Polynomial stability for wave equations with acoustic boundary conditions and boundary memory damping

Applied Mathematics and Computation 321 (2018) 593–601

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Polynomial stability for wave equations with acoustic boundary conditions and boundary memory dampingR Chan Li a, Jin Liang b, Ti-Jun Xiao c,∗ a

School of Sciences, Hangzhou Dianzi University, Zhejiang, 310018, China Department of Applied Mathematics, Shanghai Jiao Tong University, Shanghai, 200240, China c Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical Sciences, Fudan University, Shanghai, 200433, China b

a r t i c l e

i n f o

a b s t r a c t

MSC: 35L20 35B35 35B40 93D15 93C20 93D20

We study wave equations with acoustic boundary conditions, where only one memory damping acts on the acoustic boundary. Under some conditions on the memory kernel, polynomial energy decay rates are established by using higher-order energy estimates among some other techniques. © 2017 Elsevier Inc. All rights reserved.

Keywords: Acoustic boundary conditions Memory damping Wave equations Polynomial stability

1. Introduction In this paper, we investigate the stability of solutions of the following linear wave equation subject to acoustic boundary condition on one part of the boundary:

⎧ u − u = 0 , tt ⎪ ⎪ ⎪ ⎪ u = 0, ⎪ ⎪  t ⎪ ⎪ ⎨ ut − g(t − s )z(s )ds + z + ztt = 0, 0 ⎪ ⎪ ∂ν u = zt , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ u(0 ) = u0 , ut (0 ) = u1 , z(0 ) = z0 , zt (0 ) = z1 ,

in  × (0, T ), on 0 × (0, T ), on 1 × (0, T ),

(1.1)

on 1 × (0, T ), in , on 1 .

Here  is a bounded domain in with smooth boundary  = 0 ∪ 1 ,  0 ,  1 being closed, nonempty and disjoint subsets of  ; ν is outer-normal vector at x ∈  ; The function u represents the velocity potential of the fluid, and z(t, x) the normal Rn

R The work was supported partly by the NSF of China (11371095, 11571229) and the Shanghai Key Laboratory for Contemporary Applied Mathematics (08DZ2271900). ∗ Corresponding author. E-mail addresses: [email protected], [email protected] (T.-J. Xiao).

https://doi.org/10.1016/j.amc.2017.11.019 0 096-30 03/© 2017 Elsevier Inc. All rights reserved.

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C. Li et al. / Applied Mathematics and Computation 321 (2018) 593–601

displacement of the point x ∈  1 at time t; ∂ν u = zt means that the outward normal velocity at x ∈  1 equals the boundary t motion velocity; − 0 g(t − s )z(s )ds will produce a damping effect and make the system stable. The system models sound wave propagation in a domain with a portion of boundary made of viscoelastic material. Acoustic model was proposed by Morse and Ingard [12] in order to fully explain the propagation of sound waves in a concert hall, but in a nonstandard way. Beale and Rosencrans [5] improved it in a rigorous mathematical context, and Beale [3,4] analysed the model in both bounded and exterior domains, where it is assumed that each boundary point acts as a spring and does not influence each other. The study of various systems with acoustic boundary conditions has attracted a lot of interest (cf., e.g., [1,2,6–10,13–16] and references therein). For linear wave equations with the acoustic boundary condition:

