Electronic Journal of Differential Equations, Vol. 2002(2002), No. 44, pp. 1–14.
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
ftp ejde.math.swt.edu (login: ftp)
Exponential decay for the solution of semilinear
viscoelastic wave equations with localized
damping ∗
Marcelo M. Cavalcanti, Valéria N. Domingos Cavalcanti,
& Juan A. Soriano
Abstract
In this paper we obtain an exponential rate of decay for the solution
of the viscoelastic nonlinear wave equation
Z t
utt − ∆u + f (x, t, u) +
g(t − τ )∆u(τ ) dτ + a(x)ut = 0 in Ω × (0, ∞).
0
Here the damping term a(x)ut may be null for some part of the domain Ω.
By assuming that the kernel g in the memory term decays exponentially,
the damping effect allows us to avoid compactness arguments and and
to reduce number of the energy estimates considered in the prior literature. We construct a suitable Liapunov functional and make use of the
perturbed energy method.
1
Introduction
This manuscript is devoted to the study of the exponential decay of the solutions
of the viscoelastic nonlinear wave equation
utt − ∆u + f (x, t, u) +
t
Z
g(t − τ )∆u(τ ) dτ + a(x)ut = 0 in Ω × (0, ∞)
0
u=0
0
u(x, 0) = u (x);
on Γ × (0, ∞)
1
ut (x, 0) = u (x),
(1.1)
x ∈ Ω,
where Ω is a bounded domain of Rn whose boundary Γ is assumed regular. Let
x0 ∈ Rn be an arbitrary point of Rn and we set
Γ(x0 ) = {x ∈ Γ; m(x) · ν(x) > 0}
∗ Mathematics Subject Classifications: 35L05, 35L70, 35B40, 74D10.
Key words: semilinear wave equation, memory, localized damping.
c
2002
Southwest Texas State University.
Submitted March 07, 2002. Published May 22, 2002.
Partially supported by a grant from CNPq, Brazil.
1
(1.2)
2
Viscoelastic wave equations with localized damping
EJDE–2002/44
where ν represents the unit normal vector pointing towards the exterior of Ω
and
m(x) = x − x0 .
(1.3)
Let ω be a neighborhood of Γ(x0 ) in Ω and consider δ > 0 sufficiently small
such that
Q0 = {x ∈ Ω; d(x, Γ(x0 )) < δ} ⊂ ω,
0
Q1 = {x ∈ Ω; d(x, Γ(x )) < 2δ} ⊂ ω.
(1.4)
(1.5)
Here, if A ⊂ Rn , and x ∈ Rn , d(x, A) = inf y∈A |x − y|. Then, Q0 ⊂ Q1 ⊂ ω.
Next, we make some remarks about early works in connection with problem
(1.1). When g = 0 and f 6= 0 and the feedback term depends on the velocity
in a linear way, as in the present paper, Zuazua [16] proved that the energy
related to problem (1.1) decays exponentially if the damping region contains a
neighbourhood of the boundary Γ or, at least, contains a neibourhood of the
particular part given by (1.1). In the same direction and considering g = f = 0,
it is important to mention the result of Bardos , Lebeau and Rauch [2], based on
microlocal analysis, that ensures a necessary and sufficient condition to obtain
exponential decay, namely, the damping region satisfies the well known geometric control condition. The classical example of an open subset ω verifying this
condition is when ω is a neighbourhood of the boundary. Later, still considering
g = 0 and f = 0, Nakao [13, 14] extended the results of Zuazua [16] treating
first the case of a linear degenerate equation, and then the case of a nonlinear
dissipation ρ(x, ut ) assuming, as usually, that the function ρ has a polynomial
growth near the origin. More recently, Martinez [11] improved the previous results mentioned above in what concerns the linear wave equation subject to a
nonlinear dissipation ρ(x, ut ), avoiding the polynomial growth of the function
ρ(x, s) in zero. His proof is based on the piecewise multiplier technique developed by Liu [10] combined with nonlinear integral inequalities to show that the
energy of the system decays to zero with a precise decay rate estimate if the
damping region satisfies some geometrical conditions. It is important to mention that Lasiecka and Tataru [9] studied the nonlinear wave equation subject
to a nonlinear feedback acting on a part of the boundary of the system and they
were the first to prove that the energy decays to zero as fast as the solution of
some associated differential equation and without assuming that the feedback
has a polynomial growth in zero, although no decay rate has been showed.
