Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2010, Article ID 101959, 19 pages
doi:10.1155/2010/101959
Research Article
A Mixed Problem for Quasilinear Impulsive
Hyperbolic Equations with Non Stationary
Boundary and Transmission Conditions
Akbar B. Aliev
1
and Ulviya M . Mamedova
2
1
Azerbaijan Technical University, AZ 1073, Baki, Azerbaijan
2
Institute of Mathematics and Mechanics of NAS of Azerbaijan, AZ 1141, Baku, Azerbaijan
Correspondence should be addressed to Akbar B. Aliev, [email protected]
Received 10 March 2010; Revised 13 June 2010; Accepted 26 October 2010
Academic Editor: Toka Diagana
Copyright q 2010 A. B. Aliev and U. M. Mamedova. 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.
The initial-boundary value problem for a class of linear and nonlinear equations in Hilbert space
is considers. We prove the existence and uniqueness of solution of this problem. The results of
this investigation are applied to solvability of initial-boundary value problems for quasilinear
impulsive hyperbolic equations with non-stationary transmission and boundary conditions.
1. Abstract Model Initial Boundary Value Problem with
Non Stationary Boundary and Transmission Conditions for
the Impulsive Linear Hyperbolic Equations
In paper 1 there is given an abstract scheme of investigation of mixed problems for
hyperbolic equations with non stationary boundary conditions. In this direction, some results
t
u
i
t
f
i
t
,
hyperbolic equations
, 1.1
B
iν
¨u
i
t
m
k1
C
0,
stationary boundary and transmission conditions
,
u
i
0
u
0
i
, ˙u
i
0
u
1
i
,
initial conditions
,
1.2
where t ∈ 0,T,¨u
i
; D
j
kμ
are the
linear operators from H
k
to Y
j
μ
; ν 1, ,s
i
, i 1, ,m, μ 1, ,r
j
, j 1, ,m,k 1, ,m.
We will investigate this problem under the following conditions.
i Let H
i
0
⊂ H
i
,andletH
i
0
be densely in H
i
and continuously imbedded into it, i
1, 2, ,m.
In the Hilbert space H
i
, it was defined the system of the inner products ·, ·
,c
1
> 0,
u
2
H
i
t
u, u
H
i
t
,t∈
0,T
,i 1, 2, ,m.
1.3
For each u ∈ H
i
, the function t →u
2
H
i
t
ν
t
≤ c
2
v
2
X
i
ν
,c
2
> 0,
v
2
X
i
ν
t
v, v
X
i
ν
t
t acts boundedly from H
i
0
to H
i
; A
i
t is strongly continuously
differentiable.
iii The linear operators B
iν
,thatactfromH
i
1/2
to X
i
ν
, bounded, where H
i
1/2
H
i
0
,H
i
1/2
is interpolation space between H
i
into Y
j
μ
, act boundedly μ 1, ,r
j
, j
1, ,m; k 1, ,m.
Let us introduce the following designations:
H H
1
⊕···⊕H
m
,
H
0
u : u
u
1
, ,u
m
,u
i
∈ H
i
i
1/2
,i 1, ,m;
m
k1
D
j
kμ
u
k
0,μ 1, ,r
j
,j 1, ,m
,
H
1
w : w
w
1
, ,w
m
,w
i
i
⊕ X
i
1
⊕···⊕X
i
s
i
, H
m
i1
H
i
, H
1/2
H
1
, H
1/2
.
1.5
From condition v, it follows that the space
H
1/2
with the norm
,u
i
∈ H
i
1/2
,i 1, ,m
H
1
1/2
×···×H
m
1/2
. 1.7
vi Let the linear manifold
H
0
be dense in
H
1/2
, and let linear manifold H
1
be dense in
H.
viiGreen’s Identity. For arbitrary
u, v ∈
H
k1
C
i
kν
tu
k
,B
iν
v
i
X
i
ν
t
⎤
⎦
m
i1
⎡
⎣
u
i
,A
i
t
t
⎤
⎦
.