 ut −

t 0

g(t − s )zt (s )ds + z = 0,

∂ν u = zt , on 1 × (0, T ),

it is shown in [10] that the associated operator matrix generates a strongly continuous semigroup of contractions on a Hilbert space, and the semigroup is strongly stable [10] (without giving an energy decay rate). In the present paper, we are devoted to establishing energy decay rates for system (1.1), and will prove that the energy is polynomially stable under suitable conditions on the memory kernel g (Theorem 3.1). To our knowledge, there has been no work about the decay rates of acoustic wave energies when only one memory damping acting on the acoustic boundary is employed to stabilize the whole system. Uniform energy decay rates were studied in [6,15] for acoustic wave systems with both internal and boundary memory dampings (concerning the velocity potential u, rather than the normal displacements z of boundary points). t Also, polynomial stability was proved in [14] for (1.1) with the memory damping − 0 g(t − s )z(s )ds replaced by a frictional damping zt . In the case of memory case, the kernel g will affect energy decay rates, and the handling of the system is quite different from the case of frictional damping. 2. Preliminaries The following assumptions and notations will be used throughout the paper. Assumptions. (A-1) the geometric condition: there exists x0 ∈ Rn and a positive constant δ such that

( x − x0 ) · ν ( x ) ≤ 0,

( x − x0 ) · ν ≥ δ > 0,

x ∈ 0 ,

x ∈ 1 .

A typical example is

 = {x : 1 < |x| < 2}, 0 = ∂ B1 (0 ), 1 = ∂ B2 (0 ), x0 = 0. (A-2) g ∈ C 5 (R+ → R+ ) is a decreasing function satisfying



g( 0 ) > 0 ,

∞

l := 0

g(s )ds < 1,

meas{s ≥ 0; g (s ) = 0} = 0,

(2.1)

and

|g(i) (s )| ≤ c0 g(s ) (i = 1, . . . , 4 ),

|g (s )| + |g(5) (s )| ≤ −c1 g (s ) for s > 0.

Typical examples are

g(s ) = le−s ,

or g(s ) =

lα , (α > 0 ) (1 + s )α+1

satisfying the assumption (A-2). Notations: By · ,·  we denote the inner product on space L2 (), ·, ·1 the inner product on space L2 ( 1 ), || · || the norm, and || · ||1 the L2 ( 1 ) norm;

L2 ()

R = max |x − x0 |; ¯ x∈

Write

H (t ) =



t 0

g(s )ds,

H1 0 () = {u ∈ H 1 (); u|0 = 0}.

Kσ (s ) =

−g (s ) + σ, g( s )

 Gσ =

0

∞

g( s ) ds Kσ (s )

for σ ∈ (0, 1). More information about the auxiliary functions H(t), Kσ , Gσ can be found in [11]. We will use c, C to denote generic positive constants. Wellposedness: By the semigroup theory, it is not difficult to obtain the existence and uniqueness theorem.

C. Li et al. / Applied Mathematics and Computation 321 (2018) 593–601

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Theorem 2.1. For (u0 , u1 , z0 , z1 ) ∈ H 2 () ∩ H1 () × H1 () × L2 (1 ) × L2 (1 ), the system (1.1) has a unique solution (u, z) 0 0 with





u ∈ L2 0, T ; H 2 () ∩ H1 0 () ,









z ∈ L2 0, T ; L2 (1 ) ,



ut ∈ L2 0, T ; H1 0 () ,



zt ∈ L2 0, T ; L2 (1 ) ,





ztt ∈ L2 0, T ; L2 (1 ) .

Define the energy functional of solution (u, z) by

E (t ) =

1 1 ||ut ||2 + ||∇ u||2 + ||z||21 + ||zt ||21 . 2 2

For every solution of (1.1), the following identity holds

E (t2 ) − E (t1 ) =



t2

t1





t 0

g(t − s )z(s )ds, zt 1 dt.

3. Main results Theorem 3.1. Suppose that the assumptions (A-1)-(A-2) are satisfied. Then for any initial data (u0 , u1 , z0 , z1 ) ∈ H 3 () ∩ H1 () × H 2 () ∩ H1 () × L2 (1 ) × L2 (1 ), the energy of the solution of the system (1.1) decays polynomially. That is 0

0

C E (t ) ≤ , t +1

for t ≥ 0,

where C > 0 is a constant depending on ||u0 ||H 3 () and ||u1 ||H 2 () . Before giving the proof of Theorem 3.1, we make some preparations. Define a modified energy E (t ) by

E (t ) = E (t ) +

  t  1 t 1 g(t − s )|z(s ) − z(t )|2 dsdγ − g(s )ds||z(t )||21 . 2 1 0 2 0

Then

(1 − l )E (t ) ≤ E (t ).

