Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2009, Article ID 489061, 15 pages
doi:10.1155/2009/489061
Research Article
Existence of Solutions for Hyperbolic System of
Second Order Outside a Domain
Jie Xin and Xiuyan Sha
School of Mathematics and Information, Ludong University, Yantai, Shandong 264025, China
Correspondence should be addressed to Jie Xin, fdxinjie@sina.com
Received 27 June 2008; Accepted 29 April 2009
Recommended by Robert Bob Gilbert
We study the mixed initial-boundary value problem for hyperbolic system of second order outside
a closed domain. The existence of solutions to this problem is proved and the estimate for the
regularity of solutions is given. The application of the existence theorem to elastrodynamics is
discussed.
Copyright q 2009 J. Xin and X. Sha. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
1. Introduction
This paper is concerned with the exterior problem for hyperbolic system of second order. Let
K be a closed domain with smooth boundary in R3 and let the origin belong to K. Consider
the following exterior problem for the hyperbolic system of second order:
∂2t ui −
3
aijkl t, x∂j ∂l uk bi ,
i 1, 2, 3, t, x ∈ R × R3 \ K,
j,k,l1
u0, x fx,
ut, x 0,
∂t u0, x gx,
1.1
x ∈ ∂K,
where aijkl t, x ∈ CB2 0, ∞ × R3 \ K and b b1 , b2 , b3 . We assume that aijkl t, x satisfies
3
aijkl t, xeij ekl ≥ α|E|2 ,
j,k,l1
α > 0,
1.2
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Journal of Inequalities and Applications
for all symmetric matrixes E eij , where eij 1/2∂ui /∂xj ∂uj /∂xi , |E|2 3i,j1 eij2 ,
t, x ∈ R × R3 \ K.
Let v ∂t u. The system 1.1 can be written as an evolution system in the form
d
U AtU B,
dt
1.3
where
T
U u1 , u2 , u3 , ∂t u1 , ∂t u2 , ∂t u3 u, vT ,
0 I3×3
At ,
at 0 6×6
⎞
⎛
3
at ⎝ aijkl ∂j ∂l ⎠ .
j,l1
B 0, bT ,
1.4
3×3
Ikawa considered in 1 the mixed problem of a hyperbolic equation of second-order.
The existence theorem is known for the obstacle free problem in 2. Dafermos and Hrusa
proved in 3 the local existence of the Dirichlet problem for the hyperbolic system inside a
domain by energy method.
In this paper, we deal with the exterior problem for the second order hyperbolic
system. In Section 2, we show the existence of the exterior problem for the problem 1.1
by the semigroup theory. In Section 3, we prove the regularity for the solutions of the exterior
problem 1.1 and give the estimate for the regularity of solutions. In Section 4, we discuss
the application of the existence theorem to elastrodynamics.
2. Existence of the Exterior Problem for Hyperbolic System of
Second Order
Note that Ht H01 R3 \ K × L2 R3 \ K with the inner product
U1 , U2 Ht u1 , v1 , u2 , v2 Ht 3 aijkl t, x∂j ui1 , ∂l uk2 v1 , v2 .
2.1
i,j,k,l1
By 1.2 and Korn inequality cf. 4, 5, we have
Lemma 2.1. For some M > 0, we have
1 u2H 1 R3 \K v2L2 R3 \K ≤ U2Ht ≤ M u2H 1 R3 \K v2L2 R3 \K .
0
0
M
2.2
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3
Then Ht is a Hilbert space with the inner product defined as above. We define the
operator without loss of generality, we still write this operator as At in Ht by
At : D −→ Ht,
U −→ AtU,
2.3
where D H 2 R3 \ K ∩ H01 R3 \ K × H01 R3 \ K. It is obvious that At is a densely
defined operator.
Lemma 2.2. There exists a constant c > 0 such that for any U ∈ D,
AtU, UHt ≤ cU, UHt
2.4
holds.
Proof. Let U u, v ∈ D.
3 aijkl ∂j vi , ∂l uk atu, v
AtU, UHt i,j,k,l1
3
3
k i
aijkl ∂l u v νj dΓ −
vi ∂j aijkl ∂l uk dx
3 \K
i,j,k,l1 ∂K
R
i,j,k,l1
3
i
k
−
v aijkl ∂j ∂l u dx atu, v
3
i,j,k,l1 R \K
3
i
k
−
v ∂j aijkl ∂l u dx − v, atu atu, v
i,j,k,l1 R3 \K
2.5
≤ C u2H 1 R3 \K v2L2 R3 \K
0
≤
cU2Ht .
