J. Korean Math. Soc. 42 (2005), No. 6, pp. 1137–1152
GLOBAL EXISTENCE AND STABILITY FOR
EULER-BERNOULLI BEAM EQUATION WITH
MEMORY CONDITION AT THE BOUNDARY
Jong Yeoul Park∗ and Joung Ae Kim∗∗
Abstract. In this article we prove the existence of the solution to
the mixed problem for Euler-Bernoulli beam equation with memory
condition at the boundary and we study the asymptotic behavior
of the corresponding solutions. We proved that the energy decay
with the same rate of decay of the relaxation function, that is,
the energy decays exponentially when the relaxation function decay
exponentially and polynomially when the relaxation function decay
polynomially.
1. Introduction
The main purpose of this work is to study the asymptotic behavior of
the solutions of Euler-Bernoulli Beam Equation with boundary condition
of memory type. For this, we consider the following initial boundaryvalue problem:
(1.1)
utt + uxxxx + f (ut ) = 0 in Ω × R+ ,
(1.2)
u(0, t) = ux (0, t) = uxx (L, t) = 0, ∀t > 0,
Z
(1.3)
−u(L, t) +
0
(1.4)
t
g(t − τ )uxxx (L, τ )dτ = 0, ∀t > 0,
u(x, 0) = u0 (x), ut (x, 0) = u1 (x) in Ω,
where Ω = [0, L], k · k is the norm of L2 (Ω).
Received June 3, 2004.
2000 Mathematics Subject Classification: 35B40, 35L75, 74K10, 39A11.
Key words and phrases: global existence, Euler-Bernoulli beam equation, Galerkin
method, boundary value problem.
1138
Jong Yeoul Park and Joung Ae Kim
The integral equation (1.3) describe the memory effect which can be
caused, for example, by the interaction with another viscoelastic element. Frictional dissipative boundary condition for the wave equation
was studied by several authors, see for example([1, 3, 4, 5, 6, 8, 12, 13,
14])among others. In these works existence of solutions and exponential stabilization were proved for linear and for nonlinear equations. In
contrast with the large literature for frictional dissipative, for boundary
condition with memory, we have only a few works as for example([2, 3,
9, 10, 11]). The main result this paper is to show that the solutions of
system (1.1)–(1.4) decays uniformly in time with the same rate of decay
of the relaxation function. More precisely, denoting by κ the resolvent
kernel of g 0 /g(0), we show that the solution decays exponentially to zero
provided κ decays exponentially to zero. When the resolvent kernel κ decays polynomially, we show that the corresponding solution also decays
polynomially to zero.
The method used here is based on the construction of a suitable
Lyapunov functional L satisfying
d
d
c2
L(t) ≤ −c1 L(t) + c2 e−γt or
L(t) ≤ −c1 L(t)1+1/α +
dt
dt
(1 + t)α+1
for some positive constants c1 , c2 , γ, and α.
Note that, because of condition (1.2) the solution of system (1.1)–
(1.4) must belong to the following space:
V = {u ∈ H 2 (0, L) | u(0) = ux (0) = 0}
and
W = {u ∈ V ∩ H 4 (0, L) | uxx (L) = 0}.
The notation used in this paper is standard and can be found in Lions’ book[7]. In the sequel by c (sometime c1 , c2 , · · · ) we denote various
positive constants independent of t and on the initial data. The organization of this paper is as follow. In Section 2, we establish the existence
of strong solutions for system (1.1)–(1.4). In Section 3, we prove the
uniform rate of exponential decay. In Section 4, we prove the uniform
rate of polynomial decay.
2. Existence and regularity
In this section, we shall study the existence and regularity of solutions for system (1.1)–(1.4). To facilitate our analysis, we introduce the
Global existence and stability for Euler-Bernoulli beam equation
1139
following binary operators:
Z
t
(f ¤φ)(t) =
f (t − s)|φ(t) − φ(s)|2 ds,
0
Z
t
(f ∗ φ)(t) =
f (t − s)φ(s)ds,
0
where * is the convolution product.