(3.1)

A simple calculation shows that

=



t 0



g(t − s )z(s )ds, zt 1



t 0

g(t − s )(z(s ) − z(t ))ds + 



 0

t

g(t − s )z(t )ds, zt 1

  t t 1 d 1 g(t − s )|z(s ) − z(t )|2 dsdγ + g (t − s )|z(s ) − z(t )|2 dsdγ 2 dt 1 0 2 1 0

 t  1 1 d + g(s )ds||z(t )||21 − g(t )||z(t )||21 . 2 dt 2 0

=−

Hence, we see that the modified energy is nonincreasing:

E (t ) =

1 2

  1

t 0

g (t − s )|z(s ) − z(t )|2 dsdγ − g(t )||z(t )||21

≤ 0.

(3.2)

For simplicity, we denote

m ( x ) = x − x0 ,

φ (t ) = ut , u , ψ (t ) = ut , m · ∇ u , (t ) = u, z1 + zt , z1 .

Next we give a lemma about φ (t), ψ (t) and (t). Lemma 3.2. There exist positive constants c0 ,  , α , β such that

φ (t ) ≤(c0  − 1 )||∇ u||2 + ψ (t ) ≤



1 ||zt ||21 + ||ut ||2 , 4



R2 n n 3 − 1 ||∇ u||2 − ||ut ||2 + RGσ ||zt ||21 + 2δ 2 2 2 2 3  3R 2 2 + R H (t ) − 1 ||z(t )||1 + ||ztt ||1 , 2 2

(3.3) 

t 0

Kσ (t − s )g(t − s )||z(s ) − z(t )||21 ds (3.4)

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C. Li et al. / Applied Mathematics and Computation 321 (2018) 593–601

(t ) ≤

c0 α 1 G ||∇ u||2 + ||zt ||21 + σ 2 2α 2β



β

+ H (t ) +

2

−1

||z(t )||21 .



t 0

Kσ (t − s )g(t − s )||z(s ) − z(t )||21 ds (3.5)

Proof. Since  0 = ∅, by the Poincaré inequality and trace theorem, we have

||u||21 ≤ c0 ||∇ u||2 , where c0 is the Poincaré coefficient. Then, for φ (t), we get

φ (t ) = utt , u + ||ut ||2 ≤ (c0  − 1 )||∇ u||2 + For ψ (t), under the assumption (A-2), we have

ψ (t ) = utt , m · ∇ u +



m·∇

1 ||zt ||21 + ||ut ||2 . 4

|ut |2

dx 2   n n |∇ u|2 |ut |2 = ∂ν u, m · ∇ u − ||∇ u||2 − m·ν dγ + ||∇ u||2 + m·ν dγ − ||ut ||2 2 2 2 2  1  

n  2 u |∇ u|2 n ≤ ∂ν u, m · ∇ u1 − m·ν dγ + − 1 ||∇ u||2 + m · ν t dγ − ||ut ||2 2 2 2 2 1 1 ≤

R2 ||zt ||21 + 2δ

n 2



−1

 n R ||∇ u||2 − ||ut ||2 + ||ut ||21 . 2

(3.6)

2

It is easy to see that

 t   g(t − s )z(s )ds − z(t ) − ztt (t )2 dγ 1 0 2   t  t = g(t − s )(z(s ) − z(t ))ds + g(t − s )dsz(t ) − z(t ) − ztt (t ) dγ

||ut ||21 =



1

≤ 3||

 0

0

t

0



g(t − s )(z(s ) − z(t ))ds||21 + 3 H (t ) − 1

2

||z(t )||21 + 3||ztt ||21 .

