Stabilization of the wave equation on d

-

English
65 Pages
Read an excerpt
Gain access to the library to view online
Learn more

Description

Niveau: Secondaire, Lycée, Terminale
Stabilization of the wave equation on 1-d networks with a delay term in the nodal feedba ks Serge Ni aise, Julie Valein Université de Valen iennes et du Hainaut Cambrésis LAMAV, FR CNRS 2956 Institut des S ien es et Te hniques of Valen iennes F-59313 - Valen iennes Cedex 9 Fran e Julie.Valein,Serge.Ni aiseuniv-valen iennes.fr O tober 24, 2006 Abstra t In this paper we onsider the wave equation on 1-d networks with a delay term in the boundary and/or transmission onditions. We rst show the well posedness of the problem and the de ay of an appropriate energy. We give a ne essary and su ient ondition that guarantees the de ay to zero of the energy. We further give su ient onditions that lead to exponential or polynomial stability of the solution. Some examples are also given. 1 Introdu tion/Notations Time delay ee ts arise in many pra ti al problems, see for instan e [20, 28, 1? for biologi al, ele tri al engineering, or me hani al appli ations. Furthermore it is well known that they an indu e some instabilities [17, 18, 19, 30, 25?, or on the ontrary improve the performan e of the system [28, 1?.

  • nodes ofd

  • ∂2uj ∂x2

  • ient ondition whi

  • tion does

  • ontrolled nodes

  • ∂uj ∂x

  • hilbert spa

  • ?v ?

  • ient ondition


Subjects

Informations

Published by
Reads 33
Language English
Report a problem

1−d 1−d
1−d
,
1-d
applications.
net
on,
w
e
orks
equation
with
for
a
,
dela
on
y
of
term
this
in
to
the
1]
no
wn
dal


,
ks
atten
Serge
W

transmission
Julie
T
V
orks
alein
ab
Univ
13
ersit?
[20
de
engineering,
V
is


et
,
du

Hainaut
of
Cam
,
br?sis
orks
LAMA
y
V,
the
FR
v
CNRS
in
2956
w
Institut
v
des
wledge,
Sciences
w
et
et
T
some
ec
er.
hniques
11
of
24
V
for

28
F-59313
biological,
-

V
urthermore

ell
Cedex
they
9
instabilities
F
,
rance
,
Julie.V
on

impro

p
Octob
system
er

24,
trol
2006
net

pa
In
of
this
see
pap

er

w
here
e
the

dela
the
oundary
w
of
a
v
v
equation
e
w
equation
our
on
analysis
1-d
to
net
net
w
not
orks
Before
with
us
a
and
dela
of
y
e
term
,
in
12
the
14
b
26
oundary
details.
and/or

transmission
,

,
W
for
e
electrical
rst
or
sho

w
F
the
it
w
w
ell
kno
p
that
osedness

of
some
the
[17
problem
18
and
19
the
30

25
y
or
of
the
an
trary
appropriate
v
energy
the
.

W
the
e
[28
giv
1].
e
tly
a


problems
and
1-d
sucien
w
t
are

ying
that
tion
guaran
man
tees
authors,
the
[22

16
y
and
to
references
zero
there.
of
e
the
in
energy
estigate
.
eect
W
time
e
y
further
b
giv
and/or
e
stabilization
sucien
the
t
a

e
that
in
lead
e
to
net
exp
orks.
onen
o
tial
kno
or
the
p
of
olynomial
eect
stabilit
a
y
the
of
w
the
is
solution.
y
Some
done.
examples
going
are
let
also

on
denitions
net
notations
orks
out
in
Stabilization
whole
giv
w
en.
used
1
the
In
pap
tro
W

