On the solvability of the neumann boundary value problem without initial conditions for hyperbolic systems in infinite cylinders

In this paper, we study the Neumann boundary value problems without initial condition for Hyperbolic systems in cylinders. The main obtained results are the uniqueness and the existence of generalized solutions.

This paper is available online at http://stdb.hnue.edu.vn

ON THE SOLVABILITY OF THE NEUMANN BOUNDARY VALUE PROBLEM WITHOUT INITIAL CONDITIONS FOR HYPERBOLIC SYSTEMS

IN INFINITE CYLINDERS

Nguyen Manh Hung1 and Nguyen Thi Van Anh2

1National Institute of Education Management

2Hanoi National University of Education

Abstract.In this paper, we study the Neumann boundary value problem without

initial conditions for hyperbolic systems in infinite cylinders The primary results

obtained are the recognition of the uniqueness and the existence of generalized

solutions

Keywords Solvability, generalized solution, problems without initial conditions,

gronwall Bellman inequality

1 Introduction

Motivated by the fact that abstract boundary value problems for hyperbolic systems arise in many areas of applied mathematics, this type of system has received considerable attention for many years (see [2, 6, 9, 10]) Naturally, when expanding from hyperbolic

systems with initial conditions in cylinders (0, ∞) × Ω studied in [9], we consider one in

infinite cylinders (−∞, +∞)×Ω, where Ω is a bounded domain in R nwith the boundary

S = ∂Ω Base on previous achievements and direction [8], we deal with the solvability of

the Neumann boundary value problem without initial conditions for hyperbolic systems

in infinite cylinders

For a < b, set Q b a = Ω × (a, b), S b

a = S × (a, b) Let u = (u1, , u s)

be a complex-valued vector function and let us introduce some functional spaces used throughout in this paper

Received September 20, 2013 Accepted October 30, 2013.

Contact Nguyen Thi Van Anh, e-mail address: vananh89nb@gmail.com

We use H k,l (QR) the space consisting of all vector functions u : QR−→ C ssatisfying:

∥u∥2

H k,l (QR ):=

∫

QR

( ∑k

|α=0|

|D α u |2+

l

∑

j=1

|u t j |2)

dxdt,

and H k,l (e −γt , QR) is the space of vector functions with norm

∥u∥2

H k,l (e −γt ,QR):=

∫

QR

(∑k

|α=0|

|D α u |2+

l

∑

j=1

|u t j |2)

e −2γt dxdt.

In particular

∥u∥2

H k,0 (e −γt ,QR):=

∫

QR

k

∑

|α=0|

|D α

u |2

e −2γt dxdt.

Especially, we set L2(e −γt , QR) = H 0,0 (e −γt , QR)

We denote by

H k,l

(e −γt , QR) ={

u ∈ H k,l

(e −γt , QR) such that lim

t →−∞ ∥u(., t)∥ L2 (Ω) = 0}

,

soH k,l (e −γt , QR) is a linear space

Adding that, we consider an important space H k,l (e γt , QR) to be the space of vector functions with norm

∥u∥2

H k,l (e γt ,QR ):=

∫

QR

(∑k

|α=0|

|D α

u |2

+

l

∑

j=1

|u t j |2)

e 2γt dxdt < + ∞.

In particular

∥u∥2

H k,0 (e γt ,QR ):=

∫

QR

k

∑

|α|=0

|D α u |2e 2γt dxdt < + ∞.

We introduce the matrix differential operator:

L(x, t, D) =

m

∑

|p|,|q|=0

D p (a pq (x, t)D q ),

where the coefficients a pq are s × s matrices of functions with bounded complex-valued

components in QR, a pq = (−1) |p|+|q| a ∗

qp , with a ∗ qpbeing complex conjugate transportation

matrices of a pq

We recall Green’s formula (Theorem 9.47, [5]): Let B i (x, D), i = 1, , m, be a Dirichlet system of the order i − 1 Assume that Ω and the coefficients of the operators

involved are sufficiently smooth Then there exist normal boundary-value operators N i,

of order 2m − 1 − ordB i such that, for all u, v ∈ H 2m(Ω), we have:

∫

Ω

m

∑

|α|,|β|=0

a αβ (x)D β uD α vdx =

∫

Ω

( ∑m

|α|,|β|=0

(−1) |α| D α (a αβ (x)D β u)

)

vdx

−

∫

∂Ω

m

∑

j=1

(B j v)(N j u)dS.

