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Volume 2010, Article ID 101959, 19 pagesdoi:10.1155/2010/101959 Research Article A Mixed Problem for Quasilinear Impulsive Hyperbolic Equations with Non Stationary Boundary and Transmiss

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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

1 Azerbaijan Technical University, AZ 1073, Baki, Azerbaijan

2 Institute of Mathematics and Mechanics of NAS of Azerbaijan, AZ 1141, Baku, Azerbaijan

Received 10 March 2010; Revised 13 June 2010; Accepted 26 October 2010

Academic Editor: Toka Diagana

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

hyperbolic equations with non stationary boundary conditions In this direction, some results were obtained in2

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

Let H i , H i

0, X i

ν , Y μ j ν 1, 2, , s i ; i 1, 2, , m; μ 1, 2, , r j ; j 1, 2, , m be Hilbert

Spaces Consider the following abstract initial-boundary value problem:

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¨u i t A i tu i t f i t, hyperbolic equations

B iν ¨u i t m

k1

C i kν tu k t g iν t, non stationary boundary and transmission conditions

,

m

k1

,

u i 0 u0

i , ˙u i 0 u1

i , initial conditions,

1.2

where t ∈ 0, T, ¨u i d2u i /dt2, ˙u i du i /dt, A i t are the linear closed operators in H i ; B iνare

the linear operators from H i to X i

ν ; C i kν t are the linear operators from H k to X i

ν ; D kμ j are the

linear operators from H k to Y μ j ; 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

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,

c1−1u2

H i ≤ u2

H i t ≤ c1u2

H i , c1> 0,

u2

For each u ∈ H i , the function t → u2

H i t:0, T → Ris continuously diﬀerentiable,

i 1, 2, , m.

ν, it was defined the system of the inner products·, · X

νi, which generate uniform equivalent norms, that is,

c−12 v2

X i

ν ≤ v2

X i

ν t ≤ c2v2

X i

ν , c2> 0,

v2

X i

ν t v, v X i

ν t , t ∈ 0, T, ν 1, 2, , s i , i 1, 2, , m. 1.4

For each v ∈ X i

ν , the function t → v2

X i

ν t:0, T → Ris continuously diﬀerentiable

ii For each t ∈ 0, T and i 1, 2, , m, A i t is a linear closed operator in H iwhose

domain is H0i ; A i t acts boundedly from H i

0to H i ; A i t is strongly continuously

diﬀerentiable

H i

0, H i1/2 is interpolation space between H0i and H i of order 1/2 ν 1, , s i , i

1, , m see 3

ν, are

kν t is strongly continuously diﬀerentiable ν 1, , s i , i 1, , m;

k 1, , m.

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v The linear operators D j

kμ , from H 1/2 k into Y μ j, act boundedly μ 1, , r j , j

1, , m; k 1, , m.

Let us introduce the following designations:

H H1⊕ · · · ⊕ H m ,

H0

u : u u1, , u m , u i ∈ H i

0, i 1, , m;

m

k1

D j kμ u k 0, μ 1, , r j , j 1, , m

,

H 1/2

u : u u1, , u m , u i ∈ H i

1/2 , i 1, , m;

m

k1

D j kμ u k 0, μ 1, , r j , j 1, , m

,

H1w : w w1, , w m , w i u i , B i1 u i , , B is i u i , i 1, , m,

whereu1, , u m ∈ H0

,

Hi H i ⊕ X i

1⊕ · · · ⊕ X i

s i , H m

i1

Hi , H1/2 H1,H1/2

1.5

u H1/2 m

i1

u iH i

is a subspace of

H 1/2u : u u1, , u m , u i ∈ H i

1/2 , i 1, , m  H1

1/2 × · · · × H m

vi Let the linear manifold H0be dense in H 1/2, and let linear manifoldH1be dense in

H.

vii Green’s Identity For arbitrary u, v ∈ H0and t ∈ 0, T, the following identity is

valid:

m

i1

⎡

⎣A i tu i , v iH i ts i

ν1

m

k1

C i kν tu k , B iν v i

X i

ν t

⎤

⎦

i1

⎡

⎣u i , A i tv iH i ts i

ν1

B iν u i ,

m

k1

C i kν tv k

X i t

⎤

⎦.

