12
Вісник Тернопільського національного технічного університету Scientific Journal of the Ternopil National Technical University 2016, № 4 (84) ISSN 1727-7108. Web: visnyk.tntu.edu.ua Corresponding author: Mikola Yaremenko; e-mail: [email protected] ….………………………………………….. 149 UDC 517.9 SEQUENCES OF SEMIGROUPS OF NONLINEAR OPERATORS AND THEIR APPLICATIONS TO STUDY THE CAUCHY PROBLEM FOR PARABOLIC EQUATIONS Mikola Yaremenko National Technical University of Ukraine «Igor Sikorsky Kyiv Polytechnic Institute» Summary. We consider the operator function of exponential type, studied the link between these functions (semigroup) and Cauchy problem for differential parabolic equation. We establish conditions under which the semigroup is associated with Cauchy problem; we investigate semigroups sequences and their convergence to function of exponential type which is semigroup. We consider maximal dissipative operators and maximum semigroups. We study the problem of existence of the solution of nonlinear partial differential equations of parabolic type with measurable coefficients, nonlinear term which satisfies the forms bouded conditions. Key words: quasi-linear differential equations, dissipative operators, the method of forms, semigroup, maximal operators, sequence of semigroups. Received 14.11.2016 Introduction. Consider the Cauchy problem for a parabolic equation in the form for almost all with the initial condition , , where the operator is determined by the operator , which operates as follows: . The form is based on the left part of elliptic equation , where is the unknown function, a real number, and given function [8, 9]. Here a function of three variables: the dimension vector , scalar, dimension vector . Dimensional matrix of dimension satisfies ellipticity condition: for almost all [4-5, 8, 9]. Let us construct form : , which is assumed to be specified for all elements . The function is a measurable function of its arguments and ; function almost everywhere satisfies . We introduce the class of functions where . Features () (), () ( , ), p l l d ut ut ut L R dx dt 0 [0, ], t t 0 (0) u u 0 ( ) u DA : ( , ) ( , ) p l l p l l L R dx L R dx : p p p A W W 1 1 , , p p h uv A u v : p p q h W W R 1 1 , ,..., (, ) (, , ) ij ij l i j u a xu u bxu u f x x 1 ux 0 () f x (, , ) bxu u l l (,) ij a xu l l ,..., (,) l l l i ij i j i i ij l i u a xu u R 2 2 1 1 1 l x R : p p q h W W R 1 1 (,) , (,, ), p h u uv v a u bxu u v ( , ), ( , ) p l l q l l u W R dx v W R dx 1 1 (, ,) bxyz ( ) l loc b L R 1 ,, () () () bxu u x u xu x 1 2 3 ( ) ( , ): || || , l l loc PK A f L R dx hfh h a h c h 1 2 2 , c R 1 0

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Page 1: Sequences of semigroups of nonlinear operators and their ...elartu.tntu.edu.ua/bitstream/lib/20714/2/TNTUB... · maximal operators, sequence of semigroups. Received 14.11.2016 Introduction

Вісник Тернопільського національного технічного університету

Scientific Journal of the Ternopil National Technical University

2016, № 4 (84)

ISSN 1727-7108. Web: visnyk.tntu.edu.ua

Corresponding author: Mikola Yaremenko; e-mail: [email protected] ….………………………………………….. 149

UDC 517.9

SEQUENCES OF SEMIGROUPS OF NONLINEAR OPERATORS AND

THEIR APPLICATIONS TO STUDY THE CAUCHY PROBLEM FOR

PARABOLIC EQUATIONS

Mikola Yaremenko

National Technical University of Ukraine «Igor Sikorsky

Kyiv Polytechnic Institute»

Summary. We consider the operator function of exponential type, studied the link between these

functions (semigroup) and Cauchy problem for differential parabolic equation. We establish conditions under

which the semigroup is associated with Cauchy problem; we investigate semigroups sequences and their

convergence to function of exponential type which is semigroup. We consider maximal dissipative operators and

maximum semigroups. We study the problem of existence of the solution of nonlinear partial differential equations

of parabolic type with measurable coefficients, nonlinear term which satisfies the forms – bouded conditions.