refer
Time
[2
dela
3,
y
,
eects
,
arise
,
in
,
man
,
y


more
problems,
1
seen1−d R R n≥ 1
N[
R = ej
j=1
e (0, l ), l > 0,j j j
k =j, e ∩ej k
e ej j
u : R−→R, u =u uj |ej
ej
E ={e ; 1≤j ≤N} R Vj
R. v,
E = {j∈{1,...,N};v∈e }v j
v (E ) = 1, vv
(E ) ≥ 2, v Vv ext
V v∈V Eint ext v
j .v
Vext
cV =D∪N ∪V .ext ext
D
N
c cV V Vintext int
Vc
c cV =V ∪V .c int ext
1D =∅ H
 2 2∂ u ∂ uj j (x, t)− (x, t) = 0 0<x<l , t> 0, 2 2 j∂t ∂x ∀j∈{1,...,N}, u (v, t) =u (v, t) =u(v,t) ∀j, l∈E , v∈V ,t> 0, j l v int X ∂u (v) (v)j ∂u ∂u (v, t) =−(α (v, t)+α (v, t−τ )) ∀v∈V ,t> 0,v c 1 2∂nj ∂t ∂t j∈EvX
∂uj c(v, t) = 0 ∀v∈V \V ,t> 0,int int∂nj j∈Ev u (v,t) = 0 ∀v∈D,t> 0, jv ∂u jv (v,t) = 0 ∀v∈N,t> 0, ∂nj v ∂u (0) (1) u(t = 0) =u , (t = 0) =u , ∂t ∂u 0(v, t−τ ) =f (t−τ ) ∀v∈V , 0<t<τ ,v v c vv∂t

form
of
w
e
no


de,
e
b
W
me
norm.
a
a
no
ecomes
v
b
F
semi-norm
b
the
or
that
a
so
that
;
dene6
set
that
is
ose
is
supp
edges
also
v
e
of
W
of
namely
.
des,
e
no
e
trolled
no


of
either
set
the
the
wher
y
set
b
oundary
denote
W
e
in
w
if
2
exterior
v
If
F
ving
osed.
set
imp
let
e
a
b

will
the

of
transmission
the
k
e

the
a
the
where
function
,
).
of
the
subset
(her
a
or
x
le
further
extr
e
or
W6
.
and
of
that
des
is
no
initial/b
the
of
at


1.1
oundary
the
b
e
k
de.

terior
a
an
nally

and
while
of
no
des
an
no

the
ertex.
at
as

ha
oundary
of
b
the
Neumann
b
;
ertex
of
xed
des
or
no
of
the
v
at
set

y
oundary
and
b
edges
hlet
set

y
ose
denote
imp
W
will
edge
e
to
w

Clearly
set
:
w
of
a
partition
F
a
of
x

w
ans
no
e
e
de
W
a
y
vertex
b
d
denoted
al
is
emity
of
ommon
t
a
elemen
empty
single
is
the
for
,

our
interval
ulate
identify
or
with
F
we
des.

no
problem:
terior
e
in
by
of
d
set
,
the
(1)
and
d
alue
onne
a
network
des
is
no
A
exterior
Denition
of
set
or
shortness,(v) (v)
α ,α ≥ 0 τ > 0v1 2
∂uj(v,·)
∂nj
u vj
u ej j
∂u 0(v, t−τ ) =f (t−τ ) v∈V , 0<t<τv v c vv∂t
(v)
α = 0 v ∈ Vc2
(v)
α = 01
(v) (v)
α α1 2
(v) (v)
α ≤α ,∀v∈V ,c2 1
(v) (v)
α <α ,∀v∈V ,c2 1
at

y
due
W
to
for
the

dela
d
y
use
equation.

In
appropriate
the
do
absence
it
is
of
Here
dela
b
y
rates
,
t
i.
initial
e.,
e
normal
a
ard
If
w
the
out
problem
the
that
means

and
abilit
for
to
all
of
xed
5
e
giv
b
y
to
w
,
y
the
that
ab
deriv
o
.
v

e
e
problem
t
has
of
b
v
een
w

not
b
in
y
generalit
some

authors
t
in
the
some
energy
particular
the
situations,
the
for
v

abilit
Ammari
metho
and
lik
T
form
ucsnak
a

domain
Ammari,
non
Henrot
.
and
a
T
results.
ucsnak
a
[4
exp

Similarly
Ammari

and
alue
Jellouli
denotes
[5
e
,
v
6
for

w
Ammari,
of
Jellouli
.
and
giv
Khenissi
and

for
and
to
Xu,
energy
Liu
ab
and

Liu
not
[29


do
In
y
these
do
pap
estigate
ers,
its
some
but
sucien
a
t
w

ts
but
are
the
giv
Remark
en
p
in
of
order
Our
to
based
guaran
of
tee
estimates
some
without
stabilities
e
of

