RESEARCH Open Access
On the regularity of the solution for the second
initial boundary value problem for hyperbolic
systems in domains with conical points
Nguyen Manh Hung
1
, Nguyen Thanh Anh
1*
and Phung Kim Chuc
2
* Correspondence:
[email protected]
1
Department of Mathematics,
Hanoi National University of
Education, Hanoi, Vietnam
Full list of author information is
available at the end of the article
Abstract
In this paper, we deal with the second initial boundary value problem for higher
order hyperbolic systems in domains with conical points. We establish several results
on the well-posedness and the regularity of solutions.
1 Introduction
Boundary value problems in nonsmooth domains have been studied in differential
aspects. Up to now, elliptic boundary value problems in domains with point singulari-
ties have been thoroughly investi gated (see, e.g, [1,2] and the ex tensive bibliography in
this book). We are concerned w ith initial boundary value problems for hyperbolic
equations and systems in domains with co nical points. These problems with t he
Dirichlet boundary conditions were investigated in [3-5] in which the unique existence,
the regularity and the asymptotic behaviour near the conical points of the solutions are
established. The Neumann boundary problem for general second-order hyperbolic sys -

1
, , p
n
) Î N
n
, we use notations |p|=p
1
+ + p
n
,
D
p
=
∂
|p|
∂x
p
1
1
∂x
p
n
n
. For a complex-valued vector function u =(u
1
, , u
s
) defined on Q,
we denote
D

u
s
∂t
j
), |u| =(
s

j
=1
|u
j
|
2
)
1
2
.
.
Let us introduce the following functional spaces used in this paper. Let l denote a
nonnegative integer.
H
l
(Ω) - the usual Sobolev space of vector functions u defined in Ω with the norm

u

H
l
()
=

H
l−
1
2
()
= inf


v

H
l
()
: v ∈ H
l
(), v|

= u

.
H
l,0
(Q, g)(g Î ℝ)- the weighted Sobolev space of vector functions u defined in Q
with the norm

u

H
l,0
(Q,γ )

u

H
l,1
(Q,γ )
=
⎛
⎝

Q
⎛
⎝

|p|≤l
|D
p
u|
2
+ |u
t
|
2
⎞
⎠
e
−2γ t
dxdt
⎞
⎠
1


r
2(α+|p|−l)
|D
p
u|
2
dx
⎞
⎠
1
2
,
where
r = |x| =


n
k=1
x
2
k

1
2
.
H
l
α
()(α ∈ R

http://www.boundaryvalueproblems.com/content/2011/1/17
Page 2 of 18
If l ≥ 1, then
V
l−
1
2
α
(

)
, H
l−
1
2
α
(

)
denote the spaces consisting of traces of functions
from respective spaces
V
l
2
,
α
(), H
l
α
(


H
l−
1
2
α
(

)
= inf


v

H
l
α
()
: v ∈ H
l
α
(), v|

= u

.
H
l,1
α
(Q, γ )(α, γ ∈ R

⎠
e
−2γ t
dx dt
⎞
⎠
1
2
< ∞
.
From the definitions it follows the continuous imbeddings
V
l
2
,
α
() ⊂ H
l
α
()
(2:1)
and
V
l+k
2
,
α+k
() ⊂ V
l
2,α

p
|,|
q
|≤m
(−1)
|p|
D
p
(a
pq
D
q
u)
,
where a
pq
= a
pq
(x, t)arethes × s matrices with the bounded complex-valued com-
ponents in
Q
. We assume that
a
pq
=(−1)
|p|+|q|
a
∗
qp
for all |p|, |q| ≤ m,where



2
(2:4)
for all h
p
Îℂ
s
,|p|=m, and all
(
x, t
)
∈
¯
Q
.
Let v be the unit exterior normal to S . It is well known that (see, e.g., [[7], Th. 9.47])
there are boundar y operators N
j
= N
j
(x, t, D), j =1,2, ,m on S such that integration
equality


