Open Access

Averaging of the 3D non-autonomous Benjamin-Bona-Mahony equation with singularly oscillating forces

Boundary Value Problems20132013:111

DOI: 10.1186/1687-2770-2013-111

Received: 29 January 2013

Accepted: 16 April 2013

Published: 30 April 2013

Abstract

For ε ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq1_HTML.gif, we investigate the convergence of corresponding uniform attractors of the 3D non-autonomous Benjamin-Bona-Mahony equation with singularly oscillating force contrast with the averaged Benjamin-Bona-Mahony equation (corresponding to the limiting case ε = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq2_HTML.gif). Under suitable assumptions on the external force, we shall obtain the uniform boundedness and convergence of the related uniform attractors as ε 0 + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq3_HTML.gif.

MSC:35B40, 35Q99, 80A22.

Keywords

Benjamin-Bona-Mahony equation singularly oscillating forces uniform attractors translational bounded functions

1 Introduction

Let ρ [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq4_HTML.gif be a fixed parameter, Ω R 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq5_HTML.gif be a bounded domain with sufficiently smooth boundary Ω. We investigate the long-time behavior for the non-autonomous 3D Benjamin-Bona-Mahony (BBM) equation with singularly oscillating forces:
u t u t ν u + F ( u ) = f 0 ( t , x ) + ε ρ f 1 ( t / ε , x ) , x Ω , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ1_HTML.gif
(1.1)
u ( t , x ) | Ω = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ2_HTML.gif
(1.2)
u ( τ , x ) = u τ ( x ) , τ R . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ3_HTML.gif
(1.3)

Here, t R τ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq6_HTML.gif, R τ = ( τ , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq7_HTML.gif, and u = u ( t , x ) = ( u 1 ( t , x ) , u 2 ( t , x ) , u 3 ( t , x ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq8_HTML.gif is the velocity vector field, ν > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq9_HTML.gif is the kinematic viscosity, F https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq10_HTML.gif is a nonlinear vector function, f 0 ( t , x ) + ε ρ f 1 ( t / ε , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq11_HTML.gif is the singularly oscillating force.

Along with (1.1)-(1.3), we consider the averaged Benjamin-Bona-Mahony equation
u t u t ν u + F ( u ) = f 0 ( t , x ) , x Ω , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ4_HTML.gif
(1.4)
u ( t , x ) | Ω = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ5_HTML.gif
(1.5)
u ( τ , x ) = u τ ( x ) , τ R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ6_HTML.gif
(1.6)

formally corresponding to the case ε = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq2_HTML.gif in (1.1).

The function
f ε ( x , t ) = { f 0 ( x , t ) + ε ρ f 1 ( x , t / ε ) , 0 < ε < 1 , f 0 ( x , t ) , ε = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ7_HTML.gif
(1.7)

represents the external forces of problem (1.1)-(1.3) for ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq12_HTML.gif and of problem (1.4)-(1.6) for ε = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq2_HTML.gif, respectively.

The functions f 0 ( x , s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq13_HTML.gif and f 1 ( x , s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq14_HTML.gif are taken from the space L b 2 ( R , H ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq15_HTML.gif of translational bounded functions in L loc 2 ( R , H ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq16_HTML.gif, namely,
f 0 L b 2 ( R , H ) 2 : = sup t R t t + 1 f 0 ( s ) H 2 d s = M 0 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ8_HTML.gif
(1.8)
f 1 L b 2 ( R , H ) 2 : = sup t R t t + 1 f 1 ( s ) H 2 d s = M 1 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ9_HTML.gif
(1.9)

for some constants M 0 , M 1 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq17_HTML.gif.

Defining
Q ε = { M 0 + 2 M 1 ε ρ , 0 < ε < 1 , M 0 , ε = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equa_HTML.gif
as a straightforward consequence of (1.7), we have
f ε L b 2 ( R , H ) Q ε , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ10_HTML.gif
(1.10)

note that Q ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq18_HTML.gif is of the order ε ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq19_HTML.gif as ε 0 + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq20_HTML.gif.

The BBM equation is a well-known model for long waves in shallow water which was introduced by Benjamin, Bona, and Mahony ([1], 1972) as an improvement of the Korteweg-de Vries equation (KdV equation) for modeling long waves of small amplitude in two dimensions. Contrasting with the KdV equation, the BBM equation is unstable in high wavenumber components. Further, while the KdV equation has an infinite number of integrals of motion, the BBM equation only has three. For more results on the wellposedness and infinite dimensional dynamical systems for BBM equations, we can refer to [27].