(3.7)

Now we estimate the first term on the right hand of (3.7). By Hölder’s inequality, we have

|| =



t 0

|| 

g(t − s )(z(s ) − z(t ))ds||21



t

g(t − s ) 2

1

Kσ (t − s ) 2 g(t − s ) 2 (z(s ) − z(t ))ds||21 1

Kσ (t − s ) 2

1

0

1

  t g(t − s ) ds Kσ (t − s )g(t − s )(z(s ) − z(t ))2 dsdγ 0 Kσ (t − s ) 1 0  t ≤ Gσ Kσ (t − s )g(t − s )||z(s ) − z(t )||21 ds. ≤

t

(3.8)

0

Substituting (3.8) into (3.7) and combining this with (3.6), we obtain

ψ (t ) ≤





R2 n n 3 − 1 ||∇ u||2 − ||ut ||2 + RGσ ||zt ||21 + 2δ 2 2 2 2 3  3R 2 2 + R H (t ) − 1 ||z(t )||1 + ||ztt ||1 . 2 2



t 0

Kσ (t − s )g(t − s )||z(s ) − z(t )||21 ds

For (t), observe that

(t ) = u, zt 1 + ut , z1 + ||zt ||21 + ztt , z1  t   = u, zt 1 + g(t − s ) z(s ) − z(t ) ds, z(t )1 + H (t ) − 1 ||z||21 + ||zt ||21 0

≤ c0 +

α 2

β 2

||∇ u|| + 2

1 1 ||zt ||21 + || 2α 2β







t 0



||z(t )||1 + H (t ) − 1 ||z(t )||1 + ||zt ||21 . 2

2



g(t − s ) z(s ) − z(t ) ds||21

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597

Then, by (3.8) we obtain

c0 α 1 G ||∇ u||2 + ||zt ||21 + σ 2 2α 2β

(t ) ≤



+ H (t ) +

β

−1

2



t 0

Kσ (t − s )g(t − s )||z(s ) − z(t )||21 ds

||z(t )||21 + ||zt ||21 .

 ∞

Denoting h(t ) =

R(t ) :=



t 0

t

g(s )ds, we define another auxilary function R(t) by

h(t − s )||z(s ) − z(t )||21 ds + 2



t

0

h(t − s )(z(s ) − z(t ))ds, z(t )1 +

 0

t

h(t − s )||z(t )||21 ds.

For this function, we will use Lemma 4.1 in [11]. For reader’s convenience, we restate it here with a small change. Lemma 3.3. R(t) is a positive function and satisfies

R (t ) ≤ −

1 2



t

g(t − s )||z(s ) − z(t )||21 ds + 2l ||z(t )||21 .

0



(3.9)



Since g (s ) = − Kσ (s ) + σ g(s ), we have

E (t ) =

1 2



t



1 σ − Kσ (t − s ) g(t − s )||z(t ) − z(s )||21 ds − g(t )||z||21 .

(3.10)

2

0

Define a Lyapunov function F(t) by

F (t ) = N0 E (t ) + N1 φ (t ) + N2 ψ (t ) + N3 (t ) + N4 R(t ). Then, collecting (3.3), (3.4), (3.5), (3.9) and (3.10), we deduce that

F (t ) ≤

σ N0 − N4 2

+ +

3RG

2

t 0

σ

2

3R



g(t − s )||z(s ) − z(t )||21 ds +

N2 +

N0 Gσ N3 − 2β 2





t 0

n



β

−1 +

4

2

− 1 N3 −

N3 c0 α + N1 (c0  − 1 ) 2 2

 nN2 3RN2 ||ut ||2 + + N1 − ||ztt ||21 . 2 2 + N2

1

+

N2 R2 N3 + + N3 2δ 2α



||zt ||21

Kσ (t − s )g(t − s )||z(s ) − z(t )||21 ds

(H (t ) − 1 ) N2 + H (t ) + 2

N

g(t ) N0 + 2lN4 2



||z(t )||21

 ||∇ u||2

(3.11)