1.8
4AdvancesinDifference Equations
viii For all
u u
1
, ,u
m
∈
H
0
, the following inequality is fulfilled:
c
1
m
i1
u
i
2
H
i
i
H
i
t
s
i
ν1
m
k1
C
i
kν
tu
k
,B
iν
u
i
X
i
ν
t
⎤
⎦
1
, ,u
m
−→ L
t
λ
u
L
t
10
λ
u, L
t
11
λ
u, ,L
t
1s
1
λ
∈ R,where
L
t
i0
λ
u λu
i
A
i
t
u
i
,i 1, 2, ,m,
L
t
iν
λ
u λB
iν
u
i
m
j
kμ
u
0
k
0,
m
k1
D
j
kμ
u
1
k
0
i 1, 2, ,m, μ 1, 2, ,r
j
,j 1, 2, ,m
. 1.12
xi f
i
· ∈ W
1
p
0,T; H
i
,p≥ 1,i 1, ,m,
is continuous, and the function
t −→
u
1
t
,B
11
u
1
t
, ,B
1s
1
u
1
t
, ,u
m
t
,B
m1
w
A
1
t
u
1
,
m
k1
C
1
k1
t
u
k
, ,
m
k1
C
1
ks
1
ks
m
t
u
k
,t∈
0,T
,w∈H
1
.
1.15
Then the problem 1.1-1.2 is represented as the Cauchy problem
¨w A
t
w Φ
t
,
w
0
u
m
t,
Φ
t
f
1
t
,g
11
t
, ,g
1s
1
t
, ,f
m
t
1
, ,u
0
m
,B
m1
u
0
m
, ,B
ms
m
u
0
m
,
w
1
u
1
1
,B
11
u
1
1
, ,B
t
∈ C
2
0,T
; H
∩ C
1
0,T
;
H
1
, H
1/2
∩ C
0,T
; H
1
w
1
,w
2
Ht
m
i1
w
1
i
,w
2
i
H
i
t
m
i1
s
i
ν1
,w
l
i
u
l
i
,B
i1
u
l
i
, ,B
is
i
u
l
i
,i 1, 2, ,m,u
l
1
, ,u
l
m
∈
H
0
,l 1, 2.
We denote space H with inner product 1.19 by Ht.
We will prove later the following auxiliary results.
and the function t →w
2
Ht
: 0,T → R
is continuously differentiable, where w
2
Ht
w, w
Ht
.
Statement 1.4. At is a symmetric operator in Ht for each t ∈ 0,T.
Statement 1.5. At has a regular point for each t ∈ 0,T in R.
At is symmetric and RAtλIHt,forsomeλ ∈ R; therefore, for each t ∈
0,T, At is a selfadjoint operator in Htsee 4,chapterx.
Taking into account viii and Statement 1.3,weget
Atw, w
Ht
m
i1
⎡
⎣
A
i
X
i
ν
t
⎤
⎦
≥ c
1
w
2
Ht
,
1.21
that is, At is a lower semibounded selfadjoint op erator in Ht.
Thus, the operator A
0
tAtλ
0
I is selfadjoint and positive definite, where λ
0
>c
1
.
Problem 1.16 canberewrittenas
¨w
t
w
1
.
1.22
It is known that if w
0
∈H
1
and w
1
∈H
1/2
, then the problem 1.22 has a unique
solution w ∈ C
2
0,T; H ∩ C
1
0,T; H
1/2
∩C0,T; H
1
see 5, 6.
To complete the proof of the theorem, we need to show that w
0
∈H
1
and w
1
∈H
1/2
,i 1, 2, ,m.
Therefore,
w
0
u
0
1
,B
11
u
0
1
, ,B
1s
1
u
0
1
, ,u
0
m
,B
m1
u
0
m
, ,B
ms
1
i
∈ X
i
ν
ν 1, 2, ,s
i
,i 1, 2, ,m.Consequently,
w
1
u
1
1
,B
11
u
1
1
, ,B
1s
1
u
1
1
, ,u
1
m
,B
i
, ,B
is
i
u
i
,u
i
∈ H
i
1/2
,
m
k1
D
j
kμ
u
k
0,i 1, ,m,μ 1, ,r
j
,j 1, ,m
.
