Electronic Journal of Differential Equations, Vol. 2007(2007), No. 128, pp. 1–18.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu (login: ftp)
ON THE WAVE EQUATIONS WITH MEMORY IN
NONCYLINDRICAL DOMAINS
MAURO L. SANTOS
Abstract. In this paper we prove the exponential and polynomial decays
rates in the case n > 2, as time approaches infinity of regular solutions of the
wave equations with memory
Z t
b
utt − ∆u +
g(t − s)∆u(s)ds = 0 in Q
0
b is a non cylindrical domains of Rn+1 , (n ≥ 1). We show that the diswhere Q
sipation produced by memory effect is strong enough to produce exponential
decay of solution provided the relaxation function g also decays exponentially.
When the relaxation function decay polynomially, we show that the solution
decays polynomially with the same rate. For this we introduced a new multiplier that makes an important role in the obtaining of the exponential and
polynomial decays of the energy of the system. Existence, uniqueness and
regularity of solutions for any n ≥ 1 are investigated. The obtained result
extends known results from cylindrical to non-cylindrical domains.
1. Introduction
Let Ω be an open bounded domain of Rn containing the origin and having
C boundary. Let γ : [0, ∞[→ R be a continuously differentiable function. See
hypothesis (1.11)–(1.13) on γ. Consider the family of subdomains {Ωt }0≤t<∞ of
Rn given by
Ωt = T (Ω), T : y ∈ Ω 7→ x = γ(t)y
2
whose boundaries are denoted by Γt , and let the noncylindrical domain of Rn+1 be
b = ∪0≤t<∞ Ωt × {t}
Q
with lateral boundary
b = ∪0≤t<∞ Γt × {t}.
Σ
Let us consider the Hilbert space L2 (Ω) endowed with the inner product
Z
(u, v) =
u(x)v(x)dx
Ω
2000 Mathematics Subject Classification. 35K55, 35F30, 34B15.
Key words and phrases. Wave equation; noncylindrical domain; memory dissipation.
c
2007
Texas State University - San Marcos.
Submitted March 8, 2007. Published October 2, 2007.
1
2
M. L. SANTOS
EJDE-2007/128
and corresponding norm kuk2L2 (Ω) = (u, u). We also consider the Sobolev space
H 1 (Ω) endowed with the scalar product
(u, v)H 1 (Ω) = (u, v) + (∇u, ∇v).
We define the subspace of H 1 (Ω), denoted by H01 (Ω), as the closure of C0∞ (Ω) in
the strong topology of H 1 (Ω). By H −1 (Ω) we denote the dual space of H01 (Ω).
This space endowed with the norm induced by the scalar product
((u, v))H01 (Ω) = (∇u, ∇v)
is a Hilbert space; due to the Poincaré inequality
kuk2L2 (Ω) ≤ Ck∇uk2L2 (Ω) .
For 1 ≤ p < ∞, we define
kukpLp (Ω)
Z
|u(x)|p dx,
=
Ω
and for p = ∞,
kukL∞ (Ω) = esssup |u(x)|.
x∈Ω
In this work we study the existence and uniqueness of strong global solutions, as
well the exponential and polynomial decays of the energy for the wave equation
Z t
b
utt − ∆u +
g(t − s)∆u(s)ds = 0 in Q,
(1.1)
0
u=0
u(x, 0) = u0 (x),
on
X
d
,
(1.2)
ut (x, 0) = u1 (x)
in
Ω0 ,
(1.3)
where u is the transverse displacement. The method used for proving existence and
uniqueness is based on transforming our problem into another initial boundary value
problem defined over a cylindrical domain, whose sections are not time-dependent.
This is done using a suitable change of variable. Then we show the existence and
uniqueness for this new problem. Our existence result on non-cylindrical domains
will follows using the inverse transformation. That is, using the diffeomorphism
b → Q defined by
τ :Q
b → Q, (x, t) ∈ Ωt 7→ (y, t) = ( x , t)
τ :Q
(1.4)
γ(t)
b defined by
and τ −1 : Q → Q
τ −1 (y, t) = (x, t) = (γ(t)y, t).
(1.5)
Denoting by v the function
v(y, t) = u ◦ τ −1 (y, t) = u(γ(t)y, t)
the initial boundary value problem (1.1)–(1.3) becomes
Z t
−2
vtt − γ ∆v +
g(t − s)γ −2 (s)∆v(s)ds − A(t)v + a1 · ∇∂t v + a2 · ∇v = 0
(1.6)
in Q,
0
(1.7)
v|Γ = 0,
v|t=0 = v0 ,
vt |t=0 = v1
(1.8)
in Ω,
(1.9)
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WAVE EQUATIONS WITH MEMORY
3
where
A(t)v =
n
X
∂yi (aij ∂yj v)
i,j=1
and
aij (y, t) = −(γ 0 γ −1 )2 yi yj
(i, j = 1, . . . , n),
0 −1
a1 (y, t) = −2γ γ
a2 (y, t) = −γ
−2
00
y,
(1.10)
0 2
y(γ γ + (γ ) (n − 1)).
To show the existence of strong solution we will use the following hypotheses:
γ0 ≤ 0
∞
if n > 2,
γ(·) ∈ L (0, ∞),
γ0 ≥ 0
inf
0≤t<∞
if n ≤ 2,
γ(t) = γ0 > 0,
γ 0 ∈ W 2,∞ (0, ∞) ∩ W 2,1 (0, ∞).
(1.11)
(1.12)
(1.13)
b is decreasing if n > 2 and inNote that the assumption (1.11) means that Q
creasing if n ≤ 2 in the sense that when t > t0 and n > 2 then the projection of
Ωt0 on the subspace t = 0 contain the projection of Ωt on the same subspace. The
opposite holds in the case n ≤ 2.
