I ntroduction Theory

I . D . C h u e s h o v

Dissipative Systems Infinite-Dimensional

I ntroduction Theory

of InfiniteDimensional Dissipative Systems 966–7021–64–5

O R D E R

www.acta.com.ua

I . D . C h u e s h o vC h u e s h o v

U n i v e r s i t y l e c t u r e s i n c o n t e m p o r a r y m a t h e m a t i c s

Dissipativeissipative Systemsystems

of Infinite-DimensionalInfinite-Dimensional

I ntroduction ntroduction

Theory Theory

of main ideas and methods of the theory of infinite-dimensional dis - sipative dynamical systems which has been rapidly developing in re - cent years. In the examples sys tems 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 scien - tific 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 com - posed of the courses given by the author at the Department of Me chanics and Mathematics at Kharkov University during a number of years. This book con - tains 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

Constantin I. Chueshov from the Russian edition(«ACTA», 1999)

Translation edited by Maryna B. Khorolska of InfiniteDimensional

of InfiniteDimensional Dissipative Systems Dissipative Systems ISBN: 966966–70217021–6464–5

You can O R D E R O R D E R this book while visiting the website

of «ACTA» Scientific Publishing House http://www.acta.com.uawww.acta.com.ua/en/

«A CT A » 2002

C h a p t e r 2

Long-Time Behaviour of Solutions Long-Time Behaviour of Solutions Long-Time Behaviour of Solutions Long-Time Behaviour of Solutions to a Class of Semilinear Parabolic Equations to a Class of Semilinear Parabolic Equations to a Class of Semilinear Parabolic Equations to a Class of Semilinear Parabolic Equations

C o n t e n t s

. . . . § 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 solu- tions to a class of abstract nonlinear parabolic equations. A typical representative of this class is the nonlinear heat equation

considered in a bounded domain of with appropriate boundary conditions on the border . However, the class also contains a number of nonlinear partial dif- ferential 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 indepen- dent 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 pro- perties 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 re- commended for further reading.

§ 1 Positive Operators

§ 1 § 1 Positive Operators Positive Operators

§ 1 Positive Operators with Discrete Spectrum with Discrete Spectrum with Discrete Spectrum with Discrete Spectrum

This section contains some auxiliary facts that play an important role in the subse- quent considerations related to the study of the asymptotic properties of solutions to abstract semilinear parabolic equations.

Assume that is a separable Hilbert space with the inner product and the norm . Let be a selfadjoint positive linear operator with the domain . An operator is said to have a discrete spectrumdiscrete spectrumdiscrete spectrumdiscrete spectrum if in the space there exists an orthonormal basis of the eigenvectors:

, , , (1.1)

such that

, . (1.2)

The following exercise contains a simple example of an operator with discrete spec- trum.

¶^u

¶^t

--- n ¶^2u

¶^x_i²

--- +f x u( , )

i=1

å

=

W

R

¶ W

H ₍

. .

.

A H

e_k { } e_k, e_j

( )=dk j ^{A e}k= lke_k ^{k j}, =^{1 2}, , ¼ 0< l₁ £ l₂ £ ¼ l_k

klim®¥ =¥

2

C h a p t e r

Let and let be an operator defined by the equation with the domain which consists of conti- nuously differentiable functions such that (a) , (b) is absolutely continuous and (c) . Show that is a positive operator with discrete spectrum. Find its eigen- vectors and eigenvalues.

The above-mentioned structure of the operator enables us to define an operator for a wide class of functions defined on the positive semiaxis. It can be done by supposing that

,

, . (1.3)

In particular, one can define operators with . For these ope- rators are bounded. However, in this case it is also convenient to introduce the line- als if we regard as a completion of the space with respect to the norm .

Show that the space with can be identi- fied with the space of formal series such that

.

Show that for any the operator can be defined on every space as a bounded mapping from into

such that

, . (1.4)

Show that for all the space is a sepa- rable Hilbert space with the inner product

and the norm .

The operator with the domain is a positive operator with discrete spectrum in each space .

Prove the continuity of the embedding of the space into

for , i.e. verify that and .