On the other hand, when a = 0, that is, when the unique damping mechanism is given by the memory term, then, following ideas introduced by Munoz
Rivera [12], it is possible to prove that the exponential decay holds for small
initial data. Indeed, we presume that the result above mentioned is a direct
consequence of his proof ( see [12] for details ). In this context we can cite the
works of Dafermos [4], Dafermos and Nohel [4, 5], Jiang and Munoz Rivera [7],
among others.
To end, we would like to mention the works from the authors Cavalcanti,
Domingos Cavalcanti, Prates Filho and Soriano [3] in connection with the viscoelastic linear wave equation and nonlinear boundary dissipation and Aassila,
EJDE–2002/44 M. M. Cavalcanti, V. N. D. Cavalcanti & J. A. Soriano
3
Cavalcanti and Soriano [1] concerned with the wave equation subject to viscoelastic effects on the boundary and nonlinear boundary feedback.
The purpose of this paper is to obtain an exponential decay rate to the solutions of a viscoelastic nonlinear wave equation subject to a locally distributed
dissipation as in Zuazua [16]. We would like to emphasize that the additional
damping effect given by the kernel of the memory of the material allows us to
avoid the compactness arguments and the amount of estimates and fields considered in the prior literature by constructing a suitable Liapunov functional and
making use of the perturbed energy method. For simplicity, we will consider
the linear localized effect, although we could consider the nonlinear one ρ(x, ut ).
In this direction, our work complements the previous ones for the viscoelastic
wave equation.
Our paper is organized as follows. In section 2 we state the notation, assumptions and the main result and in section 3 we give the proof of the uniform
decay.
2
Notation and Statment of Results
We begin by introducing some notation that will be used throughout this work.
For the standard Lp (Ω) space we write
Z
Z
(u, v) =
u(x)v(x) dx, kukpp =
|u(x)|p dx.
Ω
Ω
Next, we give the precise assumptions on the functions a(x), f (x, t, u) and on
the memory term g(t).
(A.1) Assume that a : Ω → R is a nonnegative and bounded function such that
a(x) ≥ a0 > 0 a.e. in ω.
(2.1)
(A.2) Assume that f : Ω × [0, ∞) × R → R is an element of the space C 1 (Ω ×
[0, ∞) × R) and satisfies
|f (x, t, ξ)| ≤ C0 (1 + |ξ|γ+1 )
(2.2)
where C0 is a positive constant.
Let γ be a constant such that γ > 0 for n = 1, 2 and 0 < γ ≤ 2/(n − 2) for
n ≥ 3 and let C00 be a positive constant such that
f (x, t, ξ)η ≥ |ξ|γ ξη − C00 |ξkη|;
∀η ∈ R.
(2.3)
Assume that there exist positive constants C1 and C2 such that
|ft (x, t, ξ)| ≤ C1 (1 + |ξ|γ+1 ),
|fξ (x, t, ξ)| ≤ C2 (1 + |ξ|γ ).
(2.4)
(2.5)
4
Viscoelastic wave equations with localized damping
EJDE–2002/44
We also assume that there is a positive constant D1 such that for all η , η̂ in Rn
and for all ξ, ξˆ in R,
ˆ
ˆ γ )|ξ − ξ||η
ˆ − η̂|.
(f (x, t, ξ) − f (x, t, ξ))(η
− η̂) ≥ −D1 (|ξ|γ + |ξ|
(2.6)
A simple example of a function f that satisfies the above conditions is given by
f (x, t, ξ) = |ξ|γ ξ + ϕ(x, t)sin(ξ), where ϕ ∈ W 1,+∞ (Ω × ∞).
Remark: The variational formulation associated with problem (1.1) leads us
to the identity
(u00 (t), v) + (∇u(t), ∇v) + (f (x, t, u(t)), v)
Z t
−
g(t − τ )(∇u(τ ), ∇v)dτ + (au0 (t), v) = 0
for all v ∈ H01 (Ω).
0
Assumption (2.2) is required to prove that the nonlinear term in the above
identity is well defined. The hypothesis (2.3) is used in the first a priori estimate
and also and in the asymptotic stability. Now, hypotheses (2.4)-(2.5) are needed
for the second a priori estimate while assumption (2.6) is necessary for proving
the uniqueness of solutions.