In this paper, firstly, we shall study the asymptotic behavior of the non-autonomous BBM equation depending on the small parameter ε, which reflects the rate of fast time oscillations in the term ε ρ f 1 ( x , t / ε ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq21_HTML.gif with amplitude of order ε ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq19_HTML.gif, then we shall consider the boundedness and convergence of corresponding uniform attractors of (1.1)-(1.3) in contrast to (1.4)-(1.6).

2 Preliminaries

Throughout this paper, L p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq22_HTML.gif ( 1 p + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq23_HTML.gif) is the generic Lebesgue space, H s ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq24_HTML.gif is the Sobolev space. We set E : = { u | u ( C 0 ( Ω ) ) 3 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq25_HTML.gif, H, V, W is the closure of the set E in the topology of ( L 2 ( Ω ) ) 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq26_HTML.gif, ( H 1 ( Ω ) ) 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq27_HTML.gif, ( H 2 ( Ω ) ) 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq28_HTML.gif respectively. ‘’ stands for the weak convergence of sequences.

Lemma 2.1 For each τ R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq29_HTML.gif, every nonnegative locally summable function ϕ on R τ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq30_HTML.gif and every β > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq31_HTML.gif, we have
τ t ϕ ( s ) e β ( t s ) d s 1 1 e β sup θ τ θ θ + 1 ϕ ( s ) d s , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ11_HTML.gif
(2.1)

holds for all t τ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq32_HTML.gif.

Proof See, e.g., [8]. □

Lemma 2.2 Let ζ : R τ R + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq33_HTML.gif fulfill that for almost every t τ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq32_HTML.gif, the differential inequality
d d t ζ ( t ) + ϕ 1 ( t ) ζ ( t ) ϕ 2 ( t ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ12_HTML.gif
(2.2)
where, for every t τ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq32_HTML.gif, the scalar functions ϕ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq34_HTML.gif and ϕ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq35_HTML.gif satisfy
τ t ϕ 1 ( s ) d s β ( t τ ) γ , t t + 1 ϕ 2 ( s ) d s M , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ13_HTML.gif
(2.3)
for some β > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq31_HTML.gif, γ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq36_HTML.gif and M 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq37_HTML.gif. Then
ζ ( t ) e γ ζ ( τ ) e β ( t τ ) + M e γ 1 e β , t τ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ14_HTML.gif
(2.4)

Proof See, e.g., [8]. □

For the non-autonomous general Benjamin-Bona-Mahony (BBM) equation,
u t u t ν u + F ( u ) = g ( t , x ) , x Ω , t R τ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ15_HTML.gif
(2.5)
u ( t , x ) | Ω = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ16_HTML.gif
(2.6)
u ( τ , x ) = u τ ( x ) , τ R . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ17_HTML.gif
(2.7)
Assume that u τ H 0 1 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq38_HTML.gif, the nonlinear vector function F ( s ) = ( F 1 ( s ) , F 2 ( s ) , F 3 ( s ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq39_HTML.gif, s R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq40_HTML.gif, we denote
f i ( s ) = F i ( s ) , F i ( s ) = 0 s F i ( r ) d r , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ18_HTML.gif
(2.8)
where
f ( s ) = ( f 1 ( s ) , f 2 ( s ) , f 3 ( s ) ) , F ( s ) = ( F 1 ( s ) , F 2 ( s ) , F 3 ( s ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ19_HTML.gif
(2.9)
In addition, F i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq41_HTML.gif ( i = 1 , 2 , 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq42_HTML.gif) is a smooth function satisfying
F i ( 0 ) = 0 , | F i ( s ) | C 1 | s | + C 2 | s | 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ20_HTML.gif
(2.10)
C 1 0 + C 2 0 | s | | f i ( s ) | C 1 + C 2 | s | , | F i ( s ) | C 1 | s | 2 + C 2 | s | 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ21_HTML.gif
(2.11)

for all s R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq43_HTML.gif, where C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq44_HTML.gif and C 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq45_HTML.gif are positive constants.

Similar to [5], by the Galerkin method and a priori estimate, we easily derive the existence of a global weak solution and a uniform attractor which shall be stated in the following theorems.

Theorem 2.3 Assume that (2.8)-(2.11) hold, g L loc 2 ( R , H ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq46_HTML.gif, u τ H 0 1 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq38_HTML.gif (or V) , then there exists a unique global weak solution u ( x , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq47_HTML.gif of the problem (2.5)-(2.7) which satisfies
u C ( ( τ , T ) ; V ) , u t L 2 ( ( τ , T ) ; V ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ22_HTML.gif
(2.12)

for all τ R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq29_HTML.gif and T > τ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq48_HTML.gif.

