General decay of solutions of a wave equation with a boundary control of memory type

Nonlinear Analysis: Real World Applications 11 (2010) 2896–2904

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Nonlinear Analysis: Real World Applications journal homepage: www.elsevier.com/locate/nonrwa

General decay of solutions of a wave equation with a boundary control of memory type Salim A. Messaoudi a,∗ , Abdelaziz Soufyane b a

Department of Mathematics & Statistics, KFUPM, Dhahran 31261, Saudi Arabia

b

College of Engineering and Applied Sciences, P.O.Box 38772, Abu Dhabi, United Arab Emirates

article

info

Article history: Received 8 May 2009 Accepted 25 October 2009 Keywords: General decay Memory type Boundary damping Wave Relaxation function

abstract In this paper we consider a semilinear wave equation, in a bounded domain, where the memory-type damping is acting on a part of the boundary. We establish a general decay result, from which the usual exponential and polynomial decay rates are only special cases. Our work allows certain relaxation functions which are not necessarily of exponential or polynomial decay and, therefore, generalizes and improves earlier results in the literature. © 2009 Elsevier Ltd. All rights reserved.

1. Introduction In [1], Santos considered

 u − µ(t )uxx = 0, (x , t ) Z ∈ (0, 1) × R+   tt t u(0, t ) = 0, u(1, t ) = − g (t − s)µ(s)ux (1, s)ds,   0 u(0) = u0 , ut (0) = u1 , x ∈ (0, 1),

∀t > 0

where µ(t ) is a nonincreasing function satisfying µ(t ) ≥ µ0 > 0. By considering the resolvent kernel of −g /g (0), the boundary condition takes the form

µ(t )ux (1, t ) = −τ {ut (1, t ) + k(0)u(1, t ) − k(t )u0 (1) + k0 (t ) ∗ u(1, t )} where τ > 0 is a constant and k is the resolvent kernel of −g 0 /g (0). He showed that the energy of the solution decays exponentially (polynomially) when k and k0 decay exponentially (polynomially). This result has been later pushed to a nonlinear n-dimensional wave equation of Kirchhoff type by Santos et al. [2]. In that paper, the authors established the existence of a global unique solution and showed, under the same conditions on k and k0 , that the solution decays uniformly with the same rate of decay k. This latter result improves an earlier one by Park et al. [3]. A similar approach has been also used by Santos and Junior [4] to establish a similar result to a biharmonic wave equation supplemented by viscoelastic damping acting on a part of the boundary. Cavalcanti et al. [5] studied the existence and the uniform decay of solutions to a semilinear wave equation with a boundary damping of memory type and a nonlinear boundary source. Also, Rivera and Andrade [6] considered a one-dimensional nonlinear wave equation subject to a nonlinear boundary memory effect and

∗

Corresponding author. Tel.: +966 3 860 4570; fax: +966 3 860 2340. E-mail addresses: [email protected] (S.A. Messaoudi), [email protected] (A. Soufyane).

1468-1218/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2009.10.013

S.A. Messaoudi, A. Soufyane / Nonlinear Analysis: Real World Applications 11 (2010) 2896–2904

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showed that this effect is strong enough to guarantee global existence and uniform decay, at least for small initial data, provided that kernel decays exponentially (or polynomially). Recently, Cavalcanti and Guesmia [7] considered the following system

 utt − 1u + F (x, t , u, ∇ u) = 0 in Ω × R+    u(x, t ) = 0 Zon Γ0 × R+ t ∂u g ( t − s) u(x, t ) = − (s)ds on Γ1 × R+    ∂ν 0  u(0) = u0 , ut (0) = u1 in Ω ,

(1)

where Ω is an open bounded set of Rn with a smooth boundary ∂ Ω = Γ0 ∪ Γ1 . Here Γ0 and Γ1 are closed, disjoint, with meas(Γ0 ) > 0 and ν is the unit outer normal vector. They established a more general decay result which depends on the value of u0 on Γ0 and the rate of the decay of k0 . In their work, they treated the cases when k0 is decaying exponentially or k0 is decaying polynomially. When u0 = 0 on Γ0 , they obtained the exponential and the polynomial decay as special cases. This result has been recently generalized to the case of a system of Timoshenko type by Messaoudi and Soufyane [8]. Cavalcanti et al. [9] studied a problem of the form

