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. Aliev1 and Ulviya M. Mamedova2
1
2
Azerbaijan Technical University, AZ 1073, Baki, Azerbaijan
Institute of Mathematics and Mechanics of NAS of Azerbaijan, AZ 1141, Baku, Azerbaijan
Correspondence should be addressed to Akbar B. Aliev, aliyevagil@yahoo.com
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
were obtained in 2.
In this paper, we offer the analogues abstract model of investigation of mixed
problem with non stationary boundary and transmission conditions for impulsive linear and
semilinear hyperbolic equations.
1.1. Statement of the Problem and Main Theorem
j
Let H i , H0i , Xνi , Yμ ν 1, 2, . . . , si ; i 1, 2, . . . , m; μ 1, 2, . . . , rj ; j 1, 2, . . . , m be Hilbert
Spaces. Consider the following abstract initial-boundary value problem:
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hyperbolic equations ,
üi t Ai tui t fi t,
Biν üi t m
i
Ckν
tuk t giν t,
1.1
non stationary boundary and transmission conditions ,
k1
m
i
Dkμ
uk t 0,
stationary boundary and transmission conditions ,
k1
ui 0 u0i ,
u̇i 0 u1i ,
initial conditions,
1.2
where t ∈ 0, T, üi d2 ui /dt2 , u̇i dui /dt, Ai t are the linear closed operators in H i ; Biν are
j
i
t are the linear operators from H k to Xνi ; Dkμ are the
the linear operators from H i to Xνi ; Ckν
j
linear operators from H k to Yμ ; ν 1, . . . , si , i 1, . . . , m, μ 1, . . . , rj , j 1, . . . , m, k 1, . . . , m.
We will investigate this problem under the following conditions.
i Let H0i ⊂ H i , and let H0i 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 ·, ·H i t , which
generate uniform equivalent norms, that is,
c1−1 u2H i ≤ u2H i t ≤ c1 u2H i ,
u2H i t u, uH i t ,
c1 > 0,
t ∈ 0, T, i 1, 2, . . . , m.
1.3
For each u ∈ H i , the function t → u2H i t : 0, T → R is continuously differentiable,
i 1, 2, . . . , m.
In the Hilbert space Xνi , it was defined the system of the inner products ·, ·Xν , which
i
generate uniform equivalent norms, that is,
c2−1 v2Xi ≤ v2Xi t ≤ c2 v2Xi ,
ν
v2Xi t v, vXνi t ,
ν
ν
ν
c2 > 0,
t ∈ 0, T, ν 1, 2, . . . , si , i 1, 2, . . . , m.
1.4
For each v ∈ Xνi , the function t → v2Xi t : 0, T → R is continuously differentiable.
ν
ii For each t ∈ 0, T and i 1, 2, . . . , m, Ai t is a linear closed operator in H i whose
domain is H0i ; Ai t acts boundedly from H0i to H i ; Ai t is strongly continuously
differentiable.
i
i
iii The linear operators Biν , that act from H1/2
to Xνi , bounded, where H1/2
H0i , H i 1/2 is interpolation space between H0i and H i of order 1/2 ν 1, . . . , si , i 1, . . . , m see 3.
i
t, that act from H k to Xνi , are
iv For each t ∈ 0, T, the linear operators Ckν
i
bounded; Ckν t is strongly continuously differentiable ν 1, . . . , si , i 1, . . . , m;
k 1, . . . , m.
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3
j
j
k
into Yμ , act boundedly μ 1, . . . , rj , j v The linear operators Dkμ , from H1/2
1, . . . , m; k 1, . . . , m.
Let us introduce the following designations:
H 1 ⊕ · · · ⊕ H m,
H
0 u : u u1 , . . . , um , ui ∈ H i , i 1, . . . , m;
H
0
m
0, μ 1, . . . , rj , j 1, . . . , m ,
k1
1/2 H
j
Dkμ uk
i
u : u u1 , . . . , um , ui ∈ H1/2
, i 1, . . . , m;
m
1.5
j
Dkμ uk
0, μ 1, . . . , rj , j 1, . . . , m ,
k1
H1 w : w w1 , . . . , wm , wi ui , Bi1 ui , . . . , Bisi ui , i 1, . . . , m,
0 ,
where u1 , . . . , um ∈ H
m
H
Hi ,
Hi H i ⊕ X1i ⊕ · · · ⊕ Xsi i ,
H1/2 H1 , H1/2 .
i1
1/2 with the norm
From condition v, it follows that the space H
uH
1/2 m
ui H i
i1
1.6
1/2
is a subspace of
i
1
m
H1/2 u : u u1 , . . . , um , ui ∈ H1/2
, i 1, . . . , m H1/2
× · · · × H1/2
.