Differentiating (1.3), we arrive to the Volterra integral equations:
uxxx (L, t) +
1 0
1
g ∗ uxxx (L, t) =
ut (L, t).
g(0)
g(0)
Applying the Volterra’s inverse operator, we get
uxxx (L, t) =
1
{ut (L, t) + κ ∗ ut (L, t)},
g(0)
where the resolvent kernel satisfy
κ+
Denoting by τ =
1
g(0) ,
1 0
1 0
g ∗κ=−
g.
g(0)
g(0)
we obtain
(2.1) uxxx (L, t) = τ {ut (L, t) + κ(0)u(L, t) − κ(t)u(L, 0) + κ0 ∗ u(L, t)}.
Since we are interested in relaxation function of exponential or polynomial type and identity (2.1) involve the resolvent kernel κ, we want to
know if κ has the same properties. The following lemma answers this
question. Let h be a relaxation function and κ its resolvent kernel, that
is
(2.2)
κ(t) − κ ∗ h(t) = h(t).
Lemma 2.1. If h is a positive continuous function, then κ also is a
positive continuous function. Moreover,
(1) If there exist positive constants c0 and γ with c0 < γ such that
h(t) ≤ c0 e−γt ,
then, the function κ satisfies
κ(t) ≤
for all 0 < ² < γ − c0 .
c0 (γ − ²) −²t
e ,
γ − ² − c0
1140
Jong Yeoul Park and Joung Ae Kim
(2) Given p > 1, let us denote by
Z t
cp = supt∈R+
(1 + t)p (1 + t − s)−p (1 + s)−p ds.
0
If there exists a positive constant c0 with c0 cp < 1 such that
h(t) ≤ c0 (1 + t)−p ,
then the function κ satisfies
c0
κ(t) ≤
(1 + t)−p .
1 − c0 cp
Proof. Note that κ(0) = h(0) > 0. Now, we take t0 = inf{t ∈ R+ :
κ(t) = 0}, so κ(t) > 0 for all t ∈ [0, t0 ). If t0 ∈ R+ , from (2.2) we get
that −κ ∗ h(t0 ) = h(t0 ) but this is contradictory. Therefore κ(t) > 0 for
all t ∈ R+
0 . Now, let us fix ², such that 0 < ² < γ − c0 and denote by
κ² (t) = e²t κ(t), h² (t) = e²t h(t).
Multiplying (2.2) by e²t , we get κ² (t) = h² (t) + κ² ∗ h² (t), hence
sup κ² (s) ≤ sup h² (s)
s∈[0,t]
s∈[0,t]
Z ∞
+
0
c0 e(²−γ)s ds × sup κ² (s)
s∈[0,t]
c0
≤ c0 +
sup κ² (s).
γ − ² s∈[0,t]
Therefore
c0 (γ − ²)
,
γ − ² − c0
which implies our first assertion. To show the second part let us consider
the following notations
κ² (t) ≤
κp (t) = (1 + t)p κ(t), hp (t) = (1 + t)p h(t).
Rt
Multiplying (2.2) by (1 + t)p , we get κp (t) = hp (t) + 0 κp (t − s)(1 + t −
s)−p (1 + t)p h(s)ds, hence
sup κp (s) ≤ sup hp (s) + c0 cp sup κp (s)
s∈[0,t]
s∈[0,t]
s∈[0,t]
≤ c0 + c0 cp sup κp (s).
s∈[0,t]
Therefore
κp (t) ≤
which proves our second assertion.
c0
,
1 − c0 cp
Global existence and stability for Euler-Bernoulli beam equation
1141
Due to this Lemma, in the remainder of this paper, we shall use (2.1)
instead of (1.3). The following lemma state an important property of
the convolution operator.
Lemma 2.2. For f, φ ∈ C1 ([0, ∞); R), we have
Z t
f (t − s)φ(s)ds · φt
0
³
1
1 dh
1
f ¤φ −
= − f (t)|φ(t)|2 + f 0 ¤φ −
2
2
2 dt
Z
t
´
i
f (s)ds |φ|2 .