Key words: quasi-linear differential equations, dissipative operators, the method of forms, semigroup,

maximal operators, sequence of semigroups.

Received 14.11.2016

Introduction. Consider the Cauchy problem for a parabolic equation in the form

for almost all with the initial condition

, , where the operator is determined by the

operator , which operates as follows: . The form

is based on the left part of elliptic equation

, where is the unknown function, – a real

number, and – given function [8, 9]. Here – a function of three variables: the

dimension vector , scalar, dimension vector . Dimensional matrix of dimension

satisfies ellipticity condition: for almost all

[4-5, 8, 9].

Let us construct form : ,

which is assumed to be specified for all elements .

The function is a measurable function of its arguments and ; function

almost everywhere satisfies .

We introduce the class of functions

where . Features

( ) ( ) , ( ) ( , ),p l ld

u t u t u t L R d xdt

0

[0, ],t t

0(0)u u 0

( )u D A : ( , ) ( , )p l l p l l

L R d x L R d x

:p p p

A W W

1 1

, ,p p

h u v A u v

:p p q

h W W R 1 1

, ,...,

( , ) ( , , )ij

i j l i j

u a x u u b x u u fx x

1 u x 0

( )f x ( , , )b x u u

l l ( , )ij

a x u l l

,...,

( , )l l

l

i ij i j i

i ij l i

u a x u u R

2 2

1 1 1

lx R

:p p q

h W W R 1 1

( , ) , ( , , ),p

h u u v v a u b x u u v

( , ), ( , )p l l q l l

u W R d x v W R d x 1 1

( , , )b x y z ( )l

locb L R

1

, , ( ) ( ) ( )b x u u x u x u x 1 2 3

( ) ( , ) : || || ,l l

locPK A f L R d x h f h h a h c h

1 2

2 , c R 1

0

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Sequences of semigroups of nonlinear operators and their applications to study the cauchy problem for parabolic equations

150 …………………………………………………….… ISSN 1727-7108. Scientific Journal of the TNTU, No 4 (84), 2016

, . Growth of the function almost everywhere satisfies the

condition: where , [8,

9].

Preliminary information.

Definition 1. A set of one-parameter nonlinear operators is called a

continuous one-parameter semigroup if the following conditions are met: for any fixed

operator is a continuous nonlinear operator which operates from in

; for any fixed set of elements is strongly continuous on ;

there exists a property of group , for at , where – the identical

motion.

Definition 2. Let function satisfy (in the classical

sense) equation where – motion of at

, . If the sequence matches evenly with ,

in the strong topology, and there is a subsequence that matches a -weak

topology to the element , the element is called a solution of

generalized parabolic equation.

We know that if is an absolutely continuous function than is differentiable for

almost all because is a reflexive Banach space, and can be written by the integral

of its derivative, which exists for almost all .

Definition 3. Own solution of the Cauchy problem for a parabolic equation is a function

, if and this function is absolutely continuous for almost all and satisfies for

almost all this generalized parabolic equation.

Denote with the space of all – significant highly continuous

functions on the interval of real axis , i.e. if , then

; and through – space of all – significant

highly integrable functions on the interval , that is, if then

and .

We assume that the operator is valid from to and is one that

generates mapping from to which can be determined by the rule

almost everywhere , this mapping is

also denoted by the letter

( )PK A 2

1 ( )PK A 2 ( , , )b x y z

, , , , ( ) ( )b x u u b x v v x u v x u v 4 5 ( )PK A

2

4 ( )PK A 5

, ,t

t 0

t 0

t ,

p l lL R d x

,p l l

L R d x ,p l l

f L R d xtf t

s t s t ,t s 0

0

( , )( ) ( ) [ , ], ,p l ln L R d x

u t D C t n N 0

( ) ( ), ,n n n

du t u t n N

dt

, ,n

n N u vn v

1

( )u D v u { ( ), }n

u t n N ( ) ( )u t D [ , ]t t0

0

( ),kn k

du t n N

dt

( , )( ) [ , ]p l l

L R d x

du t L t

dt 0

( ) ( )u t D

( )u t ( )u t

t ( , )p l l

L R d x

t

( )u t ( ) ( )u t D A t

t

( , )[ , ]

l lpL R d xC t0 ( , )