1.7
0 be dense in H
1/2 , and let linear manifold H1 be dense in
vi Let the linear manifold H
H.
0 and t ∈ 0, T, the following identity is
vii Green’s Identity. For arbitrary u, v ∈ H
valid:
m
⎡
si
m
i
⎣Ai tui , vi H i t Ckν tuk , Biν vi
i1
ν1
⎤
⎦
Xνi t
k1
⎡
si
m
m
i
⎣ui , Ai tvi H i t Biν ui , Ckν tvk
i1
ν1
k1
Xνi t
⎤
⎦.
1.8
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0 , the following inequality is fulfilled:
viii For all u u1 , . . . , um ∈ H
c1
m
ui 2H i
i1
≤
m
⎡
si
Biν ui 2Xi
ν
ν1
⎣Ai tui , ui H i t i1
si
m
ν1
⎤
i
Ckν
tuk , Biν ui
Xνi t
k1
m
⎦ ≤ c2 ui 2 i ,
H
i1
1.9
1/2
where c1 ∈ R, c2 > 0.
ix For each t ∈ 0, T, an operator pencil
Lt λ : u u1 , . . . , um −→ Lt λu
Lt10 λu, Lt11 λu, . . . , Lt1s1 λu, . . . , Ltm0 λu, Ltm1 λu, . . . , Ltmsm λu ,
1.10
0 to H, has a regular point λ λ0 ∈ R, where
which acts boundedly from H
Lti0 λu λui Ai tui ,
Ltiν λu λBiν ui m
i
Ckν
tuk ,
i 1, 2, . . . , m,
ν 1, 2, . . . , si , i 1, 2, . . . , m.
1.11
k1
i
,
x u0i ∈ H0i , u1i ∈ H1/2
m
k1
j
Dkμ u0k 0,
m
k1
j
Dkμ u1k 0
i 1, 2, . . . , m, μ 1, 2, . . . , rj , j 1, 2, . . . , m .
1.12
xi fi · ∈ Wp1 0, T; H i, p ≥ 1, i 1, . . . , m,
giν · ∈ Wp1 0, T; Xνi ,
p ≥ 1, ν 1, . . . , si , i 1, . . . , m.
1.13
Definition 1.1. The function t → u1 t, . . . , um t is called a solution of problem 1.1-1.2 if
0 is continuous, and the function
the function t → ut u1 t, . . . , um t from 0, T to H
t −→ u1 t, B11 u1 t, . . . , B1s1 u1 t, . . . , um t, Bm1 um t, . . . , Bmsm um t
1.14
from 0, T to H is twice continuously differentiable and 1.1-1.2 are satisfied.
Theorem 1.2. Let conditions (i)–(xi) are satisfied, then the problem 1.1-1.2 has a unique solution.
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Proof. We define the operator At in the Hilbert space H in the following way:
DAt H1 ,
Atw A1 tu1 ,
m
m
1
1
Ck1
tuk , . . . , Cks
tuk , . . . , Atm um ,
1
k1
m
k1
m
Ck1
tuk , . . . ,
k1
m
1.15
m
Cks
tuk
m
t ∈ 0, T, w ∈ H1 .
,
k1
Then the problem 1.1-1.2 is represented as the Cauchy problem
ẅ Atw Φt,
w0 w0 ,
1.16
ẇ0 w1 ,
where wt u1 t, B11 u1 t, . . . , B1s1 u1 t, . . . , um t, Bm1 um t, . . . , Bmsm um t,
Φt f1 t, g11 t, . . . , g1s1 t, . . . , fm t, gm1 t, . . . , gmsm t ,
w0 u01 , B11 u01 , . . . , B1s1 u01 , . . . , u0m , Bm1 u0m , . . . , Bmsm u0m ,
1.17
w1 u11 , B11 u11 , . . . , B1s1 u11 , . . . , u1m , Bm1 u1m , . . . , Bmsm u1m .
It is obvious that if u1 t, . . . , um t is the solution of problem 1.1-1.2, then wt is
the solution of the problem 1.16. On the contrary, if
wt ∈ C2 0, T; H ∩ C1 0, T; H1 , H1/2 ∩ C0, T; H1 1.18
is the solution of problem 1.16, then wt u1 t, B11 u1 t, . . . , B1s1 u1 t, . . . , um t,
Bm1 um t, . . . , Bmsm um t and u1 t, . . . , um t is the solution of problem 1.1-1.2.
Let us define the system of inner product in Hilbert space H in the following way:
w1 , w2
Ht
m wi1 , wi2
i1
H i t
si m Biν u1i , Biν u2i
i1 ν1
Xνi t
,
t ∈ 0, T,
1.19
l
0 , l 1, 2.