0
The proof of this lemma follows by differentiating the term f ¤φ.
The first-order energy of system (1.1)–(1.4) is given by
´
1³
E(t, u) =
kut (t)k2 + kuxx (t)k2 − τ κ0 ¤u(L, t) + τ κ(t)|u(L, t)|2 .
2
We summarize the well-posedness of (1.1)–(1.4) in the following theorem.
Theorem 2.1. Let κ ∈ C2 (R+ ) be such that κ, −κ0 , κ00 ≥ 0 and
f ; R → R is continuously differentiable function there exist ρ a positive
constant such that
f (0) = 0 and (f (r) − f (s), r − s) ≥ ρ|r − s|2 , ∀r, s ∈ R.
If u0 ∈ W, u1 ∈ L2 (Ω) satisfying the compatibility condition uxxx (L, 0) =
τ ut (L, 0), then there is only one solution u of system (1.1)–(1.4) satisfying
(2.3)
u ∈ L∞ (0, ∞; V ), u0 ∈ L∞ (0, ∞; V ), u00 ∈ 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 V which is orthonormal in L2 (Ω) and
we represent by Vm the subspace of V generated by the first m vector.
Standard results on ordinary differential equations guarautee that there
exists only one local solution um (t) = Σm
j=1 gj,m (t)wj , of the approximate
system,
(2.4)
m
m
(um
tt , w) + (uxx , wxx ) + (f (ut ), w)
³
m
m
= − τ um
t (L, t) + κ(0)u (L, t) − κ(t)u (L, 0)
´
0
+ κ ∗ um (L, t), w(L, t) ,
for all w ∈ Vm with the initial data
0 1
(um (0), um
t (0)) = (u , u ).
1142
Jong Yeoul Park and Joung Ae Kim
The extension of these solutions to the whole interval [0, T ], 0 < T < ∞,
is a consequence of the first estimate which we are going to prove below.
A Priori Estimate I.
Replacing w by um
t (t) in (2.4), using Lemma 2.2 and from hypothesis
of f , we obtain
´
1 d³ m
2
0
m
m
2
kut (t)k2 + kum
(t)k
−
τ
κ
¤u
(L,
t)
+
τ
κ(t)|u
(L,
t)|
xx
2 dt
2
+ ρkum
t (t)k
2
m
m
≤ − τ |um
t (L, t)| + τ κ(t)(u (L, 0), ut (L, t))
τ
τ
+ κ0 (t)|um (L, t)|2 − κ00 ¤um (L, t)
2
2
τ m
τ 2
τ
2
≤ − |ut (L, t)| + κ (t)|um (L, 0)|2 + κ0 (t)|um (L, t)|2
2
2
2
τ 00 m
− κ ¤u (L, t).
2
Using κ, −κ0 , κ00 ≥ 0, we get
d
2
m
E(t, um ) + ρkum
t (t)k ≤ cE(0, u ).
dt
Integrating it over [0, t] and taking into account the definition of the
initial data of um , we conclude that
Z
(2.5)
2
kum
t (t)k
+
2
kum
xx (t)k
+ρ
0
t
2
kum
t (s)k ds
≤ c, ∀t ∈ [0, T ], ∀m ∈ N.
A Priori Estimate II.
2
First of all, we are estimating um
tt (0) in the L -norm. Considering
t = 0 and w = um
tt (0) in (2.4) and using the compatibility condition we
obtain
2
m
m
m
m
kum
tt (0)k + (uxxxx (0), utt (0)) + (f (ut (0)), utt (0)) = 0.