We assume further that Ω and{a pq } satisfy Green’s formula and we define N j , j =

1, , m- the system of operators on the boundary SR Denote by

B(u, v)(t) =

m

∑

|p|,|q|=0

(−1) |p|∫

Ω

a pq D q uD p vdx,

and

H N m(Ω) ={

u ∈ H m (Ω) : N j u = 0 on S for all j = 1, , m}

.

We assume further that the form (−1) m B(., )(t) is H m

N- uniformly elliptic with

respect to t and that means there exists a constant µ0 > 0 independent of t and u such

that:

(−1) m B(u, u)(t) ≥ µ0∥u(., t)∥2

H m(Ω)

for all u ∈ H m

N (Ω), and a.e t ≥ h.

We consider the hyperbolic system in the cylinder QR

(−1) m −1 L(x, t, D)u − u tt = f (x, t) in QR, (1.1)

N j u 

SR

Definition 1.1 Let f ∈ L2(e −γt , QR), a complex-valued vector function u ∈

H m,1 (e −γt , QR) is called a generalized solution of problem (1.1) - (1.2) if and only if

for any T > 0 the equality:

(−1) m −1∫ T

−∞

B(u, η)(t)dt +

∫

Q T

−∞

u t η t dxdt =

∫

Q T

−∞

holds for all η ∈ H m,1 (e γt , QR), η(x, t) = 0 with t ≥ T

2 The uniqueness of a generalized solution of a problem (1.1) -(1.2)

Theorem 2.1 If γ > 0 and ∂a pq

∂t < µ1e 2γt , ∀t ∈ R, ∀|p|, |q| ≤ m, the problem (1.1)-(1.2) has no more than one solution.

Proof Assume u1(x, t) and u2(x, t) to be two generalized solutions of problem (1.1) -(1.2), set u(x, t) = u1(x, t) − u2(x, t) For any T > 0, b ≤ T , denote:

u(x, t) = u1(x, t) − u2(x, t),

η(x, t) =







t

∫

b u(x, τ )dτ, −∞ ≤ t ≤ b,

So we get η(x, T ) = 0, η(x, t) ∈ H m,1 (e γt , Q T

−∞ ), and η t (x, t) = u(x, t), ∀(x, t) ∈ Q b

−∞.

Then we use η as a test function and because u = η t, according to the definition of the generalized solution, we have:

(−1) m −1

m

∑

|p|,|q|=0

(−1) |p|∫

Q b

−∞

a pq D q η t D p ηdxdt +

∫

Q b

−∞

η tt η t dxdt = 0. (2.1)

Adding the equation (2.1) with its complex conjugate we get:

(−1) m −1 2Re∫ b

−∞

B(η t , η)(t)dt + 2Re

∫

Q b

−∞

η tt η t dxdt = 0. (2.2)

We transform the first term using integration by parts and the hypotheses of the coefficients and for the second term we use integration by parts, then replacing the obtained equalities into (2.2), we get:

∥η t (., b) ∥2

L2 (Ω)+ lim

h →−∞(−1) m B(η, η)(h) = ( −1) m−1 ∑m

|p|,|q|=0

(−1) |p|∫

Q b

−∞

∂a pq

∂t D

q ηD p ηdxdt.

Noting the asumption, we then have the fact that the coeficients a pq are continuous

with respect to the time variable and η ∈ H m,1 (e γt , Q T

−∞), so there exists the limit

lim

h →−∞(−1) m B(η, η)(h) By using a uniformly elliptic condition we imply:

lim

h →−∞(−1) m

B(η, η)(h) ≥ µ0 lim

h →−∞ ∥η(., h)∥2

H m(Ω),

and thus

∥η t (., b) ∥2

L2 (Ω)+µ0 lim

h →−∞ ∥η(., h)∥2

H m(Ω) ≤ (−1) m −1

m

∑

|p|,|q|=0

(−1) |p|∫

Q b

−∞

∂a pq

∂t D

q ηD p ηdxdt.