For simplicity, we will consider the asymptotic stability for the simple case
when f (x, t, ξ) = |ξ|γ ξ, although the general case follows exactly making use of
the same procedure.
To obtain the existence of regular and weak solutions we assume that kernel
g satisfies the following.
(A.3) The function g : R+ → R+ is in W 2,1 (0, ∞)∩W 1,∞ (0, ∞). Now, to obtain
the uniform decay, we suppose that
Z ∞
1−
g(s) ds = l > 0,
(2.7)
0
and there exist ξ1 , ξ2 > 0 such that
g(0) > 0
and
− ξ1 g(t) ≤ g 0 (t) ≤ −ξ2 g(t);
∀t ≥ 0.
(2.8)
Note that (2.8) implies
g(0)e−ξ1 t ≤ g(t) ≤ g(0)e−ξ2 t ;
∀t ≥ 0.
(2.9)
Consequently from (2.9) we have that the kernel g(t) is between two exponential
functions.
We observe that given {u0 , u1 } ∈ H01 (Ω) ∩ H 2 (Ω) × H01 (Ω), problem (1.1)
possesses a unique solution in the class
1
2
0
∞
1
00
∞
2
u ∈ L∞
loc (0, ∞; H0 (Ω) ∩ H (Ω)), u ∈ Lloc (0, ∞; H0 (Ω)), u ∈ Lloc (0, ∞; L (Ω))
EJDE–2002/44 M. M. Cavalcanti, V. N. D. Cavalcanti & J. A. Soriano
5
which can easily obtained making use, for instance, of the Faedo-Galerkin
method. Now, if {u0 , u1 } ∈ H01 (Ω) × L2 (Ω) and considering standard arguments of density, we can prove that problem (1.1) has a unique solution
u ∈ C 0 ([0, ∞); H01 (Ω)) ∩ C 1 ([0, ∞); L2 (Ω)).
(2.10)
To see the proof of pexitence and uniqueness of a similar problem to the viscoelastic wave equation subject to nonlinear boundary feedback, we refer the
reader to [3].
The energy related to problem (1.1) is
E(t) =
1 0
1
1
ku (t)k22 + k∇u(t)k22 +
ku(t)kγ+2
γ+2 .
2
2
γ+2
(2.11)
Now, we are in a position to state our main result.
Theorem 2.1 Given {u0 , u1 } ∈ H01 (Ω) × L2 (Ω) and assuming that Hypotheses
(A.1)-(A.3) hold, then, the unique solution of problem (1.1) in the class given
in (2.10) decays exponentially, that is, there exist C, β positive constants such
that
E(t) ≤ Ce−βt
(2.12)
R∞
provided that 0 g(s)ds is sufficiently small.
3
Uniform decay
In this section we prove the exponential decay for regular solutions of problem
(1.1), and by using standard arguments of density we also can extend the same
result to weak solutions. From (1.1) we deduce that
Z
Z t
0
0
2
E (t) ≤ −
a(x)|u (x, t)| dx +
g(t − τ )(∇u(τ ), ∇u0 (t))dτ.
(3.1)
Ω
0
A direct computation shows that
Z
t
g(t − τ )(∇u(τ ), ∇u0 (t))dτ
0
1 0
1
d 1
(g ∇u)(t) − (g ∇u)0 (t)+ {
2
2
dt 2
Z
t
1
g(s) dsk∇u(t)k22 } − g(t)k∇u(t)k22 ,
2
0
(3.2)
Rt
2
where (g v)(t) = 0 g(t − τ )kv(t) − v(τ )k2 dτ . Define the modified energy as
=
e(t) =
1 0
1
ku (t)k22 + (1 −
2
2
t
Z
g(s) ds)k∇u(t)k22 +
0
1
1
ku(t)kγ+2
γ+2 + (g ∇u)(t).
γ+2
2
(3.3)
From (2.7) we deduce that e(t) ≥ 0. Moreover,
E(t) ≤ l−1 e(t);
∀t ≥ 0.
(3.4)
6
Viscoelastic wave equations with localized damping
EJDE–2002/44
Consequently, the uniform decay of E(t) is a consequence of the decay of e(t).
On the other hand, from (2.11), (3.1), (3.2) and (3.3) we deduce
Z
1
1
e0 (t) ≤ −
a(x)|u0 (x, t)|2 dx + (g 0 ∇u)(t) − g(t)k∇u(t)k22 .
