I. D. Chueshov
Title:
Introduction to the Theory
of InfiniteDimensional
Dissipative Systems
«A CTA » 200
Author:
ISBN:
966–7021
966
7021–64
64–5
I. D. Chueshov
I ntroduction to the Theory
of Infinite-Dimensional
D issipative
S ystems
Universitylecturesincontemporarymathematics
This book provides an exhaustive introduction to the scope
of main ideas and methods of the
theory of infinite-dimensional dissipative dynamical systems which
has been rapidly developing in recent years. In the examples
systems generated by nonlinear
partial differential equations
arising in the different problems
of modern mechanics of continua
are considered. The main goal
of the book is to help the reader
to master the basic strategies used
in the study of infinite-dimensional
dissipative systems and to qualify
him/her for an independent scientific research in the given branch.
Experts in nonlinear dynamics will
find many fundamental facts in the
convenient and practical form
in this book.
The core of the book is composed of the courses given by the
author at the Department
of Mechanics and Mathematics
at Kharkov University during
a number of years. This book contains a large number of exercises
which make the main text more
complete. It is sufficient to know
the fundamentals of functional
analysis and ordinary differential
equations to read the book.
Translated by
You can O R D E R this book
while visiting the website
of «ACTA» Scientific Publishing House
http://www.acta.com.ua
www.acta.com.ua/en/
Constantin I. Chueshov
from the Russian edition («ACTA», 1999)
Translation edited by
Maryna B. Khorolska
Chapter
2
Long-Time Behaviour of Solutions
to a Class of Semilinear Parabolic Equations
Contents
....§1
Positive Operators with Discrete Spectrum . . . . . . . . . . . . . . 77
....§2
Semilinear Parabolic Equations in Hilbert Space . . . . . . . . . . 85
....§3
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
....§4
Existence Conditions and Properties of Global Attractor . . 101
....§5
Systems with Lyapunov Function . . . . . . . . . . . . . . . . . . . . . 108
....§6
Explicitly Solvable Model of Nonlinear Diffusion . . . . . . . . . 118
....§7
Simplified Model of Appearance of Turbulence in Fluid . . . 130
....§8
On Retarded Semilinear Parabolic Equations. . . . . . . . . . . . 138
....
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
In this chapter we study well-posedness and the asymptotic behaviour of solutions to a class of abstract nonlinear parabolic equations. A typical representative of
this class is the nonlinear heat equation
¶-----u=n
¶t
d
¶2 u
- + f (x, u)
å --------¶x
i=1
2
i
considered in a bounded domain W of R d with appropriate boundary conditions on
the border ¶ W . However, the class also contains a number of nonlinear partial differential equations arising in Mechanics and Physics that are interesting from the
applied point of view. The main feature of this class of equations lies in the fact that
the corresponding dynamical systems possess a compact absorbing set.
The first three sections of this chapter are devoted to the questions of existence
and uniqueness of solutions and a brief description of examples. They are independent of the results of Chapter 1. In the other sections containing the discussion of
asymptotic properties of solutions we use general results on the existence and properties of global attractors proved in Chapter 1. In Sections 6 and 7 we present two
quite simple infinite-dimensional systems for which the asymptotic behaviour of the
trajectories can be explicitly described. In Section 8 we consider a class of systems
generated by infinite-dimensional retarded equations.
The list of references at the end of the chapter consists only of the books recommended for further reading.
§ 1 Positive Operators
with Discrete Spectrum
This section contains some auxiliary facts that play an important role in the subsequent considerations related to the study of the asymptotic properties of solutions
to abstract semilinear parabolic equations.
Assume that H is a separable Hilbert space with the inner product ( , ) and
the norm
. Let A be a selfadjoint positive linear operator with the domain D ( A) .
An operator A is said to have a discrete spectrum if in the space H there exists
an orthonormal basis { e k } of the eigenvectors:
. .
.
( ek , ej ) = dk j ,
A ek = lk ek ,
k, j = 1, 2, ¼ ,
(1.1)
such that
0 < l1 £ l2 £ ¼ ,
lim l k = ¥ .
k®¥
(1.2)
The following exercise contains a simple example of an operator with discrete spectrum.
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E x e r c i s e 1.1 Let H = L 2 ( 0 , 1 ) and let A be an operator defined by the
equation Au = - u ² with the domain D ( A) which consists of continuously differentiable functions u ( x) such that (a) u (0 ) = u (1) = 0 ,
(b) u ¢( x ) is absolutely continuous and (c) u ² Î L 2 ( 0 , 1 ) . Show
that A is a positive operator with discrete spectrum. Find its eigenvectors and eigenvalues.
The above-mentioned structure of the operator A enables us to define an operator
f ( A) for a wide class of functions f ( l ) defined on the positive semiaxis. It can be
done by supposing that
¥
¥
ì
ü
D ( f ( A) ) = í h =
ck ek Î H :
ck2 [ f ( lk ) ] 2 < ¥ ý ,
î
þ
k=1
k=1
å
å
¥
f (A) h =
åc
k
f ( lk ) ek ,
h Î D ( f ( A) ) .
(1.3)
k=1
In particular, one can define operators A a with a Î R . For a = - b < 0 these operators are bounded. However, in this case it is also convenient to introduce the lineals D ( A a ) if we regard D ( A -b ) as a completion of the space H with respect to the
norm A -b . .
E x e r c i s e 1.2 Show that the space F-b = D ( A -b ) with b > 0 can be identified with the space of formal series å c k e k such that
¥
åc
2 -2 b
k lk
< ¥.
k=1
E x e r c i s e 1.3 Show that for any b Î R the operator A b can be defined on
every space D ( A a ) as a bounded mapping from D ( A a ) into
D ( A a - b ) such that
Ab D ( Aa ) = D ( Aa - b ) ,
A
b1 + b2
=A
b1
×A
b2
.
(1.4)
E x e r c i s e 1.4 Show that for all a Î R the space F a º D ( A a ) is a separable Hilbert space with the inner product ( u , v ) a = ( Aa u , Aa v )
and the norm u a = A a u .
E x e r c i s e 1.5 The operator A with the domain F 1 + s is a positive operator
with discrete spectrum in each space F s .
Prove the continuity of the embedding of the space F a into
F b for a > b , i.e. verify that Fa Ì F b and u b £ C u a .
E x e r c i s e 1.6
E x e r c i s e 1.7
Prove that F a is dense in F b for any a > b .
Positive Operators with Discrete Spectrum
Let f Î F s for s > 0 . Show that the linear functional F ( g ) º
º ( f , g ) can be continuously extended from the space H to F-s
and ( f , g ) £ f s × g -s for any f Î F s and g Î F-s .
E x e r c i s e 1.8
E x e r c i s e 1.9 Show that any continuous linear functional F on F s has the
form: F ( f ) = ( f , g ) , where g Î F-s . Thus, F-s is the space
of continuous linear functionals on F s .
The collection of Hilbert spaces with the properties mentioned in Exercises 1.7–1.9
is frequently called a scale of Hilbert spaces. The following assertion on the compactness of embedding is valid for the scale of spaces { F s } .
Theorem 1.1.
Let s1 > s2 . Then
Then the space F s is compactly embedded into F s , i.e.
1
2
every sequence bounded in F s is compact in F s .
1
2
Proof.
It is well known that every bounded set in a separable Hilbert space is weakly
compact, i.e. it contains a weakly convergent sequence. Therefore, it is sufficient to
prove that any sequence weakly tending to zero in F s converges to zero with re1
spect to the norm of the space F s . We remind that a sequence { fn } in F s weakly
2
converges to an element f Î F s if for all g Î F s
lim ( fn , g ) s = ( f , g ) s .
n®¥
Let the sequence { fn } be weakly convergent to zero in F s and let
1
fn s £ C ,
n = 1, 2, ¼ .
(1.5)
1
Then for any N we have
N -1
2
fn s
2
£
å
2 s2
lk
k=1
1
( fn , e k ) 2 + -------------------------2 ( s1 - s 2 )
N
¥
2 s1
k ( fn ,
ål
ek ) 2 .
(1.6)
k=N
Here we applied the fact that for k ³ N
2 s2
2s
1
£ ------------------------l 1.
2 ( s1 - s2 ) k
lN
lk
Equations (1.5) and (1.6) imply that
N -1
2
fn s
2
£
2 s2
k ( fn ,
ål
- 2 ( s1 - s 2 )
ek ) 2 + C lN
.
k=1
We fix e > 0 and choose N such that
N -1
2
fn s
2
£
2 s2
k ( fn ,
ål
k=1
ek ) 2 + e .
(1.7)
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Long-Time Behaviour of Solutions to a Class of Semilinear Parabolic Equations
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Let us fix the number N . The weak convergence of fn to zero gives us
lim ( fn , ek ) = 0 ,
k = 1, 2, ¼, N - 1 .
n®¥
Therefore, it follows from (1.7) that
fn s £ e .
2
lim
2
n®¥
By virtue of the arbitrariness of e we have
lim
n®¥
fn s = 0 .
2
Thus, Theorem 1.1 is proved.
E x e r c i s e 1.10 Show that the resolvent Rl ( A) = ( A - l ) -1 , l ¹ lk , is
a compact operator in each space F s .
We point out several properties of the scale of spaces { Fs } that are important for
further considerations.
E x e r c i s e 1.11 Show that in each space F s the equation
l
Pl u =
å (u, e ) e
k
k,
u Î Fs ,
-¥ < s < ¥
k=1
defines an orthoprojector onto the finite-dimensional subspace generated by the set of elements { ek , k = 1 , 2 , ¼ , l } . Moreover,
for each s we have
lim Pl u - u s = 0 .
l®¥
E x e r c i s e 1.12 Using the Hölder inequality
1/p æ
1/q
æ
pö
qö
ak bk £ ç
ak÷
bk÷ ,
ç
è k
ø è k
ø
k
prove the interpolation inequality
å
å
å
Aq u £ Au q × u 1 - q ,
--1- + 1
--- = 1 ,
p q
0 £ q £ 1,
ak , bk > 0 ,
u Î D ( A) .
E x e r c i s e 1.13 Relying on the result of the previous exercise verify that
for any s1 , s2 Î R the following interpolation estimate holds:
1- q
q
u s ( q) £ u s
us
,
1
2
where s ( q ) = qs1 + ( 1 - q ) s2 , 0 £ q £ 1 , and u Î F max ( s , s ) .
1
2
Prove the inequality
q
- ----------2
2
1- q u 2 ,
us
( q ) £ e u s1 + C q e
s2
where 0 < q < 1 and e is a positive number.
Positive Operators with Discrete Spectrum
Equations (1.3) enable us to define an exponential operator exp ( - t A ) , t ³ 0 , in
the scale { Fs } . Some of its properties are given in exercises 1.14–1.17.
E x e r c i s e 1.14 For any a Î R and t > 0 the linear operator exp ( - t A )
F and possesses the property e -t A u a £
maps F a into
s ³ 0 s
-t l1
£ e
ua.
Ç
E x e r c i s e 1.15 The following semigroup property holds:
exp ( -t1 A ) × exp ( -t2 A ) = exp ( -( t1 + t2 ) A ) ,
t1 , t2 ³ 0 .
E x e r c i s e 1.16 For any u Î Fs and s Î R the following equation is valid:
lim e -t A u - e -t A u s = 0 .
t®t
(1.8)
E x e r c i s e 1.17 For any s Î R the exponential operator e -t A defines a dissipative compact dynamical system ( Fs , e -t A ) . What can you say
about its global attractor?
Let us introduce the following notations. Let C ( a , b ; F a ) be the space of strongly
continuous functions on the segment [ a , b ] with the values in Fa , i.e. they are continuous with respect to the norm . a = A a . . In particular, Exercise 1.16 means
that e -t A u Î C ( R + , Fa ) if u Î Fa . By C 1 ( a , b ; Fa ) we denote the subspace of
C ( a , b ; Fa ) that consists of the functions f ( t ) which possess strong (in Fa ) derivatives f ¢ ( t ) lying in C ( a , b ; F a ) . The space C k ( a , b ; Fa ) is defined similarly
for any natural k . We remind that the strong derivative (in Fa ) of a function f ( t ) at
a point t = t0 is defined as an element v Î F a such that
lim
h®0
--1- ( f ( t0 + h ) - f ( t0 ) ) - v = 0 .
h
a
E x e r c i s e 1.18 Let u0 Î Fs for some s . Show that
e -t A u0 Î C k ( d , + ¥ ; Fa )
for all d > 0 , a Î R , and k = 1 , 2 , ¼ . Moreover,
dk
-------k- e -t A u 0 = ( -A ) k e -t A u0 ,
dt
k = 1, 2, ¼ .
Let L 2 ( a , b ; F a ) be the space of functions on the segment [ a , b ] with the values
in Fa for which the integral
b
u
2
L 2 ( a , b ; Fa )
=
ò u (t)
2
a dt
a
exists. Let L ¥ ( a , b ; F a ) be the space of essentially bounded functions on [ a , b ]
with the values in Fa and the norm
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u
L¥ ( a , b ; D ( Aa ) )
= e s s sup Aa u ( t )
.
t Î [ a , b]
We consider the Cauchy problem
dy
------ + A y = f ( t ) ,
dt
t Î (a, b) ;
y ( a ) = y0 ,
(1.9)
where y Î F a and f ( t ) Î L 2 ( a , b ; Fa - 1 ¤ 2 ) . The weak solution (in F a ) to this
problem on the segment [ a , b ] is defined as a function
y ( t ) Î C ( a , b ; Fa ) Ç L 2 ( a , b ; Fa + 1 ¤ 2 )
(1.10)
such that d y ¤ d t Î L 2 ( a , b ; Fa - 1 ¤ 2 ) and equalities (1.9) hold. Here the derivative
y ¢ ( t ) º d y ¤ d t is considered in the generalized sense, i.e. it is defined by the equality
b
b
ò j ( t ) y ¢ ( t ) d t = -ò j ¢ ( t ) y ( t ) d t ,
a
j Î C0¥ ( a , b ) ,
a
C 0¥ ( a ,
b ) is the space of infinitely differentiable scalar functions on ( a , b )
where
vanishing near the points a and b .
E x e r c i s e 1.19 Show that every weak solution to problem (1.9) possesses the
property
t
t
y (t)
2
a
+
ò y (t)
a
2
dt
--a+1
2
£ y0
a
+
ò f (t)
a
2
dt .
--a -1
2
(1.11)
(Hint: first prove the analogue of formula (1.11) for yn ( t ) = Pn y ( t ) ,
then use Exercise 1.11).
E x e r c i s e 1.20 Prove the theorem on the existence and uniqueness of weak
solutions to problem (1.9). Show that a weak solution y ( t ) to this
problem can be represented in the form
t
y (t) =
e -( t - a ) A y
0
+
òe
-( t - t ) A
f (t) dt .
(1.12)
a
E x e r c i s e 1.21 Let y 0 Î F a and let a function f ( t ) possess the property
f ( t1 ) - f ( t2 ) a £ C t1 - t2 q
for some 0 < q £ 1 . Then formula (1.12) gives us a solution to problem (1.9) belonging to the class
C ( [ a , b ] ; Fa ) Ç C 1 ( ] a , b ] ; Fa ) Ç C ( ] a , b ] ; F a + 1) .
Such a solution is said to be strong in Fa .
Positive Operators with Discrete Spectrum
The following properties of the exponential operator e -t A play an important part
in the further considerations.
Lemma 1.1.
Let Q N be the orthoprojector onto the closure of the span of elements
{ ek , k ³ N + 1 } in H and let PN = I - Q N , N = 0 , 1 , 2 , ¼ . Then
1) for all h Î H , b ³ 0 and t Î R the following inequality holds:
b l t
A b PN e -t A h £ l N e N h ;
(1.13)
2) for all h Î D ( A b ) , t > 0 and a ³ b the following estimate is valid:
A a QN e -t A h
a - böa - b
-t l
a -b
æ ------------+ lN + 1 e N + 1 Ab h ,
è t ø
£
(1.14)
in the case a - b = 0 we supose that 0 0 = 0 in (1.14).
Proof.
Estimate (1.13) follows from the equation
N
ål
A b PN e -t A h 2 =
2 b -2 t lk
(h,
k e
ek ) 2 .
k=1
In the proof of (1.14) we similarly have that
Aa Q
N
e -t A h 2
£
max
l ³ l N +1
¥
2
( l a - b e -t l )
å
l2k b ( h , ek ) 2 .
k=N+1
This gives us the inequality
A a QN e -t A h
1 £ -----------t a -b
max
( m a - b e -m ) A b h .
m ³ t l N +1
Since max { m g e -m ; m ³ 0 } is attained when m = g , we have that
m
t
ì
g -l
( lN + 1 t ) e N + 1 ,
ï
max ( m g e -m ) = í
³ l N +1 t
ï g g e -g ,
î
if l N + 1 t ³ g ;
if l N + 1 t < g .
Therefore,
max
m ³ l N +1 t
( m g e -m ) £ ( g g + ( lN + 1 t ) g ) e
-lN + 1 t
.
This implies estimate (1.14). Lemma 1.1 is proved.
In particular, we note that it follows from (1.14) that
Aa e -t A h £
-b
æa
------------- ö
è t ø
a -b
a -b
+ l1
×e
-t l1
Ab h ,
a ³ b.
(1.15)
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E x e r c i s e 1.22 Using estimate (1.15) and the equation
t
e -t A u
- e -s A u
ò
= - A e -t A u d t ,
t ³ s,
u Î Fb ,
s
prove that
e -t A u - e -s A u q £ Cq , s t - s s - q u s ; t , s > 0 , (1.16)
provided q < s £ 1 + q , where a constant Cq , s does not depend
on t and s (cf. Exercise 1.16).
E x e r c i s e 1.23 Show that
a a
A a e -t A £ æ ----ö e -a ,
è tø
t > 0,
a > 0.
(1.17)
Lemma 1.2.
Let f ( t ) Î L ¥ ( R , Fa - g ) for 0 £ g < 1 . Then there exists a unique solution v ( t ) Î C ( R , F a ) to the nonhomogeneous equation
d----v+ A v = f (t) ,
dt
t ÎR ,
(1.18)
that is bounded in F a on the whole axis. This solution can be
represented in the form
t
v (t) =
òe
-( t - t ) A
f (t) d t .
(1.19)
-¥
We understand the solution to equation (1.18) on the whole axis as a function v ( t ) Î
Î C ( R , Fa ) such that for any a < b the function v ( t ) is a weak solution
(in Fa - 1 ¤ 2 ) to problem (1.9) on the segment [ a , b ] with y0 = v ( a ) .
Proof.
If there exist two bounded solutions to problem (1.18), then their difference w ( t ) is a solution to the homogeneous equation. Therefore, w ( t ) =
= exp { -( t - t0 ) A} w ( t 0 ) for t ³ t0 and for any t0 . Hence,
Aa w ( t )
£ e
-( t - t0 ) l1
Aa w ( t 0 )
£ Ce
-( t - t0 ) l1
.
If we tend t0 ® - ¥ here, then we obtain that w ( t ) = 0 . Thus, the bounded solution to problem (1.18) is unique. Let us prove that the function v ( t ) defined
by formula (1.19) is the required solution. Equation (1.15) implies that
Aa e -( t - t ) A £
g ög
g
-( t - t ) l1
æ ---------+ l1 e
e s s sup A a - g f ( t )
è t - tø
t ÎR
for t > t and 0 < g < 1 . Therefore, integral (1.19) exists and it can be uniformly estimated with respect to t as follows:
Semilinear Parabolic Equations in Hilbert Space
Aa v ( t )
1+ k
£ ----------- e s s sup Aa - g f ( t ) ,
l11- g t Î R
where k = 0 for g = 0 and
¥
k=
gg
òs
-g e -s d s
for
0 < g < 1.