By using the Cauchy inequality, we have:

(−1) m −1

m

∑

|p|,|q|=0

(−1) |p|∫

Q b

−∞

∂a pq

∂t D

q

ηD p ηdxdt ≤ µ

1m ∗

∫ b

−∞ ∥η(., t)∥2

H m(Ω)e 2γt dt.

We have yields:

∥η t (., b) ∥2

L2 (Ω)+ µ0 lim

h →−∞ ∥η(., h)∥2

H m(Ω) ≤ µ1m ∗

∫ b

−∞ ∥η(., t)∥2

H m(Ω)e 2γt dt. (2.3) Now denote by:

v p (x, t) =

∫ h t

D p u(x, τ )dτ, −∞ ≤ t ≤ b.

Hence, we can see that

D p η(x, t) =

∫ t

b

D p u(x, τ )dτ = v p (x, b) − v p (x, t), lim

h →−∞ D

p η(x, h) = v p (x, b),

lim

h →−∞ ∥η(., h)∥2

H m(Ω) =

m

∑

|p|=0

∫

Ω

|v p (x, b) |2

dx,

and from the equality (2.3) we have:

∥η t (., b) ∥2

L2 (Ω)+ µ0

m

∑

|p|=0

∫

Ω

|v p (x, b) |2dx ≤ µ1m ∗

∫ b

−∞ ∥η(., t)∥2

H m(Ω)e 2γt dt.

This then leads to

∥η t (., b) ∥2

L2 (Ω)+ µ0

m

∑

|p|=0

∥v p (x, b) ∥2

L2 (Ω) ≤ µ1m ∗

m

∑

|p|=0

∫

Q b

−∞

e 2γt |D p η(x, t) |2dxdt

≤ 2µ1m ∗ e 2γb

m

∑

|p|=0

∥v p (., b) ∥2

L2 (Ω)+ 2µ1m ∗

m

∑

|p|=0

∫ b

−∞

e 2γt ∥v p (., t) ∥2

L2 (Ω)dt

=⇒∥η t (., b) ∥2

L2 (Ω)+ (µ0− 2µ1m ∗ e 2γb)

m

∑

|p|=0

∥v p (x, b) ∥2

L2 (Ω)

≤ 2µ1m ∗

µ0− 2µ1m ∗ e 2γb

∫ b

−∞

(

e 2γt ∥η t (., t) ∥2

L2 (Ω)+ (µ0− 2µ1m ∗ e 2γb )e 2γt

m

∑

|p|=0

∥v p (x, b) ∥2

L2 (Ω)

)

dt.

So, there exists a positive number C > 0, C > µ0

2µ1m ∗ such that

∥η t (., b) ∥2

L2 (Ω)+

m

∑

|p|=0

∥v p (x, b) ∥2

L2 (Ω) ≤ C

∫ b

−∞

e 2γt(

∥η t (., t) ∥2

L2 (Ω)+

m

∑

|p|=0

∥v p (x, b) ∥2

L2 (Ω)

)

dt.

J (t) = ∥η t (x, t) ∥2

L2 (Ω)+ (µ0− 2µ1m ∗ e 2γb)

m

∑

|p|=0

∥v p (x, t) ∥2

L2 (Ω),

we have:

J (b) ≤ C

∫ b

−∞

e 2γt J (t)dt, for a.e b ≤ 1

2γ ln

1

C .

By performing a check similar to the proof of the Gronwall- Bellman inequality (see [4], page 624-625), we will prove that

J (t) ≡ 0 on (−∞, 1

2γ ln

1

C ].

In fact, taking ζ(t) = ∫t

−∞ e 2γs J (s)ds, we have ζ ′ (t) = (

∫0

−∞ e 2γs J (s)ds +

∫t

0 e 2γs J (s)ds) ′ = e 2γt J (t), then we have:

ζ ′ (t) ≤ Ce 2γt ζ(t) for a.e t ≤ 1

2γ ln

1

C .