The above method was introduced by Dal Passo and Ughi [14] to study certain
class of parabolic equations in non cylindrical domains.
We only obtained the exponential and polynomial decays of solution for our
problem for the case n > 2. The main difficulty to prove the exponential and
polynomial decays for the case n ≤ 2 are in the Lemma 3.3, 3.4 and 3.5, where
b To control those terms
appears the boundary terms, since we worked directly in Q.
we used the hypothesis (1.11). Therefore the case n ≤ 2 is an important open
problem.
The equation (1.1) can be seen as a model of propagation of seismic waves, where
the function g represents the medium of propagation of waves. In the considered
case, the medium is elastic.
The wave equations with dissipation was studied by several authors. All of them
consider essentially two types of dissipative mechanisms (or a combination of them):
(a) The frictional dissipation, obtained by introducing a frictional damping that
can be acting either on the boundary or in a neighborhood of the boundary;
(b)The viscoelastic dissipation given by the memory effects as in [11, 16, 17, 18].
The frictional damping is the simplest dissipative mechanism when one is working
either in the whole domain Ω or over a strategic part of the domain (locally). It
was proved by [1, 2, 3, 4, 7, 12, 13, 19, 20] that the first-order energy decays
exponentially to zero as time goes to infinity.
Finally, the memory effect produces a suitable dissipative mechanism which depends on the ralaxation function (see [16, 17, 18]). They proved that the energy
decays uniformly exponentially or algebraically with the same rate of decay as the
relaxation function.
In a non cylindrical domain, the problem of existence, uniqueness and exponential decay of the solutions for the wave equations with memory and weak damping
was studied by Ferreira and Santos [5]. They proved that the energy decays uniformly exponentially to zero as time goes to infinity.
4
M. L. SANTOS
EJDE-2007/128
The main result of this paper is to extend the result obtained by Ferreira and
Santos [5]. That is, to remove the term ut of the equation
Z t
g(t − s)∆u(s)ds + ut = 0
utt − ∆u +
0
in [5].
R
The main technical difficulty it is to control the term Ωt |ut |2 dx of the total energy of the system (1.1)–(1.3). To solve this problem we introduced a new multiplier
(g ∗ u)t , that makes key role to obtain the exponential and polynomial decays.
The present paper extends the results from cylindrical to non cylindrical domains.
To see the dissipative properties of the system we have to construct a suitable
functional whose derivative is negative and is equivalent to the first order energy.
This functional is obtained using the multiplicative technique following Komornik
[6], Lions [8] and Rivera [10].
The notation we use in this paper are standard and can be found in Lion’s book
[8, 9]. In the sequel by C (sometimes C1 , C2 , . . . ) we denote various positive constants which do not depend on t or on the initial data. This paper is organized as
follows. In section 2 we prove the existence, regularity and uniqueness of solutions.
We use Galerkin approximation, Aubin-Lions theorem, energy method introduced
by Lions [8] and some technical ideas to show existence regularity and uniqueness
of regular solution for the problem (1.1)–(1.3). Finally, in the section 3 and 4, we
establish the results on the exponential and polynomial decays of the regular solution to the problem (1.1)–(1.3). We use the technique of the multipliers introduced
by Kormornik [6], Lions [8] and Rivera [11] coupled with some technical lemmas
and some technical ideas.
2. Existence and Regularity
In this section we shall study the existence and regularity of solutions for the
system (1.1)–(1.3). For this we assume that the kernel g : R+ → R+ is in W 2,1 (0, ∞)
and satisfy
γ1−2 −
Z
g, −g 0 ≥ 0,
(2.1)
g(s)γ −2 (s)ds = β > 0,
(2.2)
∞
0
where γ1 = sup0<t<∞ γ(t). Note that (2.2) implies
Z t
1
−2
β ≤ γ(t) −
g(s)γ −2 (s)ds ≤ 2 .
γ
0
0
On the other hand, since all the dissipation of the system is contained only in the
memory term, we also have to require that g 6= 0, and this explains (2.2).
Typical examples of functions γ and g satisfying (1.11)–(1.13) and (2.1)–(2.2)
are
γ(t) = e−σ0 t + σ1 , g(t) = σ2 e−σ3 t
where σi , i = 0, 1, 2, 3, are positive constants. To simplify our analysis, we define
the binary operator
Z t
Z
∇u(t)
−2
g
=
g(t − s)γ (s)
|∇u(t) − ∇u(s)|2 dx ds.
γ(t)
0
Ω
EJDE-2007/128
WAVE EQUATIONS WITH MEMORY
5
With this notation we have the following statement.
Lemma 2.1. For v ∈ C 1 (0, T : H 1 (Ω)) and g ∈ C 1 (0, ∞) we have
Z Z t
Z
g(t − s)
1 g(t)
1
∇v
∇vds
·
∇v
dx
=
−
|∇v|2 dx + g 0 t
2 (s)
2 (0)
γ
2
γ
2
γ
Ω 0
Ω
Z t
Z
1 d
∇v
g(t − s)
−
g
−(
ds)
|∇v|2 dx
2
2 dt
γ
γ (s)
0
Ω
Z t
Z
0
γ (s)
g(t − s) 3
|∇u|2 dx ds.
+
γ (s) Ω
0
The proof of this lemma follows by differentiating the term
Z t
Z
∇u(t)
g(t − s)
g
−
|∇u|2 dxds.
γ(t)
γ 2 (s) Ω
0
The well-posedness of system (1.7)-(1.9) is given by the following theorem.