Z t    g (t − τ )1u(τ )dτ = 0, in Ω × (0, ∞) utt − 1u +    0  u = 0, on Γ0 × (0, ∞) Z t ∂u ∂u   − g (t − τ ) (τ )dτ + h(ut ) = 0, on Γ1 × (0, ∞)    ∂ν ∂ν 0  u(x, 0) = u0 (x), ut (x, 0) = u1 (x), x ∈ Ω ,

(2)

for g, h specific functions and established uniform decay rate results under quite restrictive assumptions on both the damping function h and the kernel g. In fact, the function g had to behave exactly like e−mt and the function h had a polynomial behavior near zero. For more general assumptions on g and h, Cavalcanti et al. [10] proved the uniform stability of (2), provided that g (0) and kg kL1 (0,∞) are small enough. They also established explicit decay rate results for some special cases. This latter result of Cavalcanti et al. [10] has been recently improved by Messaoudi and Mustafa [11], where no growth assumption on h near zero has been imposed. Stabilization of wave equations or wave systems by frictional boundary damping has been studied by many researchers. Different mechanisms have been utilized to stabilize such systems and several decay and stability results have been obtained. In this regard we mention, among many others, the work of Alabau-Boussouira [12], Cavalcanti et al. [13,14], Conrad and Rao [15], Gorain [16], Guesmia and Messaoudi [17], Komornik and Zuazua [18], Komornik [19], Komornik and Rao [20], Lasiecka [21], Lasiecka and Tataru [22], and Zuazua [23]. In this paper, we are concerned with the following problem

 utt − 1u + f (u) = 0 in Ω × R+    u(x, t ) = 0 Zon Γ0 × R+ t ∂u (s)ds on Γ1 × R+ u ( x , t ) = − g ( t − s)    ∂ν 0  u(x, 0) = u0 (x), ut (x, 0) = u1 (x), x ∈ Ω ,

(3)

where Ω is an open bounded domain of Rn with a smooth boundary ∂ Ω = Γ0 ∪ Γ1 , ν is the unit outer normal vector, and f ∈ C 1 (R) is a function satisfying uf (u) ≥ bF (u) ≥ 0,

for b > 2,

F (u) =

u

Z

f (ξ )dξ

(4)

0

with F (u) ≤ d |u|p ,

∀u ∈ R ,

(5)

for some constant d > 0 and p ≥ 1 such that (n − 2)p ≤ n. The partition Γ0 and Γ1 are closed, disjoint, with meas(Γ0 ) > 0 and satisfying

Γ0 = {x ∈ ∂ Ω : ν.m(x) ≤ 0} Γ1 = {x ∈ ∂ Ω : ν.m(x) > 0} .

(6)

where m(x) = x − x0 , for some x0 ∈ Rn . Remark 1.1. An example of a function satisfying (4) and (5) is f (u) = |u|γ −2 u, γ > 2. This work is divided into three sections. In Section 2 we state, without proof, an existence result of solutions to system (3) and present some material needed for the proof of our main result. In particular, we establish some relations between the relaxation function g and the corresponding resolvent kernel k similar to, but more general than, those usually found in the literature. In Section 3 our main result is stated and proved.

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2. Preliminaries and notations In this section we introduce some notations and discuss the existence of solutions to system (3). First, we exploit (3)3 to ∂u estimate the boundary term ∂ν . Defining the convolution product operator by

(g ∗ ϕ)(t ) =

t

Z

g (t − s)ϕ(s)ds 0

and differentiating equation (3)3 , we obtain

  ∂u 1 ∂u 1 + g0 ∗ =− ut on Γ1 × R+ . ∂ν g (0) ∂ν g (0) Applying Volterra’s inverse operator, we get

∂u 1 =− (ut + k ∗ ut ) on Γ1 × R+ , ∂ν g (0) where the resolvent kernel k satisfies k+

1 g (0)

(g 0 ∗ k) = −

1 g (0)

on Γ1 × R+ .

g0

Denoting by η = g (10) , we arrive at

∂u = −η(ut + k(0)u − k(t )u0 + k0 ∗ u) on Γ1 × R+ . ∂ν Reciprocally, if u0 = 0 on Γ1 , (7) implies (3)3 .