1.8

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viii For all u u1, , u m ∈ H0, the following inequality is fulfilled:

c1

m

i1

u i2

H is i

ν1

B iν u i2

X i ν

i1

⎡

⎣A i tu i , u iH i ts i

ν1

m

k1

C i kν tu k , B iν u i

X i

ν t

⎤

⎦ ≤ c2

m

i1

u i2

H i 1/2 ,

1.9

where c1∈ R, c2> 0.

ix For each t ∈ 0, T, an operator pencil

Lt λ : u u1, , u m −→ Lt λu

L t10λu, L t

11λu, , L t

1s1λu, , L t

m0 λu, L t

m1 λu, , L t

L t i0 λu λu i A i tu i , i 1, 2, , m,

L t iν λu λB iν u im

k1

C i kν tu k , ν 1, 2, , s i , i 1, 2, , m. 1.11

x u0

i ∈ H i

0, u1i ∈ H i

1/2 ,m

k1D j kμ u0k 0,m

k1D j kμ u1k 0

i 1, 2, , m, μ 1, 2, , r j , j 1, 2, , m. 1.12

xi f i · ∈ W1

p 0, T; H i , p ≥ 1, i 1, , m,

g iν · ∈ W1

p

0, T; X ν i

Definition 1.1 The function t → u1t, , u m t is called a solution of problem 1.1-1.2 if

the function t → ut u1t, , u m t from 0, T to H0is continuous, and the function

t −→ u1t, B11u1t, , B 1s1u1t, , u m t, B m1 u m t, , B ms m u m t 1.14 from0, 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:

D At H1,

m

k1

C k11 tu k , ,

m

k1

C1ks

1tu k , , A t m u m ,

m

k1

C m k1 tu k , ,

m

k1

C m ks

m tu k

, t ∈ 0, T, w ∈ H1.

1.15

¨

where wt u1t, B11u1t, , B 1s1u1t, , u m t, B m1 u m t, , B ms m u m t,

Φt f1t, g11t, , g 1s1t, , f m t, g m1 t, , g ms m t,

w0u01, B11u01, , B 1s1u01, , u0m , B m1 u0m , , B ms m u0m

,

.

1.17

It is obvious that ifu1t, , u m t is the solution of problem 1.1-1.2, then wt is

the solution of the problem1.16 On the contrary, if

w t ∈ C20, T; H ∩ C1

is the solution of problem 1.16, then wt u1t, B11u1t, , B 1s1u1t, , u m t,

B m1 u m t, , B ms m u m t and u1t, , u m t is the solution of problem 1.1-1.2

w1, w2

Htm

i1

w1i , w2i

H i tm

i1

s i

ν1

B iν u1i , B iν u2i

X i

1, , w l

m , 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 ∈ H0, l 1, 2.

We will prove later the following auxiliary results

Statement 1.3 There exists such c3> 0, that

c−13 w2

Ht ≤ c3w2

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and the function t → w2

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 RAt λI Ht, for some λ ∈ R; therefore, for each t ∈

0, T, At is a selfadjoint operator in Ht see 4, chapter x

i1

⎡

ν1

m

k1

X i

ν t

⎤

⎦

≥ c1w2

Ht ,

1.21

Thus, the operatorA0t Atλ0I is selfadjoint and positive definite, where λ0> c1

¨

w t A0twt − λ0w t Ft,

w 0 w0, w˙0 w1.

1.22

solution w ∈ C20, T; H ∩ C10, T; H 1/2 ∩ C0, T; H1 see 5,6

0,m

k1D j kμ u0k 0i 1, 2, , m; μ 1, 2, , r j,

j 1, 2, , m and B iν are bounded operators from H i

1/2 to X i

ν , ν 1, 2, , s i , i 1, 2, , m.