0
The continuity of the function v ( t ) in Fa follows from the following equation
that can be easily verified:
t
v (t) =
- ( t - t0 ) A
e
v (t
0)
+
òe
-( t - t ) A
g (t) d t .
t0
This also implies (see Exercise 1.18) that v ( t ) is a solution to equation (1.18).
Lemma 1.2 is proved.
§2
Semilinear Parabolic Equations
in Hilbert Space
In this section we prove theorems on the existence and uniqueness of solutions to
an evolutionary differential equation in a separable Hilbert space H of the form
d-----u
+ A u = B (u, t) ,
u t = s = u0 ,
(2.1)
dt
where A is a positive operator with discrete spectrum and B ( . , . ) is a nonlinear
continuous mapping from D ( Aq) ´ R into H , 0 £ q < 1 , possessing the property
B ( u1 , t ) - B ( u 2 , t ) £ M ( r ) Aq ( u1 - u2 )
(2.2)
for all u 1 and u2 from the domain F q = D ( A q ) of the operator A q and such that
Aq uj £ r . Here M ( r ) is a nondecreasing function of the parameter r that does
not depend on t and . is a norm in the space H .
A function u ( t ) is said to be a mild solution (in F q ) to problem (2.1) on the
half-interval [ s , s + T ) if it lies in C ( s , s + T ¢ ; F q ) for every T ¢ < T and for all
t Î [ s , s + T ) satisfies the integral equation
t
u (t) = e
-( t - s ) A
u0 +
òe
-( t - t ) A
B (u (t) , t) d t .
(2.3)
s
The fixed point method enables us to prove the following assertion on the local
existence of mild solutions.
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Theorem 2.1.
Let u0 Î F q . Then there exists T * depending on q and u0 q such that
problem (2.1) possesses a unique mild solution on the half-interval
[ s , s + T * ) . Moreover, either T * = ¥ or the solution cannot be continued
in F q up to the moment t = s + T * .
Proof.
On the space C s , q º C ( s , s + T ; F q ) we define the mapping
t
G [ u] (t) = e
-( t - s ) A
u0 +
òe
-( t - t) A
B (u(t) , t) d t .
s
Let us prove that G [ u ] ( t ) Î C ( s , s + T ; F q ) for any T > 0 . Assume that t1 , t2 Î
Î [ s , s + T ] and t1 < t2 . It is evident that
t2
G [ u ] ( t2 ) = e
- ( t2 - t1 ) A
G [ u ] ( t1 ) +
òe
- ( t2 - t ) A
B (u (t) , t) dt .
(2.4)
t1
By virtue of (1.8) we have that if t2 ® t1 , then
G [ u ] ( t1 ) - e
- ( t2 - t1 ) A
G [ u] (t
1) q
® 0.
Therefore, it is sufficient to estimate the second term in (2.4). Equation (1.15) implies that
t2
òe
- ( t2 - t ) A
B ( u (t ) , t) d t
q
t1
t2
£
ò
£
q
q -öq
æ -----------+ l1 d t ×
è t2 - t ø
t
max
Î [ s , s + T]
B (u (t) , t)
£
t1
ì qq
ü
£ t2 - t1 1 - q í ------------ + lq1 t2 - t1 q ý
î1 - q
þt
max
Î [ s , s + T]
B (u (t) , t)
(2.5)
(if q = 0 , then the coefficient in the braces should be taken to be equal to 1). Thus,
G maps C s , q = C ( s , s + T ; F q ) into itself. Let v0 ( t ) = e -( t - s ) A u 0 . In Cs , q
we consider a ball of the form
ì
U = í u ( t ) Î Cs , q :
î
u - v0 C
s, q
º
ü
u ( t ) - v0 ( t ) q £ 1 ý .
[s, s + T ]
þ
max
Let us show that for T small enough the operator G maps U into itself and is con£ 1 + u0 q for u Î U , equation (2.2) gives
tractive. Since u C
s, q
Semilinear Parabolic Equations in Hilbert Space
max
t Î [s, s + T ]
B (u (t) , t)
£
max
t Î [s, s + T ]
B(0 , t) +
+ ( 1 + u 0 q ) M ( 1 + u0 q ) º C ( T0 , u0 q )
for all T £ T0 , where T0 is a fixed number. Therefore, with the help of (2.5) we find
that
G [ u ] - v0 C
s,
q
£ T 1 - q × C 1 ( T0 , q , u0 q ) .
Similarly we have
G [ u] - G [ v] C
s,
q
£ T 1 - q × C2 ( T0 , q ) M ( 1 + u q ) u - v C
s, q
for u , v Î U . Consequently, if we choose T1 such that
T 11 - q C1 ( T0 , q , u0 q ) £ 1
and
T11 - q C2 ( T0 , q ) M ( 1 + u0 q ) < 1 ,
we obtain that G is a contractive mapping of U into itself. Therefore, G possesses
a unique fixed point in U Ì Cs , q . Thus, we have constructed a solution on the segment [ s , s + T1 ] . Taking s + T1 as an initial moment, we can construct a solution
on the segment [ s + T1 , s + T1 + T2 ] with the initial condition u0 = u ( s + T1 ) .
If we continue our reasoning, then we can construct a solution on some maximal halfinterval [ s , s + T * ) . Moreover, it is possible that T * = ¥ . Theorem 2.1 is proved.
E x e r c i s e 2.1 Let u0 Î F q and let T * = T * ( q , u0 ) be such that [ s , s + T * ) is
the maximal half-interval of the existence of the mild solution u ( t )
u(t) q =
to problem (2.1). Then we have either T *< ¥ and lim
t ® s +T*
= ¥ , or T * = ¥ .
E x e r c i s e 2.2 Using equations (1.16) and (2.5), prove that for any mild solution u ( t ) to problem (2.1) on [ s , s + T * ) the estimate
u (t) - u (t) a £ C (q, a, T) t - t q - a ,
t , t Î [ s , s + T ] , (2.6)
is valid, provided u 0 Î F q , 0 £ a £ q , and T £ T * .
E x e r c i s e 2.3 Let u0 Î F q and let u ( t ) be a mild solution to problem (2.1)
on the half-interval [ s , s + T * ) . Then
u ( t ) Î C ( s , s + T , F q ) Ç C ( s + d , s + T , F1 - s ) Ç
Ç C 1 ( s + d , s + T , F-s )
for any s > 0 , 0 < d < T , and T < T * . Moreover, equations (2.1)
are valid if they are understood as the equalities in F-s and F q , respectively.
It is frequently convenient to use the Galerkin method in the study of properties of
mild solutions to the problem of the type (2.1). Let Pm be the orthoprojector in H
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onto the span of elements { e 1 , e2 , ¼ , em } . Galerkin approximate solution
of the order m with respect to the basis { e k } is defined as a continuously differentiable function
m
um ( t ) =
2
åg
k ( t ) ek
(2.7)
k=1
with the values in the finite-dimensional space Pm H that satisfies the equations
d
---- u ( t ) + A um ( t ) = Pm B ( um ( t ) , t ) ,
dt m
t > s,
um
t=s
= Pm u 0 .
(2.8)
It is clear that (2.8) can be rewritten as a system of ordinary differential equations for the functions gk ( t ) .
E x e r c i s e 2.4 Show that problem (2.8) is equivalent to the problem of finding a continuous function um( t ) with the values in PmH that satisfies the integral equation
t
um ( t ) = e
-( t - s) A
Pm u0 +
òe
-( t - t ) A
Pm B ( um ( t ) , t ) d t . (2.9)
s
E x e r c i s e 2.5 Using the method of the proof of Theorem 2.1, prove the local
solvability of problem (2.9) on a segment [ s , s + T ] , where the parameter T > 0 can be chosen to be independent of m . Moreover,
the following uniform estimate is valid:
max
[s, s + T]
um ( t ) q < R ,
m = 1, 2, 3, ¼ ,
(2.10)
where R > 0 is a constant.
The following assertion on the convergence of approximate functions to exact ones
holds.
Theorem 2.2.
Let u0 Î F q . Assume that there exists a sequence of approximate solutions um ( t ) on a segment [ s , s + T ] for which estimate (2.10) is valid. Then
there exists a mild solution u ( t ) to problem (2.1) on the segment [ s , s + T ]
and
max
[ s, s + T]
1
u ( t ) - u m ( t ) q £ C æ ( 1 - Pm ) u 0 q + ----------------ö ,
1- q ø
è
l
m+1
where C = C ( q , R , T ) is a positive constant independent of s .
(2.11)
Semilinear Parabolic Equations in Hilbert Space
Proof.
Let n > m . We use (2.9), (1.14) and (1.17) to find that for q > 0 we have
u n ( t ) - um ( t ) q £ ( Pn - Pm ) u0 q +
t
+
q -ö
t - tø
ò æè ----------
q
+ lqm + 1 e
-lm + 1 ( t - t )
B ( un( t ) , t ) d t +
s
t
+
e -q
ò
q - ö q B (u (t) , t) - B (u (t) , t) dt .
æ ---------n
m
è t - tø
s
Therefore, equations (2.2) and (2.10) give us that
un ( t ) - u m ( t ) q £ ( Pn - Pm ) u0 q +
+ æ max B ( 0 , t ) + M ( R ) Rö Jm ( t , s ) +
è [ s , s + T]
ø
t
+ M ( R ) q q e -q
ò (t - t)
-q
un ( t ) - um ( t ) q d t ,
(2.12)
s
where
t
Jm ( t , s ) =
ò
-lm + 1 ( t - t )
q - ö q + lq
æ ---------dt .
m+1 e
è t - tø
s
It is evident that Jm ( t , s ) £ Jm ( t , - ¥ ) . By changing the variable in the integral
x = l m + 1 ( t - t ) , we obtain
Jm ( t , -¥ ) =
l-m1++1q
æ
ç1 + q q
ç
è
¥
ò
0
ö
- 1+ q
x -q e -x d x÷ º l m + 1 ( 1 + k ) .
÷
ø
Thus, equation (2.12) implies
a1 ( q , R )
- +
un ( t ) - um ( t ) q £ ( Pn - Pm ) u0 q + ---------------------l1m- +q 1
t
+ a2 ( q , R )
ò (t - t)
-q
un ( t ) - um ( t ) q d t .
s
Hence, if we use Lemma 2.1 which is given below, we can find that
1
u n ( t ) - u m ( t ) q £ C æ ( Pn - Pm ) u0 q + ----------------ö
1- q ø
è
l
m+1
(2.13)
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for all t Î [ s , s + T ] , where C > 0 is a constant depending on q , R , and T . It is also
evident that estimate (2.13) remains true for q = 0 . It means that the sequence of
approximate solutions { um ( t ) } is a Cauchy sequence in the space C ( s , s + T ; F q ) .
Therefore, there exists an element u ( t ) Î C ( s , s + T ; F q ) such that equation
(2.11) holds. Estimates (2.10) and (2.11) enable us to pass to the limit in (2.9) and
to obtain equation (2.3) for u ( t ) . Theorem 2.2 is proved.
2
E x e r c i s e 2.6 Show that if the hypotheses of Theorem 2.2 hold, then the estimate
a2
a1
- ( 1 - Pm ) u0 q + --------------u ( t ) - u m ( t ) a £ --------------1
q
a
[ s, s + T]
lm- +a 1
lm + 1
max
is valid with 0 £ a £ q . Here a1 and a2 are constants independent
of q , R , and T .
The following assertion provides a simple sufficient condition of the global solvability
of problem (2.1).
Theorem 2.3.
Assume that the constant M ( r ) in (2.2) does not depend on r , i.e. the
mapping B ( u , t ) satisfies the global Lipschitz condition
B ( u1 , t ) - B ( u2 , t )
£ M u1 - u2
(2.14)
q
for all uj Î F q with some constant M > 0 . Then problem (2.1) has a unique
mild solution on the half-interval [ s , + ¥ ) , provided u0 Î F q . Moreover,
for any two solutions u 1 and u2 the estimate
u1( t ) - u2 ( t ) q £ a1 e
a2 ( t - s )
u1 ( s ) - u2 ( s ) q ,
t ³ s ,
(2.15)
holds, where a1 and a2 are constants that depend on q , l 1 , and M only.
only.
The proof of this theorem is based on the following lemma (see the book by Henry [3],
Chapter 7).
Lemma 2.1.
Assume that j ( t ) is a continuous nonnegative function on the interval
( 0 , T ) such that
t
j ( t ) £ c0 t
- g0
ò
+ c1 ( t - t )
-g1
j (t) dt ,
t Î (0, T) ,
(2.16)
0
where c 0 , c1 ³ 0 and 0 £ g 0 , g 1 < 1 . Then there exists a constant K =
= K ( g1 , c 1 , T ) such that
c0 -g
- t 0 K ( g1 , c1 , T ) .
j ( t ) £ ------------1 - g0
(2.17)
Semilinear Parabolic Equations in Hilbert Space
Proof of Theorem 2.3.
Let u ( t ) be a solution to problem (2.1) on the maximal half-interval of its existence [ s , s + T ) . Assume that T < ¥ . Condition (2.14) gives us that
B (u, t)
£ B (0, t) + M u
q
£ M0 ( T ) + M u
q
for all u Î F q and t Î [ 0 , T ] . Therefore, from (2.3) and (1.15) we find that
t
u( t) q £
u0 q +
ò
q ö q + lq ( M ( T ) + M u ( t ) ) d t .
æ ---------1
0
q
è t - tø
s
Hence, for t Î [ s , s + T ] we have that
t
u ( t ) q £ C 0 ( u 0 q , T, q ) + C 1 ( T , q )
ò (t - t)
-q
u( t) q d t .
s
Therefore, Lemma 2.1 implies that the value u ( t ) q is bounded on [ s , s + T ) which
is impossible (see Exercise 2.1). Thus, the solution exists for any half-interval
[ s , s + T ) . For the proof of estimate (2.15) we note that, as above, inequalities (2.3)
and (1.15) for the function w ( t ) = u 1( t ) - u 2 ( t ) give us that
t
w( t) q £ w (s) q +
ò
q - ö q + lq M w ( t )
æ ---------1
è t - tø
q dt .
s
If we apply Lemma 2.1, we find that
w ( t ) q £ C ( q , l1 , M ) w ( s )
for
s £ t £ s +1.
Therefore, the estimate
w ( t ) q £ C n + 1 w ( s ) q £ C exp { ( t - s ) ln C } w ( s ) q
holds for t = s + n + s , where n is natural and 0 £ s £ 1 . Thus, Theorem 2.3
is proved.
E x e r c i s e 2.7 Using estimate (1.14), prove that if the hypotheses of Theorem 2.3 hold, then the inequality
QN ( u 1 ( t ) - u 2 ( t ) ) q £
a3
ì -l
(t - s)
a (t - s) ü
-e 2
+ -------------£ íe N + 1
ý u1 ( s ) - u2 ( s )
1- q
lN + 1
î
þ
q
(2.18)
is valid for any two solutions u1( t ) and u 2 ( t ) . Here Q N = I - PN
and PN is the orthoprojector onto the span of { e 1 , ¼ eN } , the number a2 is the same as in (2.15) and a3 depends on l 1 , q , and M .
Let us consider one more case in which we can guarantee the global solvability
of problem (2.1). Assume that condition (2.2) holds for q = 1 ¤ 2 and
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Long-Time Behaviour of Solutions to a Class of Semilinear Parabolic Equations
B ( u ) = - B0 ( u ) + B1( u , t ) ,
(2.19)
where B 1 ( u , t ) satisfies the global Lipschitz condition (2.14) with q = 1 ¤ 2 and
B 0 ( u ) is a potential operator on the space V = F1 ¤ 2 . This means that there exists
a Frechét differentiable functional F ( u ) on V such that B0 ( u ) = F ¢ ( u ) , i.e.
lim
v 1¤ 2 ® 0
1 - F(u + v) - F(u) - (B (u) , v) = 0 .
-------------0
v 1¤ 2
Theorem 2.4.
Let (2.2) be valid with q = 1 ¤ 2 and let decomposition (2.19) take place.
Assume that the functional F ( u ) is bounded below on V = F 1 ¤ 2 . Then problem (2.1) has a unique mild solution u ( t ) Î C ( [ s , s + T ] ; D ( A1 / 2 ) ) on an
arbitrary segment [ s , s + T ] .
Proof.
Let um( t ) be an approximate solution to problem (2.1) on a segment [ s , s + T ] ,
where T does not depend on m (see Exercise 2.5):
d
---- u ( t ) + Aum ( t ) = Pm B ( um ( t ) , t ) ,
(2.20)
u m ( s ) = Pm u0 .
dt m
d
Multiplying (2.20) by u· m ( t ) = ---- u m ( t ) scalarwise in the space H , we find that
dt
ü
dì
--- A1 / 2 um 2 + F ( um ) ý = ( B1 ( um , t ) , u· m ) £
u· m 2 + ---- í 1
dt î2
þ
--- u· m
£ 1
2
2
+ B1 ( 0 , t )
2
+ M 2 A1 / 2 um 2 .
Since F ( u ) is bounded below, we obtain that
d
---- W ( u m ( t ) ) £ a W ( um ( t ) ) + b + B 1 ( 0 , t ) 2
dt
with constants a and b independent of m , where
--- A1 / 2 u 2 + F ( u ) .
W ( u) = 1
2
Therefore, Gronwall’s lemma gives us that
t
ò
W ( um ( t ) ) £ æ W ( um ( s ) ) + --b-ö e a ( t - s ) + 2 e a ( t - t ) B 1 ( 0 , t ) 2 d t
aø
è
s
for all t in the segment [ s , s + T ] of the existence of approximate solutions. Firstly,
this estimate enables us to prove the global existence of approximate solutions
(cf. Exercise 2.1). Secondly, by virtue of the continuity of the functional W on
V = F 1 ¤ 2 Theorem 2.2 enables us to pass to the limit m ® ¥ on an arbitrary segment [ s , s + T ] and prove the global solvability of limit problem (2.1). Theorem 2.4
is proved.
Examples
Let u ( t ) be a mild solution to problem (2.1) such that
u ( t ) q £ C T for s £ t £ s + T . Use Lemma 2.1 to prove that
E x e r c i s e 2.8
u ( t ) a £ CT t q - a u0 q ,
t Î [ s, s + T]
(2.21)
for all q £ a < 1 , where CT = CT ( a , q ) is a positive constant.
E x e r c i s e 2.9 Let u ( t ) and v ( t ) be solutions to problem (2.1) with the initial conditions u0 , v0 Î F q and such that u ( t ) q + v ( t ) q £ CT
for t lying in a segment [ s , s + T ] . Then
u ( t ) - v ( t ) a £ CT ( q , a ) t q - a u0 - v0 q ,
t Î [ s, s + T] ,
q £ a < 1.
Thus, if B ( u , t ) º B ( u ) and the hypotheses of Theorem 2.3 or 2.4 hold, then equation (2.1) generates a dynamical system ( F q , St ) with the evolutionary operator St
which is defined by the equality St u0 = u ( t ) , where u ( t ) is the solution to problem
(2.1). The semigroup property of St follows from the assertion on the uniqueness
of solution.
E x e r c i s e 2.10 Show that Theorem 2.4 holds even if we replace the assumption of semiboundedness of F ( u ) by the condition F ( u ) ³
³ - a A1 / 2 u 2 - b for some a < 1 ¤ 2 and b > 0 .
§3
Examples
Here we consider several examples of an application of theorems of Section 2. Our
presentation is brief here and is organized in several cycles of exercises. More detailed considerations as well as other examples can be found in the books by Henry,
Babin and Vishik, and Temam from the list of references to Chapter 2 (see also Sections 6 and 7 of this chapter).
We first remind some definitions and notations. Let W be a domain in R d
( d ³ 1 ) . The Sobolev space H m ( W ) of the order m ( m = 0 , 1 , 2 , ¼ ) is defined
by the formula
H m ( W ) = { f Î L2 ( W ) : D j f Î L2 ( W ) ,
j £ m} ,
where j = ( j1 , j2 , ¼ , jd ) , jk = 0 , 1 , 2 , ¼ , j = j1 + ¼ + jd ,
j
j
j
D j f = ¶x1 ¶x2 ¼ ¶x d f ( x ) ,
1
2
d
x = ( x1 , ¼ , xd ) .