Lu
¯
vdx=

|p|,|q|≤m


)
and for all t Î [0, ∞). The order of the operator N
j
is 2m -
j for j = 1, 2, , m.
Hung et al. Boundary Value Problems 2011, 2011:17
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Page 3 of 18
In this paper, we consider the following problem:
u
tt
+ Lu =
f
in Q
,
(2:6)
N
j
u =0onS, j =1, , m
,
(2:7)
u
|
t=0
= u
t
|
t=0
=0 on


Q
f ¯η dx d
t
(2:9)
holds for all h(x, t) Î H
m,1
(Q) satisfying h(x, t) = 0 for all t ≥ T for some positive real
number T.
3 Th e unique solvability and the regularity in time
First, we introduce some notations which will be used in the proof of Theorems 3.3
and 3.4. For each vector function u,v defined in Ω and each nonnegative integer k,
|
u|
k,
=
⎛
⎝



|p|=k
|D
p
u|
2
dx
⎞
⎠
1

k,
,(u, v)

τ
=(u(·, τ ), v(·, τ ))

,
B
t
k
(t, u, v)=

|
p
|,|
q
|≤m


∂
k
a
pq
∂t
k
(·, t)D
q
u(·, t)D
p
v(·, t ) dx, B

(
τ , v, u
).
(3:1)
Next, we introduce the followi ng Gronwall-Bellman and interpolation inequalities as
two fundamental tools to establish the theorems on the unique existence and the regu-
larity in time.
Lemma 3.1 ([8], Lemma 3.1) Assume u, a, b are real-valued continuous on an inter-
val [a, b], b is nonnegative and integrable on [a, b], a is nondecreasing satisfying
u
(τ ) ≤ α(τ )+

τ
a
β(t)u(t) dt for all a ≤ τ ≤ b
.
Hung et al. Boundary Value Problems 2011, 2011:17
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Page 4 of 18
Then
u
(τ ) ≤ α(τ ) exp


τ
a
β(t) dt

for all a ≤ τ ≤ b
.

, if f Î L
2
(Q, s) for some nonnegative
real number s, the problem (2.6)-(2.8) has a unique generalized solution u in the space
H
m,1
(Q, g + s) and

u

2
H
m,1
(Q,γ +σ )
≤ C


f


2
L
2
(Q,σ )
,
(3:4)
where C is a constant independent of u and f.
Proof. The uniqueness is proved by similar way as in [[4], Th. 3.2]. We omit the
detail here. Now we prove the exist ence by Galerkin approximating method. Suppose
{ϕ

N
k
=
1
are the solution of the system of the following ordinary differential
equations of second order:
(u
N
tt
, ϕ
l
)

t
+ B(t, u
N
, ϕ
l
)=(f , ϕ
l
)

t
, l =1, , N
,
(3:5)
with the initial conditions
c
N
k

N
t
)

t
.
(3:7)
Now adding this equality to its complex conjugate, then using (3.1) and the integra-
tion by parts, we obtain
|
u
N
t
|
2
0,
τ
+ B(τ , u
N
, u
N
)=B
τ
t
(u
N
, u
N
)+2Re(f , u
N

2
0,
τ
+ B
0
(τ ,u
N
, u
N
)+ρ|u|
2
0,
τ
= B
τ
t
(u
N
, u
N
)
−

|p|, |q|≤m
|
p
| + |
q
| < 2m − 1


u
N
t
(·, τ )|
2
0,
+ μ|u
N
(·, τ )|
2
m,
+ ρ


u(·, τ )


2
0
,

.
We denote by I, II, III, IV the terms from the first, second, third, and forth, respec-
tively, of the righ t-hand sides of (3.9 ). We will give estimations for these terms. Firstly,
we separate I into two terms

|p|=|q|=m

Q
τ

p
u
N
dx dt ≡ I
1
+ I
2
.
Put
μ
1
=sup{|
∂
a
pq
∂t
(x, t)| :


p


=


q


= m,(x, t) ∈
Q} and m

u
N
|
2
0,Q
τ
) ≤ m

μ
1
|u
N
|
2
m,Q
τ
.
By the Cauchy inequality and the interpolation inequality (3.3), fo r an arbitra ry posi-
tive number ε
1
, we have
I
2
≤ ε
1
|u
N
|
2
m,