(7)

Since we are interested in relaxation functions of more general decay, we would like to know if the resolvent kernel k, involved in (7), inherits some properties of the relaxation function involved in (3)3 . The following Lemma answers this question. Let h : [0, +∞) → R+ be continuous. Let k be its resolvent, that is k(t ) = h(t ) + (k ∗ h)(t ).

(8)

It is well known that k is continuous and positive (see [7]). Lemma 2.1. Suppose that h(t ) ≤ c0 e−

Rt 0

γ (ζ )dζ

where γ : [0, +∞) → R+ , is a nonincreasing function satisfying, for some positive constant ε < 1, +∞

Z c1 = 0

Rs 1 e− 0 (1−ε)γ (ζ )dζ ds < . c0

Then k satisfies k(t ) ≤

c0 1 − c0 c1

e−ε

Rt 0

γ (ζ )dζ

.

Proof. Let δ(t ) = εγ (t ) and denote by t K (t ) = k(t )e 0 δ(ζ )dζ ,

t H (t ) = h(t )e 0 δ(ζ )dζ .

R

R

t By multiplying (8) by e 0 δ(ζ )dζ , we obtain

R

K (t ) = H (t ) +

t

Z

[e

Rt 0

δ(ζ )dζ −

e

R t −s

δ(ζ )dζ

K (t − s)h(s)]ds

Rs

γ (ζ )dζ

s K (t − s)e 0 γ (ζ )dζ h(s)]ds

0

0

= H (t ) +

t

Z

[e

Rt

t −s

δ(ζ )dζ −

e

0

R

0

≤ c0 + c0 sup K (r ) 0≤r ≤t

Z 0

t

s e− 0 [γ (ζ )−εγ (ζ +t −s)]dζ ds.

R

S.A. Messaoudi, A. Soufyane / Nonlinear Analysis: Real World Applications 11 (2010) 2896–2904

2899

By using the fact that γ is nonincreasing we arrive at K (t ) ≤ c0 + c0 sup K (r )

t

Z

0 ≤r ≤t

s e− 0 (1−ε)γ (ζ )dζ ds

R

0

+∞

Z

≤ c0 + c0 sup K (r ) 0 ≤t ≤T

s e− 0 (1−ε)γ (ζ )dζ ds,

R

∀t ≤ T .

0

which gives sup K (r ) ≤ c0 + c0 sup K (r ) 0≤t ≤T

+∞

Z

0≤t ≤T

s e− 0 (1−ε)γ (ζ )dζ ds

R

0

≤ c0 + c1 c0 sup K (r ). 0 ≤t ≤T

Consequently, sup K (r ) ≤ 0≤t ≤T

c0 1 − c0 c1

,

∀T > 0

hence c0

K (t ) ≤

1 − c0 c1

.

Therefore k(t ) ≤

c0 1 − c0 c1

e−ε

Rt 0

γ (ζ )dζ

. 

Remark 2.1. The result of [7] is only a special case. Corollary 2.2. Suppose that h(t ) ≤ c0 e−γ t for γ > c0 . Then, there exists a positive constant ε < 1 such that k(t ) ≤ β e−εγ t

(9)

where β > 0 is a constant. Proof. It is easy to verify that +∞

Z

1

e−(1−ε)γ s ds =

(1 − ε)γ

0

<

1 c0

if ε is chosen small. Thus (9) is a direct result of the lemma.