The space H m ( W ) is a separable Hilbert space with the inner product
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Long-Time Behaviour of Solutions to a Class of Semilinear Parabolic Equations
å ò D u(x) D v (x ) d x .
(u, v)m =
j
j
j £ m W
m
Below we also use the space H0 ( W ) which is constructed as the closure in H m ( W )
of the set C 0¥( W ) of infinitely differentiable functions with compact support. For
more detailed information we refer the reader to the handbooks on the theory of Sobolev spaces.
E x a m p l e 3.1 (nonlinear heat equation)
ì ¶ u = n D u + f (t, x, u, Ñu) , x Î W , t > 0 ,
ï t
(3.1)
í
ï u ¶ W = 0 , u t = 0 = u0 .
î
Here W is a bounded domain in R d , D is the Laplace operator, and n is a positive constant. Assume that f ( t , x , u , p ) is a continuous function of its variables which satisfies the Lipschitz condition
d
æ
ö
f ( t , x , u , p ) - f ( t , x , u , q ) £ K ç u1 - u2 2 +
pj - qj 2÷
è
ø
j=1
å
1/2
(3.2)
with an absolute constant K . It is clear that the operator B ( u , t ) defined by
the formula
B ( u , t ) ( x ) = f ( t , x , u ( x ) , Ñu ( x ) ) ,
can be estimated as follows:
B ( u1 , t ) - B ( u2 , t ) £ K æ u1 - u2 2 +
è
å¶
xj ( u 1 - u2 )
j
2ö 1 / 2
ø
.
(3.3)
Here and below . is the norm in the space H = L2 ( W ) . It is well-known that
the operator A = - D with the Dirichlet boundary condition on ¶ W is a positive
operator with discrete spectrum. Its domain is D ( A ) = H 2 ( W ) Ç H 01 ( W ) . Moreover, D ( A1 / 2 ) = H 01 ( W ) . We also note that
d
A1 / 2 u 2 = ( Au , u ) =
åò
j=1 W
¶u- ö 2
æ ------dx ,
è ¶x j ø
u Î D ( A1 / 2 ) = H01 ( W ) .
Therefore, equation (3.3) for B ( u , t ) gives us the estimate
1- + 1ö 1 / 2 A1 / 2 u ,
B ( u1 , t ) - B ( u 2 , t ) £ K æ ----èl
ø
1
where l 1 is the first eigenvalue of the operator - D with the Dirichlet boundary
condition on ¶ W . Therefore, we can apply Theorem 2.3 with q = 1 ¤ 2 to problem (3.1). This theorem guarantees the existence and uniqueness of a mild solution to problem (3.1) in the space C ( R + , H 01 ( W ) ) .
Examples
E x e r c i s e 3.1 Assume that f ( t , x , u , p ) º f ( t , x , u ) is a continuous function of its arguments satisfying the global Lipschitz condition with
respect to the variable u . Prove the global theorem on the existence
of mild solutions to problem (3.1) in the space H = L2 ( W ) .
E x a m p l e 3.2
Let us consider problem (3.1) in the case of one spatial variable:
ì ¶ u = n ¶ 2 u - g (x , u) + f (t, x, u, ¶ u) ,
x
x
ï t
í
u t = 0 = u0 .
ï u x = 0 = u x = 1 = 0,
î
t > 0, x Î (0, 1)
(3.4)
Assume that g ( x , u ) is a continuously differentiable function with respect to
the variable u and gu¢ ( x , u ) £ h ( u ) , where h ( u ) is a function bounded on
every compact set of the real axis. We also assume that the function f ( t , x , u , p )
is continuous and possesses property (3.2). For any element u Î D ( A1 / 2 ) =
= H 01 ( 0 , 1 ) the following estimates hold:
max u ( x ) £ u ¢
[0, 1]
L2 ( 0 , 1)
and
u
L2 ( 0 , 1 )
£ u¢
L2 ( 0 , 1)
,
where u' = ¶ x u . Therefore, it is easy to find that the inequality
B ( u1 , t ) - B ( u 2 , t ) £ M ( r ) A1 / 2 ( u1 - u2 )
(3.5)
is valid for
B ( u , t ) = -g ( x , u ( x ) ) + f ( t , x , u ( x ) , u ¢ ( x ) ) ,
provided A1 / 2 uj º uj¢ £ r . Here . is the norm in the space H =
= L2 ( 0 , 1 ) and M ( r ) = sup { h ( x ) : x < r } + K 2 . Equation (3.5) and Theorem 2.1 guarantee the local solvability of problem (3.4) in the space H 01 ( W ) .
Moreover, if the function
y
F (x, y) =
ò g (x, x) dx
0
is bounded below, then we can use Theorem 2.4 to obtain the assertion on the
existence of mild solutions to problem (3.4) on an arbitrary segment [ 0 , T ] .
It should be noted that the reasoning in Example 3.2 is also valid for several spatial
variables. However, in order to ensure the fulfilment of the estimate of the form (3.5)
one should impose additional conditions on the growth of the function h ( u ) .
For example, we can require that the equation
h ( u ) £ C1 + C2 u p
be fulfilled, where p £ 2 ¤ ( d - 2 ) if d > 2 and p is an arbitrary number if d = 2 .
In this case the inequality of the form (3.5) follows from the Hölder inequality and
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the continuity of the embedding of the space H 1 ( W ) into L q ( W ) , where q =
= 2 d ¤ ( d - 2 ) if d = dim W > 2 and q is an arbitrary number if d = 2 , q ³ 1 .
The results shown in Examples 3.1 and 3.2 also hold for the systems of parabolic equations. For example, a system of reaction-diffusion equations
2
ì ¶ t u = n D u + f ( t , u , Ñu ) ,
ï
(3.6)
í ¶u
= 0 , u t = 0 = u0 ( x ) ,
ï -----¶n ¶ W
î
can be considered in a smooth bounded domain W Ì R d . Here u = ( u1 , u2 , ¼ ,
u m ) and f ( t , u , x ) is a continuous function from R+ ´ R ( m + 1 ) d into R m such
that
f (t, u, x) - f (t, v, h) £ M ( u - v + x - h ) ,
where M is a constant, u , v
Î Rm ,
x, h
Î Rm d
(3.7)
and n is an outer normal to ¶ W .
E x e r c i s e 3.2 Prove the global theorem on the existence and uniqueness of
mild solutions to problem (3.6) in F 1 ¤ 2 = [ H 1( W ) ] m º H 1( W ) ´ ¼ ´
´ H 1( W ) .
E x a m p l e 3.3 (nonlocal Burgers equation).
ì ut - n ux x + ( w , u ) ux = f ( t ) , 0 < x < l , t > 0
íu
î x = 0 = u x = l = 0 , u t = 0 = u0 .
(3.8)
Here f ( t ) is a continuous function with the values in L 2 ( 0 , l ) ,
l
w Î L2 ( 0 , l ) ,
(w, u) =
ò w ( x) u ( x , t) d x
0
and n is a positive parameter. Exercises 3.3–3.6 below answer the question
on the solvability of problem (3.8).
E x e r c i s e 3.3 Prove the local existence of mild solutions to problem (3.8)
in the space F1 ¤ 2 = H 01 ( 0 , l ) . Hint:
A = -n ¶x2 x ,
D ( A) = H 2 ( 0 , l ) Ç H 01 ( 0 , l ) ,
B ( u ) = - ( w , u ) ux + f ( t ) .
E x e r c i s e 3.4
Consider the Galerkin approximate solutions to problem (3.8)
m
um ( t ) º um ( t , x ) =
2
--- ×
l
pk
å g (t) sin -------l x .
k
k=1
Write out the system of ordinary differential equations to determine
{ gk ( t ) } . Prove that this system is locally solvable.
Examples
E x e r c i s e 3.5
Prove that the equations
d
1
--- ----- um ( t ) 2 + n ¶ x u m ( t ) 2 = ( f ( t ) , u m ( t ) )
2 dt
(3.9)
and
d
---- ¶ u ( t ) 2 + n ¶ x2 x u m ( t ) 2 £
dt x m
£ --2- æ ( w , u ) 2 ¶ x um ( t ) 2 + f ( t ) 2ö
ø
nè
(3.10)
are valid for any interval of the existence of the approximate solution
um ( t ) . Here . is the norm in L2 ( 0 , l ) .
E x e r c i s e 3.6 Use equations (3.9) and (3.10) to prove the global existence
of the Galerkin approximate solutions to problem (3.8) and to obtain
the uniform estimate of the form
¶x um ( t ) £ C ( ¶x u 0 , T ) ,
t Î [ 0, T]
(3.11)
for any T > 0 .
Thus, Theorem 2.2 guarantees the global existence and uniqueness of weak solutions to problem (3.8) in F 1 ¤ 2 = H 01 ( 0 , l ) .
E x a m p l e 3.4 (Cahn-Hilliard equation).
ì u + n ¶ 4 u - ¶ 2 ( u3 + a u2 + b u ) = 0 , x Î ( 0 , l ) , t > 0 ,
x
x
ï t
ï
3
3
í ¶x u x = 0 = ¶x u x = 0 = 0 , ¶x u x = l = ¶x u x = l = 0 ,
ï
ïu
= u0 ( x ) ,
î t=0
(3.12)
where n > 0 , a , and b Î R are constants. The result of the cycle of Exercises
3.7–3.10 is a theorem on the existence and uniqueness of solutions to problem
(3.12).
E x e r c i s e 3.7
Prove that the estimate
¶ x2 ( u × v )
2
£ C( u
2
+ ¶x2 u 2 ) ( v
2
+ ¶ x2 v 2 )
is valid for any two functions u ( x ) and v ( x ) smooth on [ 0 , l ] . Use
this estimate to ascertain that problem (3.12) is locally uniquely
solvable in the space
ì
ü
= ux
= 0ý
V = í u Î H 2 ( 0 , l ) : ux
x=0
x=l
î
þ
(Hint: n ¶ x4 + 1 ® A , V = D ( A1 / 2 ) ).
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Long-Time Behaviour of Solutions to a Class of Semilinear Parabolic Equations
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Let us consider the Galerkin approximate solution um ( t ) to problem (3.12):
2
m
2
--- ×
l
1- g ( t ) +
u m ( t ) º um ( t , x ) = ---0
l
E x e r c i s e 3.8
pk
å g (t) cos -------l x .
k
k=1
Prove that the equality
l
d
---- W ( u m ( t ) ) +
dt
òæè¶ [ n ¶
x
2
x
2
um ( t , x ) - p ( u m ( t , x ) ) ]ö d x = 0 (3.13)
ø
0
holds for any interval of the existence of approximate solutions
{ um ( t ) } . Here p ( u ) = u 3 + a u 2 + b and
l
W (u) =
òæèn--2- u
2
x
--- u 3 + b
--- u 2ö d x .
--- u 4 + a
+1
3
2 ø
4
(3.14)
0
In particular, equation (3.13) implies that approximate solutions exist for any segment [ 0 , T ] . For u0 Î V they can be estimated as follows:
¶ x um ( t ) + max
x Î [ 0 , l]
um ( t , x )
£ CT ,
t Î [ 0, T] ,
(3.15)
where the number CT does not depend on m .
E x e r c i s e 3.9
Using (3.15) show that the inequality
d
---- ¶ 2 u ( t ) 2 + ¶ x4 u m ( t ) 2 £ C T ( 1 + ¶x2 um( t ) 2 ) ,
dt x m
t Î (0, T ]
holds for any interval ( 0 , T ) and for any approximate solution
£
u m ( t ) to problem (3.12). (Hint: first prove that u ¢ 2 4
L (0, l)
£ C max u ( x ) × u ¢ 2
).
[0 , l]
L (0, l)
E x e r c i s e 3.10 Using Theorem 2.2 and the result of the previous exercise,
prove the global theorem on the existence and uniqueness of weak
solutions to the Cahn-Hilliard equation (3.12) in the space V (see
Exercise 3.7).
E x a m p l e 3.5 (abstract form of two-dimensional system
of Navier-Stokes equations)
In a separable Hilbert space H we consider the evolutionary equation
du
------ + n Au + b ( u , u ) = f ( t ) ,
dt
u t = 0 = u0 ,
(3.16)
Examples
where A is a positive operator with discrete spectrum, n is a positive parameter and b ( u , v ) is a bilinear mapping from D ( A1 / 2 ) ´ D ( A1 / 2 ) into F-1 ¤ 2 =
= D ( A- 1 / 2 ) possessing the property
(b (u, v) , v) = 0
for all
u , v Î D ( A1 / 2 )
(3.17)
and such that for all u , v Î D ( A ) the estimates
b ( u , v ) £ C d A1 ¤ 2 - d u × A1 ¤ 2 + d v ,
--0< d< 1
2
(3.18)
and
Ab b ( u , v ) £ Cd , b A( 1 ¤ 2) + d u × A1 ¤ 2 + b v ,
--- , 0 < d £ 1
--- (3.19)
0£b£1
2
2
hold. We also assume that f ( t ) is a continuous function with the values in H .
E x e r c i s e 3.11 Prove that Theorem 2.1 on the local solvability with q =
= 1 ¤ 2 + b is applicable to problem (3.16), where b is a number
from the interval ( 0 , 1 ¤ 2 ) .
Let { ek } be the basis of eigenvectors of the operator A , let 0 < l 1 £ l2 £ ¼ be
the corresponding eigenvalues and let Pm be the orthoprojector onto the span
of { e 1 , ¼ , e m } . We consider the Galerkin approximations of problem (3.16):
ìd
ï d----t um ( t ) + n A um ( t ) + Pm b ( um ( t ) , u m ( t ) ) = Pm f ( t ) ,
í
ï u (0) = P u .
m 0
î m
(3.20)
E x e r c i s e 3.12 Show that the estimates
t
um ( t ) 2 + n
ò
0
t
1
A1 / 2 u m ( t ) 2 d t £ u0 2 + ---------l1 n
ò f (t)
2
d t (3.21)
0
and
2
-l n t
-l n t
1- æ F
---ö ( 1 - e 1 )
um( t ) 2 £ um ( 0 ) 2 e 1 + ----l è nø
(3.22)
1
are valid for an arbitrary interval of the existence of solutions to problem (3.20). Here F = sup A-1 / 2 f ( t ) . Using these estimates, prove
t ³ 0
the global solvability of problem (3.20).
E x e r c i s e 3.13 Show that the estimate
d
---- A1 / 2 u m ( t ) 2 + n Aum ( t ) 2 £
dt
2- ( b ( u ( t ) , u ( t ) ) 2 + f ( t ) 2 )
£ --n
m
m
holds for a solution to problem (3.20).
(3.23)
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Long-Time Behaviour of Solutions to a Class of Semilinear Parabolic Equations
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Using the interpolation inequality (see Exercise 1.13) and estimate (3.18), it is easy
to find that
2
Therefore, equation (3.23) implies that
2
2
------ Au 2 .
----ö u 2 A1 / 2 u 4 + n
b ( u , u ) 2 £ C u × A1 / 2 u 2 Au £ æ C
è nø
4
d
---- A1 / 2 u m ( t ) 2 £ sm ( t ) A1 / 2 um ( t ) 2 + --2- f ( t ) 2 ,
n
dt
(3.24)
where sm ( t ) = ( 2 C 2 ¤ n 3 ) um ( t ) 2 × A1 / 2 u m ( t ) 2 . Hence, Gronwall’s inequality gives
us that
ìt
A1 / 2 u
m(t)
2
£
ü
ï 2
sm ( t ) d t ý + --n
ï
ï
î0
þ
ï
A1 / 2 u 0 2 exp í
ò
t
ò
0
ìt
ü
ï
ï
f ( t ) exp í sm( x ) d x ý d t .
ï
ï
î0
þ
ò
2
It follows from equation (3.21) that the value òt sm ( t ) d t is uniformly bounded with
0
respect to m on an arbitrary segment [ 0 , T ] . Consequently, the uniform estimates
A1 / 2 u m( t ) £ C T ,
t Î [ 0, T] ,
are valid for any T > 0 and m = 1 , 2 , ¼ .
u0 Î D ( A1 / 2 )
(3.25)
E x e r c i s e 3.14 Using equations (3.23) and (3.25), prove that if u 0 Î
Î D ( A1 / 2 ) , then
d 1/2
--- Aum ( t ) 2 £ C T ,
---A um( t ) 2 + n
2
dt
for any T > 0 .
t Î [0, T]
--- and let sup Ab f ( t ) 2 £ C T . Prove that
E x e r c i s e 3.15 Let 0 < b < 1
2
[ 0, T]
d
---- A1 ¤ 2 + b um( t ) 2 + n A1 + b u m ( t ) 2 £
dt
£ C1 Ab b ( um ( t ) , um ( t ) )
2
+ CT .
E x e r c i s e 3.16 Use the results of Exercises 3.14 and 3.15 and inequality
(3.19) to prove the global existence of mild solutions to problem
(3.16) in the space D ( A1 ¤ 2 + b ) , provided that u0 Î D ( A1 ¤ 2 + b )
and f ( t ) is a continuous function with the values in D ( A b ) . Here
b is an arbitrary number from the interval ( 0 , 1 ¤ 2 ) .
Existence Conditions and Properties of Global Attractor
§ 4
Existence Conditions and Properties
of Global Attractor
In this section we study the asymptotic properties of the dynamical system generated by the autonomous equation
du
------ + Au = B ( u ) ,
u t = 0 = u0 ,
(4.1)
dt
where, as before, A is a positive operator with discrete spectrum and B ( u ) is
a nonlinear mapping from F q into H such that
B ( u1 ) - B ( u2 ) £ M ( r ) Aq ( u1 - u2 )
(4.2)
for all u1 , u 2 Î F q = D ( A q ) possessing the property Aq u j £ r , 0 £ q < 1 .
We assume that problem (4.1) has a unique mild (in F q ) solution on R+ for any
u0 Î F q . The theorems that guarantee the fulfilment of this requirement and also
some examples are given in Sections 2 and 3 of this chapter. It should be also noted
that in this section we use some results of Chapter 1 for the proof of main assertions.
Further triple numeration is used in the references to the assertions and formulae
of Chapter 1 (first digit is the chapter number).
Let ( F q , St ) be a dynamical system with the evolutionary operator St defined
by the formula St u 0 = u ( t ) , where u ( t ) is a mild solution to problem (4.1).
As shown in Chapter 1, for the system ( F q , St ) to possess a compact global attractor, it should be dissipative. It turns out that the condition of dissipativity is not only
necessary, but also sufficient in the class of systems considered.
Lemma 4.1.
Let ( F q , St ) be dissipative and let B 0 = { u : Aq u £ R0 } be its absorbing
ball. Then the set Ba = { u : Aa u £ Ra } is absorbing for all a Î ( q , 1 ) ,
where
aa
Ra = ( a - q ) a - q R 0 + ------------ sup { B ( u ) : u Î F q , Aq u £ R0 } .
1- a
(4.3)
Proof.
Using equation (2.3) and estimate (1.17), we have
t+1
Aa u ( t
+ 1) £ (a - q)
a- q
Aq u ( t )
+
a ö
t + 1- tø
ò æè -------------------
a
B (u (t) ) d t ,
t
where u ( t ) = St u 0 . Let B be a bounded set in F q . Then St y q £ R0 for all
t ³ tB and y Î B . Therefore, the estimate
B(u (t) )
£ sup { B ( u ) : u Î B 0 } ,
t ³ tB
holds for u ( t ) = St y . Hence, Aa u ( t + 1 ) £ R a for all t ³ tB . This implies the
assertion of the lemma.
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Long-Time Behaviour of Solutions to a Class of Semilinear Parabolic Equations
Theorem 4.1.