(·, τ )|
2
m
,

+ C
2
|u
N
(·, τ )|
2
0
,

,
where C
2
= C
2
(ε
2
) is a nonnegative constant independent of u
N
, f and τ. For the
terms III and IV, by the Cauchy inequality, we have
III ≤
(μ − ε
2
)ρ
2

and
IV ≤ ε
3
|u
N
t
|
2
0,Q
τ
+
1
ε
3
|f |
2
0,Q
τ
,
Hung et al. Boundary Value Problems 2011, 2011:17
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Page 6 of 18
where ε
3
> 0, arbitrary. Combining the above estimations we get from (3.9) that
|u
N
t
(·, τ )|
2

1
+
(μ − ε
2
)ρ
2
m

μ
1
+ ε
1

|u
N
|
2
0,Q
τ
+

m

μ
1
+ ε
1
μ − ε
2
+ ε

2
)ρ
2
m

μ
1
+ ε
1
ρ
− C
2
for ρ>C
2
.
We have
dg
dρ
=
ρ
2
− 2C
2
ρ −
C
1
A
A
(
ρ − C

2
+
C
1
A
.
We put
γ
0
=
1
2
inf
ε
1
>0
0<ε
2
<μ
max{
m

μ
1
+ ε
1
μ − ε
2
, g(ρ
0

1
μ − ε
2
+ ε
3
< 2γ
1
and
C
1
(ε
1
, ε
2
)+
(μ − ε
2
)ρ
2
m

μ
1
+ ε
1
ρ − C
2
(
ε
1


τ
0
|||u(·, t)|||
2

dt + C

τ
0
|f (·, t)|
2
0,
dt for all τ ≤ 0
,
(3:13)
where
C =
1
ε
3
. By the Gronwall-Bellman inequality (3.2), we receive from (3.13) that
|||u
N
(·, τ )|||
2

≤ Ce
2γ
1

τ
0
|e
−σ t
f (·, t)|
2
0,
dt
.
Hence, it follows from (3.14) that
|||u
N
(·, τ )|||
2

≤ Ce
2(γ
1
+σ )τ

τ
0
|e
−σ t
f (·, t)|
2
0,
dt ≤ Ce
2(γ
1

e
−2(γ +σ )τ
|||u
N
(·, τ )|||
2

dτ ≤ C


f


2
L
2
(Q,σ )
.
(3:16)
It is clear that |||.|||
Q,g+s
is a norm in H
m,1
(Q, g + s) which is equivalent to the norm

.

H
m,1
(

7]), we can conclude that the sequence
{u
N
}
∞
N
=
1
possesses a subsequence convergent to
a vector function u Î H
m,1
(Q, g + s) which is a generalized solution of problem (2.6)-
(2.8). Moreover, it follows from (3.17) that the inequality (3.4) holds. □
Theorem 3.4. Let h be a nonnegative integer. Assume that all the coefficients a
pq
together with their derivatives with respect to t up to the order h are bounded on
Q
. Let
g
0
be the number as in Theorem 3.3 which was defined by formula (3.11). Let the vector
function f satisfy the following conditions for some nonnegative real number s
(i)
f
t
k ∈ L
2
(
Q, kγ
0

)
γ + σ
)
for k = 0, 1, , h and
h

k
=
0

u
t
k

2
H
m,1
(Q,(k+1)γ +σ)
≤ C
h

k
=
0


f
t
k


<g. We will prove by induction that


u
N
t
k
(·, τ )


2
H
m
()
≤ Ce
2
(
(k+1)γ
1
+σ
)
τ
k

j
=0


f
t

l
)

t
+
h

k
=
0

h
k

B
t
h−k (t, u
N
t
k
, ϕ
l
)=(f
t
h , ϕ
l
)

t
, l =1, , N

h+2
, u
N
t
h+1
)

t
+
h

k
=
0

h
k

B
t
h−k (t, u
N
t
k
, u
N
t
h+1
)=(f
t

∂
∂t
B
t
h−k (t, u
N
t
k
, u
N
t
h
) − B
t
h−k+1 (t, u
N
t
k
, u
N
t
h
)