Theorem 2.4 Assume that the external force g L loc 2 ( R , H ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq46_HTML.gif and (2.8)-(2.11) hold, then the processes { U ( t , τ ) , t τ } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq49_HTML.gif generated by the global solution possess uniform attractors A g ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq50_HTML.gif in H 0 1 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq51_HTML.gif for the non-autonomous system (2.5)-(2.7).

3 Some lemmas

Lemma 3.1 The functions f 0 ( x , s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq13_HTML.gif and f 1 ( x , s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq14_HTML.gif are taken from the space L b 2 ( R , H ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq15_HTML.gif of translational bounded functions in L loc 2 ( R , H ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq16_HTML.gif, then the processes { U f ε ( t , τ ) , t τ , t , τ R } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq52_HTML.gif generated by system (1.1)-(1.3) have a uniformly (w.r.t. σ = f ε Σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq53_HTML.gif) compact attractor A ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq54_HTML.gif for any fixed ε ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq1_HTML.gif.

Proof As a similar argument in Section 2, we choose g ( t , x ) = f ε ( t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq55_HTML.gif in Theorem 2.4, since f 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq56_HTML.gif and f 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq57_HTML.gif are translational bounded in L loc 2 ( R , H ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq16_HTML.gif, then for any fixed ε ( 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq58_HTML.gif, f ε ( t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq59_HTML.gif is translational bounded in L loc 2 ( R , H ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq16_HTML.gif and we can easily deduce the existence of uniformly compact attractors A ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq54_HTML.gif. □

We can briefly describe the structure of the uniform attractor as follows: if the functions f 0 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq60_HTML.gif and f 1 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq61_HTML.gif are translational bounded, problem (1.1)-(1.3) generates the dynamical processes { U ε ( t , τ ) , t τ , τ R } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq62_HTML.gif acting on V which is defined by U ε ( t , τ ) u τ ε = u ε ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq63_HTML.gif, t τ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq32_HTML.gif, where u ε ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq64_HTML.gif is the solution to (1.1)-(1.3). The processes { U ε ( t , τ ) , t τ , τ R } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq62_HTML.gif have a uniformly (w.r.t. t R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq65_HTML.gif) absorbing set
B ε : = { u ε V | u ε V C Q ε } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ23_HTML.gif
(3.1)

which is bounded in V for any fixed ε ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq1_HTML.gif.

On the other hand, A ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq54_HTML.gif is also bounded in V for each fixed ε since A ε B 1 ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq66_HTML.gif. Assuming f 0 , f 1 L tc 2 ( R , H ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq67_HTML.gif, the external force f ε ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq68_HTML.gif appearing in equation (1.1) belongs to L tc 2 ( R , H ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq69_HTML.gif also. Moreover, if ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq12_HTML.gif and f ˆ ε H ( f ε ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq70_HTML.gif, then
f ˆ ε ( t ) = f ˆ 0 ( t ) + ε ρ f ˆ 1 ( t ε ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ24_HTML.gif
(3.2)
for some f 0 ˆ H ( f 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq71_HTML.gif and f 1 ˆ H ( f 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq72_HTML.gif. In this case, to describe the structure of the uniform attractor A ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq54_HTML.gif, we consider the family of equations
u ˆ t + A u ˆ t + ν A u ˆ + F ( u ˆ ) = f ˆ ε ( t ) , f ˆ ε H ( f ε ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ25_HTML.gif
(3.3)
For every external force f ˆ ε H ( f ε ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq73_HTML.gif, equation (3.3) generates a class of processes { U f ˆ ε ( t , τ ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq74_HTML.gif on V, which shares similar properties to those of the processes { U f ε ( t , τ ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq75_HTML.gif, corresponding to the original equation (1.1) with the external force f ε ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq68_HTML.gif. Moreover, the map
( u τ , f ˆ ε ) U f ˆ ε ( t , τ ) u τ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ26_HTML.gif
(3.4)

is ( V × H ( f ε ) , V ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq76_HTML.gif-continuous.

Lemma 3.2 If the function f 0 ( t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq77_HTML.gif in (1.4) is taken from the space L b 2 ( R , H ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq15_HTML.gif of translational bounded functions in L loc 2 ( R , H ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq16_HTML.gif, then the processes { U f 0 ( t , τ ) , t τ , τ R } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq78_HTML.gif generated by system (1.4)-(1.6) have a uniformly (w.r.t. σ = f 0 Σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq79_HTML.gif) compact attractor A 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq80_HTML.gif.