Corollary 2.3. Suppose that h( t ) ≤

c0

(1 + t )p

for c0 < p − 1. Then, there exists a positive constant ε < 1 such that k(t ) ≤

β (1 + t )εp

(10)

where β > 0 is a constant. Proof. We take γ (ζ ) = 1+ζ . It is easy to verify that p

e

−(1−ε)p

dζ 0 1+ζ

Rs

=

1

(1 + ζ )(1−ε)p

and +∞

Z

e 0

−(1−ε)p

dζ 0 1+ζ

Rs

Z ds = 0

+∞

1

(1 + ζ )(1−ε)p

ds =

1

(1 − ε)p − 1

if ε is chosen small. Thus (10) is a direct result of the lemma.



<

1 c0

2900

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Example 2.1. If we take

γ (ζ ) = aζ p ,

−1 < p < 0

and assume that h(t ) ≤ c0 e

a t p+1 − p+ 1

then with an appropriate choice of a > 0, one can easily see that, for some positive constant ε < 1, +∞

Z c1 =

− a(p1+−ε) sp+1 1

e

ds <

0

1 c0

.

Consequently, we get k(t ) ≤ β e

− pε+a1 t p+1

.

Based on Lemma 2.1, we will use the boundary relation (7) instead of (3)3 . Let us define

(g o ϕ)(t ) :=

t

Z

g (t − s) |ϕ(t ) − ϕ(s)|2 ds

(11)

0

and

(g ϕ)(t ) :=

t

Z

g (t − s)(ϕ(t ) − ϕ(s))ds.

(12)

0

By using Hölder’s inequality, we have

|(g ϕ)(t )|2 ≤

t

Z

 |g (s)| ds (|g | oϕ)(t ).

(13)

0

Lemma 2.4 (see [1–4,7]). If g , ϕ ∈ C 1 (R+ ) then 1

1

1 d

2

2

2 dt

(g ∗ ϕ)ϕt = − g (t ) |ϕ(t )|2 + g 0 o ϕ −



g oϕ−

t

Z



g (s)ds |ϕ(t )|2



.

(14)

0

The well-posedness of system (3) is presented in the following theorem. Let V = v ∈ H 1 (Ω ) : v = 0 on Γ0 .





Theorem 2.5. Let k ∈ W 2,1 (R+ ) ∩ W 1,∞ (R+ ), u0 ∈ (H 2 (Ω ) ∩ V ), and u1 ∈ V with

∂ u0 + ηu0 = 0 on Γ1 . ∂ν

(15)

Assume that (4)–(6) hold. Then there exists a unique strong solution u of system (3) such that u ∈ L∞ (R+ ; H 2 (Ω ) ∩ V ), ut ∈ L (R ; V ), ∞

+

ut ∈ L∞ (R+ ; L2 (Ω ))

utt ∈ L (R+ ; L2 (Ω )). ∞

Proof. This theorem can be proved, using the Galerkin method and following exactly the procedure of [2,4].



3. Decay of solutions In this section we study the asymptotic behavior of the solutions of system (3) when the resolvent kernel k satisfy k(0) > 0, where γ : R

+

k(t ) ≥ 0,

k0 (t ) ≤ 0,

k00 (t ) ≥ γ (t )(−k0 (t )),

(16)

+

→ R is a function satisfying the following conditions Z +∞ γ (t ) > 0, γ 0 (t ) ≤ 0, and γ (t )dt = +∞. 0

(17)

S.A. Messaoudi, A. Soufyane / Nonlinear Analysis: Real World Applications 11 (2010) 2896–2904

2901

Example 3.1. Let k(t ) =

e− t

(1 + t )a

,

t > 0, a > 0.

Direct computations show that k00 (t ) = γ (t )(−k0 (t )),

with γ (t ) = 1 +

a

+

t +1+a

a( a + 1 ) . (t + 1)(t + 1 + a)

By multiplying Eq. (3)1 by ut and integrating over Ω , using integration by parts, the boundary conditions, and (14), one can easily find that the first order energy of system (3) is given by (see Lemma 3.2 and its proof) 1

E (t ) :=

Z

2

Ω

(|ut |2 + |∇ u|2 )dx +

Z

F (u)dx +

Ω

η

Z

2

Γ1

(k(t ) |u|2 − k0 o u)dΓ1 .