, wil uli , Bi1 uli , . . . , Bisi uli , i 1, 2, . . . , m,ul1 , . . . , ulm ∈ H
where wl w1l , . . . , wm
We denote space H with inner product 1.19 by Ht.
We will prove later the following auxiliary results.
Statement 1.3. There exists such c3 > 0, that
c3−1 w2H ≤ w2Ht ≤ c3 w2H ,
t ∈ 0, T,
1.20
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and the function t → w2Ht : 0, T → R is continuously differentiable, where w2Ht w, wHt .
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 RAt λI Ht, for some λ ∈ R; therefore, for each t ∈
0, T, At is a selfadjoint operator in Ht see 4, chapter x.
Taking into account viii and Statement 1.3, we get
Atw, wHt
⎡
si
m
m
i
⎣Ai tui , ui H i t Ckν tuk , Biν ui
i1
ν1
k1
⎤
⎦
Xνi t
1.21
≥ c1 w2Ht ,
that is, At is a lower semibounded selfadjoint operator in Ht.
Thus, the operator A0 t Atλ0 I is selfadjoint and positive definite, where λ0 > c1 .
Problem 1.16 can be rewritten as
ẅt A0 twt − λ0 wt Ft,
w0 w0 ,
1.22
ẇ0 w1 .
It is known that if w0 ∈ H1 and w1 ∈ H1/2 , then the problem 1.22 has a unique
solution w ∈ C2 0, T; H ∩ C1 0, T; H1/2 ∩ C0, T; H1 see 5, 6.
To complete the proof of the theorem, we need to show that w0 ∈ H1 and w1 ∈ H1/2 .
j
0
By conditions of the theorem u0i ∈ H0i , m
k1 Dkμ uk 0i 1, 2, . . . , m; μ 1, 2, . . . , rj ,
i
to Xνi , ν 1, 2, . . . , si , i 1, 2, . . . , m.
j 1, 2, . . . , m and Biν are bounded operators from H1/2
Therefore,
w0 u01 , B11 u01 , . . . , B1s1 u01 , . . . , u0m , Bm1 u0m , . . . , Bmsm u0m ∈ H1 .
i
On the other hand, u1i ∈ H1/2
and
m
k1
1.23
j
Dkμ u1k 0 i 1, 2, . . . , m, μ 1, 2, . . . , rj , j 1, 2, . . . , m, therefore, Biν u1i ∈ Xνi ν 1, 2, . . . , si , i 1, 2, . . . , m. Consequently,
w1 u11 , B11 u11 , . . . , B1s1 u11 , . . . , u1m , Bm1 u1m , . . . , Bmsm u1m ∈ J,
J
i
,
w : w w1 , . . . , wm , wi ui , Bi1 ui , . . . , Bisi ui , ui ∈ H1/2
m
k1
j
Dkμ uk
0, i 1, . . . , m, μ 1, . . . , rj , j 1, . . . , m .
1.24
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From the definition of interpolation spaces see 3, chapter 1, 7, chapter 1, we get
the following inclusion:
m i
1/2 H1/2
⊕ X1i ⊕ · · · ⊕ Xsi i .
H1 ⊂ H1/2 ⊂ H
1.25
i1
By virtue of definition, the powers of positive selfadjoint operator see 8, chapter 2,
7, chapter 1, we have that DA01/2 t H1/2 and
c−1 wH1/2 ≤ A01/2 tw
Ht
≤ cwH1/2 ,
1.26
c > 0.
Assume that w ∈ DA0 H1 , then
2
1/2
A0 tw
Ht
A0 tw, wHt
m
⎡
⎤
si
m
i
⎣Ai tui , ui H i t Ckν tuk , Biν ui
i1
λ0
ν1
m
ui , ui H i t i1
si
k1
⎦
Xνi t
1.27
Biν ui , Biν ui Xνi t .
ν1
By virtue of conditions ii, viii, 1.26, and 1.27, we get
w2H1/2 ≤ c
m
i1
ui 2H i .
1.28
1/2
0 is dense in H
1/2 ; therefore, there exists a
Let w1 ∈ J. By virtue of condition vi, H
p
p
p
p
0 and
sequence u u1 , . . . , um , such that u ∈ H
p
1
u − u 1
m
H1/2
⊕···⊕H1/2
−→ 0,
at p −→ ∞.
1.29
−→ 0
at p, q −→ ∞.
1.30
Hence it follows, that
p
q u − u m
1
H1/2
⊕···⊕H1/2
Then from 1.28 and 1.30 it follows that {wp } is fundamental in H1/2 , that is,
p
w − wq p
p
p
H1/2
−→ 0,
at p, q −→ ∞,
p
p
p
where wp u1 , B11 u1 , . . . , B1s1 u1 , . . . , um , Bm1 um , . . . , Bmsm um , p 1, 2,. . ..