Theorem 2.2. Let us take v0 ∈ H01 (Ω) ∩ H 2 (Ω), v1 ∈ H01 (Ω) and let us suppose
that assumptions (1.11)–(1.13) and (2.1)–(2.2) hold. Then there exists a unique
solution v of the problem (1.7)-(1.9) satisfying
v ∈ L∞ (0, ∞ : H01 (Ω) ∩ H 2 (Ω)),
vt ∈ L∞ (0, ∞ : H01 (Ω)),
vtt ∈ L∞ (0, ∞ : L2 (Ω)).
Proof. The main idea is to use the Galerkin method. To do this let us take a basis
{wj }j∈N to H01 (Ω) ∩ H 2 (Ω) which is orthonormal in L2 (Ω) and we represent by Vm
the space generated by w1 , w2 , . . . , wm . Let us denote by
v0m =
m
X
(v0 , wj )wj ,
v1m =
m
X
(v1 , wj )wj .
j=1
j=1
Note that for any (v0 , v1 ) ∈ (H01 (Ω) ∩ H 2 (Ω)) × H01 (Ω), we have v0m → v0 strong in
H01 (Ω) ∩ H 2 (Ω) and v1m → v1 strong in H01 (Ω).
Standard results on ordinary differential equations imply the existence of a local
solution v m of the form
m
X
v m (t) =
gjm (t)wj ,
j=1
to the system
Z
Z
Z Z t
m
vtt
wj dy − γ −2
∆v m wj dy +
g(t − s)γ −2 (s)∇v m (s) · ∇wj dsdy
Ω
Ω
Ω 0
Z
Z
Z
m
m
+
A(t)v wj dy +
a1 · ∇vt wj dy +
a2 · ∇v m wj dy = 0, (j = 1, . . . , m),
Ω
Ω
Ω
(2.3)
m
v (x, 0) =
v0m ,
vtm (x, 0)
=
v1m .
(2.4)
The extension of this solution to the whole interval [0, ∞) is a consequence of the
a priori estimate below.
6
M. L. SANTOS
EJDE-2007/128
0
(t), summing up the prodFirst estimate. Multiplying the equation (2.3) by gjm
uct result in j = 1, 2, . . . , m, and using the Lemma 2.1 we get
Z
Z
Z
1 d m
£1 (t, v m ) +
A(t)v m vtm dy +
a1 · ∇vtm vtm dy +
a2 · ∇v m vtm dy
2 dt
Ω
Ω
Ω
1 0 ∇v m
1 g(t)
m 2
= − 2 k∇v kL2 (Ω) + g 2 γ (0)
2
γ
Z t
Z
0
γ
− 3 k∇v m k2L2 (Ω) +
g(t − s)γ 0 (s)γ −3 (s)ds
|∇v m |2 dx,
γ
0
Ω
where
m
m 2
£m
1 (t, v ) = kvt kL2 (Ω) +
1
−
γ 2 (t)
Z
0
t
g(t − s) ∇v m
ds k∇v m k2L2 (Ω) + g
.
2
γ (s)
γ
Taking into account (1.11), (1.13), (2.1) and (2.2) we obtain
1 d m
£ (t, v m ) ≤ C(|γ 0 | + |γ 00 |)£m
(2.5)
1 (t).
2 dt 1
Integrating the inequality (2.5), taking account (1.13) and using Gronwall’s Lemma
we get
m
£m
∀m ∈ N, ∀t ∈ [0, T ].
(2.6)
1 (t, v ) ≤ C,
Second estimate. From equation (2.3) we get
m
kvtt
(0)k2L2 (Ω) ≤ C,
∀m ∈ N.
(2.7)
Differentiating the equation (2.3) with respect to the time, we obtain
Z
Z
Z
Z
g(0)
γ0
m
vttt
wj dy − γ −2
∆vtm wj dy + 2 3
∆v m wj dy − 2
∆v0m wj dy
γ
γ
(0)
Ω
Ω
Ω
Ω
Z Z t
Z
d
(2.8)
(A(t)v m )wj dy
+
g 0 (t − s)γ −2 (s)∇v m (s) · ∇wj dsdy +
Ω 0
Ω dt
Z
Z
d
d
+
(a1 · ∇vtm )wj dy +
(a2 · ∇v m )wj dy = 0.
dt
dt
Ω
Ω
00
Multiplying (2.8) by gjm
(t), summing up the product result in j = 1, 2, . . . , m and
using similar arguments as (2.6) we obtain
Z t
m
m
£m
(t,
v
)
+
(2.9)
kvss
(s)k2L2 (Ω) ds ≤ C, ∀t ∈ [0, T ], ∀m ∈ N.
1
t
0
The first and second a priori estimates allow us to obtain a subsequence of (vm )
which from now on will be also denoted by (vm ) and a function v : Ω × (0, ∞) → R
satisfying:
vm → v
weak star in L∞ (0, ∞; H01 (Ω))
vtm → v
weak star in L∞ (0, ∞; H01 (Ω))
m
vtt
→ vtt
weak star in L∞ (0, ∞; L2 (Ω)).
The above convergence allows us to pass to the limit in the problem (2.3)-(2.4).
Letting m → ∞ in the equation (2.3) we conclude that
Z t
−2
vtt − γ ∆v +
g(t − s)γ −2 (s)∆v(s)ds − A(t)v + a1 · ∇∂t v + a2 · ∇v = 0
0
EJDE-2007/128
WAVE EQUATIONS WITH MEMORY
7
in L∞ (0, ∞ : L2 (Ω)). Therefore, using the elliptic regularity, we have that
v ∈ L∞ (0, ∞ : H01 (Ω) ∩ H 2 (Ω)).
Uniqueness. Suppose we have two solutions v and vb in the conditions of Theorem
2.2. Then φ = v − vb satisfies the same conditions and φ(0) = 0, φt (0) = 0. Let us
prove that φ = 0 on Ω × [0, ∞[.