(18)

Theorem 3.1. Given (u0 , u1 ) ∈ (V × L2 (Ω )). Assume that (4)–(6), (16) and (17) hold, with lim k(t ) = 0.

(19)

t →∞

Then, for some t0 large enough, we have , ∀t ≥ t0 , E (t ) ≤ cE (0)e−a

Rt 0

γ (s)ds

,

if u0 = 0 on Γ1 .

(20)

Otherwise,



E (t ) ≤ c E (0) +

Z Γ1

0 2 u dΓ 1

t

Z

k2 (s)[1 + e

Rs

a t γ (ζ )dζ 0

 Rt ]ds e−a 0 γ (s)ds ,

(21)

0

where a is a fixed positive constant and c is a generic positive constant. Remark 3.1. Assumption (19) can be replaced by kkk∞ small enough as in [7]. The main idea is to construct a Lyapunov functional L(t ) equivalent to E (t ). To do this we use the multiplier techniques. The proof of Theorem 3.1 will be achieved with the help of two lemmas. Lemma 3.2. Under the assumptions of Theorem 3.1, the energy of the solution of (3) satisfies dE

≤−

dt

η 2

Z Γ1

|ut |2 dΓ1 +

η 2

Z Γ1

k2 (t ) |u0 |2 dΓ1 −

η

Z

2

Γ1

k00 oudΓ1 .

(22)

Proof. Multiplying Eq. (1)1 by ut , and integrating by parts over Ω , we obtain

Z

d 2dt

Ω

 2  |ut | + |∇ u|2 + 2F (u) dx =

Z Γ1

∂u ut d Γ 1 . ∂ν

Using (7), and Lemma 2.4, we obtain the desired result.



Remark 3.2. (a) If u0 = 0 on Γ1 , then E (t ) ≤ E (0). (b) If u0 6= 0 on Γ1 , then E (t ) ≤ E (0) +

η

2

R

Γ1

| u0 | 2 d Γ 1

Rt 0

k2 (t )dt .

Lemma 3.3. Under the assumptions of Theorem 3.1, the solution of (3) satisfies d

∂u |ut | dx + m.ν |ut | dΓ1 − ε0 (2m.∇ u + (n − ε0 )u)ut dx ≤ (2m.∇ u + (n − ε0 )u)dΓ1 ∂ν Ω Γ1 Ω Γ1 Z Z Z |∇ u|2 dx − [(b − 2)n − ε0 b] − m.ν |∇ u|2 dΓ1 − (1 − ε0 ) F (u), ∀t ≥ 0. Z



Z

2

Z

2

dt

Γ1

for some 0 < ε0 < 1.

Ω

Ω

Z

2902

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Proof. We multiply Eq. (3)1 by 2m.∇ u + (n − ε0 )u to obtain d



Z

Z

Z

2ut m.∇ ut dx + (n − ε0 ) (2m.∇ u + (n − ε0 )u)ut dx = Ω Z Z + (n − ε0 ) u1udx − (2m.∇ u + (n − ε0 )u) f (u)dx.

dt

Ω

Ω

Ω

2

Z

|ut | dx +

Ω

2(1u)m.∇ udx

Ω

By integrating by parts and using (4), (6), and the relation div(m) = n we get

 Z Z Z ∂u |ut |2 dx + (2m.∇ u + (n − ε0 )u)ut dx ≤ (m.ν) |ut |2 dΓ1 − ε0 (2m.∇ u + (n − ε0 )u) dΓ1 dt ∂ν Ω Γ1 Ω Γ1 Z Z Z Z |∇ u|2 dx − [(b − 2)n − ε0 b] − F (u)dx − (m.ν) F dΓ1 . (m.ν) |∇ u|2 dΓ1 − (1 − ε0 ) Z

d

Γ1

Ω

Ω

By recalling (6), the proof of Lemma 3.3 is completed.