From this we see

d ds

(

ζ(s)e −Ce2γs 2γ )

= e −Ce2γs 2γ (ζ ′ (s) − Ce 2γt ζ(t)) ≤ 0.

By integrating with respect to s from −∞ to t in remark that lim

s →−∞ ζ(s)e

−Ce2γs

2γ = 0, we get

ζ(t)e −Ce2γt 2γ ≤ 0 for a.e t ≤ 1

2γ ln

1

C .

Thus we obtain ζ(t) ≤ 0 and we can conclude ζ ′ (t) ≤ 0 for a.e t ≤ 1

2γ ln C1 by the above estimate From this, one has the desied estimate

So u(x, t) = 0 almost everywhere t ∈ (−∞, 1

2γ ln

1

C] Because of the uniqueness of the

solution of a problem with initial conditions for a hyperbolic system, we imply u1(x, t) =

u2(x, t) almost everywhere t ∈ R.

We note that the obtained result about the uniqueness does not change if we consider the partial differential equations in the forms:

(−1) m Lu − u tt − αu t = f, (x, t) ∈ QR, (i)

N j u

where α is a positive constant number We have the definition of generalized solutions of

the problem (i) - (ii)

Definition 2.1 Let f ∈ L2(e −γt , QR), a complex-valued vector function u ∈

H m,1 (e −γt , QR) is called a generalization of problem (i)-(ii) if and only if for any T > 0

the equality:

(−1) m −1∫ T

−∞

B(u, η)(t)dt +

∫

QR

u t (η t − αη)dxdt =

∫

Q T −∞

f ηdxdt (iii)

holds for all η ∈ H m,1 (e γt , QR), η(x, t) = 0 with t ≥ T

By the same proofs we give the theorem about the uniqueness of this problem

Theorem 2.2 If γ > 0 and ∂a pq

∂t < µ1e 2γt , ∀t ∈ R, ∀|p|, |q| ≤ m, the problem (i) - (ii) has only one solution.

3 The existence of a generalized solution of a Neumann boundary value problem for hyperbolic system with initial conditions

First, we set the hyperbolic systems (1.1) - (1.2) in Q ∞ h with initial conditions

We restate the concept of the generalized solution of (1.1)-(1.2)-(1.3’) A function

u(x, t) is called a generalized solution of the problem (1.1)-(1.2)-(1.3’) in the space

H m,1 (e −γt , Q ∞ h ), if and only if u(x, t) belongs to H m,1 (e −γt , Q ∞ h ), u(x, h) = 0, and the

equality

(−1) m −1∫

Q T

m

∑

|p|,|q|=0

(−1) |p| a

pq D q uD p η +

∫

Q T

u t η t dxdt =

∫

Q T

f ηdxdt

holds for all η belong to H m,1 (Q T h ) satisfying η(x, T ) = 0, for all T > h.

Theorem 3.1 (The existence of a generalized solution)

Let γ > γ0 = µm 2µ ∗

0 , here m ∗ =

m

∑

|p|=0

1 Assume that the operator ( −1) m B(., )(t) satisfies the elliptic uniformity condition and

(i) sup

{

∂a pq

∂t

, |a pq | : (x, t) ∈ Q ∞

h , 0 ≤ |p|, |q| ≤ m

}

≤ µ, (ii) f (x, t) ∈ L2(e −γt , Q ∞ h ) ,

Then there exists a unique generalized solution u(x, t) ∈ H m,1 (e −γt , Q ∞ h ) of problem

(1.1) - (1.2) - (1.3’) satisfying:

