2 ( Ω ¯ , d μ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq214_HTML.gif-weakly continuous and ( L 2 ( Ω ¯ , d μ ) × H w ( g ) , ( L q ( Ω ) W 1 , p ( Ω ) ) × L q ( Γ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq215_HTML.gif-weakly continuous.

Proof For any fixed t 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq216_HTML.gif and τ, t 1 τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq217_HTML.gif, τ R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq79_HTML.gif, let u τ n u τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq218_HTML.gif ( n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq54_HTML.gif) weakly in L 2 ( Ω ¯ , d μ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq1_HTML.gif and σ n σ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq219_HTML.gif weakly in H w ( g ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq202_HTML.gif as n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq54_HTML.gif, denote by u n ( t ) = U σ n ( t , τ ) u τ n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq220_HTML.gif. The same estimates for u n E n = span { ( e i , γ e i ) } i = 1 n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq221_HTML.gif given in the Galerkin approximations (in Section 3) are valid for the u n ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq81_HTML.gif here. Therefore, for some subsequence { m } { n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq222_HTML.gif and u ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq132_HTML.gif such that for any t 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq216_HTML.gif, τ t 1 t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq223_HTML.gif, ( u m ( t 1 ) , v m ( t 1 ) ) ( u ( t 1 ) , v ( t 1 ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq224_HTML.gif weakly in L 2 ( Ω ¯ , d μ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq1_HTML.gif and ( L q ( Ω ) W 1 , p ( Ω ) ) × L q ( Γ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq225_HTML.gif. And the sequence { ( u m ( s ) , v m ( s ) ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq226_HTML.gif, τ s t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq227_HTML.gif is bounded in L ( τ , t ; L 2 ( Ω ¯ , d μ ) ( ( L p ( τ , t ; W 1 , p ( Ω ) ) L q ( τ , t ; L q ( Ω ) ) ) × L q ( τ , t ; L q ( Γ ) ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq228_HTML.gif. Denote by ξ ( s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq229_HTML.gif, χ ( s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq230_HTML.gif and η ( s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq231_HTML.gif the weak limits of A ( u m ) ( s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq232_HTML.gif, f ( u m ( s ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq233_HTML.gif and f ( v m ( s ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq234_HTML.gif in L p ( τ , t ; ( W 1 , p ( Ω ) ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq235_HTML.gif, L q ( τ , t ; L q ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq236_HTML.gif and L q ( τ , t ; L q ( Γ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq237_HTML.gif, respectively. So, we get the following equation for u ( s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq238_HTML.gif:
t u , ϕ + t v , γ ϕ + η 0 + η 1 , ϕ + η 2 , γ ϕ = σ 0 , ϕ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equax_HTML.gif

for any ϕ V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq96_HTML.gif.

By the same method as the proof of Theorem 3.1, we know that η 0 = A ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq239_HTML.gif, η 1 = f ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq240_HTML.gif and η 2 = f ( v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq241_HTML.gif, which means that ( u ( s ) , v ( s ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq242_HTML.gif in V is the weak solution of (1)-(5) with the initial condition u τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq243_HTML.gif. Due to the uniqueness of the solution, we state that U σ m ( t 1 , τ ) ( u τ m , γ u τ m ) U σ 0 ( t 1 , τ ) ( u τ , γ u τ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq244_HTML.gif weakly in L 2 ( Ω ¯ , d μ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq1_HTML.gif and ( L q ( Ω ) W 1 , p ( Ω ) ) × L q ( Γ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq225_HTML.gif. For any other subsequence, { u τ m } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq245_HTML.gif and { σ m } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq246_HTML.gif satisfy u τ m u τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq247_HTML.gif weakly in L 2 ( Ω ¯ , d μ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq1_HTML.gif and σ m σ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq248_HTML.gif, by the same process, we obtain the analogous relation U σ m ( t 1 , τ ) u τ m U σ 0 ( t 1 , τ ) u τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq249_HTML.gif weakly in L 2 ( Ω ¯ , d μ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq1_HTML.gif and ( L q ( Ω ) W 1 , p ( Ω ) ) × L q ( Γ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq225_HTML.gif holds. Then it can be easily seen that for any weakly convergent initial sequence { u τ n } L 2 ( Ω ¯ , d μ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq250_HTML.gif and weakly convergent sequence { σ n } H w ( g ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq251_HTML.gif, we have U σ n ( t 1 , τ ) u τ n U σ 0 ( t 1 , τ ) u τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq252_HTML.gif weakly in L 2 ( Ω ¯ , d μ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq1_HTML.gif and ( L q ( Ω ) W 1 , p ( Ω ) ) × L q ( Γ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq225_HTML.gif. □