Proof Use a similar technique as that in Theorem 2.4, we can easily deduce the existence of a uniformly compact attractor A 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq80_HTML.gif if we choose g ( t , x ) = f 0 ( t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq81_HTML.gif. □

4 Uniform boundedness of A ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq54_HTML.gif

Firstly, we shall consider the auxiliary linear equation with a non-autonomous external force and give some useful lemmas, and then we shall prove the uniform boundedness of  A ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq54_HTML.gif.

Considering the linear equation
Y t + A Y t + ν A Y = K ( t ) , Y | t = τ = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ27_HTML.gif
(4.1)

we get the following lemma.

Lemma 4.1 Assume that K L loc 2 ( R , H ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq82_HTML.gif, then problem (4.1) has a unique solution
Y L 2 ( ( τ , T ) ; W ) C ( ( τ , T ) ; V ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ28_HTML.gif
(4.2)
t Y L 2 ( ( τ , T ) ; W ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ29_HTML.gif
(4.3)
Moreover, the following inequalities
Y ( t ) W 2 C τ t e C ν ( t s ) K ( s ) H 2 d s , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ30_HTML.gif
(4.4)
t t + 1 Y ( s ) V 2 d s C ( Y ( t ) V 2 + t t + 1 K ( s ) H 2 d s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ31_HTML.gif
(4.5)

hold for every t τ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq32_HTML.gif and some constant C > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq83_HTML.gif, independent of the initial time τ R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq29_HTML.gif.

Proof Firstly, using the Galerkin approximation method, we can deduce the existence of a global solution for (4.1), here we omit the details.

Then multiplying (4.1) by Y and AY respectively, we get
1 2 d d t ( Y 2 + Y 2 ) + ν Y 2 = ( K ( t ) , Y ) 2 ν K ( t ) 2 + ν 2 Y 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ32_HTML.gif
(4.6)
and
1 2 d d t ( Y 2 + A Y 2 ) + ν A Y 2 = ( K ( t ) , A Y ) 2 ν K ( t ) 2 + ν 2 A Y 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ33_HTML.gif
(4.7)

By the Gronwall inequality and Poincaré inequality, we can easily prove the lemma. □

Setting K ( t , τ ) = τ t k ( s ) d s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq84_HTML.gif, t τ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq32_HTML.gif, τ R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq29_HTML.gif, we have the following lemma.

Lemma 4.2 Assume that the formula
sup t τ , τ R { K ( t , τ ) H 2 + t t + 1 K ( s , τ ) H 2 d s } l 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ34_HTML.gif
(4.8)
holds for some constant l 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq85_HTML.gif, let k L loc 2 ( R , H ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq86_HTML.gif. Then the solution y ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq87_HTML.gif yields the following problem:
y t + A y t + ν A y = k ( t / ε ) , y | t = τ = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ35_HTML.gif
(4.9)
with ε ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq1_HTML.gif satisfying the inequality
y ( t ) V 2 + t t + 1 y ( s ) V 2 d s C l 2 ε 2 , t τ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ36_HTML.gif
(4.10)

where C > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq83_HTML.gif is constant independent of K.

Moreover, we also have
t t + 1 K ε ( s ) H 2 d s C . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ37_HTML.gif
(4.11)
Proof Noting that
K ε ( t ) = τ t k ( s / ε ) d s = ε τ / ε t / ε k ( s ) d s = ε K ( t / ε , τ / ε ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ38_HTML.gif
(4.12)
we can derive the following estimates from (4.8):
sup t τ K ε ( t ) H l ε , t t + 1 K ε ( s ) H 2 d s = ε 2 t t + 1 K ( s / ε , τ / ε ) H 2 d s t t + 1 K ε ( s ) H 2 d s C ε 2 sup t τ { t t + 1 K ( s , τ ) H 2 d s } C l 2 ε 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equb_HTML.gif
From Lemma 2.1, we have
τ t e C ν ( t s ) K ε ( s ) H 2 d s t 1 t e C ν ( s t ) K ε ( s ) 2 d s + t 2 t 1 e C ν ( s t ) K ε ( s ) 2 d s + t 1 t K ε ( s ) 2 d s + e C ν t 2 t 1 K ε ( s ) 2 d s + e 2 C ν t 3 t 2 K ε ( s ) 2 d s + ( 1 + e C ν + e 2 C ν + ) K ε ( s ) L b 2 ( R ; H ) 2 1 ( 1 e C ν ) K ε ( s ) L b 2 ( R ; H ) 2 1 ( 1 e C ν ) sup t τ t t + 1 K ε ( s ) H 2 d s C l 2 ε 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ39_HTML.gif
(4.13)
Hence, from the Poincaré inequality, combining (4.12) and (4.4)-(4.5), we conclude that
Y ( t ) W 2 C l 2 ε 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ40_HTML.gif
(4.14)
t t + 1 Y ( s ) V 2 d s C ( Y ( t ) V 2 + t t + 1 K ( s ) H 2 d s ) C l 2 ε 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ41_HTML.gif
(4.15)
Setting
Y ( t ) = τ t y ( s ) d s , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ42_HTML.gif
(4.16)
we deduce that for any t τ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq32_HTML.gif,
t Y ( t ) = y ( t ) = τ t t y ( s ) d s , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ43_HTML.gif
(4.17)

since y ( τ ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq88_HTML.gif.