Therefore,

i ∈ H i

1/2andm

k1D kμ j u1

k 0 i 1, 2, , m, μ 1, 2, , r j , j

1, 2, , m, therefore, B iν u1

i ∈ X i

ν ν 1, 2, , s i , i 1, 2, , m Consequently,

∈ J,

J

w : w w1, , w m , w i u i , B i1 u 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

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From the definition of interpolation spacessee 3, chapter 1, 7, chapter 1, we get the following inclusion:

H1⊂ H1/2⊂ H1/2 m

i1

H 1/2 i ⊕ X i

1⊕ · · · ⊕ X i

s i

7, chapter 1, we have that DA1/2

c−1wH1/2≤A1/2

0 tw

A1/2

0 tw2

Ht A0tw, w Ht

i1

⎡

ν1

m

k1

X i

ν t

⎤

⎦

 λ0

m

i1

u i , u iH i ts i

ν1

B iν u i , B iν u iX i

ν t

.

1.27

By virtue of conditionsii, viii, 1.26, and 1.27, we get

H1/2 ≤ cm

i1

u i2

H i 1/2

Let w1 ∈ J By virtue of condition vi, H0 is dense in H 1/2; therefore, there exists a

sequence u p u p1 , , u p m , such that u p∈ H0and

u p − u1

H1

1/2 ⊕···⊕H m 1/2

Hence it follows, that

u p − u q

H1

1/2 ⊕···⊕H m 1/2

Then from1.28 and 1.30 it follows that {w p} is fundamental in H1/2, that is,

w p − w q

H1/2

where w p u p1 , B11u p1 , , B 1s u p1 , , u p m , B m1 u p m , , B ms u p m , p 1, 2,

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Thus, there existsw ∈ H1/2such that

w p− w

H1/2

On the other hand,H1/2⊂ H1/2, therefore,

w p− w

Hence,

u p − u

H1

1/2 ⊕···⊕H m 1/2

where u u1, , u m From this, by virtue of 1.29, u u1, that is,

wu11, B11u11, , B 1s1u11, , u1m , B m1 u1m , , B ms m u1m

1.2 Proof of Auxiliary Results

Proof of Statement 3 Consider in Hilbert spaceH the equation

whereF f1, f11, , f 1s1, , f m , f m1 , , f ms m ∈ H, λ ∈ R.

equations:

L t i0 λu λu i A i tu i f i , t ∈ 0, T, i 1, 2, , m,

L t iν λu λB iν u im

k1

C i kν tu k g iν , t ∈ 0, T, ν 1, 2, , s i , i 1, 2, , m,

m

k1

D j kμ u k 0, μ 1, 2, , r j , j 1, 2, , m.

1.37

By virtue ofix, problem 1.37 has a solution u u1, , u m ∈ H0 for some λ ∈ R Thus, for each t ∈ 0, T,

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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:

¨u i t A i tu i t f i

t, u t, ˙ut,

B iν ¨u i t m

k1

C i kν tu k t g iν

t, u t, ¨ut,

m

k1

D kμ i u k t 0,

u i 0 u0

i , ˙u i 0 u1

i ,

2.1

where t ∈ 0, T, ν 1, , s i , μ 1, , r i , i 1, , m, ˙u u1, , u m , ¨u ˙u1, , ˙u m , A i t,