Assume that the dynamical system ( F q , St ) generated by problem (4.1)
is dissipative. Then it is compact and possesses a connected compact global
attractor A . This attractor is a bounded set in F a for q £ a < 1 and has
a finite fractal dimension.
dimension.
Proof.
By virtue of Theorem 1.1 the space Fa is compactly embedded into F q for
a > q . Therefore, Lemma 4.1 implies that the dynamical system ( F q , St ) is compact. Hence, Theorem 1.5.1 gives us that ( F q , St ) possesses a connected compact
global attractor A . Evidently, Aa u £ R a for all u ÎA , where Ra is defined
by equality (4.3). Thus, we should only establish the finite dimensionality of the attractor. Let us apply the method used in the proof of Theorem 1.9.1 and based on
Theorem 1.8.1. Let St u1 and St u2 be semitrajectories of the dynamical system
( F q , St ) such that St u j q £ R for all t ³ 0 , j = 1 , 2 . Then equations (2.3) and
(1.17) give us that
t
St u1 - St u2 q £ u1 - u2 q
+ M ( R ) qq
ò (t - t)
-q
St u1 - St u2 q d t .
0
Using Lemma 2.1, we find that for all uj Î F q such that St uj q £ R the following
estimate holds:
St u1 - St u2 q £ C u 1 - u2 q for 0 £ t £ 1 ,
where the constant C depends on q and R . Therefore,
St u1 - St u2 q £ C St u1 - St u2 q
for the considered u1 and u2 . Consequently,
for
0 £ t £ t £ t+1
St u 1 - St u2 q £ C S[ t ] u1 - S[ t ] u2 q £ C [ t ] + 1 u 1 - u2 q ,
where [ t ] is an integer part of the number t . Thus, the estimate of the form
(4.4)
St u1 - St u2 q £ C e a t u1 - u2 q
is valid, where the constants C and a depend on q and R . Similarly, Lemma 1.1
gives
QN ( St u1 - St u2 ) q £ e
t
+ M (R)
ò
-lN + 1 t
u1 - u2 q +
-lN + 1 ( t - t )
q - ö q + lq
æ ---------St u1 - St u2 q d t
N+1 e
è t - tø
(4.5)
0
for all u1 , u 2 Î F q such that St uj q £ R , where Q N = I - PN and PN is the orthoprojector onto the first N eigenvectors of the operator A . If we substitute equation
(4.4) in the right-hand side of inequality (4.5), then we obtain that
Existence Conditions and Properties of Global Attractor
QN ( St u 1 - St u2 ) q £ ( e
-lN + 1 t
+ C M ( R ) e a t JN ( t ) ) u1 - u 2
q
,
where
t
JN ( t ) =
ò
-l
(t - t)
q öq
æ ---------+ lqN + 1 e N + 1
dt .
è t - tø
0
After the change of variable x = l N + 1 ( t - t ) it is easy to find that JN ( t ) £
£ C 0 l-N1++1q . Therefore,
t C (R , q)
æ -l
ö
- e a t÷ u1 - u 2 q
QN ( St u1 - St u2 ) q £ ç e N + 1 + -------------------1- q
lN + 1
è
ø
(4.6)
for all u1 , u 2 Î F q such that St uj q £ R . Hence, there exist t0 > 0 and N such
that
£ d u1 - u2 q ,
QN ( St u1 - St u2 )
q
0
0
0 < d < 1,
for all u 1 , u2 Î F q such that St u j q £ R . However, the attractor A lies in the absorbing ball B 0 . Therefore, this estimate and inequality (4.4) mean that the hypotheses of Theorem 1.8.1 hold for the mapping V = St0 . Hence, the attractor A has
a finite fractal dimension. It can be estimated with the help of the parameters in inequalities (4.4) and (4.6). Thus, Theorem 4.1 is proved.
Equations (4.4) and (4.6) which are valid for any R > 0 enable us to prove the existence of a fractal exponential attractor of the dynamical system ( F q , St ) in the same
way as in Section 9 of Chapter 1.
Theorem 4.2.
Assume that the dynamical system ( F q , St ) generated by problem (4.1)
is dissipative. Then it possesses a fractal exponential attractor (inertial
( inertial
set).
Proof.
It is sufficient to verify that the hypotheses of Theorem 1.9.2 (see (1.9.12)–
(1.9.14)) hold for ( F q , St ) . Let us show that
K=
ÈS B
t ³ t0
t
a,
Ba = { u :
Aa u £ R a }
can be taken for the compact K in (1.9.12)–(1.9.14). Here a is a number from the
interval ( q , 1 ) and Ra is defined by formula (4.3). We choose the parameter
t0 = t0 ( Ba) such that St K Ì Ba for t ³ t0 . Since K is a bounded invariant set,
equation (4.4) is valid for any u 1 , u2 Î K with some constants C and a . It is also
easy to verify that K is a compact. Indeed, let { kn } Ì K . Then kn = St yn , theren
with we can assume that kn ® w , yn ® y Î Ba and either tn ® t* < ¥ or
tn ® ¥ . In the first case with the help of (4.4) we have
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Long-Time Behaviour of Solutions to a Class of Semilinear Parabolic Equations
St yn - St y
n
*
q
£ Ce
a tn
yn - y q + St y - St y .
q
n
*
Therefore, kn = St yn ® St y Î K . In the second case
n
*
w = lim St yn Î w ( Ba ) Ì St ( Ba ) Ì K .
0
n®¥ n
Here w ( Ba ) is the omega-limit set for the semitrajectories emanating from Ba .
Thus, K is a compact invariant absorbing set. In particular this means that condition
(1.9.12) is fulfilled, therewith we can take any number for g > 0 . Conditions (1.9.13)
and (1.9.14) follow from equations (4.4) and (4.6). Consequently, it is sufficient
to apply Theorem 1.9.2 to conclude the proof of Theorem 4.2.
Thus, the dissipativity of the dynamical system ( F q , St ) generated by problem (4.1)
guarantees the existence of a finite-dimensional global attractor and an inertial set.
Under some additional conditions concerning B ( u ) the requirement of dissipativity
can be slightly weakened. We give the following definition. Let a £ q . The dynami* > 0 such that
cal system ( F q , St ) is said to be Fa -dissipative if there exists R a
for any set B bounded in Fa there exists t0 = t0 ( B ) such that
*
Aa St y º St y a £ Ra
for all
y Î B Ç F q and t ³ t0 .
Lemma 4.2.
Assume that B ( u ) satisfies the global Lipschitz condition
B ( u1) - B ( u2 ) £ M Aq ( u 1 - u2 ) .
(4.7)
Let the dynamical system ( F q , St ) generated by mild solutions to problem (4.1) be F a -dissipative for some a Î ( q -1 , q ] . Then ( F q , St ) is
a compact dynamical system, i.e. it possesses an absorbing set which is
compact in F q .
Proof.
By virtue of Lemma 4.1 it is sufficient to verify that the system ( F q , St ) is
dissipative (i.e. F q-dissipative). If we use expression (2.3) and equation (1.17),
then we obtain
Aq u ( t
+ s)
q - a-ö q - a Aa u ( t ) +
£ æ ------------è s ø
t+s
ò
q -ö q B(u (t) ) d t
æ ------------------è t + s - tø
t
for positive t and s . Here u ( t ) = St u0 . Since B ( u ) £ B ( 0 ) + M Aq u , we
have the estimate
1
Aq u ( t + s ) £ æè ---öø
s
q-aì
qq ü
- +
í ( q - a ) q - a Aa u ( t ) + B ( 0 ) ---------1 - q ýþ
î
Existence Conditions and Properties of Global Attractor
s
+M
q -ö
s - tø
òæè -----------
q
Aq u ( t + t ) d t
0
for 0 £ s £ 1 . Hence, we can apply Lemma 2.1 to obtain
Aq u ( t + s )
£ C ( q , a , M ) ( 1 + Aa u ( t ) ) s a - q ,
Therefore, if St u 0 a £
equality gives us that
*
Ra
0 £ s £ 1.
for t ³ t0 ( B ) and u0 Î B Ç F q , then the latter in-
*)
St u0 q £ C ( q , a , M ) ( 1 + Ra
for t ³ 1 + t0 ( B ) ,
i.e. ( F q , St ) is a dissipative system. Lemma 4.2 is proved.
E x e r c i s e 4.1 Show that the assertion of Lemma 4.2 holds if instead of (4.7)
we suppose that B ( u ) = B1 ( u ) + B2 ( u ) , where B1 ( u ) possesses
property (4.7) and B2 ( u ) is such that
sup { B2 ( u ) :
*} < ¥ .
Aa u £ R a
The following assertion contains a sufficient condition of dissipativity of the dynamical system generated by problem (4.1).
Theorem 4.3.
Assume that condition (4.2) is fulfilled with q = 1 ¤ 2 and B ( u ) =
= -F ¢ ( u ) is a potential operator from F 1 ¤ 2 into H (the prime stands for
the Frechét derivative).
derivative). Let
F( u ) ³ -a ,
( F ¢ ( u ) , u ) - b F ( u ) ³ - g A1 / 2 u
2
-d
(4.8)
for all u Î F 1 ¤ 2 , where a , b , g , and d are real parameters, therewith
b > 0 and g < 1 . Then the dynamical system is dissipative in F 1 ¤ 2 .
Proof.
In view of Theorem 2.4 conditions (4.8) guarantee the existence of the evolutionary operator St . Let us verify the dissipativity. As in the proof of Theorem 2.4 we
consider the Galerkin approximations { um ( t ) } . It is evident that
d
1
--- ---u ( t ) 2 + A1 / 2 u m ( t ) 2 + ( F ¢ ( um ) , um ) = 0
2 dt m
and
d
---- æ 1
--- A1 / 2 um ( t ) 2 + F ( um ( t ) )ö + u· m ( t ) 2 = 0 .
ø
dt è 2
If we add these equations and use (4.8), then we obtain that
ü
d ì
--- A1 / 2 um 2 + F ( um ) ý + ( 1 - g ) A1 / 2 um 2 + b F ( u m ) £ d .
---- 1
--- u 2 + 1
2
d t íî 2 m
þ
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Let
--- u 2 + 1
--- A1 / 2 u 2 + F ( u ) + a .
V (u) = 1
2
2
Therefore, it is clear that
d
---- V ( um ( t ) ) + w V ( um ( t ) ) £ C
dt
2
with some positive constants w and C that do not depend on m . This (cf. Exercise 1.4.1) easily implies the dissipativity of the system ( F 1 ¤ 2 , St ) , moreover,
1
C-w t ) .
--- A1 / 2 St u 2 £ V ( u0 ) e -w t + --w (1 - e
2
(4.9)
Theorem 4.3 is proved.
E x e r c i s e 4.2 Show that the assertion of Theorem 4.3 holds if (4.2) is fulfilled with q = 1 ¤ 2 and
B ( u ) = - F ¢ ( u ) + B0 ( u ) ,
where F ( u ) possesses properties (4.8) and B 0 ( u ) satisfies the estimate B0 ( u ) £ C e + e A1 / 2 u for e > 0 small enough.
Let us look at the examples of Section 3 again. We assume that the function
f (t, x, u, p) º f (x , u , p)
(4.10)
in Example 3.1 possesses property (3.2) and
ò f (x, u (x) , Ñu (x) ) u (x) dx £ (l
W
1
- d)
ò u (x) dx + C
2
(4.11)
W
Î H01 ( W ) ,
for all u
where l 1 is the first eigenvalue of the operator - D with the
Dirichlet boundary condition on ¶ W . Here d and C are positive constants.
E x e r c i s e 4.3 Using the Galerkin approximations of problem (3.1) and Lemma 4.2, prove that the dynamical system generated by equation (3.1)
is dissipative in H01 ( W ) under conditions (3.2), (4.10), and (4.11).
Therefore, if conditions (3.2), (4.10) and (4.11) are fulfilled, then the dynamical system ( H01 ( W ) , St ) generated by a mild (in H01 ( W ) ) solution to problem (3.1) possesses both a finite-dimensional global attractor and an inertial set.
E x e r c i s e 4.4
Prove that equation (4.11) holds if
d
f ( x , u , Ñu ) = f0 ( x , u ) +
¶u
- ,
å a -------¶x
j
j=1
j
where a j are real constants and the function f0 ( x , u ) possesses
the property
Existence Conditions and Properties of Global Attractor
f ( x , u ) u £ ( l1 - d ) u2 + C ,
for any x Î W .
u ÎR
E x e r c i s e 4.5 Consider the dynamical system generated by problem (3.4).
In addition to the hypotheses of Example 3.2 we assume that
f ( t , x , u , ¶x u ) º 0 and the function g ( x , u ) possesses the properties
y
ò g ( x , x ) d x ³ -a ,
y
ò
y g( x , y ) - b g( x , x ) d x ³ -g
0
0
for some positive a , b , and g . Then the dynamical system generated by (3.4) is dissipative in H01 ( 0 , 1 ) .
E x e r c i s e 4.6 Find the analogue of the result of Exercise 4.5 for the system
of reaction-diffusion equations (3.6).
E x e r c i s e 4.7 Using equations (3.9) and (3.10) prove that the dynamical
system generated by the nonlocal Burgers equation (3.8) with f ( t ) º
º f Î L2 ( 0 , l ) is dissipative in H 01 ( W ) .
Let us consider a dynamical system ( V , St ) generated by the Cahn-Hilliard equation
(3.12). We remind that
ì
ü
= ux
= 0ý .
V = í u Î H 2 ( 0 , l ) : ux
x=0
x=l
î
þ
E x e r c i s e 4.8 Let the function W ( u ) be defined by equality (3.14). Show
that for any positive R and a the set
Xa , R
ì
ï
= íu Î V : W (u) £ R ,
ï
î
ü
ï
u (x) dx £ a ý
ï
0
þ
l
ò
(4.12)
is a closed invariant subset in V for the dynamical system ( V , St )
generated by problem (3.12).
E x e r c i s e 4.9 Prove that the dynamical system ( X a , R , St ) generated by
the Cahn-Hilliard equation on the set Xa , R defined by (4.12) is dissipative (Hint: cf. Exercise 3.9).
In conclusion of this section let us establish the dissipativity of the dynamical system
( D ( A1 ¤ 2 + b ) , St ) generated by the abstract form of the two-dimensional NavierStokes system (see Example 3.5) under the assumption that f ( t ) º f Î D ( A b ) .
We consider the dynamical system ( Pm H , Stm ) generated by the Galerkin approximations (see (3.20)) of problem (3.16).
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Long-Time Behaviour of Solutions to a Class of Semilinear Parabolic Equations
E x e r c i s e 4.10 Using (3.24) prove that
A1 / 2 Stm u0 2
t
2
ì
ï
´ exp í c 1
ï
î
æ
£ ç c0 +
ç
è
ò
A1 / 2 Stm u 0 2 d t÷
÷
ò
2
A1 / 2 Stm u0
0
´
ø
0
1
æ Sm u
è t 0
ö
1
ü
2
ö d t ïý
ø
ï
þ
(4.13)
for all 0 < t £ 1 , where c 0 and c 1 are constants independent of m .
E x e r c i s e 4.11 With the help of (3.21), (3.22) and (4.13) verify the property
of dissipativity of the system ( D ( A 1 ¤ 2 + b ) , St ) in the space
D ( A1 ¤ 2 ) . Deduce its dissipativity (Hint: see Exercise 3.14–3.16).
Thus, the dynamical system generated by the two-dimensional Navier-Stokes equations possesses both a finite-dimensional compact global attractor and an inertial set.
§ 5
Systems with Lyapunov Function
In this section we consider problem (4.1) on the assumption that condition (4.2)
holds with q = 1 ¤ 2 and B ( u ) is a potential operator, i.e. there exists a functional
F ( u ) on F 1 ¤ 2 = D ( A1 / 2 ) such that its Frechét derivative F ¢ ( u ) possesses the property
B ( u ) = -F ¢ ( u ) ,
u Î D ( A1 / 2 ) .
(5.1)
Below we also assume that the conditions
F ( u ) ³ -a ,
( F ¢ ( u ) , u ) - b F ( u ) ³ - g A1 / 2 u
2
-d
(5.2)
are fulfilled for all u Î D ( A1 / 2 ) , where a , d Î R , b > 0 , and g < 1 . On the one
hand, these conditions ensure the existence and uniqueness of mild (in D ( A1 / 2 ) )
solutions to problem (4.1) (see Theorem 2.4). On the other hand, they guarantee
the existence of a finite-dimensional global attractor A for the dynamical system
( F 1 ¤ 2 , St ) generated by problem (4.1) (see Theorem 4.3). Conditions (5.1) and
(5.2) enable us to obtain additional information on the structure of attractor.
Theorem 5.1.
Assume that conditions (5.1),, (5.2),, and (4.2) hold with q = 1 ¤ 2 . Then
the global attractor A of the dynamical system ( F 1 ¤ 2 , St ) generated by
Systems with Lyapunov Function
problem (4.1) is a bounded set in F 1 = D ( A ) and it coincides with the unstable manifold emanating from the set of fixed points of the system, i.e.
A = M+ ( N ) ,
(5.3)
where N = { z Î D ( A) : A z = B ( z ) } (for the definition of M+ ( N ) see Section 6 of Chapter 1).
).
The proof of the theorem is based on the following lemmata.
Lemma 5.1.
Assume that a semitrajectory u ( t ) = St u0 possesses the property
Aa u ( t ) £ Ra for all t ³ 0 , where 1 ¤ 2 < a < 1 . Then
A1 / 2 ( u ( t ) - u ( s ) ) £ C ( Ra ) t - s a - 1 ¤ 2
(5.4)
for all t , s ³ 0 .
Proof.
For the sake of definiteness we assume that t > s ³ 0 . Equation (2.3) implies that
t
u (t) - u (s) = (e
-( t - s ) A
- I) u(s) +
òe
-( t - t ) A
B(u (t) ) d t .
s
Since
t
e
-( t - s ) A
ò
- I = - A e-( t - t) A d t ,
s
equation (1.17) gives us that
t
A1 / 2 ( u ( t )
- u (s) )
£ c0
ò (t - t)
a - 3¤ 2
d t Aa u ( s ) +
s
t
+ c1
ò (t - t)
-1 / 2
d t max B ( u ( t ) ) .
t ³ 0
s
This implies estimate (5.4).
Lemma 5.2.
There exists R 1 > 0 such that the set B1 = { u :
for the dynamical system ( F1 ¤ 2 , St ) .
Au £ R 1 } is absorbing
Proof.
By virtue of Theorem 4.3 the system ( F 1 ¤ 2 , St ) is dissipative. Therefore, it
follows from Lemma 4.1 that Ba = { u : Aa u £ Ra } is an absorbing set,where
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Ra is defined by (4.3), 1 ¤ 2 < a < 1 . Thus, to prove the lemma it is sufficient to
consider semitrajectories u ( t ) possessing the property Aa u ( t ) £ Ra , t ³ 0 .
Let us present the solution u ( t ) in the form
t
u (t) = e
-( t - s ) A
u (s) +
òe
-( t - t ) A
B (u (t) ) - B (u (t) ) d t +
s
+ A-1 ( 1 - e -( t - s ) A ) B ( u ( t ) ) .
Using Lemma 5.1 we find that
t
Au ( t ) £ C0 ( t - s ) -1 / 2 R 1 ¤ 2 + C1 ( Ra )
ò
(t - t)
3
- --- + a
2
dt
+ C2 ( R1 ¤ 2 ) .
s
This implies that
Au ( t + 1 ) £ C ( Ra ) ,
t ³ 0,
provided that Aa u ( t ) £ Ra for t ³ 0 . Therefore, the assertion of Lemma 5.2
follows from Lemma 4.1.
Proof of Theorem 5.1.