∥u∥2

H m,1 (e −γt ,Q ∞

h) ≤ C∥f∥2

L2(

e −γt ,Q ∞ h

where C = const > 0 is independent of u, h and f

Proof The uniqueness is similar way to that in [9] We omit the details here Note

that the constant C in the estimate in Theorem 2.1 in [9] depends on t = 0, so if we change the initial conditions by t = h in the same proof, we also obtain the fact that the constant C is depentdent on t = h Now we give the proof to improve it Due to

the similarities as in [3], we get the approximate solutions {u N (x, t) } ∞

N =1 defined that

u N (x, t) =∑N

k=1 c N k (t)φ k (x) such that c N l (h) = 0 and dt d c N l (h) = 0, l = 1, , N and

(−1) m −1

m

∑

|p|,|q|=0

(−1) |p|∫

Ω

a pq D q u N D p φ l dx −

∫

Ω

u N tt φ l dx =

∫

Ω

f φ l dx. (3.2)

Multiplying (3.2) by dC

N

l (t)

dt and taking the sum with respect from 1 to N , integrating

with respect to t from h to τ ( τ ≥ h), then adding that to its complex conjugate and finally

applying a pq = (−1) |p|+|q| a ∗

qp , from the initial conditions of u N we conclude that

∥u N

t (., τ ) ∥2

L2 (Ω)+ (−1) m B(u N , u N )(τ )

=−(−1) m −1

m

∑

|p|,|q|=0

(−1) |p|∫

Q τ h

∂a pq

∂t D

q

u N D p u N dxdt − 2Re

∫

Q τ h

f u N

t dxdt,

From the uniformly elliptic condition of the operator (−1) m B(., )(t) and the bounded

property of the functions a pq, ∂a pq

∂t , and the Cauchy inequality we get

∥u N

t (., τ ) ∥2

L2 (Ω)+ µ0∥u N

(., τ ) ∥2

H m(Ω)

≤ δ

∫ τ

h

(

∥u N

t (., t) ∥2

L2 (Ω)+µm

∗

δ ∥u N (., t) ∥2

H m(Ω)

)

dt + 1 δ

∫ τ h

∥f(., t)∥2

L2 (Ω)dt.

We take µm δ ∗ = µ0 then

∥u N

t (., τ ) ∥2

L2 (Ω)+ µ0∥u N (., τ ) ∥2

H m(Ω)

≤ 2γ0

∫ τ

h

(

∥u N

t (., t) ∥2

L2 (Ω)+ µ0∥u N (., t) ∥2

H m(Ω)

)

dt + C

∫ τ

h

∥f(., t)∥2

L2 (Ω)dt.

Set

J N (t) = ∥u N

t (., τ ) ∥2

L2 (Ω)+ µ0∥u N (., τ ) ∥2

H m(Ω),

we get J N (τ ) ≤ 2γ0

∫τ

h J N (t)dt + C∫τ

h ∥f(., t)∥2

L2 (Ω)dt.

Using the Gronwall-Bellman inequality we have:

J N (τ ) ≤ C

τ

∫

h

∥f(., t)∥2

L2 (Ω)dt + 2γ0

τ

∫

h

e 2γ0(τ −s)

s

∫

h

∥f(., θ)∥2

L2 (Ω)dθds.

So, the following inequality is obvious

∥u N

t (., τ ) ∥2

L2 (Ω)+∥u N

(., τ ) ∥2

≤ C1

τ

∫

h

∥f(., t)∥2

L2 (Ω)dt + C22γ0

τ

∫

h

e 2γ0(τ −s)

s

∫

h

∥f(., θ)∥2

L2 (Ω)dθds,

with the constant C1, C2not depending on h.

Now multiplying both sides of this inequality by e −2γt Then integrating with

respect to τ from h to ∞ we have:

∫ ∞

h

(

∥u N

t (., τ ) ∥2

L2 (Ω)+∥u N (., τ ) ∥2

H m(Ω)

)

e −2γt dτ

≤ C1

+∞

∫

h

e −2γτ

τ

∫

h

∥f(., t)∥2

L2 (Ω)dtdτ + C2

+∞

∫

h

e −2γτ

τ

∫

h

e 2γ0(τ −s)

s

∫

h

∥f(., θ)∥2

L2 (Ω)dθdsdτ.