1.31
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Thus, there exists w
∈ H1/2 such that
p
w − w
H1/2
−→ 0,
at p −→ ∞.
1.32
at p −→ ∞.
1.33
1/2 , therefore,
On the other hand, H1/2 ⊂ H
p
w − w
H1/2
−→ 0,
Hence,
p
u − u
1
m
H1/2
⊕···⊕H1/2
−→ 0,
at p −→ ∞,
1.34
u1 , . . . , u
m . From this, by virtue of 1.29, u u1 , that is,
where u w
u11 , B11 u11 , . . . , B1s1 u11 , . . . , u1m , Bm1 u1m , . . . , Bmsm u1m w1 .
1.35
Thus, w1 ∈ H1/2 . 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 Atw F,
t ∈ 0, T,
1.36
where F f1 , f11 , . . . , f1s1 , . . . , fm , fm1 , . . . , fmsm ∈ H, λ ∈ R.
Equation 1.36 is equivalent to the following system of differential-operator
equations:
Lti0 λu λui Ai tui fi ,
Ltiν λu λBiν ui m
i
Ckν
tuk giν ,
t ∈ 0, T, i 1, 2, . . . , m,
t ∈ 0, T, ν 1, 2, . . . , si , i 1, 2, . . . , m,
k1
m
j
Dkμ uk 0,
1.37
μ 1, 2, . . . , rj , j 1, 2, . . . , m.
k1
0 for some λ ∈ R. Thus,
By virtue of ix, problem 1.37 has a solution u u1 , . . . , um ∈ H
for each t ∈ 0, T,
RλI At Ht,
where I is an identity operator in Ht, that is, A has a regular point.
1.38
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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:
üi t Ai tui t fi t, ut, u̇t ,
Biν üi t m
i
Ckν
tuk t giν t, ut, üt ,
k1
2.1
m
i
Dkμ
uk t 0,
k1
ui 0 u0i ,
u̇i 0 u1i ,
where t ∈ 0, T, ν 1, . . . , si , μ 1, . . . , ri , i 1, . . . , m, u̇ u1 , . . . , um , ü u̇1 , . . . , u̇m , Ai t,
i
i
Biν , Ckν
t and Dkμ
satisfy all conditions of Theorem 1.2.
Assume, that the nonlinear operators fi and giν satisfy the following conditions.
xi Suppose that the nonlinear operators
t, u, u̇ −→ fi t, u, u̇ : 0, T ×
m
i1
t, u, u̇ −→ giν
t, u, u̇ : 0, T ×
×
i
H1/2
m
m
H
−→ H i ,
i1
×
i
H1/2
i
i1
m
2.2
Hi
−→ Xνi
i1
satisfy the local Lipschitz conditions in the following sense: for arbitrary t1 , t2
1/2 × H,
0, T, u1 , v1 , u2 , v2 ∈ H
1
1
2
2 fi t1 , u , v − fi t2 , u , v Hi
m 1
≤ ci r |t1 − t2 | ui − u2i i1
1
1
2
2 giν t1 , u , v − giν t2 , u , v i
H1/2
1
2
vi − vi i ,
H
2.3
Xνi
m 1
≤ ciν r |t1 − t2 | ui − u2i i1
∈
i
H1/2
1
2
vi − vi i ,
H
where ci ·, ciν ∈ CR , R , ν 1, . . . , si , i 1, . . . , m,
r
m 2 l
ui i1 l1
i
H1/2
vil i .
H
2.4
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Theorem 2.1. Let conditions (i)–(x) and (xi ) be satisfied, then there exists T ∈ 0, T, such that the
problem 2.1 has a unique solution
u u1 , . . . , um ∈ C
0 ∩ C1 0, T , H
1/2 ∩ C2 0, T , H
.
0, T , H
2.5
Additionally, if
m m 0
i
Et u̇i tH i ≤ ϕ
ui tH1/2
ui i1
i
H1/2
i1
1
,
ui i
H
t ∈ 0, T ,
2.6
where ϕ· ∈ CR , R , then T T. Otherwise, there exists T0 ∈ 0, T, such that
lim Et ∞.
2.7
t → T0 −0
In the Hilbert space H, the problem 2.1 is represented as the Cauchy problem
ẅ A0 tw Ft, w, ẇ,
w0 w0 ,
2.8
ẇ0 w1 ,
where w u1 , B11 u1 , . . . , B1s1 u1 , . . . , um , Bm1 um , . . . , Bmsm um ,
w0 u01 , B11 u01 , . . . , B1s1 u01 , . . . , u0m , Bm1 u0m , . . . , Bmsm u0m ,
w1 u11 , B11 u11 , . . . , B1s1 u11 , . . . , u1m , Bm1 u1m , . . . , Bmsm u1m ,
Ft, w, ẇ λ0 w F1 t, w, ẇ,
F1 t, w, ẇ f1 t, u, u̇ , g11 t, u, u̇ , . . . , g1s1 t, u, u̇ , . . . ,
2.9
fm t, u, u̇ , gm1 t, u, u̇ , . . . , gmsm t, u, u̇ .