Multiplying the equations (1.7) by φt , summing up the product result and using
the Lemma 2.1 we get
1 d
£1 (t, φ) ≤ C(|γ 0 | + |γ 00 |)£1 (t),
2 dt
where
£1 (t, φ) =
kφt k2L2 (Ω)
1
−
+( 2
γ (t)
Z
0
t
g(t − s)
∇φ
ds)k∇φk2L2 (Ω) + g
.
γ 2 (s)
γ
Integrating with respect to the time the above inequality and applying Gronwall’s
inequality we conclude that φ = 0 on Ω × [0, ∞[.
To show the existence in non cylindrical domains, we return to our original
problem in the non cylindrical domains by using the change variable given in (1.4)
b Let v be the solution obtained from Theorem 2.2 and
by (y, t) = τ (x, t), (x, t) ∈ Q.
u defined by (1.6), then u belongs to the class
u ∈ L∞ (0, ∞ : H01 (Ωt )),
∞
ut ∈ L (0, ∞ : H01 (Ωt )),
utt ∈ L∞ (0, ∞ : L2 (Ωt )).
(2.10)
(2.11)
(2.12)
Denoting by
u(x, t) = v(y, t) = (v ◦ τ )(x, t),
from (1.6) it is easy to see that u satisfies
Z t
utt − ∆u +
g(t − s)∆u(s)ds = 0 in L∞ (0, ∞ : L2 (Ωt )).
(2.13)
0
Using regularity elliptic, we obtain
u ∈ L∞ (0, ∞ : H01 (Ωt ) ∩ H 2 (Ωt )).
(2.14)
Let u1 , u2 be two solutions to (1.1), and v1 , v2 be the functions obtained through
the diffeomorphism τ given by (1.4). Then v1 , v2 are the solutions to (1.7). By the
uniqueness result Theorem 2.2, we have v1 = v2 , so u1 = u2 . Therefore, we have
the following result.
Theorem 2.3. Let us take u0 ∈ H01 (Ω0 )∩H 2 (Ω0 ), u1 ∈ H01 (Ω0 ) and let us suppose
that assumptions (1.11)–(1.13) and (2.1)–(2.2) hold. Then there exists a unique
solution u of the problem (1.1)–(1.3) satisfying (2.10)-(2.14).
8
M. L. SANTOS
EJDE-2007/128
3. Exponential Rate of Decay
In this section we show that the solution of system (1.1)–(1.3) decays exponentially. To this end we will assume that the memory g satisfies:
g 0 (t) ≤ −C1 g(t)
(3.1)
for all t ≥ 0, with positive constant C1 . Additionally, we assume that the function
γ(·) satisfies the conditions
γ 0 ≤ 0,
t ≥ 0,
n > 2,
1
0 < max |γ 0 (t)| ≤ ,
0≤t<∞
d
(3.2)
(3.3)
where d = diam(Ω). The condition (3.3) implies that our domains is “time like” in
the sense that
|ν| < |ν|
where ν and ν denote the t-component and x-component of the outer unit normal
b To facilitate our calculations we introduce the notation
of Σ.
Z Z t
(g∇u)(t) =
g(t − s)|∇u(t) − ∇u(s)|2 ds dx.
Ωt
0
First, we prove the following two lemmas that will be used in the sequel.
Lemma 3.1. Let F (·, ·) be the smooth function defined in Ωt × [0, ∞[, (t ∈ [0, ∞[.
Then
Z
Z
Z
d
d
γ0
F (x, t)dx =
F (x, t)dx +
F (x, t)(x · ν)dΓt ,
(3.4)
dt Ωt
γ Γt
Ωt dt
where ν is the x-component of the unit normal exterior ν.
Proof. By a change variable x = γ(t)y, y ∈ Ω, we have
Z
Z
d
d
F (x, t)dx =
F (γ(t)y, t)γ n (t)dy
dt Ωt
dt Ω
Z
n Z
X
γ0
∂F n
∂F n
= (
)γ (t)dy +
xi (
)γ (t)dy
γ
∂t
Ω ∂t
i=1 Ω
Z
+n
γ 0 (t)γ n−1 (t)F (γ(t)y, t)dy.
Ω
If we return at the variable x, we get
Z
Z
Z
Z
d
∂F
γ0
γ0
F (x, t)dx =
dx +
x · ∇F (x, t)dx + n
F (x, t)dx.
dt Ωt
γ Ωt
γ Ωt
Ωt ∂t
Integrating by part in the last equality we obtain the formula (3.4).
Lemma 3.2. For any functions g ∈ C 1 (R+ ) and u ∈ C 1 ((0, T ) : H 2 (Ωt )), we have
Z Z t
g(t − s)∇u(s) · ∇ut ds dx
Ωt
0
Z
Z t
Z
i
1
1
1 dh
= − g(t)
|∇u(t)|2 dx + g 0 ∇u −
g∇u − (
g(s)ds)
|∇u|2
2
2
2 dt
Ωt
0
Ωt
0 Z Z t
γ
+
g(t − s)|∇u(t) − ∇u(s)|2 (ν · x)dΓt
2γ Γt 0
EJDE-2007/128
−
γ0
2γ
WAVE EQUATIONS WITH MEMORY
Z
9
t
Z
g(t − s)|∇u(t)|2 (ν · x)dΓt .
0
Γt
Proof. Differentiating the term g∇u and applying the lemma 3.1 we obtain
Z Z t
d
g∇u =
g 0 (t − s)|∇u(t) − ∇u(s)|2 ds dx
dt
Ωt 0
Z Z t
−2
g(t − s)∇u(s) · ∇ut ds dx
0
Ωt
t
Z
+
g(t − s)ds
Z
0
0
γ
γ
+
Ωt
Z
Γt
Z
d
|∇u(t)|2 dx
dt
t
g(t − s)|∇u(t) − ∇u(s)|2 (x · ν)dsdΓt .