The boundedness of the attractor A in D ( A ) follows from Lemma 5.2. Let us
prove (5.3). We consider the Galerkin approximation um ( t ) of solutions to problem
(4.1):
d um ( t )
------------------ + Au m ( t ) = Pm B ( u m ( t ) ) ,
dt
u m ( 0 ) = Pm u0 .
Here Pm is the orthoprojector onto the span of { e 1 , e2 , ¼ , e m } . Let
--- ( Au , u ) + F ( u ) , u Î F 1 ¤ 2 = D ( A1 / 2 ) .
V (u) = 1
2
It is clear that
(5.5)
d
---- V ( u m ( t ) ) = ( Aum ( t ) + F ¢ ( um ( t ) ) , u· m ( t ) ) = - Pm [ Aum ( t ) - B ( u m ( t ) ) ] 2 .
dt
This implies that
t
V ( um ( t ) ) - V ( um ( s )) = -
ò P [ Au
m
m(t)
- B ( um ( t ) ) ] 2 d t £
s
t
£ -
ò P ( Au
N
m( t)
- B ( um ( t ))) 2 d t
s
for t ³ s and for any N £ m , where PN is the orthoprojector onto the span of
{ e1 , ¼ , eN } . With the help of Theorem 2.2 and due to the continuity of the func-
Systems with Lyapunov Function
tional V ( u ) we can pass to the limit m ® ¥ in the latter equation. As a result,
we obtain the estimate
t
V ( u ( t )) +
ò P ( A u ( t ) - B ( u ( t )))
N
2
d t £ V ( u ( s )) ,
t ³ s,
(5.6)
s
for any solution u ( t ) to problem (4.1) and for any N ³ 1 . If the semitrajectory
u ( t ) = St u0 lies in the attractor A , then we can pass to the limit N ® ¥ in (5.6)
and obtain the equation
t
V ( u ( t )) +
ò Au(t) - B (u(t))
2
dt £ V (u (s))
(5.7)
s
for any t ³ s ³ 0 and u ( t ) Î A . Equation (5.7) implies that the functional V ( u )
defined by equality (5.5) is the Lyapunov function of the dynamical system
( F 1 ¤ 2 , St ) on the attractor A . Therefore, Theorem 1.6.1 implies equation (5.3).
Theorem 5.1 is proved.
Using (5.6) show that any solution u ( t ) to problem (4.1) with
u 0 Î F1 ¤ 2 possesses the property
E x e r c i s e 5.1
t
ò Au (t)
2
dt < ¥ ,
t > 0.
0
Prove the validity of inequality (5.7) for any solution u ( t ) to problem (4.1).
E x e r c i s e 5.2 Using the results of Exercises 5.1 and 1.6.5 show that if the
hypotheses of Theorem 5.1 hold, then a global minimal attractor
A min of the system ( F1 ¤ 2 , St ) has the form
A min = { w Î D ( A ) : A w - B ( w ) = 0 } .
E x e r c i s e 5.3 Prove that the assertions of Theorems 4.3 and 5.1 hold if we
consider the equation
du
------ + Au = B ( u ) + h ,
u t = 0 = u0
(5.8)
dt
instead of problem (4.1). Here h is an arbitrary element from H and
B ( u ) possesses properties (5.1), (5.2) and (4.2) with q = 1 ¤ 2
(Hint: Vh ( u ) = V ( u ) - ( h , u ) ).
E x e r c i s e 5.4 Let S t be an evolutionary operator of problem (5.8). Show
that for any R > 0 there exist numbers aR ³ 0 and bR > 0 such
that
A1 / 2 ( St u 1 - St u2 )
£ bR e
aR t
A1 / 2 ( u1 - u2 ) ,
provided A1/ 2 uj £ R , j = 1 , 2 (Hint: Vh ( St uj ) £ Vh ( u j ) £ CR ).
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Long-Time Behaviour of Solutions to a Class of Semilinear Parabolic Equations
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Theorem 5.1 and the reasonings of Section 6 of Chapter 1 reduce the question on the
structure of global attractor to the problem of studying the properties of stationary
points of the dynamical system under consideration. Under some additional conditions on the operator B ( u ) it can be proved in general that the number of fixed
points is finite and all of them are hyperbolic. This enables us to use the results of
Section 6 of Chapter 1 to specify the attractor structure. For some reasons (they will
be clear later) it is convenient to deal with the fixed points of the dynamical system
generated by problem (5.8).
Thus we consider the equation
2
L ( u ) º Au - B ( u ) = h ,
u Î D (A) ,
(5.9)
where as before A is a positive operator with discrete spectrum, B ( u ) is a nonlinear mapping possessing properties (5.1), (5.2) and (4.2) with q = 1 ¤ 2 , and h is an
element of H .
Lemma 5.3.
For any h Î H problem (5.9) is solvable. If M is a bounded set in H ,
then the set L-1 ( M ) of solutions to equation (5.9) is bounded in D ( A )
for h Î M . If M is compact in H , then L-1 ( M ) is compact in D ( A ) .
Proof.
Let us consider a continuous functional
--- ( Au , u ) + F ( u ) - ( h , u )
W (u) = 1
2
(5.10)
on F1 ¤ 2 = D ( A1 / 2 ) . Since F ( u ) ³ - a for all u Î F1 ¤ 2 , the functional W ( u )
possesses the property
--- A1 / 2 u 2 - a - A-1 / 2 h A1 / 2 u ³
W (u) ³ 1
2
1
³ --- A1 / 2 u
4
2
- a - A-1 / 2 h
2
.
(5.11)
In particular, this means that W ( u ) is bounded below. Let us consider the functional W ( u ) on the subspace Pm F1 ¤ 2 ( Pm is the orthoprojector onto the span
of elements e1 , e 2 , ¼ , e m , as before). By virtue of (5.11) there exists a minimum point u m of this functional in Pm F1 ¤ 2 which obviously satisfies the equation
A um - Pm B ( um ) = Pm h .
(5.12)
Equation (5.11) also implies that
1
--- A1 / 2 um 2 £ W ( um ) + a + A-1 / 2 h 2 £
4
£ inf { W ( u ) : u Î Pm H } + a + A-1 / 2 h
2
.
Systems with Lyapunov Function
Hence,
A1 / 2 um
£ F ( 0 ) + a + A-1 / 2 h
2
£ C ( 1 + A-1 / 2 h 2 )
with a constant C independent of m . Thus, equations (5.12) and (4.2) give us
that
Aum £ B ( um ) + h £ B ( 0 ) + h + M ( A1 / 2 u m ) A1 / 2 u m .
Therefore, if h £ R , then Aum £ C ( R ) . This estimate enables us to extract
a weakly convergent (in D ( A ) ) subsequence { um } and to pass to the limit as
k
k ® ¥ in (5.12) with the help of Theorem 1.1. Thus, the solvability of equation
(5.9) is proved. It is obvious that every limit (in D ( A ) ) point u of the sequence
{ um } possesses the property
Au £ CR
if
h £ R.
(5.13)
This means that the complete preimage L-1( M ) of any bounded set M in H is
bounded in D ( A ) . Now we prove that the mapping L is proper,
proper i.e. the preimage L-1( M ) is compact for a compact M . Let { u n } be a sequence from
L-1( M ) . Then the sequence { L ( un ) } lies in the compact M and therefore there
exist an element h Î M and a subsequence { nk } such that L ( unk ) - h ® 0
as k ® ¥ . By virtue of (5.13) we can also assume that { A unk } is a weakly convergent sequence in D ( A ) . If we use the equation
Au nk - B ( un ) - h ® 0 ,
k
k ® ¥,
Theorem 1.1, and property (4.2) with q = 1 ¤ 2 , then we can easily prove that
the sequence { unk } strongly converges in D ( A ) to a solution u to the equation
L ( u ) = h . Lemma 5.3 is proved.
Lemma 5.4.
In addition to (5.1), (5.2), and (4.2) with q = 1 ¤ 2 we assume that the
mapping B ( . ) is Frechét differentiable, i.e. for any u Î F1 ¤ 2 there
exists a linear bounded operator B ¢ ( u ) from F1 ¤ 2 into H such that
B ( u + v ) - B ( u ) - B ¢ ( u ) v = o ( A1 / 2 v )
(5.14)
for every v Î F1 ¤ 2 , such that A1 / 2 v £ 1 . Then the operator A - B ¢ ( u )
is a Fredholm operator for any u Î F1 ¤ 2 = D ( A 1 / 2 ) .
We remind that a densely defined closed linear operator G in H is said to be Fredholm (of index zero) if
(a) its image is closed; and
(b) dim Ker G = dim Ker G * < ¥ .
Proof of Lemma 5.4.
It is clear that the operator G º A - B ¢ ( u ) has the structure
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G = A - C A1 / 2 = A1 / 2 ( I - A-1 / 2 C ) A1 / 2 ,
where C is a bounded operator in H ( C = B ¢ ( u ) A-1 / 2 ) . By virtue of Theorem
1.1 the operator K = A-1 / 2 C is compact. This implies the closedness of the
image of G . Moreover, it is obvious that
2
dim Ker G = dim Ker ( I - K )
and
dim Ker G * = dim Ker ( I - K * ) .
Therefore, the Fredholm alternative for the compact operator gives us that
dim Ker G * = dim Ker G < ¥ .
Let us introduce the notion of a regular value of the operator L seen as an element h Î H possessing the property that for every u Î L -1 h º { v : L ( v ) = h } the
operator L¢( u ) = A - B ¢ ( u ) is invertible on H . Lemmata 5.3 and 5.4 enable us to use
the Sard-Smale theorem (for the statement and the proof see, e.g., the book by
A. V. Babin and M. I. Vishik [1]) and state that the set R of regular values of the
mapping L ( u ) = Au - B ( u ) is an open everywhere dense set in H . The following
assertion is valid for regular values of the operator L .
Lemma 5.5.
Let h be a regular value of the operator L . Then the set of solutions to
equation (5.9) is finite.
Proof.
By virtue of Lemma 5.3 the set N = { v : L ( v ) = h } is compact. Since
h Î R , the operator L¢ ( u ) = A - B ¢ ( u ) is invertible on H for u Î N . It is also
evident that L¢( u ) has a domain D ( A ) . Therefore, by virtue of the uniform
boundedness principle A L¢( u ) -1 is a bounded operator for u ÎN . Hence,
it follows from (5.14) that
A( v - w )
£
AL¢( v ) - 1 × L¢ ( v ) ( v - w ) =
= AL¢( v ) -1 B ( v ) - B ( w ) - B ¢ ( v ) ( v - w ) = o ( A1 / 2 ( v - w ) )
for any v and w in N . This implies that for every v Î N there exists a vicinity
that does not contain other points of the set N . Therefore, the compact set N
has no condensation points. Hence, N consists of a finite number of elements.
Lemma 5.5 is proved.
In order to prove the hyperbolicity (for the definition see Section 6 of Chapter 1) of
fixed points we should first consider linearization of problem (5.8) at these points.
Assume that the hypotheses of Lemma 5.4 hold and v0 Î D ( A ) is a stationary solution. We consider the problem
Systems with Lyapunov Function
du
------ + ( A - B ¢ ( v0 ) ) u = 0 ,
u t = 0 = u0 .
(5.15)
dt
Its solution can be regarded as a continuous function in F1 ¤ 2 = D ( A 1 / 2 ) which satisfies the equation
t
u ( t ) = e -t A u0 +
òe
-( t - t ) A
B ¢ ( v0 ) u ( t ) d t .
0
Î D ( A1 / 2 ) ,
If u 0
then we can apply Theorem 2.3 on the existence and uniqueness
of solution. Let Tt stand for the evolutionary operator of problem (5.15).
Prove that T t is a compact operator in every space Fa ,
0 £ a < 1, t > 0 .
E x e r c i s e 5.5
E x e r c i s e 5.6 Prove that for any r > 0 and t > 0 the set of points of the
spectrum of the operator Tt that are lying outside the disk { l :
l £ r } is finite and the corresponding eigensubspace is finite-dimensional.
E x e r c i s e 5.7 Assume that B ¢ ( v 0 ) is a symmetric operator in H . Prove that
the spectrum of the operator Tt is real.
The next assertion contains the conditions wherein the evolutionary operator St belongs to the class C 1 + a , a > 0 , on the set of stationary points.
Theorem 5.2.
Assume that conditions (5.1),, (5.2),, and (4.2) are fulfilled with q = 1 ¤ 2 .
Let B ( u ) possess a Frechét derivative in F1 ¤ 2 such that for any R > 0
B ( u + v ) - B ( u ) - B ¢ ( u ) v £ C A1 / 2 v 1 + a ,
a > 0,
(5.16)
provided that A1 / 2 v £ R , where the constant C = C ( u , R ) depends on u
and R only. Then the evolutionary operator St of problem (5.8) has a
Frechét derivative at every stationary point v0 . Moreover, ( [ St ( v0 ) ] ¢ , u ) =
= Tt u , where Tt is the evolutionary operator of linear problem (5.15)..
Proof.
It is evident that
t
St [ v0 + u ] - v0 - Tt u =
òe
0
-( t - t ) A
ì
ü
í B ( St [ v0 + u ] ) - B ( v 0 ) - B ¢ ( v0 ) Tt u ý d t .
î
þ
Therefore, using (1.17) and (5.16) we have
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t
A1 / 2 y ( t )
£
ò (2 e [ t - t] )
-1 ¤ 2
{ C A1 / 2 ( St [ v0 + u ] - v0 )
1+ a
+
0
ü
+ B ¢ ( v0 ) × A1 / 2 y ( t ) ý d t ,
þ
2
where y ( t ) = St [ v0 + u ] - v0 - Tt u . Using the result of Exercise 5.4 we obtain that
A1 / 2 ( St [ v0 + u ] - v0 ) £ bR e
aR t
A1 / 2 u
if
A1 / 2 u £ R .
Thus,
A1 / 2 y ( t )
£ C0 t e
( 1 + a ) aR t
t
1+a
A1 / 2 u
+ C1
ò (t - t)
-1 ¤ 2
A1 / 2 y ( t ) d t .
0
Consequently, Lemma 2.1 gives us the estimate
A1 / 2 ( St [ v0 + u ] - v0 - Tt u ) £ CT A1 / 2 u 1 + a ,
t Î [ 0, T] .
This implies the assertion of Theorem 5.2.
E x e r c i s e 5.8 Assume that the constant C in (5.16) depends on R only,
provided that A1/ 2 u £ R and A1/ 2 v £ R . Prove that St Î C 1 + a
for any 0 < a < 1 .
The reasoning above leads to the following result on the properties of the set of fixed
points of problem (5.8).
Theorem 5.3.
Assume that conditions (5.1),, (5.2) and (4.2) are fulfilled with q = 1 ¤ 2
and the operator B ( u ) possesses a Frechét derivative in F1 ¤ 2 such that
equation (5.16) holds with the constant C = C ( R ) depending only on R for
A1 / 2 u £ R and A1 / 2 v £ R . Then there exists an open dense (in H ) set
R such that for h Î R the set of fixed points of the system ( F1 ¤ 2 , St ) generated by problem (5.8) is finite. If in addition we assume that B ¢ ( z ) is a
symmetric operator for z Î D ( A ) , then fixed points are hyperbolic.
hyperbolic.
In particular, this theorem means that if h Î R , then the global attractor of the dynamical system generated by equation (5.8) possesses the properties given in Exercises 1.6.9–1.6.12. Moreover, it is possible to apply Theorem 1.6.3 as well as the other
results related to finiteness and hyperbolicity of the set of fixed points (see, e.g., the
book by A. V. Babin and M. I. Vishik [1]).
Systems with Lyapunov Function
E x a m p l e 5.1
Let us consider a dynamical system generated by the nonlinear heat equation
ì ¶u
¶2 u
- n ---------2- + g ( x , u ( x ) ) = h ( x ) , 0 < x < 1 , t > 0 ,
ï -----¶
t
¶x
í
ïu
=
u x = 1 = 0 , u t = 0 = u0 ( x ) ,
î x=0
(5.17)
in H01 ( 0 , 1 ) . Assume that g ( x , u ) is twice continuously differentiable with respect to its variables and the conditions
y
ò g ( x , x ) d x ³ -a ,
y
ò
y g( x , y ) - b g( x , x ) d x ³ -g
0
0
are fulfilled with some positive constants a , b , and g .
E x e r c i s e 5.9 Prove that the dynamical system generated by equation
(5.17) possesses a global attractor A = M+ ( N ) , where N is
the set of stationary solutions to problem (5.17).
E x e r c i s e 5.10 Prove that there exists a dense open set R in L2 ( 0 , 1 ) such
that for every h ( x ) Î R the set N of fixed points of the dynamical
system generated by problem (5.17) is finite and all the points are
hyperbolic.
It should be noted that if a property of a dynamical system holds for the parameters
from an open and dense set in the corresponding space, then it is frequently said that
this property is a generic property .
However, it should be kept in mind that the generic property is not the one that
holds almost always. For example one can build an open and dense set R in [ 0 , 1 ] ,
the Lebesgue measure of which is arbitrarily small ( £ e ) . To do that we should
take
R=
¥
ì
È íî x Î (0, 1) :
k=1
ü
x - rk < e × 2- k - 2 ý ,
þ
where { rk } is a sequence of all the rational numbers of the segment [ 0 , 1 ] . Therefore, it should be remembered that generic properties are quite frequently encountered and stay stable during small perturbations of the properties of a system.
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§6
Explicitly Solvable Model
of Nonlinear Diffusion
In this section we study the asymptotic properties of solutions to the following nonlinear diffusion equation
æ 1
ö
ì
ï ut - n u x x + ç š u ( x , t ) 2 d x - G÷ u + r ux = 0 ,
ç
÷
ï
è 0
ø
í
ï
ïu
î x = 0 = u x = 1 = 0 , u t = 0 = u0 ( x ) ,
ò
0 < x < 1, t > 0,
(6.1)
where n > 0 , š > 0 , G and r are parameters. The main feature of this problem is
that the asymptotic behaviour of its solutions can be completely described with the
help of elementary functions. We do not know whether problem (6.1) is related to
any real physical process.
E x e r c i s e 6.1 Show that Theorem 2.4 which guarantees the global existence
and uniqueness of mild solutions is applicable to problem (6.1) in the
Sobolev space H 01 ( 0 , 1 ) .
E x e r c i s e 6.2 Write out the system of ordinary differential equations for the
functions { gk ( t ) } that determine the Galerkin approximations
m
um ( t ) =
2
å g (t) sin p k x
(6.2)
k
k=1
of the order m of a solution to problem (6.1).
E x e r c i s e 6.3 Using the properties of the functions um ( t ) defined by equation (6.2) prove that the mild solution u ( t , x ) possesses the properties
t
1
--- u ( t ) 2 +
2
ò (n ¶ u(t)
x
2
+ š u(t)
4
--- u0
- G u(t) 2) dt = 1
2
2
(6.3)
0
and
2
G 2 - ( 1 - e -2 n p 2 t ) .
u ( t ) 2 £ u0 2 e -2 n p t + -----------------4 n š p2
Here and below
(6.4)
. is a norm in L2 ( 0 , 1 ) .
E x e r c i s e 6.4 Using equations (6.3) and (6.4) prove that the dynamical system generated by problem (6.1) in H 01 ( 0 , 1 ) is dissipative.
Explicitly Solvable Model of Nonlinear Diffusion
Therefore, by virtue of Theorem 4.1 the dynamical system ( H01 ( 0 , 1 ) , St ) generated
by equation (6.1) possesses a finite-dimensional global attractor.
Let u ( t ) = u ( x , t ) Î C ( R+ ; H 01 ( 0 , 1 ) ) be a mild solution to problem (6.1) with
the initial condition u0 ( x ) Î H 01 ( 0 , 1 ) . Then the function u ( x , t ) can be considered as a mild solution to the linear problem
ì ut - n ux x + b ( t ) u + r ux = 0 , 0 < x < 1 , t > 0 ,
í
î u x = 0 = u x = 1 = 0 , u t = 0 = u0 ( x ) ,
(6.5)
where b ( t ) is a scalar continuous function defined by the formula
1
b (t) = š
ò u (x, t)
2
dx - G .