Integrating (4.9) with respect to time variable from τ to t, we see that Y ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq89_HTML.gif is a solution to the problem
t Y ( t ) + t ( A Y ( t ) ) + ν A Y ( t ) = K ε ( t ) , q Y ( t ) | t = τ = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ44_HTML.gif
(4.18)
such that from (4.13) and (4.14), we can derive
Y ( t ) H 2 + Y ( t ) H 2 + t t + 1 Y ( s ) V 2 d s C l 2 ε 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ45_HTML.gif
(4.19)
By virtue of y ( t ) = t Y ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq90_HTML.gif, ( A Y ( t ) , Y ( t ) ) Y ( t ) V 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq91_HTML.gif, A Y ( t ) Y ( t ) W https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq92_HTML.gif, we have
t Y ( t ) + t A Y ( t ) = y ( t ) + A y ( t ) ν Y ( t ) W + K ε ( t ) C l ε . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ46_HTML.gif
(4.20)
Hence, we conclude
y ( t ) V C ( y ( t ) + A y ( t ) ) C ( ν Y ( t ) W + K ε ( t ) ) C l ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ47_HTML.gif
(4.21)
and
t t + 1 y ( s ) V 2 d s C l 2 ε 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ48_HTML.gif
(4.22)

The proof is finished. □

Now, we shall use the auxiliary linear equation and some estimates to prove the uniform boundedness of A ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq54_HTML.gif in V. For convenience, we set
F 1 ( t , τ ) = τ t f 1 ( s ) d s , t τ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ49_HTML.gif
(4.23)
and assume that
sup t τ , τ R { F 1 ( t , τ ) 2 + t t + 1 F 1 ( s , τ ) H 2 d s } l 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ50_HTML.gif
(4.24)

holds for some constants l 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq85_HTML.gif.

Theorem 4.3 The attractors A ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq54_HTML.gif of problem (1.1)-(1.3) (or (1.4)-(1.6)) are uniformly (w.r.t. ε) bounded in V, namely,
sup ε [ 0 , 1 ) A ε V < + . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ51_HTML.gif
(4.25)
Proof Let u ε ( t ) = U ε ( t , τ ) u τ ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq93_HTML.gif be the solution to (1.1)-(1.3) with the initial data u τ ε V https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq94_HTML.gif. For ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq12_HTML.gif, we consider the auxiliary linear equation
v t + A v t + ν A v = ε ρ f 1 ( t / ε ) , v | t = τ = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ52_HTML.gif
(4.26)
From Lemma 4.2, we have the estimate
v ( t ) V 2 + t t + 1 v ( s ) V 2 d s C l 2 ε 2 ( 1 ρ ) , t τ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ53_HTML.gif
(4.27)
Setting the function w ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq95_HTML.gif as
w ( t ) = u ( t ) v ( t ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ54_HTML.gif
(4.28)
which satisfies the problem
w t + A w t + ν A w + F ( w + v ) = f 0 , w | t = τ = u τ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ55_HTML.gif
(4.29)
Taking the scalar product of (4.28) with w, we obtain
1 2 d d t ( w 2 + w 2 ) + ν w 2 + ( F ( w + v ) , w ) = ( f 0 , w ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ56_HTML.gif
(4.30)
Using the inequality
v ( t ) 2 = v ( t ) H 2 C v ( t ) V 2 C l 2 ε 2 ( 1 ρ ) , t τ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ57_HTML.gif
(4.31)
we have
( ( F ( w + v ) ) , w ) C 3 ( 1 + w 2 + v 2 ) + ν 4 λ w 2 C 3 ( 1 + w 2 + l 2 ε 2 ( 1 ρ ) ) + ν 4 λ w 2 C 4 ( 1 + w 2 + l 2 ε 2 ( 1 ρ ) ) + ν 4 λ w 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ58_HTML.gif
(4.32)

where λ is the first eigenvalue of −Δ.