Γ1



Now, we introduce the Lyapunov functional. So, for N > 0 large enough, let

Z

L(t ) = NE (t ) +

Ω

[(2m.∇ u + (n − ε0 )u)ut ]dx.

(23)

It is a routine calculation to check that, for N large, we have N 2

E (t ) ≤ L(t ) ≤ 2NE (t ).

(24)

Applying Young’s inequality and Poincaré’s inequality to the boundary integral we have, for ε > 0,

Z Γ1

∂u (2m.∇ u + (n − ε0 )u)dΓ1 ≤ ε ∂ν

Z Ω

|∇ u|2 dx + ε

Z Γ1

m.ν |∇ u|2 dΓ1 + Cε

Z 2 ∂u dΓ1 . Γ1 ∂ν

By rewriting the boundary condition (7) as

∂u = −η1 (ut + k(t )u − k(t )u0 − k0 u) on Γ1 × R+ ∂ν and combining all the above relations, we arrive at



Nη

Z

Nη

Z

|ut | dΓ1 + | u0 | 2 d Γ 1 − Cε − m.ν k (t ) 2 Γ1 Γ1 Z Z Z ηN |∇ u|2 dx − (1 − ε) (m.ν) |∇ u|2 dΓ1 − k00 oudΓ1 − (1 − ε0 − ε − Cε k2 (t ))

L (t ) ≤ − 0

2

2

2

2

Γ1

− ε0

Z

|ut | dx − [(b − 2)n − ε0 b] 2

Ω

Γ1

Z Ω

Ω

F (u)dx + Cε

Z Γ1

0 k u 2 dΓ1 + Cε k2 (t )

Z Γ1

|u0 |2 dΓ1 .

(25)

At this point, we take

ε = ε0 < min



1 (b − 2)n 4

,



b

.

Once ε is fixed (hence Cε ), we pick N large enough so that Nη 2

− Cε − max |m.ν| > 0. Γ1

By using the fact that limt →∞ k(t ) = 0, Poincaré’s inequality, and (13), we arrive at

L0 (t ) ≤ −α E (t ) + β k2 (t )

≤ −α E (t ) + β

Z Γ1

Z Γ1

0 2 u dΓ1 − ηN

Z

2

k2 (t ) |u0 |2 dΓ1 − C

Γ1

Z Γ1

k00 oudΓ1 − C

k0 oudΓ1 ,

Z Γ1

k0 oudΓ1

∀t ≥ t0

(26)

for some t0 large enough and some positive constants α, β and C . We multiply both sides of (26) by γ (t ) to get

γ (t )L0 (t ) ≤ −αγ (t )E (t ) + βγ (t )

Z

2

Γ1

k2 (t ) u0 dΓ1 − γ (t )C

Z Γ1

k0 oudΓ1 ,

∀t ≥ t0 .

S.A. Messaoudi, A. Soufyane / Nonlinear Analysis: Real World Applications 11 (2010) 2896–2904

2903

A simple calculation, using the fact that γ (t ) is nonincreasing, yields

γ (t )L0 (t ) ≤ −αγ (t )E (t ) + βγ (t )

Z Γ1

k2 (t ) |u0 |2 dΓ1 + c

Z Γ1

k00 oudΓ1 ,

∀t ≥ t0 .

By using (22), we easily see that

Z

γ (t )L (t ) ≤ −αγ (t )E (t ) + c 0

k2 (t ) |u0 |2 dΓ1 − cE 0 (t ),

Γ1

∀t ≥ t0

which yields

γ (t )L (t ) + cE (t ) ≤ −αγ (t )E (t ) + c 0

0

Z Γ1

k2 (t ) |u0 |2 dΓ1 ,

∀t ≥ t0

or d dt

(γ (t )L(t ) + cE (t )) − γ 0 (t )L(t ) ≤ −αγ (t )E (t ) + c

Z Γ1

k2 (t ) |u0 |2 dΓ1 ,

∀t ≥ t0 .