Prove that is dense in for any . E x e r c i s e 1.1 H= L²(0 1, ) ^A

Au=-u² ^{D A}( )

u x( ) ^u( )⁰ =^u( )¹ =⁰

u¢( )^{x u}² Î^L2(^{0 1}, )

A

A

f A( ) ^f( )l

D f A( ( )) ^{h c}ke_k H: c_k²[f( )lk ]²< ¥

k=1

å

Î

k=1

å

î = þ

í ý

ì ü

=

f A( )^{h c}kf( )lk ^ek k=1

å

= ^h Î^{D f A}( ( ))

A^a a _ÎR _a=-_b< 0

D A( ^a) ^{D A}( ^-b) ^H

A^-b

.

E x e r c i s e 1.2 F_-b =^{D A}( ^-b) b >⁰ c_ke_k

å

c_k²lk 2b - k=1

å

E x e r c i s e 1.3 b _ÎR A^b

D A( ^a) ^{D A}( ^a)

D A( ^{a b-} )

A^bD A( ^a)= ^{D A}( ^{a b-} ) ^Ab1+b2 =^Ab1×^Ab2

E x e r c i s e 1.4 a _ÎR _Fa º D A( ^a)

u v,

( )_a=(^{Aau A}, ^av) u_a= A^au

E x e r c i s e 1.5 A F1+s

F_s

E x e r c i s e 1.6 F_a

F_b a b> F_aÌ F_b ^u_b £ ^{C u}_a

E x e r c i s e 1.7 F_a F_b a b>

P o s i t i v e O p e r a t o r s w i t h D i s c r e t e S p e c t r u m 79

Let for . Show that the linear functional can be continuously extended from the space to

and for any and .

Show that any continuous linear functional on has the

form: , where . Thus, is the space

of continuous linear functionals on .

The collection of Hilbert spaces with the properties mentioned in Exercises 1.7–1.9 is frequently called a scalescalescalescale of Hilbert spaces. The following assertion on the com- pactness of embedding is valid for the scale of spaces .

Theorem 1.1.

Let Let

Let Let .... The The Then Thennn the space the space the space the space is compactly embedded into is compactly embedded into is compactly embedded into is compactly embedded into ,,,, i.e. i.e. i.e. i.e.

every sequence bounded in is compact in every sequence bounded in every sequence bounded in is compact in is compact in every sequence bounded in is compact in ....

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 converges to zero with re- spect to the norm of the space . We remind that a sequence in weakly converges to an element if for all

.

Let the sequence be weakly convergent to zero in and let

, . (1.5)

Then for any we have

. (1.6) Here we applied the fact that for

. Equations (1.5) and (1.6) imply that

. We fix and choose such that

. (1.7)

E x e r c i s e 1.8 f ÎF_s s >^{0 F g}( ) º

f g,

( )

º ^H F_-s

f g,

( ) £ ^f _s× ^g _-s ^f ÎF_s ^g ÎF_-s

E x e r c i s e 1.9 F F_s

F f( ) =(^{f g}, ) ^g ÎF_-s F_-s F_s

F_s { }

s1>s2 F_s1 F_s2

F_s1 F_s2

F_s1

F_s2 { }^fn F_s f ÎF_s ^g ÎF_s

f_n, g

( )_s

nlim®¥ = (f g, )_s f_n

{ } F_s1

f_n

s₁ £ C ⁿ= ^{1 2}, , ¼ N

f_{n s}

2

2 l²_k^s2(^f_n, ^e_k)^{2 1}

N^2(s1-s2)

--- l²_k^s1(^f_n, ^e_k)²

k=N

å

+

k=1 N-1

å

£

k ³ N l²_k^{s2 1}

lN 2(s₁-s₂)

---l²_k^s1

£

f_{n s}

2

2 l²_k^s2(^f_n, ^e_k)² +^Cl_N^-2(s1-s2)

k=1 N-1

å

£

e > ^{0 N}

f_{n s}

2

2 l²_k^s2(^f_n, ^e_k)²+e

k=1 N-1

å

£

2

C h a p t e r

Let us fix the number . The weak convergence of to zero gives us

, .

Therefore, it follows from (1.7) that

. By virtue of the arbitrariness of we have

. Thus, Theorem 1.1 is proved.

Show that the resolvent , , is

a compact operator in each space .

We point out several properties of the scale of spaces that are important for further considerations.

Show that in each space the equation

, ,

defines an orthoprojector onto the finite-dimensional subspace ge-

nerated by the set of elements . Moreover,

for each we have

. Using the Hölder inequality

, , ,

prove the interpolation inequality

, , .

Relying on the result of the previous exercise verify that for any the following interpolation estimate holds:

,

where , , and .