the
these
system.
estimates
On
frequency
the
The


trary
the
,
represen
if
[16
osed
ma
supp
oid
also
the
is
d
y
quite
dela

the
the
that
a
is
and
if
estimate
w
stabilit
e

ha
giv
v
and
e
for
only
tial
the
our
dela
6
y
a
part
assuming
in
in
the
v
b
an
oundary/transmission


ativ
system
of
(1)
the
ma
ertex
y
Note
b
w
ecome
sho
unstable.
the
See,
y
for
an

energy
Datk
W
o,
further
Lagnese
e
and

P
sucien
olis

[19
the

y
for
zero
the
the
example
.
of
the
a
o
string.
e
Therefore
tion
it
es
is
hold,
in
e
teresting
that
to
energy
seek
es
for

stabilization
.
results
e
in
not
general
v
1-d
this
the
in
w
full
rst
y
e
study

in
sucien
particular

No
the
if
onen
represen

the
of

energy
of
W
string

.
a
that
t

past,
for
net
e
w
giv
orks
a
when
and
the
t
parameters
for
ers,
exp
b
tial
um
y
n
the
real
.
and
e
e
nd
nonnegativ
sucien
xed

are
the
where
olynomial
nally
y
the
the
abilit
.
estimate
metho
indep
is
t
on
the
use
y
observ
The
y
er
of
organized
problem
follo
damping.
After
w

ha
some
e
and
hosen
w
obtain
sho
observ
in
y

b
that
a
problem
domain
w
d.
p
use
Then
other

hniques
w
e
pro
d'Alem
e
ert

tation
of
ula
appropriate
,
and

e
y

v
sucien
the

of
h
frequency
tees
metho

but
to
e
of
often
energy
optimal

y
is
for
oted
energy
the
Note
of
that
are
observ
b
y
oth
is
nonzero.
enden
In
of
the
dela
sp
term.
ecial
pap

is
of
as
one
ws.
string
the
and
of
a
denitions

notations,
k
e
la
w
w
the
at

one
our
extremit
is
y
ell
,
osed.
this
in
problem
3,
has
e
b
v
een
the
studied
y
b
an
y
energy
Xu,
giv
Y
a
ung
and
and
t
Li
whic
[30
guaran

the
where
y
the
0
authors
the
use
.
a
4
sp
dev
ectral
to
analysis.
pro
F
of
or
regularit
the
result
w
an
a
priori
v
used
e
the
equation
y
in
In
higher
5
dimensional
e
space
e
domain,

w
sucien
e

refer
the
to
onen
[25
stabilit

of
In
system.


with
is
[30
with
,
3
25a . b
C a b a ≤ C b a ∼ b
a.b b.a
2 2L (R) ={u :R→R;u ∈L (0,l ),∀j = 1,···,N},j j
k·k 2 VL (R)
NY
1V :={φ∈ H (0,l ) : φ (v) =φ (v)∀j, k∈E ,∀v∈V ; φ (v) = 0∀v∈D},j j k v int jv
j=1
ZN ljX ˜∂φ ∂φj j˜<φ, φ> = dx.V
∂x ∂x0j=1
1 1u∈ L (R) ={u :R→R;u ∈ L (0,l ),∀j = 1,···,N}j j
Z ZN ljX
u = u (x)dx.j
R 0j=1
2A L (R)
NY X∂uj2D(A) := {u∈V ∩ H (0,l ) : (v) = 0,∀v∈V ;j int
∂njj=1 j∈Ev
∂ujv c(v) = 0,∀v∈N ∪V },ext∂njv
2∂ uj
(Au) = − ∀j = 1,···,N,∀u∈D(A).j 2∂x
2(L (R),V,a) a
ZN ljX ∂u ∂vj j
a(u,v) = dx,∀u,v∈V.
∂x ∂x0j=1
osedness
without
group
but
extension
(1)
h
to
is
similar
where
system
will
the
hold
with
sho

erator
asso
F
No
y
w
often
w
t
e
notation
in
2
tro
W

(1)
the
e
erator
the
w
is
op
ose,
from
use
spatial
Hilb
the
In


tro
indep
in
and
in
.
to
that
itself
ultaneously
b
ell
y
the
e
aim
w
that
uses,
op
future
p
or
t
whic
it
h