(27)

Again using the fact that γ (t ) is nonincreasing and setting F (t ) = γ (t )L(t ) + cE (t ) ∼ E (t )

(28)

estimate (27) gives

Z

F (t ) ≤ −aγ (t )F (t ) + c 0

k2 (t ) |u0 |2 dΓ1 ,

Γ1

∀t ≥ t0 .

(29)

Case 1: If u0 = 0 on Γ1 , then (29) reduces to dF ≤ −aγ (t )F (t ), ∀t ≥ t0 . dt A simple integration over (t0 , t ) yields F (t ) ≤ F (t0 )e

−a

Rt

t0

γ (s)ds

,

∀t ≥ t 0 .

By using (28), then we obtain for some positive constant c E (t ) ≤ cE (t0 )e

−a

Rt

γ (s)ds

t0

,

∀t ≥ t 0

using Remark 3.2, then we get E (t ) ≤ cE (0)ea

R t0

γ (s)ds −a

Rt

γ (s)ds

,

∀t ≥ t0 .

d 0 F (t ) ≤ −aγ (t )F (t ) + C1 k2 (t ),

∀t ≥ t0 ,

0

e

0

Thus the estimate (20) is proved. Case 2: If u0 6= 0 on Γ1 , then (29) gives dt

(30)

where

Z C1 = c

Γ1

|u0 |2 dΓ1 .

In this case we introduce H (t ) := F (t ) − C1 e

−a

Rt

γ (s)ds

t0

t

Z

k2 (s)e

Rs

a t γ (ζ )dζ 0 ds

.

(31)

t0

A simple differentiation of H, using (30), leads to H 0 (t ) ≤ −aγ (t )H (t ),

∀t ≥ t0 .

Again a simple integration over (t0 , t ) yields H (t ) ≤ H (t0 )e

−a

Rt

t0

γ (s)ds

,

∀t ≥ t0 ,

(32)

which implies F (t ) ≤



F (t0 ) + C1

t

Z

Rs

a γ (ζ )dζ e t0 ds

k ( s) 2

 e

−a

Rt

t0

γ (s)ds

,

∀t ≥ t0 .

t0

By using (28) and Remark 3.2, then we obtain for some positive constant c



E (t ) ≤ c E (0) + C1

t

Z

k (s) 2

0



η 2C1

This complete the proof of Theorem 3.1.

Rs

+

a γ (ζ )dζ e t0





ds ea

R t0 0

γ (s)ds −a

e

Rt 0

γ (s)ds

,

∀t ≥ t 0 .

2904

S.A. Messaoudi, A. Soufyane / Nonlinear Analysis: Real World Applications 11 (2010) 2896–2904