1.24
Advances in Difference Equations 7
From the definition of interpolation spaces see 3,chapter1, 7,chapter1,weget
the following inclusion:
t H
1/2
and
c
−1
w
H
1/2
≤
A
1/2
0
tw
Ht
≤ c
w
H
1/2
,c>0.
1.26
A
i
t
u
i
,u
i
H
i
t
s
i
ν1
m
k1
C
i
kν
t
u
ν1
B
iν
u
i
,B
iν
u
i
X
i
ν
t
.
1.27
By virtue of conditions ii, viii, 1.26,and1.27,weget
w
2
H
1/2
≤ c
m
i1
,suchthatu
p
∈
H
0
and
u
p
− u
1
H
1
1/2
⊕···⊕H
m
1/2
−→ 0, at p −→ ∞.
1.29
Hence it follows, that
u
H
1/2
−→ 0, at p, q −→ ∞,
1.31
where w
p
u
p
1
,B
11
u
p
1
, ,B
1s
1
u
p
1
, ,u
p
m
,B
m1
u
p
m
, ,B
ms
w
p
− w
H
1/2
−→ 0, at p −→ ∞.
1.33
Hence,
u
p
− u
H
1
1/2
⊕···⊕H
m
1/2
−→ 0, at p −→ ∞,
u
1
m
, ,B
ms
m
u
1
m
w
1
. 1.35
Thus, w
1
∈H
1/2
. The theorem is proved.
1.2. Proof of Auxiliary Results
Validity of Statement 1.3 follows from condition i, the Statement 1.4 from condition vii.
Proof of Statement 3. Consider in Hilbert space H the equation
λw A
t
w F,t∈
0,T
, 1.36
u
i
f
i
,t∈
0,T
,i 1, 2, ,m,
L
t
iν
λ
u λB
iν
u
i
m
k1
C
i
kν
t
u
0
for some λ ∈ R.Thus,
for each t ∈ 0,T,
R
λI A
t
H
t
, 1.38
where I is an identity operator in Ht,thatis,A has a regular point.
Advances in Difference Equations 9
2. Abstract Model of Initial Boundary Value Problem with
Non Stationary Boundary and Transmission Conditions for
the Impulsive Semilinear Hyperbolic Equations
Consider the following initial boundary value problem:
¨u
i
t
A
i
t
k1
C
i
kν
t
u
k
t
g
iν
t,
u
t
,
¨
u
t
,
m
i
, μ 1, ,r
i
, i 1, ,m,
˙
u u
1
, ,u
m
,
¨
u ˙u
1
, , ˙u
m
, A
i
t,
B
iν
, C
i
kν
t and D
i
kμ
satisfy all conditions of Theorem 1.2.
Assume, that the nonlinear operators f
i
and g
1/2
×
m
i1
H
i
−→ H
i
,
t,
u,
˙
u
−→ g
iν
t,
u,
˙
u
:
0,T
, v
1
, u
2
, v
2
∈
H
1/2
×
H,
f
i
t
1
, u
1
, v
1
− f
i
t
u
1
i
− u
2
i
H
i
1/2
v
1
i
− v
2
i
H
i
≤ c
iν
r
|
t
1
− t
2
|
m
i1
u
1
i
− u
2
i
H
i
,i 1, ,m,
r
m
i1
2
l1
u
l
i
H
i
1/2
v
l
i
∩ C
1
0,T
,
H
1/2
∩ C
2
0,T
,
H
. 2.5
Additionally, if
E
t
u
0
i
H
i
1/2
u
1
i
H
i
,t∈
0,T
, 2.6
w
0
w
0
, ˙w
0
w
1
,
2.8
where w u
1
,B
11
u
1
, ,B
1s
1
u
1
, ,u
m
,B
m1
u
u
0
m
, ,B
ms
m
u
0
m
,
w
1
u
1
1
,B
11
u
1
1
, ,B
1s
1
u
1
1
, ,u
t, w, ˙w
f
1
t,
u,
˙
u
,g
11
t,
u,
˙
u
, ,g
1s
1
t,
u,
˙
u
, ,
2
∈ 0 ,T,w
1
,w
2
∈H
1/2
,z
1
,z
2
∈H,
F
t
1
,w
1
,z
1
−F
t
2
,w
2
1/2
z
1
− z
2
H
, 2.10
where c· ∈ CR
,R
,r
2
l1
w
l
H
1/2
z
l
1
.