(3.5)
2
2
Ω
Considering the assumptions (2.1) and (2.8), from (3.5) we obtain
e0 (t) ≤ −a0 ku0 (t)k2L2 (ω) −
ξ2
1
(g ∇u)(t) − g(t)k∇u(t)k22 .
2
2
(3.6)
Then e0 (t) ≤ 0 and consequently e(t) are non-increasing functions. Having in
mind (1.4)-(1.5), we consider ψ ∈ C0∞ (Rn ) such that
0≤ψ≤1
(3.7)
ψ = 1 in Ω\Q1
ψ = 0 in Q0 .
For an arbitrary ε > 0, define the peturbed energy
eε (t) = e(t) + ερ(t)
(3.8)
where
ρ(t) = 2
Z
u0 (h · ∇z)dx + θ
Ω
Z
u0 zdx,
(3.9)
Ω
h = mψ,
Z t
z(t) = u(t) −
g(t − τ )u(τ ) dτ,
(3.10)
(3.11)
0
and θ ∈]n − 2, n[, θ >
2n
γ+2 .
For short notation, put
2n ) > 0.
k1 = min 2(θ − n + 2), 2(n − θ), (γ + 2)(θ −
γ+2
(3.12)
Proposition 3.1 There exists δ0 > 0 such that
|eε (t) − e(t)| ≤ εδ0 e(t)
Proof.
∀t ≥ 0
and
∀ε > 0.
From the Cauchy-Schwarz inequality
|ρ(t)| ≤ 2R(x0 )ku0 (t)k2 k∇z(t)k2 + θλku0 (t)k2 k∇z(t)k2 ,
(3.13)
where λ > 0 and satisfies kv|k2 ≤ λk∇vk2 for all v ∈ H01 (Ω), and
R(x0 ) = max |x − x0 |.
(3.14)
1
1
|ρ(t)| ≤ (2R(x0 ) + θλ) ku0 (t)k22 + k∇z(t)k22 .
2
2
(3.15)
x∈Ω
From (3.13) we obtain
EJDE–2002/44 M. M. Cavalcanti, V. N. D. Cavalcanti & J. A. Soriano
7
We note that
Z t
k∇z(t)k22 =k∇u(t)k22 − 2
g(t − τ )(∇u(τ ), ∇u(t))dτ
0
Z t
Z t
g(t − τ )
g(t − s)(∇u(τ ), ∇u(s))ds dτ
+
0
0
Now we estimate each Rterm on the right-hand side of this inequality.
t
Estimate for J1 := −2 0 g(t − τ )(∇u(τ ), ∇u(t))dτ . We have,
Z t
|J1 | ≤k∇u(t)k22 + (
g(t − τ )k∇u(τ )k2 )2
0
Z t
≤k∇u(t)k22 + 2kgkL1 (0,∞)
g(t − τ ) k∇u(τ ) − ∇u(t)k22 + k∇u(t)k22 dτ.
0
Having in mind that kgkL1 (0,∞) < 1, we obtain
|J1 | ≤ 3k∇u(t)k22 + 2(g ∇u)(t).
(3.17)
Rt
Rt
Estimate for J2 := 0 g(t − τ )( 0 g(t − s)(∇u(τ ), ∇u(s))ds)dτ . We infer
Z t
|J2 | ≤(
g(t − τ )k∇u(τ )k22 )2
0
Z t
≤2kgkL1 (0,∞)
g(t − τ ) k∇u(τ ) − ∇u(t)k22 + k∇u(t)k22 dτ.
0
Consequently
|J2 | ≤ 2(g ∇u)(t) + 2k∇u(t)k22 .
(3.18)
Combining (3.15)-(3.18) taking (3.4) into account, it follows that
|ρ(t)| ≤ 20(l−1 + 1)(2R(x0 ) + θλ)e(t)
which, in view of (3.8) allow us to conclude the desired result. This completes
the proof.
Proposition 3.2 There exists δ1 = δ1 (ε) such that
e0ε (t) ≤ −δ1 e(t)
∀t ≥ 0
provided that kgkL1 (0,∞) is small enough.