Since u0 ∈ W, u1 ∈ L2 (Ω), the growth hypothesis for the function f
2
imply that f (u1 ) ∈ L2 (Ω). Hence kum
tt (0)k ≤ C, ∀m ∈ N
Global existence and stability for Euler-Bernoulli beam equation
1143
Finally, differentiating (2.4) and multiplying the both sides of equation by um
tt (t), then we have
´
1 d³ m
2
m m
kutt (t)k2 + kum
+ (f 0 (um
xxt (t)k
t )utt , utt )
2 dt
2
m
m
= − τ |um
tt (L, t)| − τ κ(0)(ut (L, t), ut (L, t))
³
´
³
´
0
m
m
+ τ κ0 (t) um (L, 0), um
(L,
t)
−
τ
(κ
∗
u
(L,
t))
,
u
(L,
t)
.
t tt
tt
Noting that
Z
0
m
(κ ∗ u )t = κ
0
(t)um
0
+
0
t
κ0 (t − s)um
t (s)ds
and using Lemma 2.2, we obtain
(2.6)
´
1 d³ m
2
0
m
m
2
kutt (t)k2 + kum
(t)k
−
τ
κ
¤u
(L,
t)
+
τ
κ(t)|u
(L,
t)|
xxt
t
t
2 dt
0 m m m
+ (f (ut )utt , utt )
2
0
m
m
= − τ |um
tt (L, t)| + τ κ (t)(u (L, 0), utt (L, t))
τ
τ 00 m
2
+ κ0 (t)|um
t (L, t)| − κ ¤ut (L, t).
2
2
We also note that from assumption on the function f , we get
¯³
´¯
¯ 0 m m m ¯
m
2
(2.7)
f
(u
)u
,
u
¯
t
tt
tt ¯ ≤ ckutt (t)k .
Substitution of inequality (2.7) into (2.6), we arrive at
´
1 d³ m
2
0
m
m
2
kutt (t)k2 + kum
(t)k
−
τ
κ
¤u
(L,
t)
+
τ
κ(t)|u
(L,
t)|
xxt
t
t
2 dt
τ
2
0
2 m
2
≤ ckum
tt (t)k + (κ (t)) |u (L, 0)| .
2
Integrating with respect to the time and applying Gronwall’s inequality,
we conclude that
(2.8)
2
m
2
kum
tt (t)k + kuxxt (t)k ≤ c, ∀m ∈ N, ∀t ∈ [0, T ].
By estimates (2.5) and (2.8), we obtain
 m
is bounded in L∞ (0, T ; V ),
 (u )
m
is bounded in L∞ (0, T ; V ),
(u )
 tm
is bounded in L∞ (0, T ; L2 (Ω)).
(utt )
1144
Jong Yeoul Park and Joung Ae Kim
Therefore, we can get subsequences, if necessary, denoted by (um ), such
that
 m
→ u weakly star in L∞ (0, T ; V ),
 u
m
u
→ ut weakly star in L∞ (0, T ; V ),
 tm
utt → utt weakly star in L∞ (0, T ; L2 (Ω)).
We can use Lions-Aubin Lemma to get the necessary compactness in order to pass (2.4) to the limit. Then it is matter of routine to conclude the
existence of global solutions in [0, T]. The uniqueness is straightforward
by standard methods and Gronwall’s inequality.
3. Exponential decay
In this section, we shall study the asymptotic behavior of the solutions of system (1.1)–(1.4) when the resolvent kernel κ is exponentially
decreasing, that is, there exist positive constants b1 , b2 such that
(3.1)
κ(0) > 0, κ0 (t) ≤ −b1 κ(t), κ00 (t) ≥ −b2 κ0 (t).
Note that this conditions implies that κ(t) ≤ κ(0)e−b1 t .
Our point of departure will be to establish some ineqalities for the
strong solution of system (1.1)–(1.4).
Lemma 3.1. Any strong solution u of system (1.1)–(1.4) satisfy
d
E(t) ≤ −
dt
+
τ
τ
|ut (L, t)|2 + κ2 (t)|u(L, t)|2
2
2
τ 0
τ
2
κ (t)|u(L, t)| − κ00 ¤u(L, t) − ρkut (t)k2 .