2.11
3. Initial Boundary Value Problem with
Non Stationary Boundary and Transmission Condition for
the Impulsive Semilinear Hyperbolic Equations
Let a
1
<a
2
< ···<a
m1
. We consider in the domain 0,T ×
m
i1
a
i
,a
i1
the following mixed
problem
¨u
i
t, x
− p
i
u,
˙
u
,
t, x
∈
0,T
×
a
i
,a
i1
,i 1, 2, ,m,
u
i
t, a
i1
u
i1
u
,t>0,
¨u
i
t, a
i1
q
i
t
u
i
t, a
i1
− u
i1
t, a
i1
g
m
t, ψ
m
u,
˙
u
,t>0,
u
i
0,x
u
0
i
x
, ˙u
i
0,x
u
1
i
i
∂
2
u
i
/∂x
2
, u u
1
, ,u
m
,
˙
u
˙u
1
, , ˙u
m
, p
i
, q
j
,f
i
,g
j
,u
0
i
0, 1, ,m,
2
0
f
i
· ∈ C
1
0,T × a
i
,a
i1
×R
4
,i 1, 2, ,m,
3
0
g
j
· ∈ C
1
0,T,R,j 0, 1, ,m,
12 Advances in Difference Equations
4
0
ϕ
i
· are nonlinear functionals acting from
m
k1
m
k1
W
1
2
a
k
,a
k1
× L
2
a
k
,a
k1
the following
inequality holds
ϕ
i
u
1
, v
1
− ϕ
W
1
2
a
k
,a
k1
v
1
k
− v
2
k
L
2
a
k
,a
k1
v
1
k
L
2
a
k
,a
k1
v
1
k
L
2
a
k
,a
k1
,
c
i
·
∈ C
× L
2
a
k
,a
k1
3.5
to R and for arbitrary
u
1
, v
1
, u
2
, v
2
∈
m
k1
W
1
2
a
k
,a
≤ c
j
r
m
k1
u
1
k
− u
2
k
W
1
2
a
k
,a
k1
6
0
u
0
i
∈ W
2
2
a
i
,a
i1
,u
1
i
∈ W
1
2
a
i
,a
i1
, i 1, 2, ,m,where
u
0
j
a
j1
) be held, then there exists a T
∈ 0,T, such that the problem
3.1 has a unique solution
u u
1
, ,u
m
,where
u
i
∈ C
2
0,T
; L
2
a
i
,a
i1
∩ C
1
u
i
t, a
i
,u
i
t, a
i1
∈ C
2
0,T
,R
,i 1, 2, ,m.
3.8
Proof. Let us denote H
i
L
2
a
i
,a
i1
ν
are defined the following inner p roducts:
u, v
H
i
t
p
−1
i
t
a
i1
a
i
uvdx,
h
1
,h
2
X
1
1
t
2
,
h
1
,h
2
∈ ,i 1, 2, ,m.
3.9
From differentiability of the functions p
i
t, i 1, 2, ,m,andq
j
t, j 0, 1, ,m it
follows that the condition i is sa tisfied.
Let us define the following operators:
A
i
tu
i
−p
i
tu
i
,u
i
∈ DA
i
t W
2
−q
0
tu
1
a
1
,C
m
m1
tu
m
q
m
tu
m
a
m1
,
C
i
k1
t0, for all other i, k,
C
i
i2
tu
i
q
i
a
i1
,D
i
i1,1
u
i1
u
i1
a
i1
,i 1, 2, ,m− 1,
D
i
k1
0,k
/
i, k
/
i 1.