From xi’, it follows that, for arbitrary t1 , t2 ∈ 0, T, w1 , w2 ∈ H1/2 , z1 , z2 ∈ H,
F t1 , w1 , z1 − F t2 , w2 , z2 where c· ∈ CR , R , r 2
≤ cr |t1 − t2 | w1 − w2 H
l1 w
l
H1/2 zl H .
H1/2
z1 − z2 ,
H
2.10
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Thus, the nonlinear operator F satisfies the condition of local solvability of the Cauchy
problem for the quasilinear hyperbolic equations in Hilbert space see 6, 9. Taking this into
account, the problem 2.8 has a unique solution
w ∈ C2 0, T ; H ∩ C1 0, T ; H1/2 ∩ C 0, T ; H1 .
2.11
3. Initial Boundary Value Problem with
Non Stationary Boundary and Transmission Condition for
the Impulsive Semilinear Hyperbolic Equations
Let a1 < a2 < · · · < am1 . We consider in the domain 0, T ×
m
ai , ai1 the following mixed
i1
problem
üi t, x − pi tui t, x fi t, x, ui t, x, ui t, x, u̇i t, x, ϕi u, u̇ ,
t, x ∈ 0, T × ai , ai1 ,
i 1, 2, . . . , m,
ui t, ai1 ui1 t, ai1 ,
i 1, 2, . . . , m − 1, t > 0,
ü1 t, a1 − q0 tu1 t, a1 g0 t, ψ0 u, u̇ , t > 0,
!
üi t, ai1 qi t ui t, ai1 − ui1 t, ai1 gi t, ψi u, u̇ ,
3.1
i 1, 2, . . . , m − 1, t > 0,
üm t, am1 qm tum t, am1 gm t, ψm u, u̇ , t > 0,
ui 0, x u0i x,
u̇i 0, x u1i x,
x ∈ ai , bi , i 1, 2, . . . , m,
where u̇i ∂ui /∂t, ui ∂ui /∂x, üi ∂2 ui /∂t2 , ui ∂2 ui /∂x2 , u u1 , . . . , um , u̇ u̇1 , . . . , u̇m , pi , qj , fi , gj , u0i , u1i are some functions, ϕi and ψj are some functionals, which will
be specified below, i 1, . . . , m, j 0, 1, . . . , m.
Recently, differential equations with impulses are great interest because of the
needs of modern technology, where impulsive automatic control systems and impulsive
computing systems are very important and intensively develop broadening the scope of their
applications in technical problems, heterogeneous by their physical nature and functional
purpose see 10, chapter 1.
Assume that the following conditions are held:
10 pi t ∈ C1 0, T, qj t ∈ C1 0, T; pi t > 0, qj t > 0, t ∈ 0, T, i 1, . . . , m, j 0, 1, . . . , m,
20 fi · ∈ C1 0, T × ai , ai1 × R4 , i 1, 2, . . . , m,
30 gj · ∈ C1 0, T, R, j 0, 1, . . . , m,
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40 ϕi · are nonlinear functionals acting from
m W21 ak , ak1 × L2 ak , ak1 3.2
k1
"m
to R and for arbitrary u1 , v 1 , u2 , v2 ∈
inequality holds
1
k1 W2 ak , ak1 × L2 ak , ak1 the following
# #
#
1
1
2
2 #
#ϕi u , v − ϕi u , v #
m 1
≤ ci r
uk − u2k k1
W21 ak ,ak1 vk1 − vk2 L2 ak ,ak1 ,
3.3
i 1, 2, . . . , m,
where r m
1
k1 uk W21 ak ,ak1 u2k W 1 a
2
k ,ak1 ci · ∈ CR , R ,
vk1 L a
2
k ,ak1 vk1 L a
2
k ,ak1 ,
R 0, ∞, i 1, 2, . . . , m,
3.4
50 ψj · are nonlinear functionals acting from
m W21 ak , ak1 × L2 ak , ak1 3.5
k1
to R and for arbitrary u1 , v 1 , u2 , v2 ∈
inequality holds
"m
1
k1 W2 ak , ak1 m # #
#
1
1
1
2
2 #
#ψj u , v − ψj u , v # ≤ cj r
uk − u2k k1
W21 ak ,ak1 × L2 ak , ak1 the following
vk1 − vk2 L2 ak ,ak1 ,
3.6
where cj · ∈ CR , R , j 0, 1, . . . , m, and r—is defined as in 3.3,
60 u0i ∈ W22 ai , ai1 , u1i ∈ W21 ai , ai1 , i 1, 2, . . . , m, where
u0j aj1 u0j1 aj1 ,
u1j aj1 u1j1 aj1 ,
j 1, 2, . . . , m − 1.