Prove the inequality

, where and is a positive number.

N f_n

f_n, e_k

( )

nlim®¥ =0 ^k =^{1 2}, , ¼, ^N-¹ f_n

s₂ £ e

nlim®¥

e

f_n

s₂ nlim®¥ = 0

E x e r c i s e 1.10 R_l( )A =(^A-l)^-1 l ¹ lk

F_s F_s { }

E x e r c i s e 1.11 F_s

P_lu (u e, _k)^ek k=1

å

= ^u ÎF_s -¥< s ¥<

e_k, k=1 2, , ¼, ^l

{ }

s

P_lu-u _s

llim®¥ =0 E x e r c i s e 1.12

a_kb_k a_k^p

å

è ø

ç ÷

æ ö^1/p b_k^q

å

è ø

ç ÷

æ ö^1/q

£

å

A^qu £ Au^q× u ^{1-q 0} £ q £ ^{1 u} Î^{D A}( ) E x e r c i s e 1.13

s1_, s2 _ÎR

u _{s q( )} u_s

1 q u_s

2 1-q

£

s q( ) = qs₁+(¹-q)s₂ ⁰ £q £^{1 u} F_{max s}

1,s₂

( )

Î

u_{s q}²_{( )} e ^u_s

1

2 C_qe-¹---^-qq

u_s

2

+ 2

£ 0< q< ¹ e

P o s i t i v e O p e r a t o r s w i t h D i s c r e t e S p e c t r u m 81

Equations (1.3) enable us to define an exponential operator , , in the scale . Some of its properties are given in exercises 1.14–1.17.

For any and the linear operator maps into and possesses the property

.

The following semigroup property holds:

, .

For any and the following equation is valid:

. (1.8)

For any the exponential operator defines a dissi- pative compact dynamical system . What can you say about its global attractor?

Let us introduce the following notations. Let be the space of strongly continuous functions on the segment with the values in , i.e. they are con- tinuous with respect to the norm . In particular, Exercise 1.16 means

that if . By we denote the subspace of

that consists of the functions which possess strong (in ) deri- vatives lying in . The space is defined similarly for any natural . We remind that the strong derivative (in ) of a function at a point is defined as an element such that

. Let for some . Show that

for all , , and . Moreover,

, .

Let be the space of functions on the segment with the values in for which the integral

exists. Let be the space of essentially bounded functions on with the values in and the norm

-t A

( )

exp t ³ 0

F_s { }

E x e r c i s e 1.14 a _ÎR t >0 ^exp(-^{t A})

F_a F_s

s ³0

Ç

e^-tl1 u_a

£ E x e r c i s e 1.15

t₁A -

( )×^exp(-^t2A)

exp = ^exp(-(t₁+t₂)^A) ^t1, t₂ ³ 0 E x e r c i s e 1.16 u ÎF_s s _ÎR

e^{-t A}u-e^-tAu_s

tlim®t = 0

E x e r c i s e 1.17 sÎR ^{e-t A}

F_s,^{e-t A}

( )

C a b( , ; F_a) a b,

[ ] F_a

.

.

e^{-t A}u _ÎC₍R_{+, Fa)} u ÎF_a ^C1(^{a b}, ; F_a)

C a b( , ; F_a) ^{f t}( ) F_a

f¢( )^{t C a b}( , ; F_a) ^Ck(^{a b}, ; F_a)

k F_a ^{f t}( )

t= t₀ ^v ÎF_a

1

h---(f t( ₀+h)-^{f t}( )₀ )-^v

a

hlim®0 = 0

E x e r c i s e 1.18 u₀ ÎF_s s

e^{-t A}u₀ ÎC^k(d + ¥, ; F_a) d > ⁰ a _ÎR k=1 2, , ¼

d^k t^k

d---e^{-t A}u₀ =(-A)^{ke-t Au}₀ ^k=^{1 2}, , ¼

L²(a b, ; F_a) [^{a b}, ]

F_a

uL² a b _F

; a ,

( )

2 u t( ) _a^2dt

a

ò

=

L^¥(a b, ; F_a) [^{a b}, ]

F_a

2

C h a p t e r

. We consider the Cauchy problem

, ; , (1.9)

where and . The weak solutionweak solutionweak solutionweak solution (in ) to this problem on the segment is defined as a function

(1.10) such that and equalities (1.9) hold. Here the derivative

is considered in the generalized sense, i.e. it is defined by the equality

, ,

where is the space of infinitely differentiable scalar functions on vanishing near the points and .