We also define the nonlinear operators as follows:
F
i
t, u, vf
i
t, x, u
i
x,u
iν
t, and D
k
iμ
and the nonlinear operators
F
i
,G
i1
, and G
i2
, i 1, ,m satisfy the conditions of Theorem 2.1,andtheproblem3.1 is
represented as an abstract initial boundary-value problem in the following way:
¨u
i
t
A
i
t
u
i
t
F
i
˙
u
,
B
i2
¨u
i
t
C
i
i2
t
u
i
t
C
i
i2,2
t
u
i
G
m
t,
u,
˙
u
,
D
i
i1
u
i
D
i
i1
u
i1
0,i 1, 2, ,m− 1.
3.10
We will show that conditions of Theorem 2.1 are satisfied. Conditions i–v follow
immediately from definitions of spaces H
i
,X
i
ν
, and Y
j
μ
u
1
, ,u
m
,u
i
∈ W
2
2
a
i
,a
i1
,i 1, ,m,
u
j
a
j1
u
j1
a
j1
1
,
w
i
u
i
,u
i
a
i1
,i 2, ,m,
u ∈
H
0
.
3.11
We also define the spaces
H
1/2
u, u
,u
i
∈ W
1
2
a
i
,a
i1
,i 1, ,m,
u
j
a
j1
u
j1
a
j1
,j 1, ,m− 1
.
3.12
Statement 3.2. H
.
3.13
Advances in Difference Equations 15
Proof. Assume that u
1
,α
1
,α
0
,u
2
,α
2
, ,u
m
,α
m
∈H. Consider the following functions:
u
0
i
x
a
i1
− x
a
i1
a
i1
u
0
i1
a
i1
α
i
,i 1, 2, ,m− 1.
3.15
Let
u u
1
, ,u
m
∈
H. Consider the function
z
z
1
, ,z
m
i1
,a
i1
m
i1
L
2
a
i
,a
i1
,whereDa
i
,a
i1
i 1, ,m is a space of infinitely differentiable finite
functions. Therefore, for an arbitrary ε>0, there exist the functions h
i
∈Da
i
,a
i1
,
i 1, ,m,suchthat
m
i1
L
2
a
i
,a
i1
<ε,
3.18
where
h
i
∈ C
∞
a
i
,a
i1
,
h
i
a
i
α
i−1
,i 1, ,m.
,α
0
,h
2
,α
1
, ,h
m
,α
m
H
<ε.
3.19
The following statement is proved in the same way.
Statement 3.3.
H
0
is dense
H
1/2
.
Now, we prove that the condition vi holds.
16 Advances in Difference Equations
Let
u u
1
, ,u
ν1
m
k1
C
i
kν
tu
k
,B
iν
v
i
X
i
ν
t
⎤
⎦
m
i1
−
a
i1
i
a
i1
− u
i1
a
i1
,v
i
a
i1
u
m
a
m1
,v
m
a
i1
,v
i
a
i1
−
m
i1
a
i1
a
i
u
i
v
i
dx −
u
−
m−1
i1
u
i1
a
i1
v
i
a
i1
u
m
a
m1
v
m
a
m1
i1
− u
1
a
1
v
1
a
1
m−1
i1
u
i
a
i
v
i
a
m1
m
i1
a
i1
a
i
u
i
v
i
dx
m
i1
a
i1
a
i
u
i
B
iν
ν
i
,
m
k1
C
i
kν
t
u
k
X
i
ν
t
⎤
⎦
m
i1
a
i1
⎡
⎣
A
i
tu
i
,u
i
H
i
t
s
i
ν1
m
k1
C
i
kν
tu
k
,B
i
x
,i 1, 2, ,m, 3.23
λu
1
a
1
− q
0
t
u
1
a
1
h
10
,
λu
i
a
m1
q
m
t
u
m
a
m1
h
m0
,
3.24
where h
i
∈ L
2
a
i
,a
i1
,i 1, ,m; h
j0
∈ R, j 0, 1, ,m, λ ∈ R.