Proof. Getting the derivative of ρ(t) given in (3.9) with respect to t and
substituting u00 = ∆z − f (x, t, u) − a(x)u0 in the obtained expression,
ρ0 (t) =2(∆z(t), h · ∇z(t)) − 2(f (x, t, u(t)), h · ∇z(t))
− 2(au0 (t), h · ∇z(t)) + 2(u0 (t), h · ∇z 0 (t)) + θ(∆z(t), z(t))
− θ(f (x, t, u(t)), z(t)) − θ(au0 (t), z(t)) + θku0 (t)k22
Z t
0
g 0 (t − τ )(u0 (t), u(τ ))dτ.
− θg(0)(u (t), u(t)) − θ
0
(3.19)
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Viscoelastic wave equations with localized damping
EJDE–2002/44
Next, we will estimate some terms on the right-hand side of identity (3.19).
Estimate for I1 := 2(∆z(t), h · ∇z(t)). Employing Gauss and Green formulas,
we deduce
Z
n Z
X
∂z ∂hi ∂z
I1 = − 2
dx + (div h)|∇z|2 dx
∂xk ∂xk ∂xi
Ω
i,k=1 Ω
(3.20)
Z
Z
∂z
2
− (h · ν)|∇z| dΓ +
(2h · ∇z)dΓ.
Γ
Γ ∂ν
Estimate for I2 := −2(u0 (t), h·∇z 0 (t)). Making use of Gauss formula and noting
that u0 = 0 on Γ,
Z
Z t
0 2
0
I2 = − (div h)|u | dx−2g(0)(u (t), h·∇u(t))−2
g 0 (t−τ )(u0 (t), h·∇u(τ ))dτ.
Ω
0
(3.21)
Estimate for I3 := θ(∆z(t), z(t)). Considering Green formula and observing
that z = 0 on Γ,
I3 = −θk∇z(t)k22 .
(3.22)
Estimate for I4 := −2(f (x, t, u(t)), h · ∇z(t)). Considering assumption (2.3),
taking (3.11) and Gauss formula into account and noting that u = 0 on Γ, it
follows that
Z
2n
2
I4 ≤
ku(t)kγ+2
+
(∇ψ · m)|u|γ+2 dx
γ+2
γ+2
γ+2 Ω
Z t
Z
(3.23)
n
X
∂u
+2
g(t − τ )
|mk ||
(x, τ )|dx dτ.
|u(x, t)|γ+1 ψ
∂xk
0
Ω
k=1
Estimate for I5 := −θ(f (x, t, u(t)), z(t)). In view of assumption (2.3),
Z t
I5 ≤ −θku(t)kγ+2
+
θ
g(t − τ )(|u(t)|γ u(t), u(τ ))dτ.
γ+2
(3.24)
0
Then, combining (3.19)-(3.24) we arrive at
n Z
X
0
ρ (t) ≤ − 2
i,k=1
Ω
∂z ∂hi ∂z
dx +
∂xk ∂xk ∂xi
Z
(div h)|∇z|2 dx
Ω
Z
(div h)|u0 |2 dx − 2g(0)(u0 (t), h · ∇u(t))
Z t
−2
g 0 (t − τ )(u0 (t), h · ∇u(τ ))dτ − θk∇z(t)k22
0
Z
2n
2
+(
− θ)ku(t)kγ+2
+
(∇ψ · m)|u|γ+2 dx
γ+2
γ+2
γ+2 Ω
Z t
Z
n
X
∂u
+2
g(t − τ )
|u(x, t)|γ+1 ψ
|mk ||
(x, τ )kdx dτ
∂x
k
0
Ω
−
Ω
k=1
(3.25)
EJDE–2002/44 M. M. Cavalcanti, V. N. D. Cavalcanti & J. A. Soriano
0
− 2(au (t), h · ∇z(t)) + θ
− θ(au0 (t), z(t)) +
Z
9
t
g(t − τ )(|u(t)|γ u(t), u(τ ))dτ
0
θku0 (t)k22
Z t
− θg(0)(u0 (t), u(t)) − θ
g 0 (t − τ )(u0 (t), u(τ ))dτ
0
Z
Z
∂z
− (h · ν)|∇z|2 dΓ +
(2h · ∇z)dΓ.
∂ν
Γ
Γ
Next we handle the boundary terms. Observe that since z = 0 on Γ we have
∂z
∂z
∂xk = ∂ν νk which implies
h · ∇z = (h · ν)
∂z
∂ν
and |∇z|2 = (
∂z 2
)
∂ν
on Γ.