Lemma 4.6 [25]

(The uniform Gronwall lemma) Let x ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq253_HTML.gif, a ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq254_HTML.gif, b ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq255_HTML.gif be three positive locally integrable functions on [ t 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq256_HTML.gif, and for some r > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq257_HTML.gif and all t t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq144_HTML.gif, x ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq253_HTML.gif, a ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq254_HTML.gif, b ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq258_HTML.gif satisfy the following inequalities:
x ( t ) a ( t ) x ( t ) + b ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equay_HTML.gif
and
t t + r x ( τ ) d τ R , t t + r a ( τ ) d τ A , t t + r b ( τ ) d τ B , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equaz_HTML.gif
where R, A, B are three positive constants. Then
x ( t ) ( R r + B ) e A http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equba_HTML.gif

for all t t 0 + r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq259_HTML.gif.

4.2 The existence of uniformly absorbing sets

In this subsection, we prove the existence of uniformly (w.r.t. σ Σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq48_HTML.gif) absorbing sets for the process { U σ ( t , τ ) } σ Σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq136_HTML.gif corresponding to (1)-(5).

Theorem 4.2 Assume that f and g satisfy (H1)-(H2). Then the family of processes { U σ ( t , τ ) } σ H w ( g ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq25_HTML.gif corresponding to problem (1)-(5) has a bounded ( L 2 ( Ω ¯ , d μ ) , L 2 ( Ω ¯ , d μ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq260_HTML.gif- and ( L 2 ( Ω ¯ , d μ ) , ( L q ( Ω ) W 1 , p ( Ω ) ) × L q ( Γ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq261_HTML.gif-uniformly (w.r.t. σ H w ( g ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq24_HTML.gif) absorbing set. That is, for any bounded subset B of L 2 ( Ω ¯ , d μ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq1_HTML.gif and any τ R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq79_HTML.gif, there exist τ 1 = τ 1 ( τ , B ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq262_HTML.gif, τ 2 = τ 2 ( τ , B ) τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq263_HTML.gif and two positive constants ρ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq264_HTML.gif, ρ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq265_HTML.gif such that
u ( t ) L 2 ( Ω ) 2 + v ( t ) L 2 ( Γ ) 2 3 ρ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equ26_HTML.gif
(26)
for any t τ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq266_HTML.gif and
u ( t ) W 1 , p ( Ω ) p + u ( t ) L q ( Ω ) q + v ( t ) L q ( Γ ) q C ρ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equ27_HTML.gif
(27)

for any t τ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq267_HTML.gif, where τ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq268_HTML.gif, τ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq269_HTML.gif, ρ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq264_HTML.gif, and ρ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq265_HTML.gif are specified in (33), (41), (32) and (40), respectively.