By applying Theorem 2.1, we obtain the following result.
3.7
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13
Theorem 3.1. Let conditions (10 )–(60) be held, then there exists a T ∈ 0, T, such that the problem
3.1 has a unique solution u u1 , . . . , um , where
ui ∈ C2 0, T ; L2 ai , ai1 ∩ C1 0, T ; W21 ai , ai1 ∩ C 0, T ; W22 ai , ai1 ,
ui t, ai , ui t, ai1 ∈ C2 0, T , R ,
j
Proof. Let us denote H i L2 ai , ai1 , H0i W22 ai , ai1 , Xνi ,Yμ 1, 2, . . . , m, μ 1, 2, . . . , rj , j 1, 2, . . . , m, where si 2,rj 1.
In space H i and Xνi are defined the following inner products:
u, vH i t pi−1 t
h1 , h2 X11 t q0−1 th1 h2 ,
h1 , h2 ∈ ,
3.8
i 1, 2, . . . , m.
, ν 1, 2, . . . , si , i $ ai1
uv dx,
ai
h1 , h2 X2i t qi−1 th1 h2 ,
3.9
i 1, 2, . . . , m.
From differentiability of the functions pi t, i 1, 2, . . . , m, and qj t, j 0, 1, . . . , m it
follows that the condition i is satisfied.
Let us define the following operators:
Ai tui −pi tui , ui ∈ DAi t W22 ai , ai1 ,
B11 u1 u1 a1 , Bj1 0, B2i ui ui ai1 , i 1, 2, . . . , m, j 2, . . . , m,
m
1
tu1 −q0 tu1 a1 , Cm1
tum qm tum am1 ,
C11
i
Ck1
t 0, for all other i, k,
i
tui qi tui ai1 , i 1, 2, . . . , m,
Ci2
j
Cj2 tuj1 −qj tuj1 aj1 , j 1, 2, . . . , m − 1,
i
Ck2
t 0, for all other i, k,
i
i
ui −ui ai1 , Di1,1
ui1 ui1 ai1 , i 1, 2, . . . , m − 1,
Di1
i
Dk1
0, k /
i, k /
i 1.
We also define the nonlinear operators as follows:
Fi t, u, v fi t, x, ui x, ui x, vi x, ϕi u, v, i 1, 2, . . . , m,
G11 t, u, v g0 t, ψ0 u, v,
Gi2 t, u, v gi t, ψi u, v, i 1, 2, . . . , m,
Gi1 t, u, v 0, i 2, 3, . . . , m.
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Advances in Difference Equations
k
k
t, and Diμ
and the nonlinear operators
It is easy to verify that linear operators Ai t, Biν , Ciν
Fi , Gi1 , and Gi2 , i 1, . . . , m satisfy the conditions of Theorem 2.1, and the problem 3.1 is
represented as an abstract initial boundary-value problem in the following way:
üi t Ai tui t Fi t, u, u̇ ,
1
B11 ü1 t C11
tu1 t G0 t, u, u̇ ,
i
i
Bi2 üi t Ci2
tui t Ci2,2
tui t Gi t, u, u̇ ,
3.10
m
Bm2 üm t Cm2
tum t Gm t, u, u̇ ,
i
i
ui Di1
ui1 0,
Di1
i 1, 2, . . . , m − 1.
We will show that conditions of Theorem 2.1 are satisfied. Conditions i–v follow
j
i
immediately from definitions of spaces H i , Xνi , and Yμ and operators Ai t, Biν , Ckν
t,
j
and Dkμ , and traces theorems see 3, chapter 2, where k 1, 2, . . . , m; ν 1, 2, . . . , si ;
i 1, 2, . . . , m; μ 1, 2, . . . , rj ; j 1, 2, . . . , m.
0 and H1 are defined in the following way:
The linear manifolds H
0 u, u u1 , . . . , um , ui ∈ W 2 ai , ai1 , i 1, . . . , m,
H
2
uj aj1 uj1 aj1 , j 1, . . . , m − 1 ,
H1 w, w w1 , . . . , wm , w1 u1 , u1 a2 , u1 a1 ,
3.11
0 .
wi ui , ui ai1 , i 2, . . . , m, u ∈ H
We also define the spaces
H1/2 u, u u1 , . . . , um , ui ∈ W21 ai , ai1 , i 1, . . . , m ,
1/2 u, u u1 , . . . , um , ui ∈ W 1 ai , ai1 , i 1, . . . , m,
H
2
3.12
uj aj1 uj1 aj1 , j 1, . . . , m − 1 .