Moreover, notice that
( f 0 , w ) ν 4 w V 2 + 4 ν f 0 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ59_HTML.gif
(4.33)
and inserting (4.29)-(4.30) into (4.28), we have
1 2 d d t ( w 2 + w 2 ) + ν w 2 C 4 ( 1 + w 2 + l 2 ε 2 ( 1 ρ ) ) + ν 4 λ w 2 + ν 4 w V 2 + 4 ν f 0 2 C 4 ( 1 + w 2 + l 2 ε 2 ( 1 ρ ) ) + ν 4 w V 2 + ν 4 w V 2 + 4 ν f 0 2 = C 4 ( 1 + w 2 + l 2 ε 2 ( 1 ρ ) ) + ν 2 w V 2 + 4 ν f 0 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ60_HTML.gif
(4.34)
which implies that
d d t ( w 2 + w V 2 ) + ϕ 1 w V 2 ϕ 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ61_HTML.gif
(4.35)
where
ϕ 1 ( t ) 2 [ ν 2 C 5 ( 1 + u 2 + l 2 ε 2 ( 1 ρ ) ) ] , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ62_HTML.gif
(4.36)
ϕ 2 ( t ) 8 ν f 0 ( t ) 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ63_HTML.gif
(4.37)
Therefore using (1.8), we derive from (4.33)-(4.36) that for any t τ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq32_HTML.gif,
τ t ϕ 1 ( s ) d s ν 2 ( t τ ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ64_HTML.gif
(4.38)
t t + 1 ϕ 2 ( s ) d s C M 0 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ65_HTML.gif
(4.39)
Applying Lemma 2.2 with ζ ( t ) = w 2 + w V 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq96_HTML.gif, β = ν 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq97_HTML.gif, γ = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq98_HTML.gif, M = C M 0 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq99_HTML.gif, we have
w 2 + w V 2 C e β ( t τ ) ( u τ 2 + u τ V 2 ) + C M 0 2 , t τ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ66_HTML.gif
(4.40)
which gives
w V 2 C e β ( t τ ) ( u τ 2 + u τ V 2 ) + C M 0 2 , t τ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ67_HTML.gif
(4.41)
Recalling that u = w + v https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq100_HTML.gif, and using (4.25) and (4.37), we end up with
u ( t ) V 2 w V 2 + v V 2 C e β ( t τ ) ( u τ 2 + u τ V 2 ) + C ( l 2 + M 0 2 ) , t τ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ68_HTML.gif
(4.42)
Thus, for every 0 < ε ε 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq101_HTML.gif, the processes { U ε ( t , τ ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq102_HTML.gif have an absorbing set
B 0 : = { u V | u V 2 2 C ( l 2 + M 0 2 ) } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ69_HTML.gif
(4.43)
On the other hand, if ε 0 < ε < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq103_HTML.gif, the processes { U ε ( t , τ ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq102_HTML.gif also possess an absorbing set
B ε 0 = { u V | u V C Q ε 0 } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ70_HTML.gif
(4.44)
In conclusion, for every ε 0 [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq104_HTML.gif, the set
B : = B 0 B ε 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ71_HTML.gif
(4.45)

is an absorbing set for { U ε ( t , τ ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq102_HTML.gif which is independent of ε. Since A ε B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq105_HTML.gif, (4.24) follows and hence the proof is complete. □

5 Convergence of A ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq54_HTML.gif to A 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq106_HTML.gif

The main result of the paper reads as follows.

Theorem 5.1 Assume that f 0 , f 1 L tc 2 ( R , H ) L b 2 ( R , H ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq107_HTML.gif and (4.23) holds. Then the uniform attractor A ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq54_HTML.gif (for problem (1.1)-(1.3)) converges to A 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq80_HTML.gif (for problem (1.4)-(1.6)) as ε 0 + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq108_HTML.gif in the following sense:
lim ε 0 + dist V ( A ε , A 0 ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ72_HTML.gif
(5.1)
Next, we shall study the difference of two solutions for (1.1) with ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq12_HTML.gif and (1.4) with ε = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq2_HTML.gif which share the same initial data. Denote
u ε ( t ) : = U ε ( t , τ ) u τ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ73_HTML.gif
(5.2)
with u τ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq109_HTML.gif belonging to the absorbing set B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq110_HTML.gif which can be found in Section 4. In particular, since u τ B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq111_HTML.gif, the formula corresponding to ε = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq2_HTML.gif
u 0 ( t ) V 2 + t t + 1 u 0 ( s ) V 2 d s R 0 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ74_HTML.gif
(5.3)

holds for some R 0 = R 0 ( ρ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq112_HTML.gif, as the size of B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq113_HTML.gif depends on ρ.