Corollary 2.3. For all real λ such that |λ| > 2c, the estimate
λI − AtUHt ≥ |λ| − cUHt
holds for any U ∈ D.
2.6
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Journal of Inequalities and Applications
Proof. By 2.4,
λI − AtU, λI − AtUHt
|λ|2 U, UHt − λ U, AtUHt AtU, UHt AtU, AtUHt
≥ |λ|2 U, UHt − 2|λ|cU, UHt
|λ| − 2c2 2c|λ| − 2c U2Ht
2.7
≥ |λ| − 2c2 U2Ht .
The estimate of the resolvent operator λI − At−1 is the following.
Lemma 2.4. There exists a constant δ > 0 such that for all λ real and |λ| > δ,
λI − At : D −→ Ht
2.8
is a bijective mapping. Moreover, we have
λI − At−1 Ht
≤
1
.
|λ| − δ
2.9
Proof. Consider the system
λI − AtU P,
2.10
namely,
λu − v p
−atu λv q,
2.11
where p, q ∈ H01 R3 \ K × L2 R3 \ K Ht.
The substitution of the first relation
v λu − p
2.12
in the second of 2.11 gives
−at λ2 u λp q w ∈ L2 R3 \ K .
2.13
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5
By the well-known variation method, there exists a solution u ∈ H 2 R3 \ K ∩ H01 R3 \ K of
the elliptic system 2.13 for any w ∈ L2 R3 \ K. Defining v by 2.12, we have a solution
u, v ∈ H 2 R3 \ K ∩ H01 R3 \ K × H01 R3 \ K D
2.14
of 2.10. Therefore, λI − At is a surjection.
From 2.6, it follows that the existence of λI − At−1 and the estimate
λI − At−1 U
Ht
≤
1
UHt .
|λ| − 2c
2.15
Let δ 2c, we have 2.9.
For U u, v ∈ H p R3 \ K × H p−1 R3 \ K, we define the following norm:
U2p u2H p R3 \K v2H p−1 R3 \K .
2.16
Suppose that aijkl t, x ∈ Cp 0, ∞ × R3 \ K, we have
Corollary 2.5. For the real number λ0 > δ (λ0 fixed) and the integer p ≥ 1, where δ is as in
Lemma 2.4, there exists dp > 0 such that for any U ∈ D ∩ H p R3 \ K × H p−1 R3 \ K,
Up < dp λ0 I − AtUp−1 .
2.17
Proof. From Lemma 2.4,
λ0 I −At : D∩ H p R3 \ K ×H p−1 R3 \ K −→ Ht∩ H p−1 R3 \ K ×H p−2 R3 \ K
2.18
is a bijective continuous mapping, then λ0 I − At is a closed operator. It implies that
λ0 I − At−1 is also a closed operator. By Banach’s closed graph theorem, λ0 I − At−1 is
continuous. So for any U ∈ D ∩ H p R3 \ K × H p−1 R3 \ K, we have
Up λ0 I − At−1 λ0 I − AtU ≤ dp λ0 I − AtUp−1 .
p
2.19
Definition 2.6. Let X be a Banach space. A family {At}t∈0,T of infinitesimal generators of
C0 semigroups on X is called stable if there are constants M ≥ 1 and δ called the stability
constants such that
ρAt ⊃ δ, ∞, ∀t ∈ 0, T ,
k
λI − Atj −1 ≤ Mλ − δ−k , ∀λ > δ,
j1
2.20
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Journal of Inequalities and Applications
for every finite sequence 0 ≤ t1 ≤ t2 ≤ · · · ≤ tk ≤ T , k 1, 2, . . ..
Lemma 2.7. For t ∈ 0, T , let At be the infinitesimal generators of C0 semigroups St s on the
Banach X. The family of generators {At}t∈0,T is stable if and only if there are constants M ≥ 1
and δ such that
ρAt ⊃ δ, ∞, ∀t ∈ 0, T ,
⎧
⎫
k
⎨ ⎬
k
St sj ≤ M exp δ sj ,
j
⎩ j1 ⎭
j1
2.21
for any finite sequence 0 ≤ t1 ≤ t2 ≤ · · · ≤ tk ≤ T , k 1, 2, . . . .