From the above expressions and taking (1.2), (1.4) and (3.7) into account,
Z
Z
∂z
− (h · ν)|∇z|2 dΓ +
(2h · ∇z)dΓ
Γ
Γ ∂ν
Z
∂z
= (h · ν)( )2 dΓ
(3.26)
∂ν
Z
ZΓ
∂z
∂z
(m · ν)ψ( )2 dΓ ≤ 0.
=
(m · ν)ψ( )2 dΓ +
∂ν
∂ν
0
0
Γ\Γ(x )
Γ(x )
Then, from (3.25)-(3.26) and after some computations, we conclude that
2n
ρ0 (t) ≤(n − 2 − θ)k∇z(t)k22 + (θ − n)ku0 (t)k22 + (
− θ)ku(t)kγ+2
γ+2
γ+2
Z
Z
+n
[1 − ψ]|u0 |2 dx + (n − 2)
[ψ − 1]|∇z|2 dx
Q1
−2
i,k=1
−
Q1
n Z
X
Z
Ω
∂z
∂ψi ∂z
mi
dx +
∂xk
∂xk ∂xi
(∇ψ · m)|u0 |2 dx +
Q1 \Q0
+2
Z
0
t
g(t − τ )
Z
Z
2
γ+2
|u(x, t)|γ+1 ψ
Ω
(∇ψ · m)|∇z|2 dx
Q1 \Q0
Z
n
X
k=1
Z
(∇ψ · m)|u|γ+2 dx
Q1 \Q0
|mk ||
∂u
(x, τ )|dx dτ
∂xk
t
− 2g(0)(u0 (t), h · ∇u(t)) − 2
g 0 (t − τ )(u0 (t), h · ∇u(τ ))dτ
0
Z t
− 2(au0 (t), h · ∇z(t)) + θ
g(t − τ )(|u(t)|γ u(t), u(τ ))dτ
0
Z t
− θ(au0 (t), z(t)) − θg(0)(u0 (t), u(t)) − θ
g 0 (t − τ )(u0 (t), u(τ ))dτ.
0
(3.27)
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Viscoelastic wave equations with localized damping
EJDE–2002/44
Having in mind (3.12), (3.16) and adding and subtracting suitable terms in
order to obtain −k1 e(t) from (3.27), and supposing without loss of generality
that g(0) ≤ 1, we infer
Z t
k1
k1
ρ0 (t) ≤ − k1 e(t) − (
g(s)ds)k∇u(t)k22 + (g ∇u)(t)
2 0
2
Z t
Z t
+2
g(t − τ )k∇u(τ )k2 k∇u(t)k2 dτ + 2(
g(t − τ )k∇u(τ )k2 dτ )2
0
Z
Z 0
0 2
0
+ 2n
|u | dx + 3R(x ) max |∇ψ(x)|
|∇z|2 dx
x∈Ω
Q1
− θg(0)(u0 (t), u(t)) − θ
Z
+ R(x0 ) max |∇ψ(x)|
x∈Ω
Z
Q1 \Q0
t
g 0 (t − τ )(u0 (t), u(τ ))dτ.
0
|u0 |2 dx
Q \Q
1
Z0
2
0
+
R(x ) max |∇ψ(x)|
|u|γ+2 dx
γ+2
x∈Ω
Q1 \Q0
Z t
0
+ 2R(x0 )
g(t − τ )ku(t)kγ+1
2(γ+1) k∇u(τ )k2 dτ + 2|(u (t), h · ∇u(t))|
0
Z t
+2
|g 0 (t − τ )|ku0 (t)k2 kh · ∇u(τ )k2 dτ + 2|(au0 (t), h · ∇z(t))|
0
Z t
0
+θ
g(t − τ )ku(t)kγ+1
2(γ+1) ku(τ )k2 dτ + θ|(au (t), z(t))|
0
Z t
+ θ|(u0 (t), u(t))| + θ
|g 0 (t − τ )|ku0 (t)k2 ku(τ )k2 dτ.
0
(3.28)
Next, we analyze someRterms on the right hand side in the above the inequality.
t
Estimate for I5 := 2 0 g(t − τ )k∇u(τ )k2 k∇u(t)k2 dτ . Considering Cauchy1 2
Schwarz inequality and also employing the inequality ab ≤ 4η
a + ηb2 , for an
arbitrary η > 0, we obtain
Z t
1
2
|I5 | ≤
k∇u(t)k2 + ηkgkL1 (0,∞)
g(t − τ )k∇u(τ )k22 dτ.