Proof Taking the inner product of (1) with u, we deduce that
1 2 d d t ( u L 2 ( Ω ) 2 + v L 2 ( Γ ) 2 ) + u W 1 , p p + Ω f ( u ) u d x + Γ f ( v ) v d S = Ω σ ( t ) u d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equ28_HTML.gif
(28)
By virtue of (5), the Hölder inequality and the Young inequality, we obtain
1 2 d d t ( u L 2 ( Ω ) 2 + v L 2 ( Γ ) 2 ) + u W 1 , p ( Ω ) p + c 1 u L q ( Ω ) q + c 1 v L q ( Γ ) q 1 2 σ ( t ) L 2 ( Ω ) 2 + 1 2 u L 2 ( Ω ) 2 + k | Ω | + k | Γ | 1 2 σ ( t ) L 2 ( Ω ) 2 + 1 2 u L 2 ( Ω ) 2 + 1 2 v L 2 ( Γ ) 2 + k | Ω | + k | Γ | . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equ29_HTML.gif
(29)
Let μ = c 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq85_HTML.gif and λ = 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq270_HTML.gif, we deduce from (12) and (29) that
d d t u ( t ) L 2 ( Ω ) 2 + d d t v ( t ) L 2 ( Γ ) 2 + 2 u 1 , p p + c 1 u L q ( Ω ) q + c 1 v L q ( Γ ) q + u L 2 ( Ω ) 2 + v L 2 ( Γ ) 2 σ ( t ) L 2 ( Ω ) 2 + C . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equ30_HTML.gif
(30)
It follows from the classical Gronwall inequality and Lemma 4.5 that
u ( t ) L 2 ( Ω ) 2 + v ( t ) L 2 ( Γ ) 2 u 0 L 2 ( Ω ¯ , d μ ) 2 e τ t + τ t e s t g ( s ) L 2 ( Ω ) 2 d s + C u 0 L 2 ( Ω ¯ , d μ ) 2 e τ t + 2 sup r R r r + 1 g ( s ) L 2 ( Ω ) 2 d s + C , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equ31_HTML.gif
(31)
where we have used the following inequality:
τ t e s t g ( s ) L 2 ( Ω ) 2 d s = t 1 t e s t g ( s ) L 2 ( Ω ) 2 d s + t 2 t 1 e s t g ( s ) L 2 ( Ω ) 2 d s + ( 1 + e 1 + e 2 + + e n + ) sup r R r r + 1 g ( s ) L 2 ( Ω ) 2 d s 1 1 e 1 sup r R r r + 1 g ( s ) L 2 ( Ω ) 2 d s 2 sup r R r r + 1 g ( s ) L 2 ( Ω ) 2 d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equbb_HTML.gif
From (31), we deduce that
u ( t ) L 2 ( Ω ) 2 + v ( t ) L 2 ( Γ ) 2 3 ρ 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equbc_HTML.gif
where
ρ 1 = sup r R r r + 1 g ( s ) L 2 ( Ω ) 2 d s + C , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equ32_HTML.gif
(32)
τ 1 = τ + max { 0 , ln ( u 0 L 2 ( Ω ¯ , d μ ) 2 ρ 1 ) } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equ33_HTML.gif
(33)
Integrating (30) over [ r , r + 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq271_HTML.gif, we obtain
c 1 r r + 1 u ( s ) L q ( Ω ) q d s + c 1 r r + 1 v ( s ) L q ( Γ ) q d s + 2 r r + 1 u ( s ) 1 , p p d s u 0 L 2 ( Ω ¯ , d μ ) 2 e τ r + 3 sup r R r r + 1 g ( s ) L 2 ( Ω ) 2 d s + C . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equ34_HTML.gif
(34)
Let F ( s ) = 0 s f ( θ ) d θ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq272_HTML.gif, we deduce from (5) that there exist three positive constants α 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq273_HTML.gif, α 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq274_HTML.gif, β such that
α 1 | u | q β F ( u ) α 2 | u | q + β , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equbd_HTML.gif
and
α 1 | u | L q ( Ω ) q β | Ω | Ω F ( u ) d x α 2 | u | L q ( Ω ) q + β | Ω | , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equ35_HTML.gif
(35)
α 1 | v | L q ( Γ ) q β | Γ | Γ F ( v ) d S α 2 | v | L q ( Γ ) q + β | Γ | . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equ36_HTML.gif
(36)
Thanks to (34), we deduce from (35)-(36) that
2 r r + 1 u ( s ) W 1 , p ( Ω ) p d s + c 1 α 2 r r + 1 Ω F ( u ( s ) ) d x d s + c 1 α 2 r r + 1 Γ F ( v ( s ) ) d S d s u 0 L 2 ( Ω ¯ , d μ ) 2 e τ r + 3 sup r R r r + 1 g ( s ) L 2 ( Ω ) 2 d s + C . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equ37_HTML.gif
(37)
On the other hand, taking the inner product of (1) with u t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq275_HTML.gif, we obtain
u t L 2 ( Ω ) 2 + v t L 2 ( Γ ) 2 + d d t ( 1 p u W 1 , p ( Ω ) p + Ω F ( u ) d x + Γ F ( v ) d S ) 1 2 g ( s ) L 2 ( Ω ) 2 + 1 2 u t L 2 ( Ω ) 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Eqube_HTML.gif
which implies
u t L 2 ( Ω ) 2 + v t L 2 ( Γ ) 2 + d d t ( 2 p u W 1 , p ( Ω ) p + 2 Ω F ( u ) d x + 2 Γ F ( v ) d S ) g L 2 ( Ω ) 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equ38_HTML.gif
(38)
Combining (37) with (38), by virtue of the uniform Gronwall Lemma 4.6, we get
u ( t ) W 1 , p ( Ω ) p + Ω F ( u ( t ) ) d x + Γ F ( v ( t ) ) d S C ( u 0 L 2 ( Ω ¯ , d μ ) 2 e τ r + sup r R r r + 1 g ( s ) L 2 ( Ω ) 2 d s + 1 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equ39_HTML.gif
(39)
which implies that for any ( u 0 , γ u 0 ) B http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq276_HTML.gif and τ R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq79_HTML.gif, there exists a positive constant ρ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq265_HTML.gif such that
u ( t ) W 1 , p ( Ω ) p + u ( t ) L q ( Ω ) q + v ( t ) L q ( Γ ) q C ρ 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equbf_HTML.gif
where
ρ 2 = sup t R t t + 1 g ( s ) L 2 ( Ω ) 2 d s + 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equ40_HTML.gif
(40)
τ 2 = max { τ 1 , ln ( u 0 L 2 ( Ω ¯ , d μ ) 2 ρ 2 ) + τ } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equ41_HTML.gif
(41)