Remark 3.3. Estimates (20) and (21) are also true for t ∈ [0, t0 ] by virtue of continuity and boundedness of E (t ) and γ (t ). Remark 3.4. This result generalizes and improves the results of [1,2,4,7]. In particular, it allows kernels which satisfy k00 ≥ a(−k0 )1+q , for 0 < q < 1 instead of the usual assumption 0 < q < 1/2. It suffices to take, for example, k(t ) = 1/(1 + t )ν , for ν > 0. Direct computations yield k00 (t ) = c (−k0 (t ))1+1/(1+ν) . It is clear that 0 < 1/(1 + ν) < 1, for ν > 0. Remark 3.5. Note that the exponential and the polynomial decay estimates, given in early works [1–4,7] are only particular cases of (21). More precisely, we obtain exponential decay for γ (t ) ≡ a and polynomial decay for γ (t ) = a(1 + t )−1 , where a > 0 is a constant. Remark 3.6. We note that our result also holds for the system (1), studied by Cavalcanti and Guesmia [7]. We only considered (3) for simplicity. Acknowledgments The first author thanks KFUPM for its continuous support. This work has been partially funded by KFUPM under Project # SB090004. References [1] M.L. Santos, Asymptotic behavior of solutions to wave equations with a memory conditions at the boundary, Electron. J. Differ. Equ. 78 (2001) 1–11. [2] M.L. Santos, J. Ferreira, D.C. Pereira, C.A. Raposo, Global existence and stability for wave equation of Kirchhoff type with memory condition at the boundary, Nonlinear Anal. 54 (2003) 959–976. [3] J.Y. Park, J.J. Bae, Il Hyo Jung, Uniform decay for wave equation of Kirchhoff type with nonlinear boundary damping and memory term, Nonlinear Anal. 50 (2002) 871–884. [4] M.L. Santos, F. Junior, A boundary condition with memory for Kirchhoff plates equations, Appl. Math. Comput. 148 (2) (2004) 475–496. [5] M.M. Cavalcanti, V.N. Domingos Cavalcanti, J.S. Prates Filho, J.A. Soriano, Existence and uniform decay of solutions of a degenerate equation with nonlinear boundary memory source term, Nonlinear Anal. 38 (1999) 281–294. [6] J.E. Muñoz Rivera, D. Andrade, Exponential decay of nonlinear wave equation with a viscoelastic boundary condition, Math. Methods Appl. Sci. 23 (1) (2000) 41–61. [7] M.M. Cavalcanti, A. Guesmia, General decay rates of solutions to a nonlinear wave equation with boundary conditions of memory type, Differential Integral Equations 18 (5) (2005) 583–600. [8] S.A. Messaoudi, A. Soufyane, Boundary stabilization of solutions of a nonlinear system of Timoshenko type, Nonlinear Anal. 67 (2007) 2107–2121. [9] M.M. Cavalcanti, V.N. Domingos Cavalcanti, J.S. Prates Filho, J.A. Soriano, Existence and uniform decay rates for viscoelastic problems with nonlinear boundary damping, Differential Integral Equations 14 (1) (2001) 85–116. [10] S.A. Messaoudi, M.I. Mustafa, On the control of solutions of viscoelastic equations with boundary feedback, Nonlinear Anal. RWA 10 (2009) 3132–3140. [11] M.M. Cavalcanti, V.N. Domingos Cavalcanti, P. Martinez, General decay rate estimates for viscoelastic dissipative systems, Nonlinear Anal. 68 (1) (2008) 177–193. [12] F. Alabau-Boussouira, Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems, Appl. Math. Optim. 51 (1) (2005) 61–105. [13] M.M. Cavalcanti, V.N. Domingos Cavalcanti, M.L. Santos, Existence and uniform decay rates of solutions to a degenerate system with memory conditions at the boundary, Appl. Math. Comput. 150 (2004) 439–465. [14] M.M. Cavalcanti, V.N. Domingos Cavalcanti, I. Lasiecka, Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping—source interaction, J. Differential Equations 236 (2) (2007) 407–459. [15] F. Conrad, B. Rao, Decay of solutions of wave equations in a star-shaped domain with nonlinear boundary feedback, Asympt. Anal. 7 (1993) 159–177. [16] G.C. Gorain, Boundary stabilization of nonlinear vibrations of a flexible structure in a bounded domain in Rn , J. Math. Anal. Appl. 319 (2) (2006) 635–650. [17] A. Guesmia, S.A. Messaoudi, On the boundary stabilization of a compactly coupled system of nonlinear wave equations, Int. J. Evol. Equ. 1 (3) (2005) 211–224. [18] V. Komornik, E. Zuazua, A direct method for the boundary stabilization of the wave equation, J. Math. Pure Appl. 69 (1990) 33–54. [19] V. Komornik, On the nonlinear boundary stabilization of the wave equation, Chen. Ann. Math 14B (2) (1993) 153–164. [20] V. Komornik, B. Rao, Boundary stabilization of compactly coupled wave equations, Asympt. Anal. 14 (1997) 339–359. [21] I. Lasiecka, Stabilization of wave and plate-like equations with nonlinear dissipation on the boundary, J. Differential Equations 79 (1989) 340–381. [22] I. Lasiecka, D. Tataru, Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping, Differential Integral Equations 6 (3) (1993) 507–533. [23] E. Zuazua, Uniform stabilization of the wave equation by nonlinear boundary feedback, SIAM J. Control. Optim 28 (1990) 466–477.