=2Re(f
t
h , u
N
t
h+1

N
t
h
)+
h
−1

k=0

h
k

B
τ
t
h−k+1
(u
N
t
k
, u
N
t
h
)
−
h−1

k
=

N
t
h
and the last term of the
righthand side of (3.8) replaced by the following expression
h
−1

k
=
0

h
k

B
τ
t
h−k+1
(u
N
t
k
, u
N
t
h
) −
h
−1

together with their derivatives with respect to t up to the
order h are bounded, by the Cauchy and interpolation inequalities and the induction
assumption, we see that
|
h
−1

k=0

h
k

B
t
h−k (τ , u
N
t
k
, u
N
t
h
)|≤ε

|u
N
t
h
(·, τ )|
2

t
h
(·, τ )|
2
m,
+ |u
N
t
h
(·, τ )|
2
0,

+ Ce
2(hγ
1
+σ)τ
k

j
=0


f
t
j


2
L

h
|
2
m,Q
τ
+ |u
N
t
h
|
2
0,Q
τ

+ C
h−1

k=0


u
N
t
k


2
m,Q
τ
= ε

k
(·, t)


2
m,
dt
≤ ε

|u
N
t
h
|
2
m,Q
τ
+ |u
N
t
h
|
2
0,Q
τ

+ C
k

j=0

τ
+ |u
N
t
h
|
2
0,Q
τ

+ Ce
2(hγ
1
+σ)τ
k

j
=0


f
t
j


2
L
2
(Q,jγ
0


2
L
2
(Q)
≤ ε|u
N
t
h+1
|
2
0,Q
τ
+ Ce
2(hγ
1
+σ)τ


f
t
h


2
L
2
(
Q,hγ
0

j


2
L
2
(Q,jγ
0
+σ )
, k =0, , h
.
(3:24)
From this inequality, by again standard weakly convergent arguments, we can con-
clude that the sequence
{u
N
t
k
}
∞
N=
1
possesses a subsequence convergent to a vector func-
tion u
(k)
Î H
m,1
(Q,(k +1)g +s), moreover, u
(k)
is the kth generalized derivative in t of

(x, t) D
p
, j =1, , m
.
Let L
0
(x, t, D), N
0j
(x, t, D), be the principal homogeneous parts of L(x, t, D), N
j
(x, t,
D). It can be directly verified that the derivative D
a
can be written in the form
D
α
= r
−|α|
|α|

p
=0
P
α,p
(ω, D
ω
)(rD
r
)
p

(
0, t, D
)
= r
−2m
L
(
ω, t, D
ω
, rD
r
),
N
0,
j
(0, t, D)=r
−2m+j
N
j
(ω, t, D
ω
, rD
r
)
.
The operator pencil associated with the problem is defined by
U
(λ, t)=(L(ω, t, D
ω
, λ), N

0
is called an eigenvalue of
U
(
λ, t
)
if
there exists 
0
Î H
2m
( G)suchthat
0
≠ 0and
U(
λ
0
, t
)
ϕ
0
=
0
. I t is well known that
the spectrum of the operator
U(
λ
0
, t
)

2m,1
α
(Q,(k +2)γ + σ
)
for k = 0, 1, , h -1and
h−1

k
=
0

u
t
k

2
H
2m,1
α
(Q,(k+2)γ +σ)
≤ C
h

k
=
0


f
t

x, t
0
, D
)
u = fin
,
(4:3)
N
j
(x, t
0
, D)u = g
j
on , j =1, , m
,
(4:4)
where
f ∈ V
l
2
,
α
(), g
j
∈ V
l+j−
1
2
2
,

2
V
l
2,α
()
+
m

j=1


g
j


2
V
l+j−
1
2
2
,
α
()
+

u

2
V

|
≤ 2
−k+1
}
, k =1,2,
,
and Γ
k
= ∂Ω ∩ ∂Ω
k
, k = 0, 1 According to well known results on the regularity of
solutions of elliptic boundary problems in smooth domains (see, e.g., [12]), we have
Hung et al. Boundary Value Problems 2011, 2011:17
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Page 11 of 18

u

2
H
l+2m
(
2
)
≤ C(


f



+

u

2
L
2
(
1
∪
2
∪
3
)
)
with the constant C independent of u, f, g
j
and t
0
. By m aking chang e of variable x =
2
-k
x’ for a positive integer k, we get from (4.3), (4.4) that