Let h
2
p
i
t
u
i
xx
h
i
x
,i 1, 2, ,m,
3.26
where g Fg is a Fourier transformation of the function gx.From3.26,weobtain
u
h
i
/λ k
2
p
a
i
,a
i1
.Considering
linearity of the problem 3.23, 3.24, the solution can be represented in the form
u
i
v
i
u
i
, 3.27
where v
i
u
i
− u
i
is a solution of the following problem:
λv
i
x
− p
i
t
λv
i
a
i1
q
i
t
v
i
a
i1
− v
i1
a
i1
h
h
10
− λu
1
a
1
q
0
tu
1
a
1
,
h
i0
h
i0
− λu
i
a
i1
− q
i
t
m1
q
m
t
u
m
a
m1
.
3.30
18 Advances in Difference Equations
A general solution of a system 3.28 is found in the following form:
v
i
x
c
i1
e
−x−a
i
√
−a
1
√
λ/p
i
t
− q
0
t
λ
p
i
t
c
11
− c
12
e
−a
2
−a
1
t
λ
p
i
t
c
i1
e
−a
i1
−a
i
√
λ/p
i
t
c
i2
−
λ
p
−a
i
√
λ/p
i
t
− c
i2
−
c
i1,1
− c
i1,2
e
−a
i2
−a
i1
√
λ/p
i1
t
0,i 1, ,m− 1,
λ
c
h
m
0
.
3.32
Let Rλ be a matrix of coefficients of system 3.32.From3.32, it is clear that Rλ
R
0
λR
1
λ,wheredetR
0
λ → ∞ and det R
1
λ → 0asλ → ∞.Thus,forsufficiently
large λ, Rλ is invertible and det Rλ → ∞. Therefore, the system 3.32 has a unique
solution.
Thus, for sufficiently positive large λ,theproblem3.23-3.24 has a unique solution
u u
1
, ,u
m
∈ H
0
.
Thus, the condition ix is satisfied. The fulfillment of other conditions follows from
1
u
i
f
1i
t, x, u
i
,u
i
, ˙u
i
, ϕ
i
u,
˙
u
,
g
0
t, ψ
0
u,
˙
u
t, ψ
i
u,
˙
u
−
|
u
i
a
i1
|
τ
i
u
i
a
i1
g
i1
t, ψ
i
u,
i
,v
i
,ξ
i
,η|≤c1 |u
i
|
ρ
i
2/2
|v
i
| |ξ
i
| |η|,
Advances in Difference Equations 19
9
0
|g
0i
t, η|≤c1 |η|,
10
0
|ϕ
i
u, v|≤c1
n
i1
–10
0
be held and initial data satisfy the condition 6
0
, then the
problem 3.1 has a unique solution
u u
1
, ,u
m
,where
u
i
∈ C
2
0,T
; L
2
a
i
,a
i1
∩ C
1
t, a
i
,u
i
t, a
i1
∈ C
2
0,T
; R
,i 1, 2, ,m.
3.34
References
1 Y. Yakubov, “Hyperbolic differential-operator equations on a whole axis,” Abstract and Applied
Analysis, no. 2, pp. 99–113, 2004.
2 P. Lancaster, A. Shkalikov, and Q. Ye, “Strongly definitizable linear pencils in Hilbert space,” Integral
Equations and Operator Theory, vol. 17, no. 3, pp. 338–360, 1993.
3 J L. L ions and E. Magenes, Problemes aux Limites non Homogenes et Applications, vol. 1, Dunod, Paris,
France, 1968.
4 M. Reed and B. Simon, Methods o f Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness,
Academic Press, New York, NY, USA, 1975.
5 T. Kato, “Linear evolution equations of “hyperbolic” type. II,” Journal of the Mathematical Society of
Japan, vol. 25, pp. 648–666, 1973.
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