is
the
a
ell-p
Hilb
that
ert
e
space
bilinear
for
dened
the
space
natural
e
inner
,
pro
an

from
Its
e
for
t
shortness
enden
or
of
F

sucien
that
t
The

means
for
and
the
sim
p
.
olynomial
W
stabilit
p
y
of
of
problem
our
e
system.
to
Finally
w
w
problem
e
This
end
erator
up
a
with
ositiv
asso
selfadjoin

op
some
since
illustrativ
is
e
F
examples
hs
in
of

triple
7.
w
In
osed.
the
or
whole
purp
pap
w

,
pro
the
inner
form
the
is
with
b
ed
semi-
equipp
ert
er
the
the
b

further
damping.
.
notation
Let
means
theory
that
and
there
idea
exists
write
e
[25
F
tro
b
a
y
p
denoted
ositiv
b
e
norm

op
4
eratorNY
2X =V∩ H (0,l )j
j=1
(u,v) = (u,v) 2 +(Δu,Δv) 2 ,∀u,v∈X,X L (R) L (R)
2∂ uj
(Δu) = ∀j = 1,···,N,u∈X.j 2∂x
∂uv ∈ V z (ρ,t) = (v, t− τ ρ)c v v∂t
ρ ∈ (0, 1) t > 0

2 2∂ u ∂ uj j (x, t)− (x, t) = 0 0<x<l , t> 0,∀j∈{1,...,N},2 2 j ∂t ∂x ∂z ∂zv v τ (ρ, t)+ (ρ, t) = 0 0<ρ< 1, t> 0,∀v∈V , v c∂t ∂ρ u (v, t) =u (v, t) =u(v,t) ∀j, l∈E , v∈V ,t> 0, j l v int X ∂u (v) (v) j ∂u (v, t) =−(α (v, t)+α z (1,t)) ∀v∈V ,t> 0, v c1 2∂n ∂t j j∈Ev X ∂uj c(v, t) = 0 ∀v∈V \V ,t> 0,int int∂nj
 j∈E v u (v,t) = 0 ∀v∈D,t> 0,j v ∂ujv (v,t) = 0 ∀v∈N,t> 0, ∂njv ∂u z (0, t) = (v, t) ∀v∈V , t> 0, v c∂t (0) ∂u (1) u(t = 0) =u , (t = 0) =u , ∂t 0z (ρ, 0) =f (−τ ρ) ∀v∈V , 0<ρ< 1.v v cv
z t,ρv
t = 0 ρ = 0
z = (z )v v∈Vc
∂u ⊤U := (u, , z) ,
∂t
U
2∂u ∂ u ∂z ∂u 1 ∂zv′ ⊤ ⊤U = ( , , ) = ( , Δu,−( ) ) .v∈Vc2∂t ∂t ∂t ∂t τ ∂ρv

′U =AU,
0 1 0 ⊤U(0) = (u ,u , (f (−τ .)) ) ,v v
A
   
wu
   ΔuA w :=
1 ∂zv−( )z vτ ∂ρv
where
in
it
tro
in

tro
Note
an
the
a
auxiliary
Let
us
v
If
ariable
ha
that
v
,
where
for
y
and
and
.
k
In
inner
this
e
manner,
pro
and
w
whic
datum
(2)
ariables,
to
set
t
ort
alen
op
equiv
dened
is
satises
(1)
satises
h
(1)
is
our
a
further
F

problem

Hilb
in
ert
w
and
.
(1)

in
and
term
e
y
at
dela
initial
the
with
eliminate
v
Consequen
e
tly
the
the
equation
problem
transp
(2)
the
ma
erator
y
is
b
b
e
follo
rewritten
as
ws.
transform
the
No
rst
then
order
w
ev
system
olution
to
equation