In order to control the term ||zt ||2 , we introduce an auxiliary system. Let v = ut , w = zt . Then z, v, w satisfy the following 1 system

⎧ v − v = 0 , tt ⎪ ⎪ ⎪ ⎪ v = 0, ⎪ ⎪  t ⎪ ⎪ ⎨ vt − z(t )g(0 ) − g (t − s )z(s )ds + w + wtt = 0, 0 ⎪ ⎪ ∂ v = w , ⎪ ν t ⎪ ⎪ ⎪ ⎪ v (0 ) = u1 , vt (0 ) = u0 , ⎪ ⎩ w(0 ) = z1 , wt (0 ) = −(u1 + z0 ),

in  × (0, T ), on 0 × (0, T ), on 1 × (0, T ),

(3.12)

on 1 × (0, T ), in , on 1 .

We can also show the wellposedness of this system by the semigroup theory. Define an energy function for system (3.12) by

E1 (t ) =

1 ||vt ||2 + ||∇v||2 + ||w||21 + ||wt ||21 . 2

E1 (t ) =

d g(0 ) w, z(t )1 − g(0 )||zt ||21 + wt , dt

Then

Since

 0

t

g (t − s )z(s )ds1 .

(3.13)

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C. Li et al. / Applied Mathematics and Computation 321 (2018) 593–601

− zt , =



t 0

g (t − s )z(s )ds1

  t   t 1 d 1 g (t − s )|z(s ) − z(t )|2 dsdγ − g (t − s )|z(s ) − z(t )|2 dsdγ dt 2 1 0 2 1 0

 t 1 d 1 2 − ||z||1 g (s )ds + g (t )||z||21 , 2 dt 2 0

it follows that

wt , =



t 0

d

w, dt

g (t − s )z(s )ds1  0

t

g (t − s )z(s )ds1 −

g (0 ) d ||z||21 − zt , 2 dt



t 0

g (t − s )z(s )ds1

 t   t g (0 ) d d 1 d

w, g (t − s )z(s )ds1 − ||z||21 + g (t − s )|z(s ) − z(t )|2 dsdγ dt 2 dt 2 dt 1 0 0

  t  t 1 1 d 1 − g (t − s )|z(s ) − z(t )|2 dsdγ − ||z||21 g (s )ds + g (t )||z||21 2 1 0 2 dt 2 0  t   t g (t ) d d 1 d =

w, g (t − s )z(s )ds1 − ||z||21 + g (t − s )|z(s ) − z(t )|2 dsdγ dt 2 dt 2 dt 1 0 0   t 1 − g (t − s )|z(s ) − z(t )|2 dsdγ . 2 1 0 =

(3.14)

Define a modified energy E1 (t ) for system (3.12) by

E1 (t ) = E1 (t ) − g(0 ) w, z(t )1 − w, −



t

0

g (t − s )z(s )ds1

  t 1 1 g (t − s )|z(s ) − z(t )|2 dsdγ + g (t )||z||21 . 2 1 0 2

Now we establish a key lemma which is useful for dealing with term ||zt ||2 in (3.11). 1

Lemma 3.4. The modified energy E1 (t ) is lower bounded:

E1 (t ) ≥ −CE (0 ),

(3.15)

and E1 (t ) satisfies the equality

  t 1 1 g (t )||z||21 − g(0 )||zt ||21 − g (t − s )|z(s ) − z(t )|2 dsdγ . 2 2 1 0

E1 (t ) =

Proof. From (3.13),(3.14), it is easy to get (3.16). Next we prove (3.15). Indeed, it is easy to check that