Lemma 2.8. Let {At}t∈0,T be a stable family of infinitesimal generators of C0 semigroups St s on
the Banach space X such that DAt D is independent of t and for every U0 ∈ D, AtU0 is
continuously differentiable in X. If Bt ∈ C1 0, T ; X, then
d
Ut AtUt Bt
dt
2.22
has a unique classical solution Ut ∈ C1 0, T ; X ∩ C0, T ; D such that U0 U0 .
The proofs of Lemmas 2.7 and 2.8 are in 6. The straightforward application of the
semigroup theory to the system 1.3 gives the following proposition.
Proposition 2.9. Given U0 ∈ D and Bt ∈ C1 0, T , H01 R3 \ K × L2 R3 \ K, then there exists
one and only one solution Ut ∈ C1 0, T ; H01 R3 \ K × L2 R3 \ K ∩ C0, T ; D of 1.3 such
that U0 U0 .
Proof. Let X Ht. For given t > 0, At is an infinitesimal generator of C0 semigroups St s
on X. For any U ∈ D, it is easy to know that
St sUHt ≤ eδs UHt .
2.23
Then for any U ∈ D, t1 , t2 > 0, we have
U2Ht1 3
R3 \K
i,j,k,l1
3
R3 \K
aijkl t1 ∂j ui ∂l uk dx v, v
aijkl t2 ∂j ui ∂l uk dxv, v
i,j,k,l1
≤ U2Ht2 C|t1 − t2 |U2Ht2 ,
3
R3 \K
aijkl t1 −aijkl t2 ∂j ui ∂l uk dx
i,j,k,l1
2.24
Journal of Inequalities and Applications
7
namely,
UHt1 ≤ 1 C1 |t1 − t2 |1/2 UHt2 .
2.25
For any finite sequence 0 ≤ t1 ≤ t2 ≤ · · · ≤ tk ≤ T and any sj , j 1, 2, . . . , k,
Stk sk Stk−1 sk−1 · · · St1 s1 UHt
≤ CStk sk Stk−1 sk−1 · · · St1 s1 UHtk ≤ Ceδsk Stk−1 sk−1 · · · St1 s1 UHtk ≤ Ceδsk 1 C1 tk − tk−1 1/2 Stk−1 sk−1 · · · St1 s1 UHtk−1 ≤ Ceδsk sk−1 ···s2 s1 1C1 tk −tk−1 1/2 1C1 tk−1 −tk−2 1/2 · · · 1C1 t2 −t1 1/2 UHt1 ⎛
⎞
k
k C1 tk k/2
⎝
⎠
≤ C exp δ sj
UHt
k
j1
⎞
k
≤ C exp⎝δ sj ⎠eC1 T/2 UHt
⎛
j1
⎛
≤ M exp⎝δ
k
⎞
sj ⎠UHt ,
j1
2.26
where M ≥ 1. From Lemma 2.4, for any t ∈ 0, T , δ, ∞ ⊂ ρAt. Then by Lemma 2.7,
{At}t∈0,T is a stable family. Obviously, AtU0 is continuously differentiable in X. So
Proposition 2.9 follows from Lemma 2.8.
From Proposition 2.9, we obtain the existence of solutions to the problem 1.1.
Theorem 2.10. Given f, g ∈ D and b ∈ C1 0, T ; L2 R3 \ K, then there exists one and only one
solution ut, x of 1.1 such that
ut, x ∈ C 0, T ; H 2 R3 \ K ∩ H01 R3 \ K
∩ C1 0, T ; H01 R3 \ K ∩ C2 0, T ; L2 R3 \ K .
2.27
Proof. Let U0 f, gT , B 0, bT . By Proposition 2.9, there exists a solution Ut ∈
C1 0, T ; H01 R3 \ K × L2 R3 \ K ∩ C0, T ; D of problem 1.3 such that U0 U0 . Let
ut, x denote the forgoing three components of Ut, then ut, x is the solution of problem
1.1 and satisfies 2.27.
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Journal of Inequalities and Applications
3. Regularity of Solutions for the Exterior Problem
First, we show the energy inequalities for our problem. These inequalities play an important
role in the proof of the regularity of solutions.
Proposition 3.1. Suppose that
ut, x ∈ C 0, T ; H 2 R3 \ K ∩ H01 R3 \ K
∩ C1 0, T ; H01 R3 \ K ∩ C2 0, T ; L2 R3 \ K
3.1
is a solution of problem 1.1 and that bt, x ∈ C1 0, T ; L2 R3 \ K, then for any given t ∈ 0, T ,
we have
ut, ·H 2 R3 \K ∂t ut, ·H 1 R3 \K ∂2t ut, ·
L2 R3 \K
≤ CT u0, ·H 2 R3 \K ∂t u0, ·H 1 R3 \K
b0, ·L2 R3 \K t
0
3.2
∂s bs, ·L2 R3 \K ds ,
where CT is a constant which depends on T .