0
From where it follows that
Z Z t
2
g(t − s)∇u(s) · ∇ut ds dx
Ωt
0
Z t
Z
d
|∇u(t)|2 dx}
g(t − s)ds
= − {g∇u −
dt
0
Ωt
Z Z t
Z
+
g 0 (t − s)|∇u(t) − ∇u(s)|2 ds dx − g(t)
Ωt 0
0 Z
+
γ
γ
γ0
−
2γ
Γt
Z
Γt
|∇u(t)|2 dx
Ωt
t
Z
g(t − s)|∇u(t) − ∇u(s)|2 (x · ν)dsdΓt
0
Z
t
g(t − s)|∇u(t)|2 (ν · x)dΓt .
0
The proof is complete.
Let us introduce the functional
Z t
E(t) = kut k2L2 (Ωt ) + 1 −
g(s)ds k∇uk2L2 (Ωt ) + g∇u.
0
H01 (Ω0 )
Lemma 3.3. Let us take u0 ∈
∩ H 2 (Ω0 ), u1 ∈ H01 (Ω0 ) and let us suppose
that assumptions (1.11)–(1.13) and (2.1)–(2.2) hold. Then any regular solution of
system (1.1)–(1.3) satisfies
Z
γ0
d
E(t) −
(ν · x)(|ut |2 + |∇u|2 )dΓt
dt
Γt γ
Z
Z t
γ0
−
g(t − s)|∇u(t) − ∇u(s)|2 dsdΓt
(ν · x)
γ
Γt
0
Z
1
1
=−
g(t)|∇u|2 dx + g 0 ∇u.
2 Ωt
2
Proof. Multiplying the equation (1.1) by ut and integrating over Ωt we get
Z
Z
Z Z t
1
d
1
d
|ut |2 dx +
|∇u|2 dx −
g(t − s)∇u(s) · ∇ut ds dx = 0.
2 Ωt dt
2 Ωt dt
Ωt 0
10
M. L. SANTOS
EJDE-2007/128
Using Lemmas 3.1 and 3.2 we obtain
Z t
Z
Z
d
γ0
γ0 g(s)ds
1−
(ν · x)|∇u|2 dΓt
E(t) −
(ν · x)|ut |2 dΓt −
dt
2γ Γt
2γ
0
Γt
Z Z t
γ0
(ν · x)g(t − s)|∇u(·, t) − ∇u(·, s)|2 dsdΓt
−
2γ Γt 0
Z
1
1
= − g(t)
|∇u|2 dx + g 0 ∇u.
2
2
Ω
The proof is complete.
For the estimate of the term Ωt |ut |2 dx we introduced the functional
Z
Z
1
|g ∗ ∇u|2 dx
ϕ(t) = −
ut (g ∗ u)t dx +
2 Ω
Ωt
R
where (g ∗ u)t = g(0)u + g 0 ∗ u.
Lemma 3.4. Let us take u0 ∈ H01 (Ω0 ) ∩ H 2 (Ω0 ), u1 ∈ H01 (Ω0 ) and let us suppose
that assumptions (1.11)–(1.13) and (2.1)–(2.2) hold. Then any regular solution of
system (1.1)–(1.3) satisfies
Z
1 d
γ0
ϕ(t) −
(ν · x)|g ∗ ∇u|2 dΓt
2 dt
2γ Γt
Z
Z
Z
3g(0)
g(0)
2
2
|ut | dx +
|∇u| dx + g(t)
|∇u|2 dx
≤−
2 Ωt
2
Ωt
Ωt
Rt
Z
( |g 0 (s)|ds) 0
|g 0 (t)|2
|u0 |2 dx
|g |∇u +
+ 0
2g(0)
g(0) Ωt
Rt
Z
( 0 |g 0 (s)|ds) 0
2
− g(t)
|ut | dx +
|g |ut .
g(0)
Ωt
Proof. From the equation (1.1) and using the fact that u = 0 on the boundary we
get
Z
d
−
ut (g ∗ u)t dx
dt Ωt
Z
Z
Z
=
(−∆u + g ∗ ∆u)(g ∗ u)t dx − g(0)
|ut |2 dx −
ut (g 0 ∗ u)dx
Ωt
Ωt
Ωt
Z
Z
Z
(3.5)
1
d
|g ∗ ∇u|2 dx
= g(0)
|∇u|2 dx +
∇u · (g 0 ∗ ∇u)dx −
2 Ωt dt
Ωt
Z
ZΩt
− g(0)
|ut |2 dx −
ut (g 0 ∗ u)t dx.
Ωt
Ωt
Using Lemma 3.1 we have
Z
Z
Z
1
d
1 d
γ0
2
2
|g ∗ ∇u| dx =
|g ∗ ∇u| dx −
(ν · x)|g ∗ ∇u|2 dΓt .
2 Ωt dt
2 dt Ωt
2γ Γt
Define
Z
Z
∇u ·
I1 :=
Ωt
0
t
g 0 (t − s)∇u(s)ds dx;
(3.6)
EJDE-2007/128
WAVE EQUATIONS WITH MEMORY
then we have
Z
Z
I1 =
g(t)|∇u|2 dx +
∇u ·
Ωt
Z
≤(
t
Z
Ωt
t
g 0 (t − s)(∇u(·, t) − ∇u(·, s))ds dx
0
Z
Z
|∇u|2 dx)1/2 (
|g 0 (s)|ds)1/2 (|g 0 |∇u)1/2 + g(t)
Ωt
11
0
(3.7)
|∇u|2 dx.