− w, = − zt , ≥− ≥− Then



t 0



t 0

g (t − s )z(s )ds1 g (t − s )(z(s ) − z(t ))ds +

1 ||zt ||21 − || 4



t 0







t 0

g (t − s )z(t )ds1

−g (t − s ) −g (t − s )(z(s ) − z(t ))ds +



t 0

g (t − s )z(t )ds||21

  t 1 ||zt ||21 − 2g(0 )2 ||z||21 + 2g(0 ) g (t − s )|z(s ) − z(t )|2 dsdγ . 4 1 0   t 1 ||zt ||21 + 2g(0 ) g (t − s )|z(s ) − z(t )|2 dsdγ 4 1 0   t 1 1 −2g2 (0 )||z||21 − g (t − s )|z(s ) − z(t )|2 dsdγ + g (t )||z||21 2 1 0 2   t ≥ −C ||z(t )||21 + 2g(0 ) g (t − s )|z(s ) − z(t )|2 dsdγ

E1 (t ) ≥ E1 (t ) − g(0 ) w, z(t )1 −

1

0

  t 1 − g (t − s )|z(s ) − z(t )|2 dsdγ 2 1 0

(3.16)

C. Li et al. / Applied Mathematics and Computation 321 (2018) 593–601

≥ −C ||z(t )||21 − C1

 1



t 0

599

g(t − s )|z(s ) − z(t )|2 dsdγ

≥ −CE (0 ), where we use the assumption (A-2) and the nonincreasing property of E (t ) in the last inequality.



||2

Next, we will deal with the term ||ztt  in (3.11) by introducing another system. Let θ = vt and ξ = wt . Then 1

⎧ θ − θ = 0 , tt ⎪ ⎪ ⎪ ⎪ θ = 0, ⎪ ⎪  t ⎪ ⎪ ⎨ θt − w(t )g(0 ) − g (t − s )z(s )ds − g (0 )z(t ) + ξ + ξtt = 0, 0 ⎪ ⎪ ∂ν θ = ξt , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ θ (0 ) = u0 , θt (0 ) = u1 , ξ (0 ) = −(u1 + z0 ), ξt (0 ) = −u0 − g(0 )z0 − z1 ,

Denote

E2 (t ) =

1 1 ||θt ||2 + ||∇θ ||2 − g(0 ) ξ , w1 − ξ , 2 2



t

0

in  × (0, T ), on 0 × (0, T ), on 1 × (0, T ),

(3.17)

on 1 × (0, T ), in , on 1 .

g (t − s )z(s )1 − g (0 ) ξ , z1

1 1 1 + g (0 )||w||21 + ||ξ ||21 + ||ξt ||2 + g (0 ) w, z + w, 2 2 2   t g (t ) 1 + g(4) (t − s )|z(s ) − z(t )|2 dsdγ − ||z||21 . 2 1 0 2



t 0

g (t − s )z(s )ds1

As in Lemma 3.4, we also have a lemma for system (3.17). Lemma 3.5. E2 (t ) is lower bounded:

E2 (t ) ≥ −cE (t ),

(3.18)

and it satisfies

E2 (t ) = −g(0 )||ztt ||21 + g (0 )||w||21 −

  t g(4) (t ) 1 ||z||21 + g(5) (t − s )|z(s ) − z(t )|2 dsdγ . 2 2 1 0

After the above preparations, we now prove Theorem 3.1. Proof. Denote

F1 (t ) = F (t ) + N5 E1 (t ) + N6 E2 (t ). Then by (3.11), (3.16),(3.19), we infer that

F1 (t ) = F (t ) + N5 E1 (t ) + N6 E2 (t )  σ N0 − N4 t ≤ g(t − s )||z(s ) − z(t )||21 ds 2 0

N



N2 R2 N3 + − N5 g(0 ) + N3 + N6 g (0 ) ||zt ||21 4 2δ 2α

3RG  t Gσ σ + N2 + N3 − N0 Kσ (t − s )g(t − s )||z(s ) − z(t )||21 ds 2 2β 0 +

+

1

3R 2

n

+



 N β (H (t ) − 1 )2 N2 + N3 H (t ) + − 1 − 0 g(t ) + 2lN4 ||z(t )||21 2





2



nN2 c0 α + − 1 N2 + N3 + (c0  − 1 )N1 ||∇ u||2 + N1 − 2 2 2   t N g (t ) 5 + N5 ||z(t )||21 − g (t − s )|z(s ) − z(t )|2 dsdγ 2 2 1 0