2
2
Proof. Multiplying (1.1) by ut and integrating by parts over Ω, we
get
´ ³
´
³
´
1 d³
kut (t)k2 + kuxx (t)k2 + f (ut (t)), ut (t) = − uxxx (L, t), ut (L, t) .
2 dt
Global existence and stability for Euler-Bernoulli beam equation
1145
Substituting the boundary term, using Lemma 2.1 and assumption of f ,
we get
´
1 d³
kut (t) + kuxx (t)k2 − τ κ0 ¤u(L, t) + τ κ(t)|u(L, t)|2
2 dt
³
´
+ f (ut (t)), ut (t)
τ
τ
= − τ |ut (L, t)|2 + κ0 (t)|u(L, t)|2 − κ00 ¤u(L, t)
2
2
³
´
+ τ κ(t) u(L, 0), ut (L, t)
τ
τ
τ
≤ − |ut (L, t)|2 + κ0 (t)|u(L, t)|2 − κ00 ¤u(L, t)
2
2
2
τ 2
2
+ κ (t)|u(L, t)| .
2
Using assumption of f , we obtain
d
E(t) ≤ −
dt
−
τ
τ
τ
|ut (L, t)|2 + κ2 (t)|u(L, t)|2 + κ0 (t)|u(L, t)|2
2
2
2
τ 00
2
κ ¤u(L, t) − ρkut (t)k .
2
Let us consider the following binary operator:
Z t
(κ ¦ ϕ)(t) =
κ(t − s)(ϕ(t) − ϕ(s))ds.
0
Then applying the Hölder’s inequality for 0 ≤ µ ≤ 1, we have
i
hZ t
2
(3.2)
|(κ ¦ ϕ)(t)| ≤
|κ(s)|2(1−µ) ds (|κ|2µ ¤ϕ)(t).
0
Let us introduce the following functional:
1
Ψ(t) = δ(u(t), ut (t)) with 0 < δ < .
2
The following lemma plays an important role for the construction of the
Lyapunov functional.
Lemma 3.2. For any strong solution of system (1.1)–(1.4), we get
d
Ψ(t) ≤ − δkuxx (t)k2 + ckut (t)k2 + cku(t)k2 + c|ut (L, t)|2 + c|u(L, t)|2
dt
+ cκ(t)|u(L, 0)|2 + cκ(0)|κ0 |¤u(L, t) + cκ(t)|u(L, t)|2 .
1146
Jong Yeoul Park and Joung Ae Kim
Proof. From (1.1), it follows that
(3.3)
d
Ψ(t) = δ(u(t), utt (t)) + δkut (t)k2
dt
³
´
= δkut (t)k2 − δkuxx (t)k2 − δ f (ut (t)), u(t)
− δτ (ut (L, t), u(L, t)) + δτ κ(t)(u(L, 0), u(L, t))
³
´
− δτ κ(0)u(L, t) + κ0 ∗ u(L, t), u(L, t) .
Note that
−κ(0)u(L, t) − κ0 ∗ u(L, t)
Z t
h
i
=−
κ0 (t − s) u(L, s) − u(L, t) ds − κ(t)u(L, t)
0
´1 h
³Z t
i1
(3.4)
2
2
|κ0 (s)|ds
≤
|κ0 |¤u(L, t) + κ(t)|u(L, t)|
0
h
i1
1
2
≤ |κ(t) − κ(0)| 2 |κ0 |¤u(L, t) + κ(t)|u(L, t)|.
Using (3.3), (3.4), and Young’s inequality, it follows that
³
´
d
Ψ(t) ≤ δkut (t)k2 − δkuxx (t)k2 − δ f (ut (t)), u(t)
dt
− δτ (ut (L, t), u(L, t)) + δτ κ(t)(u(L, 0), u(L, t))
³
h
i1
´
1
2
+ δτ |κ(t) − κ(0)| 2 |κ0 |¤u(L, t) , u(L, t) + δτ κ(t)|u(L, t)|2
≤ − δkuxx (t)k2 + ckut (t)k2 + cku(t)k2 + c|ut (L, t)|2 + c|u(L, t)|2
+ cκ(t)|u(L, 0)|2 + cκ(0)|κ0 |¤u(L, t) + cκ(t)|u(L, t)|2 .