Ωt
Define
Z
I2 := −
ut (g ∗ u)t dx
Ω
Z
=−
ut (g 0 (t)u0 + g 0 ∗ ut )dx
Ωt
Z
Z
=−
g 0 (t)ut u0 dx −
g(t)|ut |2 dx
Ωt
Ωt
Z
t
Z
−
g 0 (t − s)(ut (·, s) − ut (·, t))ds dx;
ut
Ωt
0
then thanks to the Young inequality, we obtain
|I2 | ≤
g(0)
2
Z
|ut |2 dx+
Ωt
|g 0 |2
g(0)
Z
|u0 |2 dx−
Ωt
Z
Ωt
|ut |2 dx+
|g(t) − g(0)| 0
|g |ut . (3.8)
g(0)
Substituting the inequalities (3.6), (3.7) and (3.8) into (3.5) we obtain the conclusion of lemma.
For the estimate of the term
η(t) :=
1
2
Z
R
Ωt
|g∗∇u|2 dx we introduced the following functional
Z
gut dx −
Ωt
Ωt
Z
0
t
Z
1
g(s)ds |ut |2 dx −
|g ∗ ∇u|2 dx.
2 Ωt
Lemma 3.5. Let us take u0 ∈ H01 (Ω0 ) ∩ H 2 (Ω0 ), u1 ∈ H01 (Ω0 ) and let us suppose
that assumptions (1.11)–(1.13) and (2.1)–(2.2) hold. Then any regular solution of
system (1.1)–(1.3) satisfies
Z
d
γ0
η(t) +
(ν · x)|g ∗ ∇u|2 dΓt
dt
2γ Γt
Z Z t
γ0
−
(ν · x)g(t − s)|ut (·, t) − ut (·, s)|2 dsdΓt
2γ Γt 0
Z
Z t
γ0
(ν · x)(
+
g(s)ds)|ut |2 dΓt
2γ Γt
0
Z
Z
g(0)
2
0
≤ −g(t)
|ut | dx + g ut −
|∇u|2 dx
2 Ωt
Ωt
Rt
Z
( |g 0 (s)|ds) 0
+ g(t)
|∇u|2 dx + 0
|g |∇u.
2g(0)
Ωt
12
M. L. SANTOS
EJDE-2007/128
Proof. Multiplying the equation (1.1) by g ∗ ut and using similar argument as in
the lemma 3.4 we obtain
Z
d
γ0
η(t) +
(ν · x)|g ∗ ∇u|2 dΓt
dt
2γ Γt
Z Z t
γ0
(ν · x)g(t − s)|ut (·, t) − ut (·, s)|2 dsdΓt
−
2γ Γt 0
Z
Z t
γ0
(3.9)
g(s)ds)|ut |2 dΓt
+
(ν · x)(
2γ Γt
0
Z
Z
1
1 0
2
= − g(t)
|ut | dx + g ut − g(0)
|∇u|2 dx
2
2
Ωt
Ωt
Z t
Z
Z
g 0 (t − s)(∇u(·, s) − ∇u(·, t))ds dx
∇u
|∇u|2 dx +
+ g(t)
0
Ωt
Ωt
Noting that
Z
Z
∇u
Ωt
≤
t
g 0 (t − s)(∇u(·, s) − ∇u(·, t))ds dx
0
Z
1/2 Z t
1/2
|∇u|2 dx
|g 0 (s)|ds
(|g 0 |∇u)1/2 ,
Ωt
0
a2
2
b2
2
considering the inequality ab ≤
+
and using the Cauchy-Schwarz inequality
we deduce that
Z
d
γ0
η(t) +
(ν · x)|g ∗ ∇u|2 dΓt
dt
2γ Γt
Z Z t
γ0
(ν · x)g(t − s)|ut (·, t) − ut (·, s)|2 dsdΓt
−
2γ Γt 0
Z
Z t
γ0
g(s)ds)|ut |2 dΓt
+
(ν · x)(
2γ Γt
0
Z
Z
1 0
g(0)
1
2
|ut | dx + g ut −
|∇u|2 dx
≤ − g(t)
2
2
2
Ωt
Ωt
Rt 0
Z
(
|g
(s)|ds)
|g 0 |∇u.
+ g(t)
|∇u|2 dx + 0
2g(0)
Ωt
The proof is complete.
To estimate the term (1 −
Rt
0
g(s)ds)
R
Ωt
|∇u|2 dx, we introduced the functional
Z
ψ(t) =
ut udx.
Ωt
Lemma 3.6. Let us take u0 ∈ H01 (Ω0 ) ∩ H 2 (Ω0 ), u1 ∈ H01 (Ω0 ) and let us suppose
that assumptions (1.11)–(1.13) and (2.1)–(2.2) hold. Then any regular solution of
system (1.1)–(1.3) satisfies
Z t
Z
Z
d
2
ψ(t) ≤ −(1 −
g(s)ds)
|∇u| dx + g(0)
|∇u|2 dx
dt
0
Ωt
Ωt
Rt
Z
( g(s)ds)
+ 0
g∇u +
|ut |2 dx.
4g(0)
Ωt
EJDE-2007/128
WAVE EQUATIONS WITH MEMORY
13
Proof. From (1.1) we get
Z
Z
Z
Z t
d
2
g(t − s)∇u(·, s)ds dx +
|ut |2 dx.
ψ(t) = −
|∇u| dx +
∇u
dt
0
Ωt
Ωt
Ωt
2
2
Considering the inequality ab ≤ a2 + b2 , making use of the Cauchy-Schwarz inequality and using similar arguments as in the lemmas 3.4 and 3.5, follows the
conclusion of lemma.