3R



g(4) (t ) N2 − g(0 )N6 ||ztt ||21 − N6 ||z||21 2 2   t N6 + g(5) (t − s )|z(s ) − z(t )|2 dsdγ . 2 1 0 +



||ut ||2

(3.19)

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C. Li et al. / Applied Mathematics and Computation 321 (2018) 593–601

For arbitrary N2 > 0, N3 > 0, let

n−1 N2 , 2

N1 = Then

N2

n 2



−1 +

Moreover, let N6 =

=

1 , nc0

α=

N2 . 2nN3 c0

N2 N3 c0 α + N1 (c0  − 1 ) = − , 2 4n

3RN2 2 g( 0 )

N1 −

nN2 N2 =− . 2 2

+ g(10 ) . Then

3RN2 − g(0 )N6 = −1. 2 For any fixed N4 > 0, β < 2(1 − l ), let

N3 =

3R

1 1 − β2 − l

2



N2 + 2lN4 + 1 .

Then



3R β (H (t ) − 1 )2 N2 + N3 H (t ) + − 1 2 2 Let

1 N5 = g( 0 )



−

N0 g(t ) + 2lN4 ≤ −1. 2

nN32 c0 n(n − 1 )c0 N2 N2 R2 + + + N3 + N6 g (0 ) + 1 . 8 2δ N2

Then

N1 N2 R2 N3 + + − N5 g(0 ) + N3 + N6 g (0 ) = −1. 4 2δ 2α

For any fixed σ ∈ (0, 1), let

N0 =

3RGσ Gσ N2 + N3 + 1. 2 2β

Then

3RGσ Gσ N2 + N3 − N0 = −1. 2 2β By Lebesgue’s dominated convergence theorem, we have

σ Gσ → 0, as σ → 0, by noting (2.1). Therefore, there exists σ 0 > 0 such that for σ ≤ σ 0 ,

3R 2

Set σ =

N2 +

N3 2β

 N σ Gσ ≤ 4 . 4

N min{σ0 , 24 },

σ N0 − N4 ≤ −

we get

N4 . 4

Collecting these inequalities and using assumption (A-2) shows the existence of positive constants c, c1 such that



F1 (t ) ≤ − cE (t ) + 

N6 + 2 1



≤ − cE (t ) +

g (t ) g(4) (t ) N5 + N6 2 2

t

0

c1 2

  t N ||z(t )||21 − 5 g (t − s )|z(s ) − z(t )|2 dsdγ 2

1

0

g(5) (t − s )|z(s ) − z(t )|2 dsdγ



g(t )||z||21 −



 1

0

t

g (t − s )|z(s ) − z(t )|2 dsdγ

= − cE (t ) − c1 E (t ),

(3.20)

which implies that F1 (t ) + c1 E (t ) is a decreasing function. Using the lower boundness of E1 (t ) and E2 (t ) in (3.15) and (3.18), we then get the boundedness of F1 (t ) + c1 E (t ). Integrating (3.20) on (0, t), we have



t

c 0

E (s )ds ≤ F1 (0 ) + c1 E (0 ) − F1 (t ) − c1 E (t ) ≤ C (E (0 ), E2 (0 )).

C. Li et al. / Applied Mathematics and Computation 321 (2018) 593–601

601

Since E (t ) is nonincreasing (by (3.2)), we get

E (t ) ≤

c (E (0 ), E2 (0 )) . t

By (3.1), we obtain the desired estimate:

E (t ) ≤

1 c E (t ) ≤ , 1−l t

where c depends on ||u0 ||H 3 () , ||u1 ||H 2 () , ||z0 ||1 , ||z1 ||1 .



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