Statement 3.2. H1 is dense in
H L2 a1 , b1 ⊕
⊕ ⊕
m
i2
L2 ai , bi ⊕ .
3.13
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15
Proof. Assume that u1 , α1 , α0 , u2 , α2 , . . . , um , αm ∈ H. Consider the following functions:
u0i x x − ai
ai1 − x
αi−1 αi ,
ai1 − ai
ai1 − ai
x ∈ ai , ai1 , i 1, . . . , m.
3.14
From definitions of u0i x, i 1, . . . , m, we can see that
u0i ai1 u0i1 ai1 αi ,
i 1, 2, . . . , m − 1.
3.15
Consider the function
Let u u1 , . . . , um ∈ H.
z z1 , . . . , zm u1 − u01 , . . . , um − u0m .
3.16
"m
"m
It is obvious that z ∈
i1 L2 ai , ai1 . On the other hand,
i1 Dai , ai1 , ai1 i1 L2 ai , ai1 , where Dai , ai1 i 1, . . . , m is a space of infinitely differentiable finite
functions. Therefore, for an arbitrary ε > 0, there exist the functions hi ∈ Dai , ai1 ,
i 1, . . . , m, such that
"m
m
zi − hi < ε.
3.17
i1
i u0 hi from 3.17, we get
By denoting h
i
m i
ui − h
i1
L2 ai ,ai1 < ε,
3.18
i ai αi−1 , i 1, . . . , m.
i ∈ C∞ ai , ai1 , h
where h
Thus,
u1 , u1 a2 , u1 a1 , u2 , u2 a2 , . . . , um , um am1 − h1 , α1 , α0 , h2 , α1 , . . . , hm , αm H < ε.
3.19
The following statement is proved in the same way.
0 is dense H
1/2 .
Statement 3.3. H
Now, we prove that the condition vi holds.
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Advances in Difference Equations
0 , then
Let u u1 , . . . , um , v v1 , . . . , vm ∈ H
⎡
si
m
m
i
⎣Ai tui , vi H i t Ckν tuk , Biν vi
i1
ν1
m
⎤
⎦
Xνi t
k1
$a
i1
−
ui vi dx −u1 a1 , v1 a1 ai
i1
m ui ai1 − ui1 ai1 , vi ai1 um am1 , vm am i1
m !
ui ai , vi ai − ui ai1 , vi ai1 i1
m $
ai1
−
i1
−
m−1
ui vi dx − u1 a1 , v1 a1 ui ai1 vi ai1 i1
ai
m−1
ui1 ai1 vi ai1 um am1 vm am1 i1
m m−1
m
ui ai vi ai − ui ai vi ai ui ai vi ai − ui ai1 vi ai1 − u1 a1 v1 a1 i1
i2
i1
um am1 vm am1 m $ ai1
i1
ai
ui vi dx m $ ai1
i1
ai
ui vi dx.
3.20
Similary, we obtain the following identity:
m
⎡
⎣ui , Ai tvi H i t i1
si
ν1
Biν νi ,
m
⎦
i
Ckν
tuk
Xνi t
k1
⎤
m $ ai1
i1
ai
ui vi dx.
3.21
Thus, by virtue of 3.20-3.21, the condition vi holds.
From 3.20 or 3.21, putting vi ui , we also obtain the identity
m $ ai1
i1
ai
u2i dx
⎡
m
si
m
i
⎣
Ckν tuk , Biν ui
Ai tui , ui H i t i1
that is, condition viii is satisfied, c1 c2 1.
ν1
k1
Xνi t
⎤
⎦,
3.22
Advances in Difference Equations
17
Now, we verify fulfillment of condition ix. To that end, we consider the mixed
problem
λui − pi tui hi x,
i 1, 2, . . . , m,
λu1 a1 − q0 tu1 a1 h10 ,
!
λui ai1 qi t ui ai1 − ui1 ai1 hi0 , i 1, 2, . . . , m − 1,
3.23
3.24
λum am1 qm tum am1 hm0 ,
where hi ∈ L2 ai , ai1 ,i 1, . . . , m; hj0 ∈ R, j 0, 1, . . . , m, λ ∈ R.
Let hi x be the extend of function hi x to R. We consider the system of the differential
equations
i x,
λ
ui − pi t
uixx h
i 1, 2, . . . , m.