Show that every weak solution to problem (1.9) possesses the property

. (1.11) (Hint: first prove the analogue of formula (1.11) for , then use Exercise 1.11).

Prove the theorem on the existence and uniqueness of weak solutions to problem (1.9). Show that a weak solution to this problem can be represented in the form

. (1.12)

Let and let a function possess the property

for some . Then formula (1.12) gives us a solution to pro- blem (1.9) belonging to the class

. Such a solution is said to be strong in .

uL^¥(a b, ;D A( ^a)) e s s A^au t( )

t Î[a bsup, ]

=

y d

t

---d +A y =^{f t}( ) ^t Î(^{a b}, ) ^{y a}( ) =^y₀

y ÎF_a ^{f t}( ) Î^L2(^{a b}, ; ^F_a-1 2¤ ) F_a a b,

[ ]

y t( ) Î^{C a b}( , ; F_a)Ç^L2(^{a b}, ; F_a+1 2¤ ) y

d ¤dt Î^L2(^{a b}, ; ^F_a-1 2¤ ) y¢( )^t º^dy¤^dt

j( )^{t y}¢( )^{t dt}

a

ò

ò

C₀^¥(a b, ) (^{a b}, )

a b

E x e r c i s e 1.19

y t( ) _a^{2 y}( )t _a 1 2--- +

2 ^dt

a

ò

+ ^y0a f( )t _a 1

2--- -

2 ^dt

a

ò

+

£

y_n( )t =^Pny t( ) E x e r c i s e 1.20

y t( )

y t( ) ^e-(t-a)Ay0 e^-(t-t)Af( ) tt ^d

a

ò

+

=

E x e r c i s e 1.21 y₀ ÎF_a ^{f t}( ) f t( )₁ -^{f t}( )2 _a £ ^{C t}1-t₂^q 0< q £ ¹

C([a b, ]; F_a)Ç^C1(]^{a b}, ]; F_a)Ç^C(]^{a b}, ]; F_a+1) F_a

P o s i t i v e O p e r a t o r s w i t h D i s c r e t e S p e c t r u m 83

The following properties of the exponential operator play an important part in the further considerations.

Lemma 1.1.

Let be the orthoprojector onto the closure of the span of elements

in and let , . Then

1) for all , and the following inequality holds:

; (1.13)

2) for all , and the following estimate is valid:

, (1.14) in the case we supose that in (1.14).

Proof.

Estimate (1.13) follows from the equation

. In the proof of (1.14) we similarly have that

. This gives us the inequality

.

Since is attained when , we have that

Therefore,

. This implies estimate (1.14). Lemma 1.1 is proved.

In particular, we note that it follows from (1.14) that

, . (1.15)

e^{-t A}

Q_N e_k, k ³ N+1

{ } ^{H P}_N=^I-^Q_N ^N= ^{0 1 2}, , , ¼

h ÎH b ³ ^{0 t} _ÎR A^bP_Ne^{-t A}h lN

b e^lNt h

£ h ÎD A( )^{b t} >⁰ a ³ b A^aQ_Ne^{-t A}h a b-

---t

è ø

æ ö^{a b-} +l_N^{a b}₊^-₁ ^{e-tlN+1 Abh}

£

a b- =^{0 00} =⁰

A^bP_Ne^{-t A}h ² lk

2be^-2tlk(h e, _k)²

k=1

å

=

A^aQ_Ne^{-t A}h ² (l^{a b- e-tl})² lk

2b(h e, _k)²

k=N+1

å

£

A^aQ_Ne^{-t A}h 1 t^{a b-}

--- (m^{a b- e-m}) ^Abh

m³maxtl_N+1

£ m^ge-m; m ³ ⁰

{ }

max m=g

m^ge-m

( )

m ³maxl_N+1t (lN+1t)^ge-lN+1t, ^if lN+1t³ g^; g^ge-g, if lN+1t< g^. îï

íï ì

=

m^ge-m

( ) £ (g^g+(l_N+1t)^g)^e-lN+1t

m³maxl_N+1t

A^ae^{-t A}h a b- ---t

è ø

æ ö^{a b-} +l1a b- ×^{e-tl1 Abh}

£ a ³ b

2

C h a p t e r

Using estimate (1.15) and the equation

, , ,

prove that

; , (1.16)

provided , where a constant does not depend on and (cf. Exercise 1.16).