(3)
or
space
set
all
e
let
with
w
the
as
w
5
eNY
2 1 VcD(A) :={(u,w, z)∈ (V ∩ H (0, l ))×V ×H (0, 1) :j
j=1
X ∂uj (v) (v)
(v) =−(α w(v)+α z (1))∀v∈V ;v c1 2∂nj
j∈EvX
∂uj c(v) = 0∀v∈V \V ;int int∂nj
j∈Ev
∂ujv(v) = 0∀v∈N ; z (0) =w(v)∀v∈V },v c∂njv
V Vc c
2 2 VcH :=V ×L (R)×L (0, 1) ,
   * + Z Zu u˜ N l 1jX X∂u ∂u˜j j   w , w˜ = ( +w w˜ )dx+ z (ρ)z˜ (ρ)dρ.j j v v
∂x ∂x0 0z z˜ j=1 v∈Vc
D(A) H
⊤(f, g,h) ∈H D(A),
   * + Z Zu f N l 1jX X∂u ∂fj j   0 = w , g = ( +w g )dx+ z (ρ)h (ρ)dρ,j j v v
∂x ∂x0 0z h j=1 v∈Vc
⊤(u, w, z) ∈D(A).
Vcu = 0 w = 0 z ∈D(0, 1) (0, 0, z)∈D(A),
Z 1X
z (ρ)h (ρ)dρ = 0.v v
0v∈Vc
2D(0, 1) L (0, 1), h = 0.
N NY Y
2D(0, l ) L (0, l ), u = 0j j
j=1 j=1
NY
z = 0 w∈ D(0, l ) g = 0.j
j=1
ZN ljX ∂u ∂fj j
0 = dx,∀(u,w, z)∈D(A).
∂x ∂x0j=1
is
is
for
dense
e
in
y

get
pro
des
inner
o
and

.
w
usual
of
the
in
with
b
ed
The
equipp
e
space
is
As
in
.
Lemma
ert
um
b
b
of
no
taking
tak
Hilb
is
the
that

with
,
all
as
ab
manner
v
same
orthogonalit
and

the
then
w
to
namely
dense
in
domain
tro
is
of
where
ts
the
elemen
n
that
Since

2.1
e
er
Let
No
Pro
e
of.
w
.
all
W
to
rst
orthogonal
e
e
dense
w
and
e
see
y
6
Inw = 0 z = 0,
ZN ljX ∂u ∂fj j
dx = 0,∀(u, 0, 0)∈D(A).
∂x ∂x0j=1
(u, 0, 0)∈D(A) u∈D(A).
D(A) V < ., . >V
f = 0
(v) (v)
α ≤α ,∀v∈V .c2 1
A C0
H

(v) (v) (v)(v)τ α ≤ξ ≤τ (2α −α ),∀v∈V .v v c2 1 2
H
   * + Z Zu u˜ N l 1jX X∂u ∂u˜j j (v)   w , w˜ = ( +w w˜ )dx+ ξ ( z (ρ)z˜ (ρ)dρ)j j v v
∂x ∂x0 0z z˜ j=1 v∈VcH
H
U ∈ H0
U ∈C([0, +∞),H) U ∈D(A)0
1U ∈C([0, +∞), D(A))∩C ([0, +∞),H).
A
⊤A U = (u,w, z) ∈D(A)
   * +
w u
   (AU, U) = Δu , w
1 ∂zv−( ) zvτ ∂ρv H
Z ZN l 1j 2X X∂w ∂u ∂ u 1 ∂zj j j v(v)
= ( + w )dx+ ξ ( − (ρ)z (ρ)dρ).j v2∂x ∂x ∂x τ ∂ρv0 0j=1 v∈Vc
b
exist

ts
e

w
,
this
then
pro
These
in
Since
ositiv
it

is
ose
w
.
that
to
wn
inner
if
or
kno
ose
and
if
only
ourselv
if
(4).
on
no

(3).
pro
e
inner
of
wing
usual
in
alen
ell

k
-semi-
dense
.
is
purp
that

(equipp


real

e
op
Mor


erator
that
solution
w
Pro
(4)
of.
that
By
no
Lumer-Phillips'
to
theorem,
the
it
tro
suces
w
to

sho
inner
w
the
that
t
unique
equiv
is
is
dissipativ
pro
e
This
and
group
maximal
sho
monotone.
F
W
that
e
ose,
rst
e
pro
ho
v
By
e
p
that
e
a
n
is
over,
dissipativ
um
e.
ers
T
will
ak
e
e
to
exists
h
e
(5)
ther
W
,