¤
Let us introduce the Lyapunov functional
(3.5)
L(t) = N E(t) + Ψ(t), with N > 0.
Using Young’s inequality and taking N large enough we find that
(3.6) q0 E(t) ≤ L(t) ≤ q1 E(t), for some positive constants q0 and q1 .
We will show later that the functional L satisfies the inequality of the
following Lemma.
Lemma 3.3. Let f be a real positive function of class C1 . 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 .
Global existence and stability for Euler-Bernoulli beam equation
1147
Proof. First, let us suppose that γ0 < γ1 . Define F (t) by
c0
F (t) = f (t) +
e−γ1 t .
γ1 − γ0
Then
γ1 c0 −γ1 t
F 0 (t) = f 0 (t) −
e
≤ −γ0 F (t).
γ1 − γ0
Integrating from 0 to t, we arrive to
³
c0 ´ −γ0 t
e
.
F (t) ≤ F (0)e−γ0 t ⇒ f (t) ≤ f (0) +
γ1 − γ0
Now, we shall assume that γ0 ≥ γ1 . In this conditions, we get
f 0 (t) ≤ −γ1 f (t) + c0 e−γ1 t ⇒ (eγ1 t f (t))0 ≤ c0.
Integrating from 0 and t, we obtain
f (t) ≤ (f (0) + c0 t)e−γ1 t .
Since t ≤ (γ1 − ²)e(γ1 −²)t for any 0 < ² < γ1 , we conclude that
f (t) ≤ (f (0) + c0 (γ1 − ²))e−²t .
This completes the proof.
Finally, we shall show the main result of this section.
Theorem 3.1. Let us suppose that the initial data (u0 , u1 ) ∈ W ×
L2 (Ω) and that the resolvent κ satisfies the conditions (3.1). Then there
exist positive constants α1 and γ1 such that
E(t) ≤ α1 e−γ1 t E(0) for all t ≥ 0.
Proof. We will suppose that (u0 , u1 ) ∈ (H 4 (Ω)∩W )×W and satisfies
the compatibility conditions uxxx (L, 0) = τ ut (L, 0); our conclusion will
follow by standard density arguments. Using Lemmas 3.1 and Lemma
3.2, we get
³τ
´
d
τ
τ
L(t) ≤ −
N − c |ut (L, t)|2 + N κ2 (t)|u(L, t)2 | + N κ0 (t)|u(L, t)|2
dt
2
2
2
τ
00
2
− N κ ¤u(L, t) − (N ρ − δ)kut (t)k − δkuxx (t)k2
2
+ cku(t)k2 + c|u(L, t)|2 + cκ(t)|u(L, 0)|2
+ cκ(0)|κ0 |¤u(L, t) + cκ(t)|u(L, t)|2 .
Then, choosing N large enough and N ρ > δ,
τ
2N
> c, we obtain
d
L(t) ≤ −q2 E(t) + cκ2 (t)E(0), where q2 > 0 is a small constant.
dt
1148
Jong Yeoul Park and Joung Ae Kim
Here we use (3.1) to conclude the following estimates for the corresponding two terms appearing in Lemma 3.1.
τ
− κ00 ¤u(L, t) ≤ c1 κ0 ¤u(L, t),
2
τ 0
κ |u(L, t)|2 ≤ − c1 κ|u(L, t)|2 .
2
Finally, from (3.1) and (3.6), we conclude that
d
q2
L(t) ≤ − L(t) + cE(0)exp(−2b1 t).
dt
q1
Using the exponential decay of the resolvent kernel κ and Lemma 3.3,
we conclude
L(t) ≤ {L(0) + c}e−γ1 t for all t ≥ 0.
Use of (3.6) now completes the proof.