Show that

, , . (1.17)

Lemma 1.2.

Let for . Then there exists a unique solu- tion to the nonhomogeneous equation

, , (1.18)

that is bounded in on the whole axis. This solution can be represented in the form

. (1.19)

We understand the solution to equation (1.18) on the whole axis as a function such that for any the function is a weak solution (in ) to problem (1.9) on the segment with .

Proof.

If there exist two bounded solutions to problem (1.18), then their diffe- rence is a solution to the homogeneous equation. Therefore,

for and for any . Hence, .

If we tend here, then we obtain that . Thus, the bounded so- lution to problem (1.18) is unique. Let us prove that the function defined by formula (1.19) is the required solution. Equation (1.15) implies that

for and . Therefore, integral (1.19) exists and it can be uniform- ly estimated with respect to as follows:

E x e r c i s e 1.22

e^{-t A}u-e^{-s A}u ^{A e-tAud}t

s

ò

-

= ^t ³ ^{s u} ÎF_b

e^{-t A}u-e^{-s A}u _q £ ^C_{q s,} ^t-^{ss q- u}_s ^{t s}, > ⁰

q s< £ ¹+q ^C_{q s,}

t s

E x e r c i s e 1.23

A^ae^{-t A} a ----t è øæ ö^a_e-a

£ ^t > ⁰ a> ⁰

f t( ) Î^L¥(R, F_{a g-} ) ⁰ £ g< ¹ v t( ) _Î^C₍R_{, Fa)}

v d

t

d---+A v= ^{f t}( ) ^t _ÎR

F_a

v t( ) ^e-(t-t)Af( ) tt ^d

¥ -

ò

=

v t( ) Î C(R, F_a)

Î ^a< ^{b v t}( )

F_a-1 2¤ [a b, ] ^y0=v a( )

w t( ) ^{w t}( ) =

t-t₀

( )^A

-

{ }^{w t}( )0

= exp ^t ³ ^t0 ^t0

A^aw t( ) £ ^{e-(t-t0) l1 Aaw t}( )0 £ ^{C e-(t-t0) l1}

t₀®-¥ ^{w t}( ) =⁰

v t( )

A^ae^-(t-t)A g t-t ---

è ø

æ ö^g l1

+ g ^{e-(t-t) l1 e s s Aa g- f}( )t

t _ÎR

£ sup t> t ⁰< <g ¹

t

S e m i l i n e a r P a r a b o l i c E q u a t i o n s i n H i l b e r t S p a c e 85

,

where for and

for .

The continuity of the function in follows from the following equation that can be easily verified:

.

This also implies (see Exercise 1.18) that is a solution to equation (1.18).

Lemma 1.2 is proved.

§ 2 Semilinear Parabolic

§ 2 § 2 Semilinear Parabolic Semilinear Parabolic

§ 2 Semilinear Parabolic Equations Equations Equations Equations in Hilbert Space in Hilbert Space in Hilbert Space 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 of the form

, , (2.1)

where is a positive operator with discrete spectrum and is a nonlinear continuous mapping from into , , possessing the property

(2.2) for all and from the domain of the operator and such that

. Here is a nondecreasing function of the parameter that does not depend on and is a norm in the space .

A function is said to be a mild solutionmild solutionmild solution (in mild solution ) to problem (2.1) on the half-interval if it lies in for every and for all

satisfies the integral equation

. (2.3)

The fixed point method enables us to prove the following assertion on the local existence of mild solutions.

A^av t( ) ¹+^k l¹₁^-g

--- e s s A^{a g-} f( )t

t _ÎsupR

£ k=0 g= ⁰

k g^{g s-ge-sds}

0

¥

ò

= ⁰< <g ¹

v t( ) F_a

v t( ) ^{e-(t-t0)Av t}( )0 ^e-(t-t)Ag( ) tt ^d

t₀

ò

+

=

v t( )

H u

d t

---d +A u= ^{B u t}( , ) ^u_t=s=^u0

A B₍

. .