The following lemma is the key to obtain exponential decay.
Lemma 3.7. Let f be a real positive function of class C 1 . If there exists positive
constants γ0 , γ1 and C0 such that
f 0 (t) ≤ −γ0 f (t) + C0 e−γ1 t ,
then there exist positive constants γ and C such that
f (t) ≤ (f (0) + C)e−γt .
Proof. First, suppose that γ0 < γ1 . Define
F (t) := f (t) +
Then
F 0 (t) = f 0 (t) −
C0
e−γ1 t .
γ1 − γ0
γ1 C0 −γ1 t
e
≤ −γ0 F (t).
γ1 − γ0
Integrating from 0 to t we arrive at
C0 −γ0 t
e
.
γ1 − γ0
Now, we shall assume that γ0 ≥ γ1 . In this conditions we get
F (t) ≤ F (0)e−γ0 t
⇒
f (t) ≤ f (0) +
f 0 (t) ≤ −γ1 f (t) + C0 e−γ1 t
⇒
{eγ1 t f (t)}0 ≤ C0 .
Integrating from 0 to t we obtain
f (t) ≤ f (0) + C0 t e−γ1 t .
Since t ≤ (γ1 − )e(γ1 −)t for any 0 < < γ1 we conclude that
f (t) ≤ {f (0) + C0 (γ1 − )}e−t .
This completes the proof.
Let us introduce the functional
L(t) = N1 E(t) + N2 η(t) + ψ(t) + ϕ(t),
(3.10)
with N1 > N2 > 0 and > 0 small enough. It is not difficult to see that L(t)
verifies
k0 E(t) ≤ L(t) ≤ k1 E(t),
(3.11)
for k0 and k1 positive constants. Now we are in a position to show the main result
of this paper.
Theorem 3.8. Let us take u0 ∈ H01 (Ω0 ), u1 ∈ L2 (Ω0 ) and let us suppose that
assumptions (1.12), (1.13), (2.1), (2.2), (3.2) and (3.3) hold. Then any regular
solution of system (1.1)–(1.3) satisfies
E(t) ≤ Ce−ξt E(0),
where C and ξ are positive constants.
∀t ≥ 0
14
M. L. SANTOS
EJDE-2007/128
Proof. We shall prove this result for strong solutions, that is, for solutions with
initial data u0 ∈ H01 (Ω0 ) ∩ H 2 (Ω0 ), u1 ∈ H01 (Ω0 ). Our conclusion will follows by
standard density arguments. Taking N1 , N2 large enough, with N1 > N2 , > 0
small enough and using the lemmas (3.3), (3.4), (3.5) and (3.6), we conclude that
there exist positive constants α0 and C0 such that
d
L(t) ≤ −α0 L(t) + C0 g 2 (t)E(0).
dt
Using the lemma 3.7 we obtain
L(t) ≤ {L(0) + C}e−α1 t
where C and α1 are positive constants. From equivalence relation (3.11) our conclusion follows.
4. Polynomial Rate of Decay
In this section we assume that the memory g satisfies:
1
g 0 (t) ≤ −C1 g 1+ p (t)
Z ∞
1
α :=
g 1− p (s)ds < ∞
(4.1)
(4.2)
0
for some p > 1 and t ≥ 0, with positive constant C1 . The following lemmas will
play an important role in the sequel.
1
Lemma 4.1. Suppose that g and h are continuous functions, g ∈ L1+ q (0, ∞) ∩
L1 (0, ∞) and g r ∈ L1 (0, ∞) for some 0 ≤ r < 0. Then
Z t
|g(t − s)h(s)|ds
0
q nZ t
1
o q+1
o q+1
nZ t
1+ 1−r
q
≤
|g(t − s)|
|h(s)|ds
.
|g(t − s)|r |h(s)|ds
0
0
Proof. Without loss of generality we can suppose that g, h ≥ 0. Note that for any
fixed t we have
Z t
m
X
g(t − si )h(si )∆si .
g(t − s)h(s)ds = lim
k∆si k→0
0
i=1
Letting
r
Im
:=
m
X
g r (t − sj )h(sj )∆sj ,
j=1
we may write
m
X
g(t − si )h(si )∆si =
i=1
m
X
ϕi θ i ,
i=1
where
r
ϕi = (g 1−r (t − si )Im
),
θi =
g r (t − si )h(si )∆si .
r
Im
1
Since the function F (z) := |z|1+ q is convex, it follows that
m
m
m
X
X
X
F(
g(t − si )h(si )∆si ) = F (
ϕi θ i ) ≤
θi F (ϕi ),
i=1
i=1
i=1
EJDE-2007/128
WAVE EQUATIONS WITH MEMORY
15
so, we have
m
X
g(t − si )h(si )∆si
1+ q1
≤
r q1
|Im
|
i=1
m
X
g 1+
1−r
q
(t − si )h(si )∆si .
(4.3)
i=1
In view of
lim
k∆si k→0
r
Im
=
Z
t
g r (t − s)h(s)ds,
0
letting k∆si k → 0 in (4.3), we get
Z
Z
Z
1+q t 1+ 1−r t
1+ q1
t r
q
g(t − s)h(s)ds
g
g (t − s)h(s)ds
≤
,
0
0
0
from which our result follows.
Lemma 4.2. Let w ∈ C(0, T ; H01 (Ωt )) and g be a continuous function satisfying
hypothesis (4.1)–(4.2). Then for 0 < r < 1 we have
Z
(1−r)p
t r
1 1
g∇w ≤ 2
g dskwkC(0,T ;H01 (Ωt )) 1+(1−r)p g 1+ p ∇w 1+(1−r)P ,
0
while for r = 0 we have
Z
t
1 p
g∇w ≤ 2
kw(s)k2H 1 (Ωt ) ds + tkw(t)k2H 1 (Ωt ) g 1+ p 1+p .