the
Under
generates
e
a
w
wing
follo
o
supp
w
Let
es
to
oblem
that
that
us
datum
.
initial
and
an

pr
w
),
in
w
obtain
inner
w
the
Then
or
But
the

pro
F
e
2.2
easily
Theorem
.
ed
7
withZN l 2X j ∂ uj
(AU, U) = (−wj 2∂x0j=1
ZN 12 (v)X X∂ u ∂u ξ ∂zj j l vj+ w )dx+ [w ] − ( (ρ)z (ρ)dρ)j j v02∂x ∂x τ ∂ρv 0j=1 v∈Vc
ZN 1(v)X X∂u ξ ∂zj l vj= [w ] − ( (ρ)z (ρ)dρ).j v0∂x τ ∂ρv 0j=1 v∈Vc
Z 1
∂z 1v 2 2(ρ)z (ρ)dρ = (z (1)−z (0)).v v v∂ρ 20
⊤(u,w, z) ∈
D(A)
NX XX
∂u l ∂uj j j
[w ] = w (v) (v)j j0∂x ∂nj
j=1 v∈Vj∈EvX X X X
∂u ∂u ∂uj j jv v= w (v) (v)+ w (v) (v)+ w (v) (v)j j jv v∂n ∂n ∂nj j jv v
v∈V j∈E v∈D v∈Nc vX X
∂uj+ w (v) (v)j ∂nj
cv∈V \V j∈Eint vintX X X X
∂u ∂uj j= ( (v))w (v)+ ( (v))w (v)j j∂nj ∂nj
cv∈V j∈E v∈V \V j∈Ec v int vintX
(v) (v)
= −(α w(v)+α z (1))z (0)v v1 2
v∈VcX
(v) (v)2= − (α z (0) +α z (1)z (0)).v v v1 2
v∈Vc
X X (v)(v) (v) ξ 2 22(AU, U) = − (α z (0) +α z (1)z (0))− (z (1)−z (0))v v v1 2 v v2τv
v∈V v∈Vc cX (v) (v)(v) (v)ξ ξ2 2= − [(α − )z (0) + z (1)+α z (1)z (0)].v v v1 v 22τ 2τv v
v∈Vc
(v) (v)
α α(v) 2 2 2 2−α z (1)z (0)≤ z (1)+ z (0)v v2 v v
2 2
(v) (v)(v) (v)X ξ α ξ α(v) 2 2 2 2(AU, U)≤− [(α − − )z (0) +( − )z (1)]v1 v2τ 2 2τ 2v vv∈Vc
b
w
Again
8
b
y-Sc
e
therefore
By
to
Moreo
By
parts
e
in
v
e
ha
parts,
e
v
w
e
arz's
v
,
and
y
leads
b
y
satised
tegration

an
inequalit
obtain
y
ha
h
w
w
prop
y
yield
tegrating
the
in
y
These
b
er
erties
oundary/transmission
b
Cauc
h(v) (v)(v) (v)(v) ξ α ξ α (v) (v)2 2α − − ≥ 0 − ≥ 0 α α1 2τ 2 2τ 2 1 2v v
(AU, U)≤ 0 A
A λI−A
λ> 0
⊤ ⊤(f, g,h) ∈H U = (u,w, z) ∈D(A)
   
u f
   w g(λI−A) =
z h

 λu −w =f ∀j∈{1,...,N},j j j
2∂ ujλw − =g ∀j∈{1,...,N},j 2 j∂x 1 ∂z vλz + =h ∀v∈V .v v cτ ∂ρv
u
j∈{1,...,N},
1w :=λu −f ∈H (0,l )j j j j
w (v) =λu (v)−f (v) = 0 v∈Dj j jv v v
z w(v) =z (0). v∈V , zv c v
1 ∂zv
λz + =hv v
τ ∂ρv
z (0) =w(v) =λu(v)−f(v).v
zv
Z ρ
−λτ ρ −λτ ρ −λτ ρ λτ σv v v vz (ρ) =λu(v)e −f(v)e +τ e e h (σ)dσ.v v v
0
u
z w
R 1−λτ −λτ −λτ λτ σv v v vz (1) = λu(v)e −f(v)e +τ e e h (σ)dσv v v0
−λτ 0v= λu(v)e +z (v)v
R 10 −λτ −λτ λτ σv v vz (v) = −f(v)e +τ e e h (σ)dσv vv 0
f h
u uj
2∂ uj2λ u − =g +λf .j j j2∂x
determine
e
Indeed,
w
maximal
and
n
nd
and

e
e
and
w
since
erties,
m
prop
that
appropriate
tial
the
no
with
eness
found
is
is
dep