4. Polynomial rate of decay
The proof of the existence of global solutions for (1.1)–(1.4) with
resolvent kernel κ decaying polynomially is essentially the same as in
Section 3. Here our attention will be focused on the uniform rate of
decay when the resolvent κ decays polynomially such as (1 + t)−p . In
this case, we will show that the solution also decays polynomially with
the same rate. We shall use the following hypotheses:
0 < κ(t) ≤ b0 (1 + t)−p ,
(4.1)
−b1 κ
b3 (−κ0 )
p+1
p
p+2
p+1
≤ κ0 (t) ≤ −b2 κ
p+1
p
≤ κ00 (t) ≤ b4 (−κ0 )
,
p+2
p+1
,
where p > 1 and bi , i = 0, 1, · · · , 4, are positive constants.
Also we assume that
Z ∞
1
(4.2)
|κ0 (t)|r dt < ∞ if r >
.
p+1
0
The following lemmas will play an important role in the sequel.
Global existence and stability for Euler-Bernoulli beam equation
1149
Lemma 4.1. Let m and h be integrable functions, 0 ≤ r < 1 and
q > 0. Then, for t ≥ 0 :
Z t
|m(t − s)h(s)|ds
0
µZ
t
≤
1+ 1−r
q
|m(t − s)|
q µZ
1
¶ q+1
¶ q+1
t
r
|h(s)|ds
.
|m(t − s)| |h(s)|ds
0
0
Proof. Let
r
1− q+1
v(s) = |m(t − s)|
q
r
1
|h(s)| q+1 , w(s) = |m(t − s)| q+1 |h(s)| q+1 .
Then using Hölder’s inequality with δ =
we arrive to the conclusion.
q
q+1
for v and δ ∗ = q + 1 for w,
Lemma 4.2. Let p > 1, 0 ≤ r < 1 and r ≥ 0. Then for r > 0,
¡ 0
¢ 1+(1−r)(p+1)
|κ |¤u(L, t) (1−r)(p+1)
1
³Z t
´
´
³
(1−r)(p+1)
1+ 1
|κ0 (s)|r dskuk2L∞ (0,T ;L2 (0,L))
≤2
|κ0 | p+1 ¤u(L, t) ,
0
and for r = 0,
¡ 0
¢ p+2
|κ |¤u(L, t) p+1
1
¶ p+1
µZ t
´
³
1+ 1
2
2
ku(s)kL2 (0,L) ds + tku(s)kL2 (0,L)
≤2
|κ0 | p+1 ¤u(L, t) .
0
Proof. The above inequality is a immediate consequence of Lemma
4.1 with m(s) = |κ0 (s)|, h(s) = |u(x, t) − u(x, s)|2 , q = (1 − r)(p + 1),
and t fixed.
Lemma 4.3. Let α > 0, β ≥ α + 1, and f ≥ 0 be differentiable
1
c¯2
function satisfying f 0 (t) ≤ −c¯11 f (t)1+ α + (1+t)
β f (0) for t ≥ 0 and some
f (0) α
positive constants c¯1 , c¯2 . Then there exists a constant c¯3 > 0 such that
c¯3
for t ≥ 0, f (t) ≤ (1+t)
α f (0).
Proof. Let t ≥ 0 and
F (t) = f (t) +
2c¯2
(1 + t)−α f (0).
α
Then
F 0 = f 0 − 2c¯2 (1 + t)−(α+1) f (0)
−c¯1 1+ 1
−(α+1)
α − c¯ (1 + t)
f (0),
≤
2
1 f
α
f (0)
1150
Jong Yeoul Park and Joung Ae Kim
where we used β ≥ α + 1. Hence
´
1
1
−c
−c ³ 1+ 1
−(α+1)
1+ α
1+ α
α + (1 + t)
f
(0)
≤
.
F0 ≤
f
1
1 F
f (0) α
F (0) α
F (0)
c
Integration yields F (t) ≤ (1+ct)
α ≤ (1+t)α f (0), hence f (t) ≤
for some c¯3 , which proves Lemma 4.3.
c¯3
(1+t)α f (0)
Theorem 4.1. Assume that (u0 , u1 ) ∈ W × L2 (0, L) and that the
resolvent κ satisfies (4.1). Then there exists a positive constant c for
which
c
E(t) ≤
E(0).