Proof. Put Ut u, ∂t u, then Ut ∈ D and satisfies
d
Ut AtUt Bt,
dt
d
Ut, UtHt
dt
3.3
U t, UtHt Ut, U tHt Ut, UtḢt
AtUt Bt, UtHt Ut, AtUt BtHt Ut, UtḢt ,
where U t d/dtUt, Ut, UtḢt 3
i,j,k,l1 ∂t aijkl ∂j u
i
, ∂l uk . Obviously,
Ut, UtḢt ≤ CUt2Ht .
3.4
AtUt, UtHt Ut, AtUtHt ≤ CUt2Ht .
3.5
By 2.4,
Journal of Inequalities and Applications
9
Thus
d
Ut2Ht ≤ C Ut2Ht BtHt UtHt ,
dt
d
UtHt ≤ C UtHt BtHt .
dt
3.6
Applying Gronwall’s inequality, we get
UtHt ≤ e
Ct
U0H0 t
0
3.7
BsHs ds .
Without loss of generality, we assume that ∂t ut, x ∈ C0, T ; H 2 R3 \ K ∩ H01 R3 \
K ∩ C1 0, T ; H01 R3 \ K. Then we see
U t ∂t u, ∂2t u ∈ D,
3.8
d U t AtU t A tUt B t.
dt
Applying 3.7 for U t, we get
t
Ct U t
A sUs B s Hs ds .
U 0 H0 ≤e
Ht
3.9
0
By 2.17 and 2.2,
Ut2 U t1 ≤ d2 λ0 I − AtUtHt CU tHt
≤ d2 λ0 UtHt U tHt BtHt CU tHt
≤ CT U0H0 t
0
t
0
3.10
BsHs ds BtHt U 0H0
A sUs
ds Hs
t
0
B s
ds .
Hs
Obviously,
U 0
≤
≤
C
·
A0U0
B0
b0,
U0
2 R3 \K .
H0
H0
H0
L
H0
3.11
Also we have
BtHt ≤ C
t
0
∂s bs, ·L2 R3 \K ds b0, ·L2 R3 \K ,
3.12
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Journal of Inequalities and Applications
and for all t ∈ 0, T ,
t
0
BsHs ds ≤ T
t
0
∂s bs, ·L2 R3 \K ds b0, ·L2 R3 \K .
3.13
Inserting these estimates to the above inequality, we get
Ut2 U t1
≤ CT U02 b0, ·L2 R3 \K t
0
∂s bs, ·L2 R3 \K ds t
0
Us2 ds .
3.14
An application of Gronwall’s inequality implies
t
Ut2 U t1 ≤ CT U02 b0, ·L2 R3 \K ∂s bs, ·L2 R3 \K ds .
3.15
0
Namely,
ut, ·H 2 R3 \K vt, ·H 1 R3 \K ∂t ut, ·H 1 R3 \K ∂t vt, ·L2 R3 \K
ut, ·H 2 R3 \K ∂t ut, ·H 1 R3 \K ∂t ut, ·H 1 R3 \K ∂2t ut, ·
L2 R3 \K
≤ CT u0, ·H 2 R3 \K ∂t u0, ·H 1 R3 \K b0, ·L2 R3 \K
t
0
3.16
∂s bs, ·L2 R3 \K ds .
This completes the proof of 3.2.
Theorem 3.2. For h > 2, suppose that aijkl t, x ∈ CBh 0, T × R3 \ K, f ∈ H h R3 \ K, g ∈
H h−1 R3 \ K, and
b ∈ Cβ 0, T ; H h−2−β R3 \ K ,
0 ≤ β ≤ h − 2,
1
2
3
.
∂h−1
t b ∈ L 0, T ; L R \ K
3.17
Journal of Inequalities and Applications
11
If the compatibility conditions of order h − 1 are satisfied, then problem 1.1 has a solution u such that
ut, x ∈ Cβ 0, T ; H h−β R3 \ K ,
0 ≤ β ≤ h,
sup∂ ut, ·L2 R3 \K ≤ C f H h R3 \K g H h−1 R3 \K sup sup ∂α br, ·L2 R3 \K
α
|α|≤h
0≤r≤t |α|≤h−2
t ∂h−1
br,
·
r
L2 R3 \K
0
3.18
∀t ≥ 0.
dr ,
Proof. At first we prove
ut, x ∈ Ch−2 0, T ; H 2 R3 \ K ∩ Ch−1 0, T ; H 1 R3 \ K ∩ Ch 0, T ; L2 R3 \ K .