2η
0
Having in mind that kgkL1 (0,∞) < 1, the last inequality yields
Z t
|I5 | ≤(2η)−1 k∇u(t)k22 + 2η(g ∇u)(t) + 2η(
g(s)ds)k∇u(t)k22
0
Z t
≤(2η)−1 k∇u(t)k22 + 4ηe(t) + 2η(
g(s)ds)k∇u(t)k22 .
(3.29)
0
Rt
Estimate for I6 := 2( 0 g(t − τ )k∇u(τ )k2 dτ )2 . Considering the Cauchy-Schwarz
inequality it holds that
Z t
|I6 | ≤ kgkL1 (0,∞)
g(t − τ )k∇u(τ )k22 dτ ≤ 4(g ∇u)(t) + 4k∇u(t)k22 (3.30)
0
EJDE–2002/44 M. M. Cavalcanti, V. N. D. Cavalcanti & J. A. Soriano
where in the last inequality we used kgkL1 (0,∞)
R < 1.
Estimate for I7 := 3R(x0 ) maxx∈Ω k∇ψ(x)| Q1 \Q0 |∇z|2 dx.
Setting
A = 3R(x0 ) max |∇ψ(x)|
11
(3.31)
x∈Ω
and taking (3.16) into account, we obtain, as in (3.29) and (3.30), the estimate
|I7 | ≤(3A + A(4η)−1 )k∇u(t)k22 + 4ηAe(t) + 2A(g ∇u)(t)
Z t
+ 2Aη(
g(s)ds)k∇u(t)k22 ,
(3.32)
o
where η > 0 is an arbitrary number.
R
Estimate for I8 := 2A(γ + 2)−1 Q1 \Q0 |u|γ+2 dx. Considering k0 > 0 such that
kvkγ+2 ≤ k0 k∇vk2 for all v ∈ H01 (Ω), and taking (3.4) into account, we deduce
|I8 | ≤ 2(γ+2)/2 Ak0γ+2 (γ + 2)−1 l−γ/2 [E(0)]γ/2 k∇u(t)k22 .
(3.33)
Rt
Estimate for I9 := 2R(x0 ) 0 g(t − τ )ku(t)kγ+1
2(γ+1) k∇u(τ )k2 dτ . Considering k1 >
0 such that kvk2(γ+1) ≤ k1 k∇vk2 for all v ∈ H01 (Ω), making use of the inequality
1 2
ab ≤ ηa2 + 4η
b for an arbitrary η > 0 and also the Cauchy-Schwarz one, we
obtain
2(γ+1)
R2 (x0 )2γ η −1 l−γ [E(0)]γ k∇u(t)k22 + 2ηl−1 e(t)
Z t
+ 2η(
g(s)ds)k∇u(t)k22 .
|I9 | ≤k1
(3.34)
o
Estimate for I10 := 2|(u0 (t), h · ∇u(t))|. We have
|I10 | ≤ 4ηe(t) + R2 (x0 )η −1 k∇u(t)k22 .
(3.35)
Rt
Estimate for I11 := 2 0 |g 0 (t − τ )|ku0 (t)k2 kh · ∇u(τ )k2 dτ . Analogously we have
done before and taking the assumption (2.8) into account, we arrive at
|I11 | ≤ 2ηe(t) + 2ξ1 R2 (x0 )η −1 (g ∇u)(t) + 2ξ1 R2 (x0 )η −1 k∇u(t)k22 .
(3.36)
Estimate for I12 := 2|(au0 (t), h·∇z(t))|. From Cauchy-Schwarz inequality, mak1 2
ing use of the inequality ab ≤ ηa2 + 4η
b (η > 0 arbitrary) and considering (3.16),
(3.17) and (3.18), we deduce
|I12 | ≤ (2η)−1 kakL∞ (Ω) ku0 (t)k2L2 (Q1 \Q0 ) + 4ηR(x0 )(6l−1 + 3)e(t).