(1 + t)p+1
Proof. We will suppose that (u0 , u1 ) ∈ H 2 (0, L) ∩ W × W and satisfies the compatibility condition; our conclusion will follows by standard
density arguments. We define the functional L as in (3.5) and we have
the equivalence to the energy term E as given in (3.6) again.
From the Lemmas 3.1 and 3.2, we conclude that
³
d
L(t) ≤ − c1 |ut (L, t)|2 + κ00 ¤u(L, t)
dt
(4.3)
´
+ ku(t)k2 + kuxx (t)k2 + c2 κ2 (t)E(0).
From hypothesis (4.1), we obtain
³
d
1+ 1
L(t) ≤ − c1 |ut (L, t)|2 + (−κ0 ) p+1 ¤u(L, t)
dt
(4.4)
´
+ ku(t)k2 + kuxx (t)k2 + c2 κ2 (t)E(0).
From Lemma 4.2, we get
1
1+ p+1
(−κ0 )
(4.5)
¤u(L, t)
c
≥ R
((−κ0 )¤u(L, t))
1
1
t 0 r
(1−r)(p+1)
(1−r)(p+1)
( 0 |κ | ds)
E(0)
1
1+ (1−r)(p+1)
On the other hand, since the energy is bounded, we have
(4.6)
1
³
´1+
(1−r)(p+1)
2
2
2
|ut (L, t)| + ku(t)k + kuxx (t)k
³
´
1
≤ cE(0) (1−r)(p+1) |ut (L, t)|2 + ku(t)k2 + kuxx (t)k2 .
.
Global existence and stability for Euler-Bernoulli beam equation
1151
Substituting (4.5) and (4.6) into (4.4), we arrive at
h¡
1
¢1+
d
c
(1−r)(p+1)
L(t) ≤ −
|ut (L, t)|2 + ku(t)k2 + kuxx (t)k2
1
dt
E(0) (1−r)(p+1)
i
1
¡ 0
¢1+
(1−r)(p+1)
+ cκ2 (t)E(0).
+ |κ |¤u(L, t)
Taking into account (3.6), we conclude that
(4.7)
1
d
c
1+
L(t) ≤ −
L(t) (1−r)(p+1) + cκ2 (t)E(0).
1
dt
L(0) (1−r)(p+1)
Applying the Lemma 4.3 with f = L and β = 2p, we have:
(4.8)
L(t) ≤
c
(1 +
t)(1−r)(p+1)
Since (1 − r)(p + 1) > 1,
Z ∞
Z
(4.9)
E(s)ds ≤ c
0
L(0).
∞
L(s)ds ≤ cL(0),
0
tku(t)k2L2 (0,L) ≤ ctL(t) ≤ cL(0),
(4.10)
Z
(4.11)
0
t
Z
ku(s)k2L2 (0,L) ds
≤c
∞
L(t)dt ≤ cL(0).
0
In this conditions applying Lemma 4.2 for r = 0, we get
³
´1+ 1
c
p+1
1+ 1
(−κ0 ) p+1 ¤u(L, t) ≥
(−κ)¤u(L,
t)
.
1
E(0) p+1
Using the same arguments as in the derivation of (4.7), we have
1
d
c
1+ p+1
L(t) ≤ −
+ cκ2 (t)E(0).
1 L(t)
dt
p+1
L(0)
Applying the Lemma 4.3, we obtain
L(t) ≤
c
L(0).
(1 + t)p+1
Finally, from (3.6) we obtain E(t) ≤
present proof.
c
E(0),
(1+t)p+1
which complete the
1152
Jong Yeoul Park and Joung Ae Kim
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Jong Yeoul Park and Joung Ae Kim
Department of Mathematics
College of Science
Pusan National University
Pusan 609-735, Korea
E-mail : [email protected]
[email protected]