0
We consider the function
ì
ï
v ( x , t ) = u ( x , t ) exp í
ï
î
ü
r2
r ï
b ( t ) d t + ------- t - ------- x ý .
4n
2n ï
0
þ
t
ò
(6.6)
Then it is easy to check that v ( x , t ) is a mild solution to the heat equation
ìv = n v , x Î ( 0, 1) , t > 0 ,
xx
ï t
ï
í
ì r ü
ïv
- xý .
ï x = 0 = v x = 1 = 0 , v t = 0 = u 0 ( x ) exp í - -----î 2n þ
î
(6.7)
The following assertion shows that the asymptotic properties of equation (6.7)
completely determine the dynamics of the system generated by problem (6.1).
Lemma 6.1.
Every mild (in H 01 ( 0 , 1 ) ) solution u ( x , t ) to problem (6.1) can be rewritten in the form
w (x, t)
u ( x , t ) = ---------------------------------------------------------------- ,
ì
ü1 / 2
t
ï
ï
í1 + 2 š w (t) 2 dt ý
ï
ï
0
î
þ
(6.8)
ò
where w ( x , t ) has the form
ì
r ü
r2
w ( x , t ) = v ( x , t ) exp í æè G - -------öø t + ------- x ý
n
n þ
4
2
î
and v ( x , t ) is the solution to problem (6.7).
(6.9)
119
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Long-Time Behaviour of Solutions to a Class of Semilinear Parabolic Equations
Proof.
Let w ( x , t ) have the form (6.9). Then we obtain from (6.6) that
ì
ï
w ( x , t ) = u ( x , t ) exp í š
ï
î
2
t
ò
0
ü
ï
u (t) d t ý .
ï
þ
2
(6.10)
Therefore,
ì
ü
ì
ü
t
t
ï
ï
ï
ï
d
1 ---exp í 2 š u ( t ) 2 d t ý .
w ( x , t ) 2 = u ( x , t ) 2 exp í 2 š u ( t ) 2 d t ý = -----š
2
d
t
ï
ï
ï
ï
0
0
î
þ
î
þ
ò
ò
Hence,
ì
ï
exp í 2 š
ï
î
t
ò
0
ü
ï
u(t) d t ý = 1 + 2 š
ï
þ
2
t
ò w( t)
2
dt .
0
This and equation (6.10) imply (6.8). Lemma 6.1 is proved.
Now let us find the fixed points of problem (6.1). They satisfy the equation
-n ux x + ( š u
2
- G ) u + r ux = 0 ,
u x=0 = u x=1 = 0.
ìr ü
Therefore, u ( x ) = w ( x ) exp í ------- x ý , where w ( x ) is the solution to the problem
î2 n þ
r2
-n wx x + æ š u 2 - G + -------ö w = 0 , w x = 0 = w x = 1 = 0 .
è
4 nø
However, this problem has a nontrivial solution w ( x ) if and only if
w ( x ) = C sin p n x
and
š u
2
r2
- G + ------- + n ( p n ) 2 = 0 ,
4n
ìr ü
where n is a natural number. Since u = w exp í ------- x ý , we obtain the equation
î2 n þ
1 r
--- x
r2
š C 2 e n sin2 p n x d x + n ( p n ) 2 - G + ------- = 0
4n
ò
0
which can be used to find the constant C . After the integration we have
r2
2 n2 n2 p2
--n- ( e r / n - 1 ) -------------------------------------- = G - ------- - n ( p n ) 2 .
š C2 r
4n
r2 + 4 n2 n2 p2
The constant C can be found only when the parameter n possesses the property
G - r 2 ¤ ( 4 n ) - n ( p n ) 2 > 0 . Thus, we have the following assertion.
Explicitly Solvable Model of Nonlinear Diffusion
Lemma 6.2.
r2
Let g = g ( G , n , r ) º G - -----. If g £ n p 2 , then problem (6.1) has a unique
4n
fixed point u0 ( x ) º 0 . If ( p n ) 2 < g £ p 2 ( n + 1 ) 2 for some n ³ 1 , then
the fixed points of problem (6.1) are
r
------ x
u0 ( x ) º 0 ,
u±k ( x ) = ±m k e 2n sin p k x ,
k = 1, 2, ¼, n ,
(6.11)
where
1
m k º m k ( G , r , n ) = -----------------2
4n pk
2 r dk ( G , r , n )
,
---------------------------------------r
š æè exp æè ---öø - 1öø
n
(6.12)
dk ( G , r , n ) = [ 4 G n - r2 - 4 ( p n n ) 2 ] × [ r2 + 4 ( p n n ) 2 ] .
E x e r c i s e 6.5
Show that every subspace
r
ì ------- x
ü
H N = Lin í e 2 n sin p k x : k = 1 , 2 , ¼ , N ý
î
þ
(6.13)
is positively invariant for the dynamical system ( H 01 ( 0 , 1 ) , St ) generated by problem (6.1).
Theorem 6.1.
r2
Let g º G - ------- < n p 2 . Then for any mild solution u ( t ) to problem (6.1)
4n
the estimate
¶x u ( t )
£ C ( r , n ) e -( n p
2 - g) t
¶x u0 ,
t ³ 0,
(6.14)
is valid. If g ³ n p 2 and n p 2 N 2 > g , then the subspace HN - 1 defined by
formula (6.13) is exponentially attracting:
dist
1
H0 ( 0 , 1 )
( St u0 , HN - 1 ) £ C ( r , n ) e -( n p
2 N2 - g ) t
¶x u0 .
(6.15)
Here St is the evolutionary operator in H01 ( 0 , 1 ) corresponding to (6.1)..
Proof.
Let w ( x , t ) be of the form (6.9). Then
r
- ------ x
wr ( t ) º e 2n w ( t ) =
¥
åe
2
-( n ( p k ) - g ) t
Ck , r ek ( x ) ,
k=1
where e k ( x ) =
2 sin p k x and
1
Ck , r =
ò
0
r
- ------ x
e 2n u0 ( x ) ek ( x ) d x .
(6.16)
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Long-Time Behaviour of Solutions to a Class of Semilinear Parabolic Equations
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Assume that g < n p 2 . Then it is obvious that
¥
¶ x wr ( t )
2
å (p k)
=
2
C k2, r exp { -2 ( n ( p k ) 2 - g ) t } £
k=1
r
æ - ------ x ö 2
£ exp { -2 ( n p 2 - g ) t } ¶ x ç e 2n u0÷ .
è
ø
2
(6.17)
However, equation (6.8) implies that
r
æ - ------ x
ö
¶x ç e 2n u ( t )÷
è
ø
£
¶ x wr ( t ) .
Therefore, estimate (6.14) follows from (6.17) and from the obvious equality
u £ 1 ¤ p ¶x u . When g ³ n p 2 and N > ( 1 ¤ p ) g ¤ n the function w ( t ) in (6.9)
can be rewritten in the form
r
------ x
w ( t ) = hN ( t ) + e 2n wr , N ( t ) ,
(6.18)
where
r
N -1
------ x
hN ( t ) =
2 e 2n
åe
2
-( n ( p k ) - g ) t
C k , r sin p k x Î H N - 1
k=1
and wr , N ( t ) can be estimated (cf. (6.17)) as follows:
r
¶x wr , N ( t )
2
æ - ------ x ö
£ exp { -2 ( n ( p N ) 2 - g ) t } ¶ x ç e 2n u0÷
è
ø
2
.
(6.19)
Thus, it is clear that
dist
1
H0 ( 0 , 1 )
( St u0 , HN -1 ) £ C ( r , n ) ¶x wr , N ( t ) .
Consequently, estimate (6.15) is valid. Theorem 6.1 is proved.
In particular, Theorem 6.1 means that if g º G - r 2 ¤ ( 4 n ) < n p 2 , then the global attractor of problem (6.1) consists of a single zero element, whereas if g ³ n p 2 , the
attractor lies in an exponentially attracting invariant subspace HN , where N0 =
0
= [ ( 1 ¤ p ) g ¤ n ] and [ . ] is a sign of the integer part of a number.
The following assertion shows that the global minimal attractor of problem
(6.1) consists of fixed points of the system. It also provides a description of the corresponding basins of attraction.
Theorem 6.2.
r2
Let g º G - ------ > n p 2 . Assume that the number p -1 g ¤ n is not integer.
4n
Suppose that N0 is the greatest integer such that n ( p N0 ) 2 < g . Let
Explicitly Solvable Model of Nonlinear Diffusion
1
Cj ( u0 ) =
2
ò
r
- ------ x
e 2n u 0 ( x ) sin p j x d x
0
and let u ( t ) be a mild solution to problem (6.1)..
(a) If Cj ( u0 ) = 0 for all j = 1 , 2 , ¼ , N0 , then
¶x u ( t )
£ C ( n , r ) exp { -( n p 2 ( N0 + 1 ) 2 - g ) t } ¶x u0 ,
t ³ 0 . (6.20)
(b) If Cj ( u0 ) ¹ 0 for some j between 1 and N0 , then there exist positive
numbers C = C ( n , r , g ; u 0 ) and b = b ( g , n ) such that
¶x ( u ( t ) - us k ) £ C e -b t ,
t ³ 0,
(6.21)
where u±k ( x ) is defined by formula (6.11),, s = sign Ck ( u 0 ) , and k is
the smallest index between 1 and N0 such that Ck ( u0 ) ¹ 0 .
Proof.
In order to prove assertion (a) it is sufficient to note that the value hN ( t ) is
identically equal to zero in decomposition (6.18) when N = N0 + 1 . Therefore,
(6.20) follows from (6.19).
Now we prove assertion (b). In this case equation (6.18) can be rewritten in the
form
w ( t ) = gk ( t ) + h k ( t ) + wr , N +1 ( t ) ,
0
(6.22)
where
r
gk ( t ) =
hk ( t ) =
2
------ x
2 e 2n e -[ n ( p k ) - g ] t Ck ( u0 ) sin p k x ,
N0
r
------ x
2n
2e
å
2
e -[ n ( p j ) - g ] t C j , r sin p j x ,
j=k+1
and hk ( t ) º 0 if k = N 0 . Moreover, the estimate
¶x wr ,
N0 + 1 ( t )
£ C (r, n) e
2
-( n p 2 ( N0 + 1 ) - g ) t
¶x u 0
(6.23)
is valid for wr , N + 1 ( t ) . It is also evident that
0
¶x hk ( t ) £ C ( r , n ) e ( g -n p
2 (k + 1)2) t
¶x u0 .
(6.24)
Since
w ( t ) 2 - gk ( t ) 2 £ ( w ( t ) + gk ( t ) ) w ( t ) - gk ( t )
£ ( 2 gk ( t ) + hk ( t ) + wr ,
N0 + 1 ( t )
) ( hk ( t ) + wr ,
£
N0 + 1 ( t )
) ,
using (6.23) and (6.24) we obtain
w ( t ) 2 - gk ( t ) 2 £ C exp { 2 [ g - n p 2 ( k 2 + ( k + 1 ) 2 ) ] t } ¶x u 0 2 .
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Long-Time Behaviour of Solutions to a Class of Semilinear Parabolic Equations
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Integration gives
t
2š
ò g (t)
2
k
d t = 4 š [ Ck ( u0 ) ]
2
0
1 r
t
--- x
2
n
e sin p k x d x
ò
òe
0
2
2 (g - n (p k) ) t
dt =
0
C k ( u 0 )ö 2 2 ( g - n ( p k ) 2 ) t
= 2 æ -----------------(e
- 1) ,
è mk ø
2
where m k is defined by formula (6.12). Hence,
t
1 + 2š
ò
C k ( u0 )ö 2 2 ( g - n ( p k ) 2 ) t
+ ak ( t ) ,
w ( t ) 2 d t = 1 + 2 æ -----------------e
è mk ø
(6.25)
0
where
ì
k2 + ( k + 1 ) 2 ü
ak ( t ) £ C 1 + exp í 2 æ g - n p 2 --------------------------------- ö t ý
è
ø
2
î
þ
¶x u0
2
,
k £ N0 - 1 .
Let
gk ( t )
-.
vk ( t ) = ------------------------------------------------------------t
æ
ö 1/2
ç 1 + 2 š w ( t ) 2 d t÷
ç
÷
è
ø
0
ò
We consider the case when 1 £ k £ N 0 - 1 . Equations (6.23)–(6.25) imply that
¶ x ( u ( t ) - vk ( t ) )
£
¶ x wr ,
N0 + 1( t )
ì -( n p 2 ( N0 + 1 )
£ C ( r , n ) ¶x u0 í e
î
2
¶ x hk ( t )
+ ------------------------------------------------------------t
æ
ö 1/2
ç 1 + 2 š w ( t ) 2 d t÷
ç
÷
è
ø
0
- g) t
£
ò
+
exp { ( g -n p 2 ( k + 1 ) 2 ) t }
ü
+ -------------------------------------------------------------------------------------------------------------- ý .
2
1/2
2
þ
k ( u 0 )ö
æ1 + 2 æ C
----------------- e 2 ( g - n ( p k ) ) t + a k ( t )ö
è mk ø
è
ø
Therefore,
¶x ( u ( t ) - vk ( t ) )
£
2
2
2
ì - ( n p2 ( N0 + 1 ) 2 - g ) t
ü
+ e -n p ( ( k + 1 ) - k ) t G ( t ) -1 / 2 ý ,
£ C ( r , n , u0 ) í e
î
þ
Explicitly Solvable Model of Nonlinear Diffusion
where
C k ( u0 )ö 2
2 2
G ( t ) = ( 1 + ak ( t ) ) e -2 ( g - n p k ) t + 2 æ -----------------.
è mk ø
It follows from (6.25) that G ( t ) > 0 for all t ³ 0 and G ( 0 ) = 1 . Moreover, the
above-mentioned estimate for ak ( t ) enables us to state that
Ck ( u0 )ö 2
> 0.
lim G ( t ) = 2 æ ----------------è mk ø
t®¥
This implies that there exists a constant Gmin = Gmin ( g , k , n , u0 ) > 0 such that
G ( t ) ³ Gmin for all t ³ 0 . Consequently,
¶x ( u ( t ) - vk ( t ) )
£ C ( r , n , g ; u0 ) e -a t ,
(6.26)
where a = a ( k , n , g ) = min { n p 2 ( N 0 + 1 ) 2 - g , n p 2 ( 2 k + 1 ) } . Now we consider
the value vk ( t ) . Evidently,
vk ( t ) = jk ( t , u0 ) us k ( x ) ,
where
2 Ck ( u 0 ) exp { ( g -n p 2 k 2 ) t }
jk ( t , u 0 ) = ------------------------------------------------------------------------------------------------------------------------------------- .
1/2
Ck ( u0 )ö 2
ö
2 2
m k æè 1 + 2 æè ----------------m k ø exp { 2 ( g -n p k ) t } + ak ( t )ø
Simple calculations give us that
jk ( t , u0 ) - 1 £ C ( r , n , g , u0 ) e -b t
with the constant b = n p 2 ( 2 k + 1 ) . This and equation (6.26) imply (6.21), provided k £ N0 - 1 . We offer the reader to analyse the case when k = N0 on his / her
own. Theorem 6.2 is proved.
Theorem 6.2 enables us to obtain a complete description of the basins of attraction of
each fixed point of the dynamical system ( H 01 ( 0 , 1 ) , St ) generated by problem (6.1).
Corollary 6.1.
Let g º G - r 2 ¤ ( 4 n ) > n p 2 . Assume that the number p -1 g ¤ n is not integer. Let N 0 be the greatest integer possessing the property n ( p N0 ) 2 < g .
We denote
1
Cj ( u ) =
2
ò
r
- ------- x
e 2 n u ( x ) sin p j x d x
0
and define the sets
D k = { u Î H01 ( 0 , 1 ) : Cj ( u ) = 0 , j = 1 , ¼ , k - 1 , Ck ( u ) > 0 } ,
D -k = { u Î H01 ( 0 , 1 ) : Cj ( u ) = 0 , j = 1 , ¼ , k - 1 , Ck ( u ) < 0 }
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for k = 1 , 2 , ¼ , N0 . We also assume that
D 0 = { u Î H 01 ( 0 , 1 ) : Cj ( u ) = 0 , j = 1 , 2 , ¼ , N0 } .
Then for any l = 0 , ±1 , ±2 , ¼ , ±N0 we have that
= 0 Û v ÎDl ,
lim St v -ul 1
H0 ( 0 , 1 )
t®¥
where { u l } are the fixed points of problem (6.1) which are defined by
equalities (6.11).
The next assertion gives us a complete description of the global attractor of problem
(6.1).
Theorem 6.3.
Assume that the hypotheses of Theorem 6.2 hold and N0 is the same as
in Theorem 6.2.. Then the global attractor A of the dynamical system
( H01 ( 0 , 1 ) , St ) generated by equation (6.1) is the closure of the set
r
ì
ü
------ x
N0
ï
ï
x k e 2n sin p k x
2
k=1
ï
- : xj Î R ý ,
A = ïí v ( x ) = ------------------------------------------------------------------------------------ak j 1 / 2
N0
æ1 + 2 š
ï
ï
x x -----------------ö
è
ï
ï
k , j = 1 k j n k + n jø
î
þ
å
å
(6.27)
where nk = g - n ( p k ) 2 , k = 1 , 2 , ¼ , N0 , and
ak j = 2
1 r
--- x
e n sin p k x × sin p j x d x ,
ò
k , j = 1 , 2 , ¼ , N0 .
0
Every complete trajectory { u ( t ) : t Î R } which lies in the attractor and
does not coincide with any of the fixed points u k , k = 0 , ±1 , ¼ , ± N0 , has
the form
å
å
N0
nk t
r
------ x
2n
e
sin p k x
x e
2
k=1 k
- ,
u ( t ) = ---------------------------------------------------------------------------------------------------------------1/2
ak j
N0
(n + n ) t
æ1 + 2 š
xk xj ----------------- e k j ö
è
ø
nk + nj
k, j = 1
(6.28)
where x k are real numbers, k = 1 , 2 , ¼ , N0 , t Î R .
E x e r c i s e 6.6
Show that for all x Î R
a (t, x) = 2 š
å
N0
k,
N0
the function
ak j ( n + n ) t
k
j ,
xk xj ---------------nk + nj e
j=1
t ³ 0
(6.29)
is nonnegative and it is monotonely nondecreasing with respect to t
(Hint: at¢ ( t , x ) ³ 0 and a ( t , x ) ® 0 as t ® - ¥ ).
Explicitly Solvable Model of Nonlinear Diffusion
Proof of Theorem 6.3.
Let h belong to the set A given by formula (6.27). Then by virtue of Lemma 6.1
we have
w (x, t)
St h = -------------------------------------------------------------- ,
1/2
t
æ
ö
ç 1 + 2 š w ( t ) 2 d t÷
ç
÷
è
ø
0
(6.30)
ò
where
r
w(t) =
------ x
2
----------------------------e 2n
1 + a (0, x)
N0
åx
ke
nk t
sin p k x
k=1
and the value a ( t , x ) is defined according to (6.29). Simple calculations show that
t
2š
ò w(t)
2
a (t , x ) - a( 0 , x )
d t = --------------------------------------------- .
1 + a(0, x)
0
Therefore, it is easy to see that St h = u ( t ) for t ³ 0 , where u ( t ) has the form (6.28).