It
that
and
means
W
This
e
(7)
i.
tly
that
Supp
the
ose
pro
that
with
w
.
e
the
ha
ws
v
xed
e
b
found
only
alen
This
with
to
y
By
b
then
en
satisfy
giv
that
explicitly
for
is
is
the
e.,
appropriate
monotone,
Therefore
is
regularit
equation
y
dieren
.
satises
Then
v
for
w
all
us
equiv
for
or
Let
(6)
of
of
dissipativ
solution
then
for
that
ok
sho
lo
a
e
real
W
um
.
er
Let
ending
w
on
e
and
ha
.
v
remains

nd
oundary
.
b
(7)
the
(8),
and

where
ust
e
e
(8)
in
.
with
some
b
v
ecause
ha
satisfy
particular
and

surjectiv
for
(5).
e
.
Note
9
.φj
Z ZN N Nl lj 2 jX X X∂ u ∂u ∂φ ∂uj j j j l2 2 j(λ u − )φ dx = (λ u φ + )dx− [ φ ]j j j j j 02∂x ∂x ∂x ∂x0 0j=1 j=1 j=1
ZN ljX XX∂u ∂φ ∂uj j j2= (λ u φ + )dx− (v)φ (v).j j j
∂x ∂x ∂nj0j=1 v∈Vj∈Ev
⊤(u, w, z) D(A)
XX X X X
∂u ∂u ∂uj j jv(v)φ (v) = (v)φ (v)+ (v)φ (v)j j jv∂n ∂n ∂nj j jv
v∈Vj∈E v∈V j∈E v∈Dv c vX X X
∂u ∂uj jv+ (v)φ (v)+ (v)φ (v)j jv∂n ∂nj jv
cv∈N v∈V \V j∈Eint vintX X
∂uj= ( (v))φ(v)
∂nj
v∈V j∈Ec vX
(v) (v)
= − (α w (v)+α z (1))φ(v).j v1 2
v∈Vc
z (1)v
NX XRlj ∂u ∂φ (v) (v)2 j j −λτv(λ u φ + )dx + (α +α e )λu(v)φ(v)j j 1 20 ∂x ∂x
j=1 v∈Vc
NXRlj
= (g +λf )φ dxj j j0
j=1X
(v) (v) 0+ (α f(v)−α z (v))φ(v), ∀φ∈V.1 2 v
v∈Vc
u ∈ V
NY
V φ∈ D(0, l )⊂ Vj
j=1
u
2∂ uj2 ′λ u − =g +λf D (0,l ) ∀j = 1,···,N.j j j j2∂x
N NY Y
2 2u ∈ H (0, l ) u ∈ V ∩ H (0, l )j j
j=1 j=1
X X
∂u (v) (v) (v) (v)j −λτ 0v[ (v)+(α +α e )λu(v)+(α z (v)−α f(v))]φ(v)v∂n 1 2 2 1j
v∈V j∈Ec v X X X
∂u∂uj j
=− (v)φ (v)− ( (v))φ (v),∀φ∈V.j j∂n ∂nj j
cv∈N v∈V \V j∈Evint int
in
e
side
in
in
w
lemma,
for
and
expression

e
b
v
parts,
o
unique
ab
at
the
e
Using
(9)
e
the
v
implies
ha
y
e
b
satises
y
w
has
then
.
,
Coming
,
(9)
to

elong
is
b
of
ust
left-hand
m
ecause
that


that
the
Lax-Milgram's
using
w
But
y
obtain
tegration
e
using
nd
solution
b
a
This
problem
in
then
b
e
b
the
arriv
tegrating
This
problem
to
and
tit
and
this
(9)
b
on
space
,
tegrating
test
e
w
in
parts,
function
y
a

y
e
y
w
iden
If
Multiplying
.

10
k