 □

From Theorem 4.2, the compactness of the Sobolev embedding W 1 , p ( Ω ) L 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq277_HTML.gif, the compactness of the Sobolev trace embedding W 1 , p ( Ω ) L 2 ( Γ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq278_HTML.gif and Lemma 4.2, we have the following result.

Corollary 4.1 The family of processes { U σ ( t , τ ) } σ H w ( g ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq25_HTML.gif generated by (1)-(5) with initial data u 0 L 2 ( Ω ¯ , d μ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq75_HTML.gif has an ( L 2 ( Ω ¯ , d μ ) , L 2 ( Ω ¯ , d μ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq279_HTML.gif-uniform (w.r.t. σ H w ( g ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq24_HTML.gif) attractor A 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq280_HTML.gif, which is compact in L 2 ( Ω ¯ , d μ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq1_HTML.gif and attracts every bounded subset of L 2 ( Ω ¯ , d μ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq1_HTML.gif in the topology of L 2 ( Ω ¯ , d μ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq1_HTML.gif. Moreover,
A 2 = ω τ , H w ( g ) ( B 0 ) = σ H w ( g ) K σ ( s ) , s R , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equbg_HTML.gif

where B 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq179_HTML.gif is the ( L 2 ( Ω ¯ , d μ ) , L 2 ( Ω ¯ , d μ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq281_HTML.gif-uniformly (w.r.t. σ H w ( g ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq24_HTML.gif) absorbing set in L 2 ( Ω ¯ , d μ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq1_HTML.gif and K σ ( s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq180_HTML.gif is the section at t = s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq172_HTML.gif of kernel K σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq181_HTML.gif of the process { U σ ( t , τ ) } σ H w ( g ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq25_HTML.gif with symbol σ H w ( g ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq24_HTML.gif.

4.3 The existence of ( L 2 ( Ω ¯ , d μ ) , L q ( Ω ¯ , d μ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq282_HTML.gif-uniform attractor

The main purpose of this subsection is to give an asymptotic a priori estimate for the unbounded part of the modular ( | u | , | v | ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq283_HTML.gif for the solution ( u , v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq284_HTML.gif of problem (1)-(5) in the L q ( Ω ¯ , d μ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq28_HTML.gif-norm.

Theorem 4.3 The family of processes { U σ ( t , τ ) } σ H w ( g ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq25_HTML.gif corresponding to problem (1)-(5) with initial data u 0 L 2 ( Ω ¯ , d μ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq75_HTML.gif has an ( L 2 ( Ω ¯ , d μ ) , L q ( Ω ¯ , d μ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq282_HTML.gif-uniform (w.r.t. σ H w ( g ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq24_HTML.gif) attractor A q http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq285_HTML.gif, which is compact in L q ( Ω ¯ , d μ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq28_HTML.gif and attracts every bounded subset B of L 2 ( Ω ¯ , d μ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq1_HTML.gif in the topology of L q ( Ω ¯ , d μ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq28_HTML.gif. Moreover,
A q = ω τ , H w ( g ) ( B 0 ) = σ H w ( g ) K σ ( s ) , s R , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_Equbh_HTML.gif

where B 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq179_HTML.gif is the ( L 2 ( Ω ¯ , d μ ) , L q ( Ω ¯ , d μ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq282_HTML.gif-uniformly (w.r.t. σ H w ( g ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq24_HTML.gif) absorbing set and K σ ( s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq180_HTML.gif is the section at t = s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq172_HTML.gif of kernel K σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq181_HTML.gif of the process { U σ ( t , τ ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq170_HTML.gif with symbol σ H w ( g ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq24_HTML.gif.

Proof We need only prove that the process { U σ ( t , τ ) } σ H w ( g ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq25_HTML.gif satisfies the assumption (ii) of Lemma 4.4.

From (H3), we deduce that for any σ H w ( g ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-128/MediaObjects/13661_2012_Article_382_IEq24_HTML.gif,
sup t R σ ( t ) L 2 ( Ω