(3.4)

Denote by I1, I2 the terms from the first and second respectively of the right-hand sides

of above inequalty We will give estimations for these terms

First,

I1 = C1

+∞

∫

h

∥f(., t)∥2

L2 (Ω)

+∞

∫

t

e −2γτ dτ dt = C1

2γ ∥f∥2

L2(e −γt ,Q ∞

h),

and

I2 = C2

+∞

∫

h

∥f(., θ)∥2

L2 (Ω)

+∞

∫

θ

e −2γ0s

+∞

∫

s

e 2(γ0−γ)τ dτ dsdθ = C2

4γ(γ − γ0)∥f∥2

L2(e −γt ,Q ∞

h),

here C1, C2 are constants not depending on h, f , u N Combining the above estimate we get:

∥u N ∥2

H m,1 (e −γt ,Q ∞

h) ≤ C∥f∥2

L2(e −γt ,Q ∞

h).

From this, in the same manner as in Theorem 2.1 (see [9]), we can conclude that there exists a generalized solution of the problem satisfying (3.1)

4 The existence of a generalized solution of the problem (1.1) -(1.2)

A generalized solution of a problem (1.1) - (1.2) can be approximated by a sequence

of solutions of problems with initial conditions (1.1) - (1.2) - (1.3’) in cylinder Q ∞ h Consider in the real lineR, we use Theory 5.5 ([5]), we then see that there exists a

test function θ ∈ C ∞(R) such that θ(t) = 0, ∀t ≤ 0, θ(t) = 1∀t ≥ 1, θ(t) ∈ [0, 1], ∀t ∈ [0, 1] Moreover, we can suppose that all derivatives of θ(t) are bounded Let h ∈ (−∞, 0]

be an integer

Set f h (x, t) = θ(t − h)f(x, t) =⇒ f h (x, t) ∈ L2(e −γt , QR), h ∈ Z.

f h (., t) =

{

f (., t) if t ≥ h + 1

0 if t < h

And we have∥f h ∥2

L2(e −γt ,R)≤ ∥f∥2

L2(e −γt ,QR)

We consider the following problem in the cylinder Q ∞ h :

(−1) m−1 Lu − u tt = f h in Q ∞ h ,

N j u = 0, j = 1, , m on S h ∞ ,

u

t=h = u t

t=h = 0 on Ω.

It is easy to see that there exists a number γ0 > 0 such that for each γ > γ0 the

above problem has a unique generalized solution called u h in H m,1 (e −γt , Q ∞ h ) which the following estimate satisfies:

∥u h ∥2

H m,1 (e −γt ,QR) ≤ C∥f∥2

L2(e −γt ,QR).

Let consider genenalized solutions u h and u k of problems in cylinders Q ∞ h and

Q ∞ k with f (x, t) is replaced by f h (x, t) and f k (x, t) respectively If h > k, u h can be

understood in H m,1 (e −γt , Q ∞ k ) with u h (x, t) = 0, ∀k ≤ t ≤ h.

Setting v kh = u k − u h , f kh = f k − f h , so v kh is the generalized solution of the following problem:

(−1) m −1 Lv − v tt = f kh in Q ∞ k ,

N j v = 0, j = 1, , m on S k ∞ ,

v

t=k = v t

t=k = 0 on Ω.

Then, we get∥u kh ∥2

H m,1 (e −γt ,QR)≤ C∥f kh ∥2

L2(e −γt ,QR)

From the definition of f h we can see that{f h } −∞

h=0 is the Cauchy sequence in the

space L2(e −γt , QR), it follows that{u h } −∞

h=0is a Cauchy sequence in the completed space

H m,1 (e −γt , QR) and thus, u h converge to u in H m,1 (e −γt , QR)

We transform the first term using integration by parts and the hypotheses of the coefficients and for the second term we use integration by parts, then replacing the obtained equalities into (2.2),... only one solution.

3 The existence of a generalized solution of a Neumann boundary value problem for hyperbolic system with initial conditions< /b>

First, we set the. .. Theorem 2.1 in [9] depends on t = 0, so if we change the initial conditions by t = h in the same proof, we also obtain the fact that the constant C is depentdent on t = h Now we give the proof to improve