B iν , C i

kν t and D i

kμsatisfy all conditions of Theorem1.2

xi Suppose that the nonlinear operators

t, u, ˙u

−→ f i

t, u, ˙u

i1

H 1/2 i

i1

H i

−→ H i ,

t, u, ˙u

−→ g iν

t, u, ˙u

i1

H i

1/2

i1

H i

ν

2.2

0, T, u1, v1, u2, v2 ∈ H 1/2× H,

f i

t1, u1, v1

− f i

t2, u2, v2

H i

≤ c i r

|t1− t2| m

i1

u1

i − u2

i

H i 1/2

v1

i − v2

i

H i

,

g iν

t1, u1, v1

− g iν

t2, u2, v2

X i ν

≤ c iν r

|t1− t2| m

i1

u1

i − u2

i

H i 1/2

v1

i − v2

i

H i

,

2.3

where c i ·, c iν ∈ CR, R, ν 1, , s i , i 1, , m,

rm

i1

2

l1

u l

i

H i 1/2

v l

i

H i

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Theorem 2.1 Let conditions (i)–(x) and (xi) be satisfied, then there exists T∈ 0, T, such that the

problem2.1 has a unique solution

u u1, , u m ∈ C

0, T

, H0

∩ C1

0, T

, H 1/2

∩ C2

0, T

, H

Additionally, if

E t m

i1

u i t H i

1/2 ˙u i t H i

i1

u0

i

H i 1/2

u1

i

H i

, t∈0, T

where ϕ · ∈ CR, R, then T T Otherwise, there exists T0∈ 0, T, such that

lim

¨

w A0tw Ft, w, ˙w,

w 0 w0, w˙0 w1,

2.8

where w u1, B11u1, , B 1s1u1, , u m , B m1 u m , , B ms m u m,

,

Ft, w, ˙w λ0w F1t, w, ˙w,

F1t, w, ˙w f1

t, u, ˙u

, g11

t, u, ˙u

, , g 1s1

t, u, ˙u

, ,

f m

t, u, ˙u

, g m1

t, u, ˙u

, , g ms m

t, u, ˙u

.

2.9

Fromxi’, it follows that, for arbitrary t1, t2∈ 0, T, w1, w2∈ H1/2 , z1, z2∈ H,

Ft1, w1, z1

− Ft2, w2, z2

|t1− t2| w1− w2

H1/2

z1− z2

H

where c· ∈ CR, R, r 2

l1w lH  z lH

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Thus, the nonlinear operatorF satisfies the condition of local solvability of the Cauchy problem for the quasilinear hyperbolic equations in Hilbert spacesee 6,9 Taking this into

w ∈ C2

0, T

;H∩ C1

0, T

;H1/2

∩ C0, T

;H1

3 Initial Boundary Value Problem with

Non Stationary Boundary and Transmission Condition for

the Impulsive Semilinear Hyperbolic Equations

Let a1 < a2 < · · · < a m1 We consider in the domain0, T × m

i1a i , a i1 the following mixed problem

¨u i t, x − p i tu

i t, x f i

t, x, u i t, x, u

i t, x, ˙u i t, x, ϕ i

u, ˙u

,

t, x ∈ 0, T × a i , a i1, i 1, 2, , m,

u i t, a i1 u i1t, a i1, i 1, 2, , m − 1, t > 0,

¨u1t, a1 − q0tu

1t, a1 g0

t, ψ0

u, ˙u

, t > 0,

¨u i t, a i1 q i tui t, a i1 − u

i1t, a i1! g i

t, ψ i

u, ˙u

,

i 1, 2, , m − 1, t > 0,

¨u m t, a m1 q m tu

m t, a m1 g m

t, ψ m

u, ˙u

, t > 0,

u i 0, x u0

i x, ˙u i 0, x u1

i x, x ∈ a i , b i , i 1, 2, , m,

3.1

where ˙u i ∂u i /∂t, ui ∂u i /∂x, ¨u i ∂2u i /∂t2, u i ∂2u i /∂x2, u u1, , u m , ˙u

 ˙u1, , ˙u m , p i , q j , f i , g j , u0

i , u1

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 purposesee 10, chapter 1

Assume that the following conditions are held:

10 p i t ∈ C10, T, q j t ∈ C10, T; p i t > 0, q j t > 0, t ∈ 0, T, i 1, , m, j

0, 1, , m,

20 f i · ∈ C10, T × a i , a i1 × R4, i 1, 2, , m,

30 g j · ∈ C10, T, R, j 0, 1, , m,

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