0
0
0
Proof. ¿From hypotheses on w and Lemma 4.1 we get
Z t
g∇w =
g(t − s)h(s)ds
0
Z
Z
(1−r)p
t r
1 t 1+ 1
≤
g (t − s)h(s)ds 1+p(1−r)
g p (t − s)h(s)ds 1+p(1−r)
0
(4.4)
0
1
(1−r)p
1
≤ {g r ∇w} 1+p(1−r) {g 1+ p ∇w} 1+p(1−r)
where
Z
h(s) =
|∇w(t) − ∇w(s)|2 ds.
Ωt
For 0 < r < 1, we have
Z Z t
Z t
g r ∇w =
g r (t − s)|∇w(t) − ∇w(s)|2 ds dx ≤ 4
g r (s)kwk2C(0,T :H 1 Ωt )) ,
Ωt
0
(
0
from which the first inequality of Lemma 4.2 follows. To prove the last part, let us
take r = 0 in Lemma 4.1 to get
Z Z t
1∇w =
|∇w(t) − ∇w(s)|2 ds dx
Ωt
0
≤ 2tkw(t)k2H 1 (Ωt ) + 2
0
Z
0
t
kw(s)k2H 1 (Ωt ) .
0
Substitution of the above inequality into (4.4) yields the second inequality. The
proof is complete.
16
M. L. SANTOS
EJDE-2007/128
Lemma 4.3. Let f be a non-negative C 1 function satisfying
1
k1
f 0 (t) ≤ −k0 [f (t)]1+ p +
,
(1 + t)p+1
for some positive constants k0 , k1 and p > 1. There exists a positive constant C1
such that
pf (0) + 2k1
f (t) ≤ C1
(1 + t)p
Proof. Let h(t) :=
2k1
p(1+t)p+1
and F (t) := f (t) + h(t). Then
F 0 (t) = f 0 (t) −
2k1
(1 + t)p+1
1
≤ −k0 [f (t)]1+ p −
k1
(1 + t)p+1
1
o
n
1
1
p1+ p
1+ p
1+ p
+
≤ −k0 [f (t)]
.
1 [h(t)]
2k0 k1p
From which it follows that there exists a positive constant C1 such that
1
1
F 0 (t) ≤ −C1 {[f (t)]1+ p + [h(t)]1+ p },
which gives the required inequality.
Theorem 4.4. Let us take u0 ∈ H01 (Ω0 ), u1 ∈ L2 (Ω0 ) and let us suppose that
assumptions (1.12), (1.13), (2.1) (2.2), (4.1) and (4.2) hold. Then any regular
solution of system (1.1)–(1.3) satisfies
E(t) ≤ CE(0)(1 + t)−p ,
where C is a positive constant and p > 1.
Proof. We shall prove this result for strong solutions, that is, for solutions with
initial data u0 ∈ H01 (Ω0 ) ∩ H 2 (Ω0 ), u1 ∈ H01 (Ω0 ). Our conclusion will follows by
standard density arguments. From the Lemmas 3.3, 3.4, 3.5 and 3.6 we get
Z t
Z
nZ
o
d
2
L(t) ≤ −C0
|ut | dx + 1 −
g(s)ds
|∇u|2 dx − g 0 ∇u
dt
Ωt
0
Ωt
Z
+ C1 g 2 (t)
|u0 |2 dx.
Ω0
Using hypothesis (4.1) we have
Z t
Z
nZ
o
d
L(t) ≤ −C0
|ut |2 dx + 1 −
g(s)ds
|∇u|2 dx
dt
Ωt
0
Ωt
Z
1
1+ p
2
2
− C1 g
(t)∇u + C2 g (t)
|u0 | dx.
Ω0
for some positive constants C0 , C1 and C2 . Let us define the functional
Z
Z
N (t) =
|ut |2 dx +
|∇u|2 dx.
Ωt
Ωt
Since the total energy is bounded, Lemma 4.2 implies
N (t) ≥ C2 N (t)
(1+(1−r)p)
(1−r)p
,
EJDE-2007/128
WAVE EQUATIONS WITH MEMORY
1
g 1+ p (t)∇u ≥ C2 { g∇u}
(1+(1−r)p)
(1−r)p
17
.
It is not difficult to see that for N1 , N2 large enough, with N1 > N2 , and small
enough the inequality
CE(t) ≤ L(t) ≤ C3 {N (t) + g∇u} ≤ C4 E(t)
holds. From this follows that
Z
(1+(1−r)p)
d
L(t) ≤ −C5 L(t) (1−r)p + C2 g 2 (t)
|u0 |2 dx.
dt
Ω0
Using Lemma 4.3, we obtain
L(t) ≤ C{L(0) + C6 }
1
(1 + t)p(1−r)
where C and C6 are positive constants independent on the initial data. From which
it follows that the energy decay to zero uniformly.
Using Lemma 4.2 for r = 0 we get
(1+p)
N (t) ≥ C2 N (t) p ,
1
1
g 1+ p (t)∇u ≥ C2 g∇u p .
Repeating the same reasoning as above, we obtain
L(t) ≤ C{L(0) + C6 }
1
(1 + t)p
From which our result follows. The proof is now complete.
Acknowledgements. The author is thankful to the anonymous referee of this
paper for his/her valuable suggestions which improved this paper.
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Mauro de Lima Santos
Faculdade de Matemática, Universidade Federal do Pará, Campus Universitario do
Guamá,, Rua Augusto Corrêa 01, Cep 66075-110, Pará, Brasil
E-mail address: ls@ufpa.br