It follows that St A = A and that a complete trajectory u ( t ) lying in A has the form
(6.28). In particular, this means that A Ì A . To prove that A = A it is sufficient to
verify using Theorem 6.1 and the reduction principle (see Theorem 1.7.4) that for
any element h Î HN there exists a semitrajectory v ( t ) Ì A such that
0
lim ¶ x ( St h - v ( t ) ) = 0
t®¥
uniformly with respect to h from any bounded set in HN . To do this, it should be
0
kept in mind that for
N0
h=
2
å
r
------ x
xk e 2n sin p k x Î HN
0
k=1
St h has the form (6.30) with
w (t) =
2
N0
r
------ x
e 2n
åx
ke
nk t
sin p k x .
k=1
Therefore, it is easy to find that
w (t)
St h = ------------------------------------------------------------------ ,
(1 + a (t, x) - a (0, x) )1/2
where a ( t , x ) is given by formula (6.29). Hence, if we choose
w (t)
v ( t ) = ---------------------------------------- Î A ,
(1 + a (t, x) )1/2
we find that
(6.31)
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St h - v ( t ) = y ( t , h ) St h ,
(6.32)
where
a (0, x) 1/2
y ( t , h ) = 1 - æè 1 - ----------------------------öø .
1 + a (t, x)
Using the obvious inequality
1 - (1 - x)1/2 £ x ,
0 £ x £ 1,
we obtain that
a (0, x)
y ( t , h ) £ ---------------------------- ,
1 + a (t, x)
provided that h has the form (6.31). It is evident that
t
a ( t , x ) ³ a ( 0 , x ) + c0
òe
2 nN t
0
N0
dt ×
åx
2
k
.
k=1
0
Therefore,
æ t 2n t ö
N0
y (t, h) £ C ç e
d t÷
ç
÷
è0
ø
ò
-1
.
Consequently, the dissipativity property of St in H10 ( 0 , 1 ) and equations (6.32) give
us that
æ t 2n t ö
N0
¶x ( St h - v ( t ) ) £ C ç e
d t÷
ç
÷
è0
ø
ò
-1
,
t ³ tB
for all h Î B Ì HN0 , where B is an arbitrary bounded set. Thus, A = A and therefore Theorem 6.3 is proved.
Show that the set A coincides with the unstable manifold
M+ ( 0 ) emanating from zero, provided that the hypotheses of Theorem 6.3 hold.
E x e r c i s e 6.7
E x e r c i s e 6.8 Show that the set A = M + ( 0 ) from Theorem 6.3 can be described as follows:
N0
ì
r
------ x
ï
A = ív = 2
h k e 2n sin p k x :
ï
k=1
î
å
æ
ç2 š
è k,
ü
ak j
N0ö ï
h k h j ----------------- < 1 , h Î R ÷ ý .
nk + nj
øï
j=1
þ
N0
å
Explicitly Solvable Model of Nonlinear Diffusion
Therewith, the global attractor A has the form
N0
N0
ì
ü
r
------ x
ak j
ï
ï
2n
hk e
h k h j --------------sin p k x : 2 š
A = ív = 2
nk + nj £ 1 ý .
ï
ï
k=1
k, j = 1
î
þ
å
E x e r c i s e 6.9
å
Prove that the boundary
ì
N0
r
------ x
ï
2n
¶A = ív = 2
hk e
sin p k x :
ï
k=1
î
å
æ
ç2 š
è k,
ü
ak j
N öï
- = 1 , h Î R 0÷ ý
h k h j ---------------nk + nj
øï
j=1
þ
N0
å
of the set A is a strictly invariant set.
E x e r c i s e 6.10 Show that any trajectory g lying in ¶A has the form { u ( t ),
t Î R } , where
N0
å
hk e
nk t
r
------ x
e 2n sin p k x
k=1
u ( t ) = ------------------------------------------------------------------------------------------------,
N0
1/2
ak j
æ
ö
(n + n ) t
-e k j ÷
šç
h k h j ---------------n
+
n
k
j
è
ø
h ÎR
N0
.
å
k, j = 1
E x e r c i s e 6.11 Using the result of Exercise 6.10 find the unstable manifold
M+( u k ) emanating from the fixed point uk , k = ±1 , ±2 , ¼ , ±N0 .
E x e r c i s e 6.12 Find out for which pairs of fixed points { u k , uj } , k , j =
= ±1 , ±2 , ¼ , ±N0 , there exists a heteroclinic trajectory connecting them, i.e. a complete trajectory { uk j ( t ) : t Î R } such that
uk =
lim u k j ( t ) ,
t ® -¥
uj =
lim uk j ( t ) .
t ® +¥
E x e r c i s e 6.13 Display graphically the global attractor A on the plane generated by the vectors e 1 = e [ r ¤ ( 2 n ) ] x sin p x and e2 = e [ r ¤ ( 2 n ) ] x ´
´ sin 2 p x for N0 = 2 .
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E x e r c i s e 6.14 Study the structure of the global attractor of the dynamical
system generated by the equation
ì
æ 1
ö
ï
ï u - n u + ç š ( u ( x , t ) 2 + v ( x , t ) 2 ) d x - G÷ u - a v = 0, 0 < x < 1 , t > 0 ,
xx ç
÷
ï t
è 0
ø
ï
ï
í
æ 1
ö
ï
2
2
ç
÷
ï vt - n vx x + ç š ( u ( x , t ) + v ( x , t ) ) d x - G÷ v + a u = 0, 0 < x < 1 , t > 0 ,
ï
è 0
ø
ï
ïu
= u x = 1 = v x = 0 = v x = 1 =0
î x=0
ò
ò
in H01 ( 0 , 1 ) ´ H 01 ( 0 , 1 ) , where n , š , G , a are positive parameters.
§ 7
Simplified Model of Appearance
of Turbulence in Fluid
In 1948 German mathematician E. Hopf suggested (see the references in [3]) to consider the following system of equations in order to illustrate one of the possible scenarios of the turbulence appearance in fluids:
u = m ux x - v * v - w * w - u * 1 ,
ì t
ï v = mv + v u + v a + w b ,
*
*
*
xx
í t
ï w = mw + w u -v b + w a ,
*
*
*
xx
î t
(7.1)
(7.2)
(7.3)
where the unknown functions u , v , and w are even and 2 p -periodic with respect
to x and
2p
1( f * g ) ( x ) = -----2p
ò f (x - y) g (y) dy .
0
Here a ( x ) and b ( x ) are even 2 p -periodic functions and m is a positive constant.
We also set the initial conditions
u t = 0 = u 0 ( x ) , v t = 0 = v0 ( x ) , w t = 0 = w0 ( x ) .
(7.4)
As in the previous section the asymptotic behaviour of solutions to problem
(7.1)–(7.4) can be explicitly described.
Let us introduce the necessary functional spaces. Let
2
( R ) : f ( x ) = f ( -x ) = f ( x + 2 p ) } .
H = { f Î Lloc
Simplified Model of Appearance of Turbulence in Fluid
Evidently H is a separable Hilbert space with the inner product and the norm defined by the formulae:
2p
( f, g) =
ò f (x) g(x) dx ,
f 2 = ( f, f )
0
There is a natural orthonormal basis
ì 1
ü
1
1
í ----------- , ------- cos x , ------- cos 2 x , ¼ ý
p
p
p
2
î
þ
in this space. The coefficients Cn ( f ) of decomposition of the function f Î H with
respect to this basis have the form
2p
1C0 ( f ) = ---------2p
E x e r c i s e 7.1
ò
2p
f (x) dx ,
0
1Cn ( f ) = -----p
ò f (x) cos n x dx .
0
Let f , g Î H . Then f * g Î H and
1- f × g .
f * g £ ---------2p
(7.5)
The Fourier coefficients Cn of the functions f * g , f , and g obey
the equations
1 - C ( f ) C (g) ,
C0 ( f * g ) = ---------0
0
2p
1 - C ( f ) C ( g) . (7.6)
Cn ( f * g ) = ---------n
n
2 p
Let pm be the orthoprojector onto the span of elements
{ cos k x , k = 0 , 1 , ¼ , m } in H . Show that
E x e r c i s e 7.2
pm ( f * g ) = ( pm f ) * g = f * ( pm g ) .
(7.7)
Let us consider the Hilbert space H = H 3 º H ´ H ´ H with the norm ( u ; v ; w ) H =
= ( u 2 + v 2 + w 2 ) 1 / 2 as the phase space of problem (7.1)–(7.4). We define
an operator A by the formula
A ( u , v , w ) = ( u - m ux x ; v - m vx x ; w - m wx x ) ,
(u ; v ; w) Î D (A)
on the domain
3
2
(R) ] Ç H ,
D ( A) = [ Hloc
2
where H loc
( R ) is the second order Sobolev space.
E x e r c i s e 7.3 Prove that A is a positive operator with discrete spectrum. Its
eigenvalues { ln }¥
have the form:
n=0
l3 k = l3k + 1 = l 3k + 2 = 1 + m k 2 ,
k = 0, 1, 2, ¼ ,
131
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Long-Time Behaviour of Solutions to a Class of Semilinear Parabolic Equations
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while the corresponding eigenelements are defined by the formulae
1 - ( 1 ; 0 ; 0 ) , e = ---------1 - ( 0 ; 1 ; 0 ) , e = ---------1 - (0 ; 0 ; 1) ,
e 0 = ---------1
2
2p
2p
2p
1
1
e 3 k = ------- ( cos k x ; 0 ; 0 ) , e 3 k + 1 = ------- ( 0 ; cos k x ; 0 ) ,
p
p
1 - ( 0 ; 0 ; cos k x ) ,
e 3 k + 2 = -----p
2
ü
ï
ï
ï
ý (7.8)
ï
ï
ï
þ
where k = 1 , 2 , ¼
Let
b1 ( u , v , w ) = -v * v - w * w - u * 1 ,
b2 ( u , v , w ) = v * u + v * a + w * b ,
b3 ( u , v , w ) = w * u - v * b + w * a .
Equation (7.5) implies that bj ( u , v , w ) Î H , provided a , b , u , v , and w are the
elements of the space H , j = 1 , 2 , 3 . Therefore, the formula
B ( u , v , w ) = ( b1( u , v , w ) + u ; b2 ( u , v , w ) + v ; b3 ( u , v , w ) + w )
gives a continuous mapping of the space H into itself.
E x e r c i s e 7.4
Prove that
B ( y1) - B ( y2 )
H
£ C æ 1 + a + b + y1
è
H
+ y2 Hö y1 - y2
ø
H
,
where yj = ( uj ; vj ; wj ) Î H , j = 1 , 2 .
Thus, if a , b Î H , then problem (7.1)–(7.4) can be rewritten in the form
dy
------ + A y = B ( y ) ,
y t = 0 = y0 ,
dt
where A and B satisfy the hypotheses of Theorem 2.1. Therefore, the Cauchy problem (7.1)–(7.4) has a unique mild solution y ( t ) = ( u ( t ) , v ( t ) , w ( t )) in the space
H on a segment [ 0 , T ] , provided that a , b Î H . In order to prove the global existence theorem we consider the Galerkin approximations of problem (7.1)–(7.4).
The Galerkin approximate solution ym ( t ) of the order 3 m with respect to basis
(7.8) can be presented in the form
ym ( t ) = æ u ( m ) ( t ) ; v ( m ) ( t ) ; w ( m ) ( t )ö =
è
ø
m -1
å æèu (t) ; v (t) ; w (t)öø cos k x ,
k
k
k
(7.9)
k=0
where uk ( t ) , vk ( t ) , and wk ( t ) are scalar functions. By virtue of equations (7.7)
it is easy to check that the functions u ( m ) ( t ) , v ( m ) ( t ) , and w ( m ) ( t ) satisfy equations (7.1)–(7.3) and the initial conditions
u( m ) ( 0 ) = pm u0 ,
v ( m ) ( 0 ) = pm v0 ,
w ( m ) ( 0 ) = pm w0 .
(7.10)
Simplified Model of Appearance of Turbulence in Fluid
Thus, approximate solutions exist, locally at least. However, if we use (7.1)–(7.3) we
can easily find that
ü
d ì
1
--- × ---- í u ( m ) ( t ) 2 + v ( m ) ( t ) 2 + w ( m ) ( t ) 2 ý +
2 dt î
þ
ì
+ m í u x( m )
î
2
+ vx( m )
2
ü
+ wx( m ) 2 ý =
þ
= -( u( m) * 1 , u( m ) ) + ( v( m ) * a , v ( m ) ) + ( w ( m ) * a , w( m ) ) .
Therefore, inequality (7.5) leads to the relation
d
---- y ( t ) H2 £ C ( 1 + a ) ym ( t ) H2 .
dt m
This implies the global existence of approximate solutions ym ( t ) (see Exercise 2.1).
Therefore, Theorem 2.2 guarantees the existence of a mild solution to problem
(7.1)–(7.4) in the space H = H 3 on the time interval of any length T . Moreover, the
mild solution y ( t ) = ( u ( t ) ; v ( t ) ; w ( t ) ) possesses the property
max y ( t ) - ym ( t ) ® 0 ,
[0, T ]
m®¥
for any segment [ 0 , T ] . Approximate solution ym ( t ) has the structure (7.9).
E x e r c i s e 7.5 Show that the scalar functions { uk( t ) ; v k( t ) ; wk( t ) } involved
in (7.9) are solutions to the system of equations:
ì
ï
ï
í
ï
ï
î
u· k + m k 2 u k = - vk2 - wk2 - u 0 d k 0 ,
v· k + m k 2 vk = vk uk + vk a k + wk bk ,
w· k + m k 2 wk = wk u k -vk bk + wk ak .
(7.11)
(7.12)
(7.13)
Here k = 1 , 2 , ¼ , d k0 = 0 for k ¹ 0 , d 00 = 1 , the numbers ak =
= Ck ( a ) and bk = Ck ( b ) are the Fourier coefficients of the functions a ( x ) and b ( x ) .
Thus, equations (7.1)–(7.3) generate a dynamical system ( H , St ) with the evolutionary operator St defined by the formula
St y0 = ( u ( t ) ; v ( t ) ; w ( t ) ) ,
where ( u ( t ) ; v ( t ) ; w ( t ) ) is a mild solution (in H ) to the Cauchy problem (7.1)–
(7.4), y0 = ( u0 , v0 , w0 ) . An interesting property of this system is given in the following exercise.
Let Lk be the span of elements { e3 k , e3 k + 1 , e 3 k + 2 } , where
k = 0 , 1 , 2 , ¼ and { en } are defined by equations (7.8). Then the
subspace Lk of the phase space H is positively invariant with respect to St ( St Lk Ì Lk ) .
E x e r c i s e 7.6
133
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Long-Time Behaviour of Solutions to a Class of Semilinear Parabolic Equations
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Therefore, the phase space H of the dynamical system ( H , St ) falls into the orthogonal sum
2
¥
H=
åÅL
k
k=0
of invariant subspaces. Evidently, the dynamics of the system ( H , St ) in the subspace Lk is completely determined by the system of three ordinary differential
equations (7.11)–(7.13).
Lemma 7.1.
Assume that a , b Î H and
2p
1C0 = ---------2p
ò a (x) dx < 0 .
(7.14)
0
Then the dynamical system ( H , St ) is dissipative.
Proof.
Let us fix N and then consider the initial conditions y0 = ( u0 , v0 , w0 )
from the subspace
N
HN =
å
k=0
N
Å Lk =
å Å Lin {e
3 k,
e3 k + 1 , e3 k + 2 } ,
k=0
where { e n } are defined by equations (7.8). It is clear that H N is positively invariant and the trajectory
N
y (t) = (u (t) ; v (t) ; w (t) ) =
å (u (t) ; v (t) ; w (t) ) cos k x
k
k
k
k=0
of the system is a function satisfying (7.1)–(7.4) in the classical sense. Let
pm be the orthoprojector in H onto the span of elements { cos k x : n = 0 , 1 , ¼,
˜
m } . We introduce a new variable u ( t ) = u ( t ) + a m instead of the function
u ( t ) . Here a m = ( pm - p0 ) a . Equations (7.1)–(7.3) can be rewritten in the
form
ì˜
˜
˜
m
m
ï ut - m ux x = - v * v - w * w + ( - u + a ) * 1 - m ax x ,
ï
˜ + v (q a) + w b + v (p a) ,
í vt - m vx x = v * u
* m
*
* 0
ï
˜
ï w -mw
x x = w * u - v * b + w * ( qm a ) + w * ( p0 a ) ,
î t
(7.15)
(7.16)
(7.17)
where qm = 1 - pm . The properties of the convolution operation (see Exercises 7.1 and 7.2) enable us to show that
Simplified Model of Appearance of Turbulence in Fluid
d ˜
˜
1
--- ---- ( u 2 + v 2 + w 2 ) + m ( u
x
2 dt
˜
˜
m
m
= ( ( - u + a ) * 1 , u ) + m ( ax ,
2
+ vx
2
+ wx 2 ) =
˜ ) + (v (q a) , v) +
u
* m
x
+ ( w * ( qm a ) , w ) + ( v * ( p0 a ) , v ) + ( w * ( p0 a ) , w ) .
(7.18)
It is clear that
˜ ) = -[ C ( u
˜ + am) 1 , u
˜ ) ]2 ,
( ( -u
*
0
1 - C ( a ) [ C ( v) ] 2 ,
( v * ( p0 a ) , v ) = ---------0
0
2p
where C0 ( f ) is the zeroth Fourier coefficient of the function f ( x ) Î H . Moreover, the estimate
qm v £ vx ,
m ³ 1,
holds. We choose m ³ 1 such that ( 1¤ 2 p ) qm a £ m ¤ 2 . Then equation (7.18)
implies that
d ˜
---- æ u 2 + v 2 +
dt è
˜
+ 2 C0 ( u ) 2 +
˜
w 2ö + m æ ux 2 + vx 2 + wx 2ö +
ø
è
ø
2
2ö
m 2
2
æ
-p- C0 ( a ) è C0 ( v ) + C0 ( w ) ø £ m ax .
If we use the inequality
f 2 £ C0 ( f ) 2 + fx 2 ,
then we find that
d ˜
---- ( u 2 + v 2 + w 2 ) + n ( u 2 + v 2 + w 2 ) £ m axm 2 ,
dt
where n = min( m , 2 ,
2 ¤ p C0 ( a ) ) . Consequently, the estimate
˜
y ( t ) H2 £
m
˜
m 2
-n t
y ( 0 ) H2 e -n t + --n ( 1 - e ) ax
(7.19)
˜
˜
˜
is valid for y ( t ) = ( u ( t ) , v ( t ) , w ( t )) , provided qm a £ m p ¤ 2 and y0 = y0 +
+ ( a m , 0 , 0 ) where y0 Î HN . By passing to the limit we can extend inequality
(7.19) over all the elements y0 Î H . Thus, the system ( H , St ) possesses an absorbing set
Bm = { ( u , v , w ) :
2
},
u + ( pm - p0 ) a 2 + v 2 + w 2 £ R m
(7.20)
where m is such that a - pm a £ m p ¤ 2 and
2
= m [ ( pm - p0 ) a ] x 2 æminæ m , 2 ,
Rm
è
è
--2- C0 ( a ) ö ö
øø
p
-1
+1.
E x e r c i s e 7.7 Show that the ball Bm defined by equality (7.20) is positively
invariant.
135
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Long-Time Behaviour of Solutions to a Class of Semilinear Parabolic Equations
E x e r c i s e 7.8 Consider the restriction of the dynamical system ( H , S t ) to
the subspace
ì
˜
H = ïí h ( x ) = ( u ( x ) ; v ( x ) ; w ( x ) ) Î H ,
ï
î
2p
ò
0
ü
ï
h (x) dx = 0 ý =
ï
þ
åL
n.
n ³ 1
˜
Show that ( H , St ) is dissipative not depending on the validity of
condition (7.14).
Lemma 7.2.
Assume that the hypotheses of Lemma 7.1 hold and let pm be the orthoprojector onto the span of elements { cos k x : k = 0 , 1 , ¼ , m } in H ,
qm = 1 - pm . Then the estimate
qm a ö
ì
2
£ y0 H2 exp í -2 æ ( m + 1 ) 2 m - -------------ym ( t ) H
è
2p ø
î
ü
tý
þ
(7.21)
holds for all m such that qm a < m 2 p ( m + 1 ) 2 . Here ym ( t ) = ( qm u ( t ) ;
qm v ( t ) ; qm w ( t ) ) and ( u ( t ) ; v ( t ) ; w ( t ) ) is the mild solution to problem
(7.1)–(7.4) with the initial condition y0 = ( u 0 ; v0 ; w0 ) .