3.19
Let φ0 f and φ1 g. We define φpi by
φpi
p−2 p − 2
3 n
j,k,l1 n0
p−2−n
∂t
p−2
aijkl ∂j ∂l φnk ∂t bi 0, x,
i 1, 2, 3, p 2, 3, . . . , h − 1,
3.20
then φp , φp1 ∈ D, p 1, 2, . . . , h − 2.
We consider the following problem:
i
∂2t vq1
−
3
k
aijkl t, x∂j ∂l vq1
j,k,l1
h−3 h − 2 3 j,k,l1 n0
n
∂h−2−n
aijkl
t
vq1 0, x φh−2 x,
i
∂nt ∂j ∂l ukq ∂h−2
t b,
i 1, 2, 3,
3.21
∂t vq1 0, x φh−1 x,
vq1 t, x 0,
x ∈ ∂K,
where
th−3
uq t, x φ0 x tφ1 x · · · φh−3 x h − 3!
here u0 ≡ 0.
t
t − rh−3
vq r, xdr,
0 h − 3!
3.22
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Journal of Inequalities and Applications
From 3.21,
3
i
k
∂2t vq1
− vqi −
aijkl t, x∂j ∂l vq1
− vqk
j,k,l1
h−3 h − 2 3 n
k
k
∂h−2−n
∂
u
a
∂
∂
−
u
ijkl
j
l
q
t
t
q−1
n
j,k,l1 n0
3 h−3 h − 2 ∂h−2−n
aijkl
t
n
j,k,l1 n0
3.23
t
∂j ∂l
t − rh−3−n k
k
dr.
vq − vq−1
0 h − 3 − n!
By 3.2, we have
vq1 − vq H 2 R3 \K
∂t vq1 − ∂t vq H 1 R3 \K ∂2t vq1 − ∂2t vq L2 R3 \K
t
≤ CT vq − vq−1 H 2 R3 \K dr,
3.24
q 2, 3, . . . ,
0
thus
2
2 ∂t vq1 − ∂t vq 1 3
vq1 − vq 2 3
v
−
∂
v
∂
t q1
t q
H R \K
H R \K
L2 R3 \K
≤C
CT tq
. 3.25
q!
This implies that vq converges to some v in C0, T ; H 2 R3 \ K ∩ C1 0, T ; H 1 R3 \ K ∩
C2 0, T ; L2 R3 \ K. Set
ut, x φ0 x tφ1 x · · · th−3
φh−3 x h − 3!
t
t − rh−3
vr, xdr,
0 h − 3!
3.26
then uq tends to u in C0, T ; H h−2 R3 \K∩Ch−1 0, T ; H 1 R3 \K∩Ch 0, T ; L2 R3 \K.
The passage to the limit of 3.21 shows
i
∂2t ∂h−2
t u −
3
k
aijkl t, x∂j ∂l ∂h−2
t u
j,k,l1
3 h−3 h − 2 j,k,l1 n0
∂h−2−n
aijkl
t
n
3.27
i
∂nt ∂j ∂l uk ∂h−2
t b,
i 1, 2, 3,
namely,
⎛
h−2
3
⎞
d
i
⎝∂2 ui −
aijkl t, x∂j ∂l uk ⎠ ∂h−2
t
t b,
dth−2
j,k,l1
i 1, 2, 3.
3.28
Journal of Inequalities and Applications
13
Taking account of the definition of φp , we see
⎞
d ⎝ 2 i
k ⎠
∂
u
−
a
t,
x∂
∂
u
ijkl
j l
t
p
dt
j,k,l1
⎛
3
p
p
∂t bi 0, x,
i 1, 2, 3, p 0, 1, 2, . . . , h − 2. 3.29
t0
Therefore u is the solution of problem 1.1 and satisfies 3.19.
We now prove 3.18 by induction. When h 2, 3.18 follows from 3.2. For h > 2,
suppose that 3.18 holds for h − 1. We show that it still holds for h.
Applying 3.2 to 3.27, we conclude from the inductive hypothesis that
h−2 ∂t u
H 2 R3 \K
∂h−1
t u
H 1 R3 \K
∂ht u
L2 R3 \K
≤ the right-hand side of 3.19. 3.30
In a similar way, we can obtain
sup
|α|≤h−2
α ∂ u 2 3
∂α1
t
t u
H R \K
H 1 R3 \K
∂α2
t u
L2 R3 \K
≤ the right-hand side of 3.19.