Estimate for I13 := θ
Rt
0
2(γ+1) γ −γ
I13 ≤ θ2 k1
2 l
(3.37)
g(t − τ )ku(t)kγ+1
2(γ+1) ku(τ )k2 dτ . Analogously, we have
(4η)−1 k∇u(t)k22 + 4ηλ2 e(t) + 2ηλ2
Z
t
g(s)dsk∇u(t)k22 ,
o
(3.38)
12
Viscoelastic wave equations with localized damping
EJDE–2002/44
where λ > 0 comes from the Poincaré inequality kvk2 ≤ λk∇vk2 for all v ∈
H01 (Ω).
Estimate for I14 := θ|(au0 (t), z(t))|. Similarly, we deduce
I14 ≤ 4ηθ2 λ2 kakL∞ (Ω) e(t) + 3(4η)−1 k∇u(t)k22 .
(3.39)
Estimate for I15 := θ|(u0 (t), u(t))|. We have
|I15 | ≤ 2ηe(t) + θ2 λ2 (4η)−1 k∇u(t)k22 .
Estimate for I16 := θ
Rt
0
(3.40)
|g 0 (t − τ )|ku0 (t)k2 ku(τ )k2 dτ . It holds that
|I16 | ≤ 2ηe(t) + θ2 λ2 ξ1 (2η)−1 (g ∇u)(t) + θ2 λ2 ξ 2 (2η)−1 k∇u(t)k22 .
(3.41)
Combining (3.28)-(3.41),
ρ0 (t) ≤ − [k1 − ηL]e(t)
−1
− [2
Z t
k1 − 4η(1 + A)](
g(s)ds)k∇u(t)k22 + M (η)(g ∇u)(t) (3.42)
o
+ N (η)k∇u(t)k22 + K(η)ku0 (t)k2L2 (ω) ,
where
L = 10 + 2l−1 + 4R(x0 )(6l−1 + 3) + 4λ2 + 4θλ2 kakL∞ (Ω) ,
M (η) = 2−1 k1 + 4 + 2A + 2η −1 ξ12 R2 (x0 ) + θ2 λ2 ξ12 (2η)−1 ,
N (η) =3A + 4 + (γ + 2)−1 2(γ+2)/2 k0γ+2 l−γ/2 [E(0)]l/2 + θ2 λ2 (4η)−1
+ θ2 λ2 ξ12 (2η)−1 + (4η)−1 4(1 + A)
2(γ+1) γ −γ
2(γ+1) γ −γ
+ R2 (x0 ) 4k1
2 l [E(0)]γ + 2ξ12 + 1) + θ2 k1
2 l
,
K(η) = (2η)−1 + 2η + 3−1 A.
Choosing η > 0 sufficiently small such that
k2 = k1 − ηL > 0 and 2−1 k1 − 4η(1 + A) ≥ 0,
from (3.6), (3.8), (3.9), (3.42) and considering the assumption (2.8), we obtain
e0ε (t) =e0 (t) + ερ0 (t)
−1
≤ − εk2 e(t) + ξ1 2
Z
t
g(s)dsk∇u(t)k22 − (a0 − εK)ku0 (t)k2L2 (ω) − (2−1 ξ2
0
− εM )(g ∇u)(t) − (2−1 g(0) − εN )k∇u(t)k22 .
(3.43)
Choosing ε > 0 small enough such that
a0 − εK ≥ 0,
2−1 g(0) − εN ≥ 0
and 2−1 ξ2 − εM ≥ 0,
EJDE–2002/44 M. M. Cavalcanti, V. N. D. Cavalcanti & J. A. Soriano
13
from (3.43) we deduce
e0ε (t) ≤ − εk2 − ξ1 kgkL1 (0,∞) l−1 e(t).
(3.44)
For a fixed ε > 0 sufficiently small such that the Propositions 3.1 and 3.2
hold and considering kgkL1 (0,∞) small enough, from (3.44) we conclude that
e0ε (t) ≤ −δ1 e(t) which completes the proof of Proposition 3.2.
Combining the Propositions 3.1 and 3.2, we deduce the exponential rate of
decay.
Acknowledgments. The authors would like to thank the anonymous referees
for their important remarks and comments which allow us to correct, improve,
and clarify several points in this article.
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Marcelo M. Cavalcanti (e-amil: marcelo@gauss.dma.uem.br)
Valéria N. Domingos Cavalcanti (e-mail: valeria@gauss.dma.uem.br)
Juan A. Soriano (e-mail: soriano@gauss.dma.uem.br)
Departamento de Matemática
Universidade Estadual de Maringá
87020-900 Maringá - PR, Brasil