Lemma 5.2 For every ε ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq1_HTML.gif, τ R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq29_HTML.gif, u τ B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq111_HTML.gif and u ε ( 0 ) = u 0 ( 0 ) = u τ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq114_HTML.gif, the difference
w ( t ) = u ε ( t ) u 0 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ75_HTML.gif
(5.4)
satisfies the estimate
w ( t ) V D ε 1 ρ e R ( t τ ) , t τ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ76_HTML.gif
(5.5)

for some positive constants D = D ( ρ , l ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq115_HTML.gif and R = R ( ρ , l ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq116_HTML.gif, both independent of ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq12_HTML.gif.

Proof Since the difference w ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq95_HTML.gif solves the equation
w t + A w t + ν A w + ( F ( u ε ) F ( u 0 ) ) = ε ρ f 1 ( ε / t ) , w | t = τ = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ77_HTML.gif
(5.6)
the difference
q ( t ) = w ( t ) v ( t ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ78_HTML.gif
(5.7)
fulfills the Cauchy problem
q t + A q t + ν A q + ( F ( u ε ) F ( u 0 ) ) = 0 , q | t = τ = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ79_HTML.gif
(5.8)

where v ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq117_HTML.gif is the solution to (4.25).

Taking an inner product of equation (5.8) with q in H, we obtain
1 2 d d t ( q 2 + q 2 ) + ν q 2 + ( ( F ( u ε ) F ( u 0 ) ) , q ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ80_HTML.gif
(5.9)
Noting that
( ( F ( u ε ) F ( u 0 ) ) , q ) sup i ( F i ( u ε ) + F i ( u 0 ) ) 2 u ε u 0 2 + ν 4 λ q 2 C 3 ( 1 + u ε 2 + u 0 2 ) w 2 + ν 4 λ q 2 C 3 ( 1 + u ε 2 + R 0 2 ) w 2 + ν 4 λ q 2 C 4 ( 1 + u ε 2 + R 0 2 ) q + v V 2 + ν 4 λ q 2 C 5 ( 1 + K 0 2 + R 0 2 ) v V 2 + ν 2 q V 2 + ν 4 λ q 2 = f ( t ) + ν 2 q V 2 + h ( t ) q 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ81_HTML.gif
(5.10)
where λ is the first eigenvalue of −Δ, K 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq118_HTML.gif is the upper bound for u ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq119_HTML.gif (by Lemma 3.1) and
h ( t ) = ν 4 λ , f ( t ) = C 5 ( 1 + K 0 2 + R 0 2 ) v V 2 C ( 1 + K 0 2 + R 0 2 ) l 2 ε 2 ( 1 ρ ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equc_HTML.gif
thus, it follows from (5.9) and (5.10) that
1 2 d d t ( q 2 + q 2 ) + ν 2 q V 2 C h ( t ) q 2 + f ( t ) C h ( t ) ( q 2 + q 2 ) + f ( t ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ82_HTML.gif
(5.11)
Noting that q ( τ ) = q ( τ ) V = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq120_HTML.gif, by the Gronwall inequality, we get
q 2 + q 2 2 exp { 2 C τ t h ( s ) d s } τ t f ( s ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ83_HTML.gif
(5.12)
Moreover, we can derive the following formulas:
τ t h ( s ) d s ν 4 λ ( t τ + 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ84_HTML.gif
(5.13)
and
τ t f ( s ) d s = τ t [ C ( 1 + K 0 2 + R 0 2 ) v V 2 ] d s τ t [ C ( 1 + K 0 2 + R 0 2 ) l 2 ε 2 ( 1 ρ ) ] d s = [ C ( 1 + K 0 2 + R 0 2 ) l 2 ε 2 ( 1 ρ ) ] ( t τ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ85_HTML.gif
(5.14)
Consequently,
q ( t ) V 2 C ( q 2 + q 2 ) C [ ( 1 + K 0 2 + R 0 2 ) l 2 ε 2 ( 1 ρ ) ] ( t τ + 1 ) e ν 4 λ ( t τ + 1 ) C D 1 2 ε 2 ( 1 ρ ) e ν 4 λ ( t τ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ86_HTML.gif
(5.15)
holds for some positive constants D 1 = D 1 ( ρ , l ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq121_HTML.gif. Finally, since w = q + v https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq122_HTML.gif, using (4.26) to control v V https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq123_HTML.gif, we may obtain
w ( t ) V 2 C ( q V 2 + v V 2 ) C D 1 2 ε 2 ( 1 ρ ) e ν 4 λ ( t τ ) + C l 2 ε 2 ( 1 ρ ) D 2 ε 2 ( 1 ρ ) e 2 R ( t τ ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ87_HTML.gif
(5.16)

where R is a positive constant. The proof is finished. □

Next, we want to generalize Lemma 5.2 to derive the convergence of corresponding uniform attractors. Let the external force in equation (3.3) as f ˆ = f ˆ ε H ( f ε ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq124_HTML.gif, then f ˆ 1 H ( f 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq125_HTML.gif satisfies inequality (5.22).