D A( )^q _´ R H 0 £ q< ¹ B u( ₁, t)-^{B u}( ₂, ^t) £ ^M( )r ^Aq(^u₁-^u₂)

u₁ u₂ F_q =^{D A}( )^{q Aq}

A^qu_j £ r ^M( )r r

t

.

u t( ) F_q

s s, +T)

[ ^{C s s}( , +^T¢; F_q) ^T¢< ^T t Î[s s, +T)

u t( ) ^e-(t-s)Au₀ ^{e-(t-t)AB u}( ( ) tt , ) t^d

s

ò

+

=

2

C h a p t e r

Theorem 2.1.

Let Let Let

Let .... Then there exists Then there exists Then there exists Then there exists depending on depending on depending on depending on and and and and such that such that such that such that problem

problem problem

problem (2.1) possesses a unique mild solution on the half-interval possesses a unique mild solution on the half-interval possesses a unique mild solution on the half-interval possesses a unique mild solution on the half-interval .... Moreover, either Moreover, either Moreover, either Moreover, either or the solution cannot be continued or the solution cannot be continued or the solution cannot be continued or the solution cannot be continued in up to the moment

in up to the moment in up to the moment

in up to the moment ....

Proof.

On the space we define the mapping

.

Let us prove that for any . Assume that

and . It is evident that

. (2.4)

By virtue of (1.8) we have that if then

.

Therefore, it is sufficient to estimate the second term in (2.4). Equation (1.15) im- plies that

(2.5) (if , then the coefficient in the braces should be taken to be equal to 1). Thus,

maps into itself. Let . In

we consider a ball of the form

.

Let us show that for small enough the operator maps into itself and is con- tractive. Since for , equation (2.2) gives

u₀ ÎF_q ^T* q ^u0 q

s s, +T^*)

[ ^T*=¥

F_q ^t= ^s+^T*

C_s,q º C s s( , +T; F_q)

G u[ ]( )^{t e-(t-s)Au}0 e^-(t-t)AB u( ( ) tt , ) t^d

s

ò

+

=

G u[ ]( )^t Î^{C s s}( , +^T; F_q) ^T >^{0 t}1, t₂ Î s s, +T

[ ]

Î ^t1< t₂

G u[ ]( )^t2 ^{e-(t2-t1)AG u}[ ]( )^t1 ^{e-(t2-t)AB u}( ( ) tt , ) t^d

t₁ t₂

ò

+

=

t₂®t₁,

G u[ ]( )^t1 -^{e-(t2-t1)AG u}[ ]( )^t1 _q®⁰

e^-(t2-t)AB u( ( ) tt , ) t^d

t₁ t₂

ò

£

q t₂-t ---

è ø

æ ö^q l1

+ q ^dt

t₁ t₂

ò

t₂-t₁^1-q q^q 1-q

---+lq1 t₂ -t₁^q

î þ

í ý

ì ü

B u( ( ) tt , )

t Îmax[s s, +T]

£

£

×

£

q= ⁰

G C_s,q= C s s( , +T; F_q) ^v0( )t = ^e-(t-s)Au0 ^Cs, q

U u t( ) ^Cs,q: u-v_0C

s,q u t( ) -^v0( )t _q £ ¹

s, s+T [ max ]

º

î Î þ

í ý

ì ü

=

T G U

u_C

s,q 1 u₀ + q

£ ^u Î^U

S e m i l i n e a r P a r a b o l i c E q u a t i o n s i n H i l b e r t S p a c e 87

for all , where is a fixed number. Therefore, with the help of (2.5) we find that

. Similarly we have

for . Consequently, if we choose such that

and ,

we obtain that is a contractive mapping of into itself. Therefore, possesses a unique fixed point in . Thus, we have constructed a solution on the seg- ment . Taking as an initial moment, we can construct a solution

on the segment with the initial condition .

If we continue our reasoning, then we can construct a solution on some maximal half- interval . Moreover, it is possible that . Theorem 2.1 is proved.

Let and let be such that is

the maximal half-interval of the existence of the mild solution to problem (2.1). Then we have either and

or .

Using equations (1.16) and (2.5), prove that for any mild solu- tion to problem (2.1) on the estimate

, , (2.6)

is valid, provided , , and .

Let and let be a mild solution to problem (2.1) on the half-interval . Then

for any , , and . Moreover, equations (2.1) are valid if they are understood as the equalities in and , re- spectively.