3.25
Hence, we have
% i h
%i − k2 pi tu
% i x,
λu
xx
i 1, 2, . . . , m,
3.26
where g% Fg is a Fourier transformation of the function gx. From 3.26, we obtain
% i F −1 h
% h
% i /λ k2 pi t satisfy 3.25, and
% i /λ k2 pi t, then functions u
i F −1 u
u
i ∈ W22 ai , ai1 . Considering
their constrictions on ai , ai1 satisfy the 3.23. It is clear that u
linearity of the problem 3.23, 3.24, the solution can be represented in the form
i ,
ui vi u
3.27
i is a solution of the following problem:
where vi ui − u
λvi x − pi tvi x 0,
3.28
10 ,
λv1 a1 − q0 tv1 a1 h
!
i0 ,
λvi ai1 qi t vi ai1 − vi1
ai1 h
i 1, 2, . . . , m − 1,
3.29
m0 ,
λvm am1 qm tvm
am1 h
10 h10 − λ
u1 a1 q0 t
u1 a1 ,
where h
!
i0 hi0 − λ
ui ai1 − qi t u
h
i ai1 − u
i1 ai1 ,
i 1, 2, . . . , m − 1,
m0 0 hm − λ
h
um am1 qm t
um am1 .
0
3.30
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Advances in Difference Equations
A general solution of a system 3.28 is found in the following form:
√
√
vi x ci1 e−x−ai λ/pi t ci2 e−bi −x λ/pi t ,
i 1, 2, . . . , m.
3.31
Then, for determination of ci1 , ci2 , i 1, 2, . . . , m, from 3.29, we get the following
system of the algebraic equations:
&
√
√
λ −a2 −a1 λ/pi t
0,
− q0 t
λ c11 c12 e
c11 − c12 e−a2 −a1 λ/pi t h
pi t
&
√
√
λ −ai1 −ai λ/pi t
λ ci1 e
− ci2 qi t
ci1 e−ai1 −ai λ/pi t ci2
pi t
&
√
λ i0 , i 1, 2, . . . , m − 1,
−
ci1,1 ci1,2 e−ai2 −ai1 λ/pi1 t h
pi1 t
√
√
ci1 e−ai1 −ai λ/pi t − ci2 − ci1,1 − ci1,2 e−ai2 −ai1 λ/pi1 t 0, i 1, . . . , m − 1,
3.32
√
√
m .
λ cm1 e−am1 −am λ/pm t cm2 −cm1 e−am1 −am λ/pm t cm2 h
0
Let Rλ be a matrix of coefficients of system 3.32. From 3.32, it is clear that Rλ R0 λ R1 λ, where det R0 λ → ∞ and det R1 λ → 0 as λ → ∞. Thus, for sufficiently
large λ, Rλ is invertible and det Rλ → ∞. Therefore, the system 3.32 has a unique
solution.
Thus, for sufficiently positive large λ, the problem 3.23-3.24 has a unique solution
u u1 , . . . , um ∈ H0 .
Thus, the condition ix is satisfied. The fulfillment of other conditions follows from
10 –60 .
Now, let us consider a class of nonlinear equations, for which the large solvability
theorem takes place.
Let
fi t, x, ui , ui , u̇i , ϕ u, u̇ −|ui |ρi ui f1i t, x, ui , ui , u̇i , ϕi u, u̇ ,
g0 t, ψ0 u, u̇ −|u1 a1 |τ0 u1 a1 g01 t, ψ0 u, u̇ ,
gi t, ψi u, u̇ −|ui ai1 |τi ui ai1 gi1 t, ψi u, u̇ ,
3.33
i 1, 2, . . . , m,
where ρi ≥ 0, τj ≥ 0, i 1, 2, . . . , m; j 0, 1, . . . , m and
70 f1i , g1j, ϕi and ψj , i 1, 2, . . . , m, j 1, 2, . . . , m satisfy the conditions 20 − −50 .
80 |fi t, x, ui , vi , ξi , η| ≤ c1 |ui |ρi 2/2 |vi | |ξi | |η|,
Advances in Difference Equations
19
90 |g0i t, η| ≤ c1 |η|,
100 |ϕi u, v| ≤ c1 ni1 |ui |ρi 2/2 |vi |2 |ui yi |τi 2/2 ,
where yi ai1 , i 0, 1, . . . , m, ρ maxmini1,2,...,m ρi , 2.
Theorem 3.4. Let conditions 70 –100 be held and initial data satisfy the condition 60 , then the
problem 3.1 has a unique solution u u1 , . . . , um , where
ui ∈ C2 0, T; L2 ai , ai1 ∩ C1 0, T; W21 ai , ai1 ∩ C 0, T; W22 ai , ai1 ,
ui t, ai , ui t, ai1 ∈ C 0, T; R,
2
3.34
i 1, 2, . . . , m.
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