Proof.
As in the proof of Lemma 7.1 we assume that y0 = ( u0 ; v0 ; w0 ) Î H N for
some N . If we apply the projector qm to equalities (7.15)–(7.17), then we get
the equations
ì u m - m u m = - vm vm - wm wm ,
*
*
xx
ï t
ï m
m
í vt - m vx x = v m * u m + w m * b + v m * qm a ,
ï
ï w m - m w m = wm um -vm b + wm q a ,
*
*
* m
xx
î t
where u m = qm u , v m = qm v and w m = qm w . Therefore, we obtain
d
2
1
--- ---- æ u m 2 + v m 2 + w m 2ö + m æ um
+ vxm 2 + wxm 2ö £
ø
è x
ø
2 dt è
1 - q a æ vm
£ ---------m è
2p
2
+ w m 2ö
ø
(7.22)
as in the proof of the previous lemma. It is easy to check that ( qm h ) x 2 ³
³ ( m + 1 ) 2 h 2 for every h Î pN H . Therefore, inequality (7.22) implies that
d
1
1 - q a ö y (t) 2 £ 0 .
--- ---y ( t ) H2 + æ m ( m + 1 ) 2 - ---------m ø m
H
è
2 dt m
2p
Hence, equation (7.21) is valid. Lemma 7.2 is proved.
Simplified Model of Appearance of Turbulence in Fluid
Lemmata 7.1 and 7.2 enable us to prove the following assertion on the existence
of the global attractor.
Theorem 7.1.
Let a , b Î H and let condition (7.14) hold. Then the dynamical system
( H , St ) generated by the mild solutions to problem (7.1)–(7.4) possesses a
global attractor A m . This attractor is a compact connected set. It lies in the
finite-dimensional subspace
N
HN =
å Å Lin {e
3 k,
e3 k + 1 , e3 k + 1 } ,
k=0
where the vectors { e n } are defined by equalities (7.8) and the parameter N
is defined as the smallest number possessing the property qN a <
< m 2 p ( N + 1 ) 2 . Here qN is the orthoprojector onto the subspace generated
by the elements { cos n x : n ³ N + 1 } in H .
To prove the theorem it is sufficient to note that the dynamical system is compact
(see Lemma 4.1). Therefore, we can use Theorem 1.5.1. In particular, it should be
noted that belonging of the attractor A m to the subspace HN means that
dimf A m £ 3 ( N + 1 ) , where N is an arbitrary number possessing the property
qN a < m 2 p ( N + 1 ) 2 . Below we describe the structure of the attractor and evaluate its dimension exactly.
According to Lemma 7.2 the subspace HN is a uniformly exponentially attracting
and positively invariant set. Therefore, by virtue of Theorem 1.7.4 it is sufficient
to study the structure of the global attractor of the finite-dimensional dynamical system ( HN , St ) . To do that it is sufficient to study the qualitative behaviour of the trajectory in each invariant subspace Lk , 0 £ k £ N (see Exercise 7.6). This
behaviour is completely described by equations (7.11)–(7.13) which get transformed into system (1.6.4)–(1.6.6) studied before if we take m = m k 2 + d k 0 ,
n = m k 2 - ak , and b = bk . Therefore, the results contained in Section 6 of Chapter 1
lead us to the following conclusion.
Theorem 7.2.
Let the hypotheses of Theorem 7.1 hold. Then the global minimal attrac(m)
tor A min
of the dynamical system ( H , St ) generated by mild solutions to
(m)
= { 0 } È S m , where
problem (7.1)–(7.4) has the form Amin
Sm =
È
È
k Î Jm ( a ) 0 £ j <
ì
cos k x ü
í ( uk ; rk cos j ; rk sin j ) ---------------- ý .
p þ
2 pî
Here u k = m k 2 - ak , rk = k ( a k m - m 2 k 2 ) 1 ¤ 2 , the values ak = Ck ( a ) are the
137
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Long-Time Behaviour of Solutions to a Class of Semilinear Parabolic Equations
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Fourier coefficients of the function a ( x ) , and the number k ranges over
the set of indices Jm ( a ) such that 0 < m k 2 < ak . Topologically S m is a torus
(i.e. a cross product of circumferences) of the dimension Card Jm( a ) . The
(m )
global attractor A m of the system ( H , St ) can be obtained from Amin
by attaching the unstable manifold M+ ( 0 ) emanating from the zero element of
the space H . Moreover, dim A m = 2 Card Jm ( a ) .
2
It should be noted that appearance of a limit invariant torus of high dimension possesssing the structure described in Theorem 7.2 is usually assosiated with the Landau-Hopf picture of turbulence appearance in fluids. Assume that the parameter m
gradually decreases. Then for some fixed choice of the function a ( x ) the following
picture is sequentially observed. If m is large enough, then there exists only one attracting fixed point in the system. While m decreases and passes some critical value
m 1 , this fixed point loses its stability and an attracting limit cycle arises in the system. A subsequent decrease of m leads to the appearance of a two-dimensional
torus. It exists for some interval of values of m : m 3 < m < m2 ( < m 1 ) . Then tori
of higher dimensions arise sequentially. Therefore, the character of asymptotic behaviour of typical trajectories becomes more complicated as m decreases. According
to the Landau-Hopf scenario, movement along an infinite-dimensional torus corresponds to the turbulence.
§ 8
On Retarded Semilinear
Parabolic Equations
In this section we show how the above-mentioned ideas can be used in the study of
the asymptotic properties of dynamical systems generated by the retarded perturbations of problem (2.1). It should be noted that systems corresponding to ordinary retarded differential equations are quite well-studied (see, e. g., the book by J. Hale [4]).
However, there are only occasional journal publications on the retarded partial differential equations. The exposition in this section is quite brief. We give the reader
an opportunity to restore the missing details independently.
As before, let A be a positive operator with discrete spectrum in a separable
Hilbert space H and let C ( a , b , Fq ) be the space of strongly continuous functions
on the segment [ a , b ] with the values in Fq = D ( A q ) , q ³ 0 . Further we also use
the notation C q = C ( - r , 0 ; F q ) , where r > 0 is a fixed number (with the meaning
of the delay time). It is clear that C q is a Banach space with the norm
ì
ü
v C º max í Aq v ( s ) : s Î [ -r ; 0 ] ý .
q
î
þ
On Retarded Semilinear Parabolic Equations
Let B be a (nonlinear) mapping of the space C q into H possessing the property
B ( v1 ) - B ( v2 ) £ M ( R ) v1 - v2 C ,
q
0 £ q < 1,
(8.1)
for any v1, v2 Î Cq such that vj C £ R , where R > 0 is an arbitrary number and M(R)
q
is a nondecreasing function. In the space H we consider a differential equation
du
------ + Au = B ( ut ) ,
t ³ t0 ,
(8.2)
dt
where ut denotes the element from C q determined with the help of the function
u ( t ) by the equality
ut ( s ) = u ( t + s ) ,
s Î [ -r , 0 ] .
We equip equation (8.2) with the initial condition
ut ( s ) = u ( t0 + s ) = v ( s ) ,
0
s Î [ -r , 0 ] ,
(8.3)
where v is an element from C q .
The simplest example of problem (8.2) and (8.3) is the Cauchy problem for the
nonlinear retarded diffusion equation:
ì
ï
í
ï
î
¶-----u
- Du = f 1 ( u ( t , x ) ) - f 2 ( u ( t - r , x ) ) ,
¶t
u ¶ W = 0 , u (s) s Î [ t - r, t ] = v (s) .
0
x Î W , t > t0 ,
(8.4)
0
Here r is a positive parameter, f1 ( u ) and f2 ( u ) are the given scalar functions.
As in the non-retarded case (see Section 2), we give the following definition.
A function u ( t ) Î C ( t 0 - r , t0 + T ; F q ) is called a mild (in F q ) solution
to problem (8.2) and (8.3) on the half-interval [ t0 , t0 + T ) if (8.3) holds and u ( t )
satisfies the integral equation
t
u (t) =
-( t - t0 ) A
e
v (0)
+
ò
e
- ( t - t0 ) A
B (u
t) dt
.
(8.5)
t0
The following analogue of Theorem 2.1 on the local solvability of problem (8.2)
and (8.3) holds.
Theorem 8.1.
Assume that (8.1) holds. Then for any initial condition v Î Cq there
exists T > 0 such that problem (8.2) and (8.3) has a unique mild solution
on the half-interval [ t0 , t0 + T ) .
Proof.
As in Section 2, we use the fixed point method. For the sake of simplicity we
consider the case t0 = 0 (for arbitrary t0 Î R the reasoning is similar). In the space
Cq ( 0 , T ) º C ( 0 , T ; F q ) we consider a ball
139
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Long-Time Behaviour of Solutions to a Class of Semilinear Parabolic Equations
Br = { w Î Cq ( 0 , T ) :
w - v̂ C ( 0 , T ) £ r } ,
q
where v̂ ( t ) = exp { - At } v ( 0 ) and v ( s ) Î C q is the initial condition for problem
(8.2) and (8.3). We also use the notation
w C ( 0 , T ) º max { Aq w ( t ) : t Î [ 0 , T ] } .
q
2
We consider the mapping K from Cq ( 0 , T ) into itself defined by the formula
t
[ K w ] ( t ) = v̂ ( t ) +
òe
-( t - t ) A B ( w ) d t
t
.
0
Here we assume that w ( s ) = v ( s ) for s Î [ - r , 0 ) . Using the estimate (see Exercise 1.23)
q- ö q ,
Aq e -s A £ æ ----è e sø
s > 0,
(8.6)
B ( w1 , t ) - B ( w2 , t ) d t .
(8.7)
(for q = 0 we suppose q q = 1 ) we have that
t
Aq ( K w1 ( t )
- K w2 ( t ) )
£
q -ö
ò æè ------------------e ( t - t )ø
q
0
If w Î B r , then
max Aq w ( t )
[0 , T]
£ r + Aq v ( 0 ) .
This easily implies that
wt C
q
£ r+ vC ,
q
t Î [ 0, T] ,
where, as above, wt Î C q is defined by the formula
s Î [ -r , 0 ] .
wt ( s ) = w ( t + s ) ,
Hence, estimate (8.1) for wj Î B r gives us that
B ( w1 , t ) - B ( w2 , t )
£ M ( r + v C ) w1 , t - w2 , t C .
q
q
Since w1( s ) = w2 ( s ) for s Î [ - r , 0 ) , the last estimate can be rewritten in
the form
B ( w1 , t ) - B ( w2 , t )
£ M ( r + v C ) w1 - w2 C
q
q (0, T)
for t Î [ 0 , T ] . Therefore, (8.7) implies that
q
1 - æq
--- ö T 1 - q M ( r + v C ) w 1 - w2 C ( 0 , T ) ,
Kw1 - K w2 C ( 0 , T ) £ ----------1 - q è eø
q
q
q
if wj Î B r , j = 1 , 2 . Similarly, we have that
q
ì
ü
1 - æq
--e- ö T 1 - q í B ( 0 ) + M ( r + v C ) ( r + v C ) ý
K w - v̂ C ( 0 , T ) £ ----------q
1-qè ø
q
q
î
þ
On Retarded Semilinear Parabolic Equations
for w Î B r . These two inequalities enable us to choose T = T ( q , r , v C ) > 0
q
such that K is a contractive mapping of Br into itself. Consequently, there exists
a unique fixed point w ( t ) Î C q ( 0 , T ) of the mapping K . The structure of the operator K implies that w ( + 0 ) = [ K w ] ( + 0 ) = v ( 0 ) . Therefore, the function
ì w (t) , t Î [ 0, T ] ,
u (t) = í
î v ( t ) , t Î [ -r , 0 ] ,
lies in C ( - r , T ; Fq ) and is a mild solution to problem (8.2), (8.3) on the segment
[ 0 , T ] . Thus, Theorem 8.1 is proved.
In many aspects the theory of retarded equations of the type (8.2) is similar to the
corresponding reasonings related to the problem without delay (see (2.1)). The exercises below partially confirm that.
E x e r c i s e 8.1 Prove the assertions similar to the ones in Exercises 2.1–2.5
and in Theorem 2.2.
E x e r c i s e 8.2 Assume that the constant M ( R ) in (8.1) does not depend
on R . Prove that problem (8.2) and (8.3) has a unique mild solution
on [ t0 , ¥ ) , provided v ( s ) Î Cq . Moreover, for any pair of solutions u1 ( t ) and u2 ( t ) the estimate
u1 ( t ) - u2 ( t ) q £ a1 e
a2 ( t - t0 )
v1 - v2 C
q
(8.8)
is valid, where vj ( s ) is the initial condition for u j ( t ) (see (8.3)).
For the sake of simplicity from now on we restrict ourselves to the case when the
mapping B has the form
B ( v ) = B0 ( v ( 0 ) ) + B1 ( v ) ,
v = v ( s ) Î C1 ¤ 2 ,
(8.9)
where B 0 ( . ) is a continuous mapping from F 1 ¤ 2 º D ( A 1 / 2 ) into H , B1 ( . ) continuously maps C 1 ¤ 2 into H and possesses the property B1( 0 ) = 0 . We also assume
that B 0 ( . ) is a potential operator, i.e. there exists a continuously Frechét differentiable function F ( u ) on F1 ¤ 2 such that B0 ( u ) = - F ¢ ( u ) . We require that
F ( u ) ³ -a ,
( F ¢(u) , u ) - b F(u ) ³ g u
2
-d
(8.10)
for all u Î F1 ¤ 2 , where a , b , g , and d are real parameters, b and g are positive
(cf. Section 2). As to the retarded term B1 ( v ) , we consider the uniform estimate
0
B1( v1 ) - B 2 ( v2 )
£ M
òA
1 / 2 (v (s)
1
- v2 ( s ) ) d s
-r
to be valid. Here M is an absolute constant and vj ( s ) Î C º C 1 ¤ 2 .
(8.11)
141
142
C
h
a
p
t
e
r
2
Long-Time Behaviour of Solutions to a Class of Semilinear Parabolic Equations
E x e r c i s e 8.3 Assume that conditions (8.9)–(8.11) hold. Then problem (8.2)
and (8.3) has a unique mild solution (in F1 ¤ 2 ) on any segment
[ t0 , t0 + T ] for every initial condition v from C = C1 ¤ 2 .
Therefore, we can define an evolutionary operator St acting in the space C º C1 ¤ 2
by the formula
( St v ) ( s ) = ut ( s ) º u ( t + s ) ,
s Î [ -r , 0 ] ,
(8.12)
where u ( t ) is a mild solution to problem (8.2) and (8.3).
E x e r c i s e 8.4 Prove that the operator S t given by formula (8.12) satisfies
the semigroup property: St ° St = St + t , S0 = I , t , t ³ 0 and the
pair ( C1 ¤ 2 , St ) is a dynamical system.
Theorem 8.2.
Let conditions (8.9)–(8.11) and (8.1) with q = 1 ¤ 2 hold. Assume that
the parameters in (8.10) and (8.11) satisfy the equation
r 2- ( 2 + g ) M 2 exp ì r × min ( 2 , g , b ) ü £ min ( 2 , g , b ) .
----í
ý
2g
î
þ
Then the dynamical system ( C , St ) generated by equality (8.12) is a dissipative compact system.
Proof.
We reason in the same way as in the proof of Theorem 4.3. Using the Galerkin
approximations it is easy to find that a solution to problem (8.2) and (8.3) satisfies
the equalities
d
1
--- ---u ( t ) 2 + A1 / 2 u ( t ) 2 + ( F ¢ ( u ( t ) ) , u ( t ) ) = ( B 1 ( ut ) , u ( t ) )
2 dt
and
d
1
--- ---- ( A1 / 2 u ( t ) 2 + 2 F ( u ( t ) ) ) + u· ( t ) 2 = ( B1 ( ut ) , u· ( t ) ) .
2 dt
If we add these two equations and use (8.10), then we get that
d
---- V ( u ( t ) ) + A1 / 2 u ( t ) 2 + g u ( t ) 2 + u· ( t ) 2 + b F ( u ( t ) ) £
dt
£ d + B ( u ) ( u ( t ) + u· ( t ) ) ,
1
t
(8.13)
where
--- u 2 + 1
--- A1 / 2 u 2 + F ( u ) + a .
V (u) = 1
2
2
Using (8.11) it is easy to find that there exists a constant D0 > 0 such that
(8.14)
On Retarded Semilinear Parabolic Equations
t
ò
d
--- w 2
---- V ( u ( t ) ) + w1V ( u ( t ) ) £ D 0 + 1
2
dt
A1 / 2 u ( t ) 2 d t ,
t -r
where
w 2 = --r- æè 1 + --2g-öø M 2 .
2
w 1 = min ( 2 , g , b ) ,
Consequently, the inequality
t
ò y (t) d t
· (t) + w y (t) £ D + w
y
1
0
2
t -r
is valid for y ( t ) = V ( u ( t ) ) . Therefore, we have
t
w t
w r
j· ( t ) £ D 0 e 1 + w 2 e 1
ò j (t) d t
t -r
for the function
j (t) = e
w1 t
y (t) = e
w1 t
V(u(t) ) .
If we integrate this inequality from 0 to t , then we obtain
t
D
w t
w r
j ( t ) £ j ( 0 ) + ------0- ( e 1 - 1 ) + w 2 e 1 r
w1
ò j (t) dt .
-r
Therefore, Gronwall’s lemma gives us that
V ( u ( t ) ) £ ( V ( u ( 0 ) ) + C 0 v C2 ) e
1¤ 2
-w3 t
+ C1 ,
provided that
w2 e
w 1r
r < w1 .
w r
Here C 1 and C2 are positive numbers, w 3 = w 1 - w 2 e 1 r > 0 . This implies
the dissipativity of the dynamical system ( C , St ) . In order to prove its compactness
we note that the reasoning similar to the one in the proof of Lemma 4.1 enables us
to prove the existence of the absorbing set B q which is a bounded subset of the
space C q = C ( - r , 0 ; D ( Aq ) ) for 1 ¤ 2 < q < 1 . Using the equality (cf. (8.5))
t
u(t) =
e -( t - s ) A u ( s )
+
òe
-( t - t ) A B ( u
t) dt
s
for t ³ s large enough, we can show that the equation
A1 / 2 ( u ( t ) - u ( s ) ) £ C t - s b
holds for the solution u ( t ) in the absorbing set B q . Here the constant C depends
on B q and the parameters of the problem only, b > 0 . This circumstance enables
143
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Long-Time Behaviour of Solutions to a Class of Semilinear Parabolic Equations
C
h
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us to prove the existence of a compact absorbing set for the dynamical system
( C , St ) . Theorem 8.2 is proved.
Theorem 8.2 and the results of Chapter 1 enable us to prove the following assertion
on the attractor of problem (8.2) and (8.3).
2
Theorem 8.3.
Assume that the hypotheses of Theorem 8.2 hold. Then the dynamical
system ( C , St ) possesses a compact connected global attractor A which is
a bounded set in the space Cq = C ( - r , 0 ; Fq ) for each q < 1 .
It should be noted that the finite dimensionality of this attractor can be proved
in some cases.
References
1.
2.
3.
4.
BABIN A. V., VISHIK M. I. Attractors of evolution equations. –Amsterdam:
North-Holland, 1992.
TEMAM R. Infinite-dimensional dynamical systems in Mechanics and
Physics. –New York: Springer, 1988.
HENRY D. Geometric theory of semilinear parabolic equations. Lect. Notes
in Math. 840. –Berlin: Springer, 1981.
HALE J. K. Theory of functional differential equations. –Berlin; Heidelberg;
New York: Springer, 1977.