|
p
|=2m
a
p
D

x

u +

|
p
|≤2m−
j
−1
2
(|p|−2m+j)k
b
jp
D
p
x

u=2
−(2m−j)k
g
j
on , j =1, , m
.
(4:7)
Similarly as above, from (4.6), (4.7), we have

u

2
W

−(2m−j)k
g
j



2
W
l+j−
1
2
(

1
∪
2
∪
3
)
+

u

2
L
2
(
1
∪
2

l+2m
(
2
)
≤ C




2
−2mk
f



2
W
l
(
1
∪
2
∪
3
)
+



2

)

.
(4:9)
Returning to variable x with noting that, in Ω
k+2
,2
-k-2
≤ r ≤ 2
-k-1
, from (4.9) we have

|p|≤l+2m



r
2(p+|p|−l−2m)
u



2
L(
k+2
)
≤ C
⎛
⎝



2
W
l+j
(
k+1
∪
k+2
∪
k+3
)
+



r
2(p−l−2m)
u



2
L
2
(
k+1
∪
k+2
∪
k+3

j=1


˜
g
j


2
V
l+j+1
2,α
()
+

u

2
V
0
2,α−l−2m
()
⎞
⎠
.
(4:11)
Here it is noted that

|
p



2
V
l
2,α
()
+
m

j=1


g
j


2
V
l+j−
1
2
2
,
α
()
+

u


L
(
x, t
0
, D
)
u = fin
,
(4:12)
N
j
(x, t
0
, D)u = g
j
on , j =1, , m
,
(4:13)
where
f ∈ H
0
m
+ε
(
)
,
g
j
∈ H
j−



2
H
0
m+ε
()
+
m

j=1


g
j


2
H
j−
1
2
m+ε
()
+

u

2
H

j−
1
2
m+ε
() ≡ V
j−
1
2
2
,
m+ε
(
)
. Moreover, it is obvious
that
H
0
m+ε
() ≡ V
0
2
,
m+ε
(
)
. Hence,
f ∈ V
0
2
,

m
2
,
0
() ⊂ V
0
2
,
−m
(
)
by (2. 3) and (2.2). Thus the
assertion of the lemma follows from Lemma 4.2 w ith noting that the space
V
2m
2
,
m+ε
(
)
is continuously imbedded in
H
2
m
m
+ε
()
according to (2.1).
Now consider the case
m ≥

u
(x)=

|
α
|
≤m−−1
c
α
x
α
+ v(x)
,
where
v ∈ V
m
2
,
ε
(
)
, and
c
α
=
1
α!
lim
r→0
1

|≤C

u

H
m
(

)
, |α|≤m −  − 1
.
(4:16)
Here
|| =


d
ω
,andC is a constant independent of u and t
0
.Put
w =

|
α
|
≤m−−1
c
α
x

where C is a constant independent of u and t
0
. From this we have
L(x, t
0
, D)w ∈ H
0
m+ε
()=V
0
2,m+ε
(),
N
j
(x, t
0
, D)w ∈ H
j−
1
2
m+ε
()=V
j−
1
2
2
,
m+ε
()
.

j−
1
2
2
,
m+ε
(), j =1, , m
.
(4:18)
Now we can apply Lemma 4.2 to conclude from (4.17) and (4.18) that
v ∈ V
2m
2
,
m+ε
(
)
.
Therefore,
u
= v + w ∈ H
2m
m
+
ε
(
)
with the estimate (4.14). The lemma is completely
proved.
Proof of Theorem 4.1: First, we show by induction on h that

=
(
f
(
·, t
)
− u
tt
(
·, t
)
, η
)
(4:20)
for all h Î H
m
(Ω) and a.e. t Î (0, ∞). Since f (·, t)-u
tt
(·, t) Î L
2
(Ω)fora.e.t Î (0,
∞), according to results for elliptic boundary value problem in domains with smooth
boundaries, it follows from (4.20) that
u
(·, t) ∈ H
2m
loc
(\{0}
)
for a.e. t Î (0, ∞), more-