Define
G ˆ 1 ( t , τ ) = τ t f ˆ 1 ( s ) d s , t τ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ88_HTML.gif
(5.17)
we have
sup t τ , τ R { G ˆ 1 ( t , τ ) H 2 + t t + 1 G ˆ ( s , τ ) H 2 d s } l 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ89_HTML.gif
(5.18)
For any ε [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq126_HTML.gif, we observe that u ˆ ε ( t ) = U f ˆ ε ( t , τ ) y τ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq127_HTML.gif is a solution to (3.3) with the external force f ˆ ε = f ˆ 0 + ε ρ f 1 ˆ ( / ε ) H ( f ε ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq128_HTML.gif and y τ ( f ε ) B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq129_HTML.gif. For ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq12_HTML.gif, we investigate the property of the difference
w ˆ ( t ) = u ˆ ε ( t ) u ˆ 0 ( t ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ90_HTML.gif
(5.19)
Lemma 5.3 The inequality
w ˆ ( t ) D ε 1 ρ e R ( t τ ) , t τ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ91_HTML.gif
(5.20)

holds, here D and R are defined in Lemma  5.2.

Proof As the similar discussion in the proof of Lemma 5.2, replacing u ˆ ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq130_HTML.gif, f ˆ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq131_HTML.gif and f ˆ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq132_HTML.gif by u ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq119_HTML.gif, f 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq56_HTML.gif and f 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq57_HTML.gif, respectively, noting that (5.1) still holds for u ˆ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq133_HTML.gif, and the family { U f ˆ ε ( t , τ ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq74_HTML.gif ( f ˆ ε H ( f ε ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq73_HTML.gif), is ( H × H ε ( f ε ) , H ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq134_HTML.gif-continuous, using (5.18) in place of (4.23), we can finally complete the proof of the lemma. □

Proof of Theorem 5.1 For ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq12_HTML.gif, u ε A ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq135_HTML.gif, we obtain that there exists a complete bounded trajectory u ˆ ε ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq136_HTML.gif of equation (3.3), with some external force
f ˆ ε = f ˆ 0 + ε ρ f ˆ 1 ( / ε ) H ( f ε ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ92_HTML.gif
(5.21)

such that u ˆ ε ( 0 ) = u ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq137_HTML.gif.

We choose L 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq138_HTML.gif such that
u ˆ ε ( L ) A ε B . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ93_HTML.gif
(5.22)
From the equality
u ε = U f ˆ 0 ( 0 , L ) u ˆ ε ( L ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ94_HTML.gif
(5.23)
applying Lemma 5.3 with t = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq139_HTML.gif, τ = L https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq140_HTML.gif, we obtain
u ε U f ˆ 0 ( 0 , L ) u ˆ ε ( L ) V D ε 1 ρ e R L . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ95_HTML.gif
(5.24)
On the other hand, the set A 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq80_HTML.gif attracts all sets U f ˆ 0 ( t , L ) B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq141_HTML.gif uniformly when f ˆ 0 H ( f 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq142_HTML.gif. Then, for all δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq143_HTML.gif, there exists some time T = T ( δ ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq144_HTML.gif which is independent of L such that
dist V ( U f ˆ 0 ( T L , L ) u ˆ ε ( L ) , A 0 ) δ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ96_HTML.gif
(5.25)
Choosing L = T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq145_HTML.gif and collecting (5.15)-(5.16), we readily get
dist V ( u ε , A 0 ) u ε U f ˆ 0 ( 0 , T ) u ˆ ε ( T ) V + dist V ( U f ˆ 0 ( 0 , T ) u ˆ ε ( T ) , A 0 ) D ε 1 ρ e R T + δ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_Equ97_HTML.gif
(5.26)

Since u ε A ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq135_HTML.gif and δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq143_HTML.gif is arbitrary, taking the limit ε 0 + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-111/MediaObjects/13661_2013_Article_362_IEq3_HTML.gif, we can prove the theorem. □

Declarations

Acknowledgements

All authors give their thanks to the reviewer’s suggestions, XY was in part supported by the Innovational Scientists and Technicians Troop Construction Projects of Henan Province (No. 114200510011) and the Young Teacher Research Fund of Henan Normal University (qd12104).

Authors’ Affiliations

(1)
College of Mathematics and Information Science, Pingdingshan University
(2)
College of Mathematics and Information Science, Henan Normal University
(3)
College of Education and Teacher Development, Henan Normal University

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