3.31
Set Ut {u, ∂t u}, then Ut is the solution of 1.3 and
Ut ∈ Ch−2 0, T ; H 2 R3 \ K × H 1 R3 \ K .
3.32
Now
λ0 I − AtUt λ0 Ut − U t Bt ∈ C 0, T ; H 2 R3 \ K × H 1 R3 \ K ,
3.33
then by 2.17 taking p 3, we see
Ut ∈ C 0, T ; H 3 R3 \ K × H 2 R3 \ K ,
ut, ·H 3 R3 \K ∂t ut, ·H 2 R3 \K
Ut3 ≤ λ0 I − AtUt2
≤ C Ut2 U t2 Bt2
≤ C ut, ·H 2 R3 \K ∂t ut, ·H 2 R3 \K ∂2t ut, ·
H 1 R3 \K
bt, ·H 1 R3 \K
≤ the right-hand side of 3.19.
3.34
Differentiation of 3.33 with respect to t gives
λ0 I − AtU t λ0 U t − U t B t − U t A tUt,
3.35
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Journal of Inequalities and Applications
and by the above result A tUt ∈ C0, T ; H 2 R3 \ K × H 1 R3 \ K,
the right-hand side of 3.36 ∈ C 0, T ; H 2 R3 \ K × H 1 R3 \ K ,
3.36
from which it follows that
U t ∈ C 0, T ; H 3 R3 \ K × H 2 R3 \ K ,
∂t ut, ·H 3 R3 \K ∂2t ut, ·
H 2 R3 \K
U t3 ≤ the right-hand side of 3.19.
3.37
Repeating this process, we get
Ut ∈ Ch−3 0, T ; H 3 R3 \ K × H 2 R3 \ K ,
sup ∂αt ut, ·H 3 R3 \K ≤ the right-hand side of 3.19.
3.38
|α|≤h−3
Using this, we see the right-hand side of 3.33 ∈ C0, T ; H 3 R3 \ K × H 2 R3 \ K, then by
2.17 taking p 4
Ut ∈ C 0, T ; H 4 R3 \ K × H 3 R3 \ K .
3.39
This assures that the right-hand side of 3.35 ∈ C0, T ; H 3 R3 \ K × H 2 R3 \ K, then
U t ∈ C 0, T ; H 4 R3 \ K × H 3 R3 \ K ,
∂t ut, ·H 4 R3 \K ≤ the right − hand side of 3.19.
3.40
Repeating this process, we get
Ut ∈ Ch−4 0, T ; H 4 R3 \ K × H 3 R3 \ K ,
sup ∂αt ut, ·H 4 R3 \K ≤ the right − hand side of 3.19.
3.41
|α|≤h−4
Step by step, finally, we get
Ut ∈ C 0, T ; H h R3 \ K ×H h−1 R3 \ K ∩C1 0, T ; H h−1 R3 \ K ×H h−2 R3 \ K
∩ · · · ∩ Ch−2 0, T ; H 2 R3 \ K × H 1 R3 \ K
3.42
and 3.18.
Journal of Inequalities and Applications
15
4. Application to Elastrodynamics
It is well known that the displacement u u1 , u2 , u3 ut, x of an isotropic, homogeneous,
hyperelastic material without the action of external force satisfies the following hyperbolic
system cf. 4, 5:
Lu ∂2t u − c22 Δu − c12 − c22 ∇div u Ft, x,
4.1
where F F 1 , F 2 , F 3 , and c1 , c2 are given by the Lamé constants λ, μ:
c12 λ 2μ,
c22 μ.
4.2
We assume that μ > 0, λ μ > 0.
From 5, system 4.1 can be written as
∂2t ui −
3
aijkl t, x∂j ∂l uk 0,
i 1, 2, 3,
4.3
j,k,l1
where A aijkl t, x stands for the elastic tensor.
The system 4.3 is the special case of the system 1.1. So by the existence Theorem 3.2,
we derive the existence of solutions for the initial-boundary problem to the elastrodynamic
system 4.3 outside a domain.
Acknowledgments
Projects 10626046 supported by NSFC and 20070410487 supported by China Postdoctoral
Science Foundation. The authors would like to thank Professor Tatsien Li and Professor Tiehu
Qin for helpful discussions and suggestions.
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