0
(\{0}
)
and a.e. t Î (0, ∞). Assume now (4.19) holds for h -2.Itfol-
lows from (4.20) that
B(t , u
t
h−1 ,η)=(f
t
h−1 (·, t), η) − (u
t
h+1 (·, t), η) −
h
−2

k
=
0

h − 1
k

B
t
h−1−k (t, u
t
k , η
)
(4:22)
for all h Î H

http://www.boundaryvalueproblems.com/content/2011/1/17
Page 14 of 18
for all
η ∈ C
∞
0
(\{0}
)
and a.e. t Î (0, ∞). Combining (4.22) and (4.23) we obtain
B
(
t, u
t
h−1 , η
)
=
(
F
h−1
(
·, t
)
, η
)
(4:24)
for all
C
∞
0
(\{0}

for a.e. t Î (0,
∞), and therefore, (4.19) holds for h -1.
Now we prove t he assertion of the theorem by induction on h. Let us consider first
the case h = 1. We rewrite (2.6), (2.7) in the form
L
(
x, t, D
)
u = f
1
:= f − u
tt
in Q
,
(4:25)
N
j
(x, t, D)u =0 onS, j =1, , m
.
(4:26)
Since
f
1
(·, t) ∈ L
2
() ⊂ H
0
α
(
)

2
L
2
()
+


u(·, t)


2
H
m
()

,
where C is a constant independent of u, f
1
and t. Since the trip
m − ε −
n
2
≤ Reλ ≤ 2m − α −
n
2
does not contain any eigen value of
U
(
λ, t
)

1
(·, t)


2
L
2
()
+


u(·, t)


2
H
m
()

,
(4:27)
where C is a constant independent of u, f
1
and t. Now multiplying both sides of
(4.27) with e
-2(2g+s)t
, then integrating with respect to t from 0 to ∞ and using estimates
from Theorem 3.4, we obtain

u

t
h+1 −
h
−2

k
=
0

h − 1
k

L
t
h−1−k u
t
k in Q
,
(4:29)
N
j
u
t
h−1 =
˜
g
j
:= −
h
−2

.
Moreover,
f
t
h−1 ∈ L
2
(
Q,
(
h − 1
)
γ + σ
)
⊂ L
2
(
Q,
(
h +1
)
γ + σ
)
by the assumption of the theorem and
u
t
h+1 ∈ L
2
(
Q,
(




2
H
0
α
()
≤ C



f
t
h−1 (·, t)


2
+


u
t
h+1 (·, t)


2
L
2
()

2
α
(

)
≤ C
h−2

k=0


u
t
k (·, t)


2
H
2m
α
()
, j =1, , m,
where C is the constant independent of u, f and t. Now we can repeat the arguments
above to conclude that
u
t
h−1 ∈ H
2m,1
α
(Q,(h +1)γ + σ

0
are
now defined by
μ =1,μ
1
=0 and
γ
0
=0
.
The o perator pencil associated with the problem (5.1)-(5.3) is now defined by (see
[[13], Sec. 2.3])
U(
λ, t
)
u = U
(
λ
)
u| =
(
δu + λ
(
λ + n − 2
)
u, ∂
ν
u|
∂G
),

k ∈ L
2
(
Q, σ
)
, k ≤ h
,
(ii)
f
t
k
(
x,0
)
=0,0≤ k ≤ h − 1
.
Assume further that n >4-2a. Then for an arbitrary positive real number g the pro-
blem (5.1)-( 5.3) has a unique genera lized solution u in the space H
1,1
(Q, g + s) which
has derivatives with respect to t up to the order h with
u
t
k ∈ H
2,1
α
(Q,(k +2)γ + σ
)
for k
= 0, 1, , h -1,and

L
2
(Q,σ )
,
where C is a constant independent of u and f.
Acknowledgements
This work was supported by National Foundation for Science and Technology Development (NAFOSTED), Vietnam,
under project no. 101.01.58.09.
Author details
1
Department of Mathematics, Hanoi National University of Education, Hanoi, Vietnam
2
Department of Mathematics,
Can Tho University, Can Tho City, Vietnam
Authors’ contributions
All authors typed, read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Received: 22 January 2011 Accepted: 25 August 2011 Published: 25 August 2011
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