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 be the orthoprojector in

B u( ( ) tt , )

t Î[maxs, s+T] B(0, t) 1 u₀

+ q

( )^M(¹+ ^u0 q) ^{C T}0 u₀ , q

( )

º

+ +

t Î[maxs, s+T]

£

T £ T₀ ^T0

G u[ ] -^v0_C

s,q £ T^1-q×C₁(T₀, ,q ^u0 q)

G u[ ]-^{G v}[ ]C_s,q £ T^1-q×C₂(T₀, q)^M(¹+ ^u _q) ^u-^vC_s,q

u v, Î^{U T}1

T₁^1-qC₁(T₀, ,q ^u_{0 q}) £ ^{1 T}₁^1-qC₂(^T₀, q)^M(¹+ ^u_{0 q})< ¹

G U G

UÌ C_{s, q} s s, +T₁

[ ] ^s+^T₁

s+T₁, ^s+^T1+T₂

[ ] ^u0=u s( +T₁)

s s, +T^*)

[ ^T*=¥

E x e r c i s e 2.1 u₀ÎF_q ^T*=^T*(q,^u0) [^{s s}, +^T*) u t( ) T^*<¥ ^{u t}( ) _q

t®lims T+ * =

¥^,

= ^T*=¥

E x e r c i s e 2.2

u t( ) [^{s s}, +^T*)

u t( )-^u( )t _a £ ^C(q a, , ^T) ^t-t^{q a- t}, t Î[^{s s}, +^T] u₀ ÎF_q ⁰ £ a £ q ^T £ ^T* E x e r c i s e 2.3 u₀ ÎF_q ^{u t}( )

s s, +T^*) [

u t( ) ^{C s s}( , +^T, F_q) ^{C s}( +d, ^s+^T, F1-s) C¹(s+d, ^s+^T, F_-s)

Ç Ç

Ç Î

s >^{0 0}< d< ^{T T} < ^T*

F_-s F_q

P_m H

2

C h a p t e r

onto the span of elements . Galerkin approximate solutionGalerkin approximate solutionGalerkin approximate solutionGalerkin approximate solution of the order

of the order of the order

of the order with respect to the basis is defined as a continuously diffe- rentiable function

(2.7) with the values in the finite-dimensional space that satisfies the equations

, , . (2.8)

It is clear that (2.8) can be rewritten as a system of ordinary differential equa- tions for the functions .

Show that problem (2.8) is equivalent to the problem of find- ing a continuous function with the values in that satis- fies the integral equation

. (2.9)

Using the method of the proof of Theorem 2.1, prove the local solvability of problem (2.9) on a segment , where the pa- rameter can be chosen to be independent of . Moreover, the following uniform estimate is valid:

, , (2.10)

where is a constant.

The following assertion on the convergence of approximate functions to exact ones holds.

Theorem 2.2.

Let Let Let

Let .... Assume that there exists a sequence of approximate solu- Assume that there exists a sequence of approximate solu- Assume that there exists a sequence of approximate solu- Assume that there exists a sequence of approximate solu- tions on a segment for which estimate

tions on a segment for which estimate tions on a segment for which estimate

tions on a segment for which estimate (2.10) is valid. Then is valid. Then is valid. Then is valid. Then there exists a mild solution to problem

there exists a mild solution to problem there exists a mild solution to problem

there exists a mild solution to problem (2.1) on the segment on the segment on the segment on the segment andand

andand

,,,, (2.11) where

where where

where is is a is a is a positive constant independent of a positive constant independent of positive constant independent of ....positive constant independent of e₁, e₂, ¼, ^em

{ }

m { }e_k

u_m( )t ^gk( )t ^ek k=1

å

=

P_mH d

t

d----u_m( )t +^{A u}m( )t = ^PmB u( _m( )t , ^t) ^t >^{s u}m

t=s=P_mu₀ g_k( )t

E x e r c i s e 2.4

u_m( )t ^PmH

u_m( )t ^e-(t-s)AP_m^u0 e^-(t-t)AP_mB u( _m( ) tt , ) t^d

s

ò

+

=

E x e r c i s e 2.5

s s, +T

[ ]

T > 0 ^m

u_m( )t _q

s, s+T

[ max ] < R ^m= ^{1 2 3}, , , ¼ R >0

u₀ÎF_q

u_m( )t [^{s s}, +^T]

u t( ) [^{s s}, +^T]

u t( )-^um( )t _q ^C (¹-^Pm)^u0 q

1 l1m-+q1

---

è + ø

æ ö

s, s+T £

[ max ]

C =C(q, ,^{R T}) ^s