p l lL R d x

[ , ]t0 ( , )( ) [ , ]

l lpL R d xu C t 0

:[ , ] ( , )p l l

u t L R d x0 ( , )[ , ]p l l

L R d xL L t 0 ( , )

p l lL R d x

[ , ]t0 ( , )( ) [ , ]p l l

L R d xu L t 0

:[ , ] ( , )p l l

u t L R d x0|| || || ( ) ||

t

Lu u s ds

0

( , )p l l

L R d x ( , )p l l

L R d x

( , )[ , ]p l l

L R d xC t0

( , )[ , ]p l l

L R d xL t0

( , ) ( , )[ , ] { [ , ] : ( ) ( )p l l p l l

L R d x L R d xC t u v L t v s u s 0 0 { }}s

.

Page 3: Sequences of semigroups of nonlinear operators and their ...elartu.tntu.edu.ua/bitstream/lib/20714/2/TNTUB... · maximal operators, sequence of semigroups. Received 14.11.2016 Introduction

Mikola Yaremenko

ISSN 1727-7108. Вісник ТНТУ, № 4 (84), 2016 …………………………………………………………………………………… 151

Nonlinear semigroups and local generators.

Definition 4. Semigroup is called the maximum compression semigroup if there is

no compression semigroup with a broader definition domain to which it may be extended.

Remark. Any compression semigroup can be extended to a maximum compression

semigroup.

Remark. Maximum dissipative operator does not necessarily generate a maximum

compression semi-group.

Lemma 1. Let and – two systems of areas in :

. If

and , then because of stems condition

.

Lemma 2. Let – convex closed shell . For any fixed natural number k there

is mapping that and .

If and of satisfy correlation , there is an

expansion of that .

Proof. Let us assume that set is ordered like . Using transfinite

induction, we construct the map . Suppose that – a reflection of compression that is

defined for . Let .

Thus, for systems of areas and

, then ,

as belongs to this intersection, it follows:

Denote the projection with p , thus qpf at ||||inf|||| 1

1

fqfqq

.

As inequality for the norms is true:

therefore, there is identity 0

3 B , from where we get

0

3 Bpf .

Select an item 00

Bf and let , then compression is on

}:{)( fTD t .

Using transfinite induction we have received the necessary map . The Lemma

statement is proved.

Consider the semi group

,...2,1,0,2

: jj

tTTkt

, where and

stst TTT when k

jt

2

, k

js

2

, and the set of maps - from Lemma 2.

Denote with k the set }:{ T and define canonical map for kn as

tT

:B :B ( , )p l l

L R d x

( , ) :|| || ,p l l

B f L R d x f f r ( , ) :|| ||p l l

B f L R d x f f r

|| || || ||f f f f r r

, B

B

( )t

D T

:k

U || || || ||k k

U f U q f q ,f q k kU f T f

2( )

tf D T

f 0

f 0 || || || ||

kf f f f

0 2 0 ( )t

f D T

kU

kT

2 kU f f

0 0

( )t

D T { }f

kU

kU

: ( ) :t k k

f q D T U q T q

2 ( ; ) { :|| || || ||}B f f f f

1 2 1 1 2 1

{ ( ; ), ( ; ) : , ( )}t

B f f f B f q q q D T

{ ( ; ), ( ; ) : , ( )}k k t

B f f U f B f q U q q D T ( )

( : ) ( : )ty D T

B f f f B f q q

f

( )

( : ) ( ; ) .t

k ky D T

B f f U f B f q U q B

0

( , )p l l

L R d x

1 1|| || || || , ,

p l lp f f f L R d x

0

kU f f k

U

kU

2 k kT U

{ : }k

U

,:

n k n kJ

Page 4: Sequences of semigroups of nonlinear operators and their ...elartu.tntu.edu.ua/bitstream/lib/20714/2/TNTUB... · maximal operators, sequence of semigroups. Received 14.11.2016 Introduction

Sequences of semigroups of nonlinear operators and their applications to study the cauchy problem for parabolic equations

152 …………………………………………………….… ISSN 1727-7108. Scientific Journal of the TNTU, No 4 (84), 2016

at , where , nT

. Noting that ,

, , kT

, , , obtain approval

. Therefore, we can put for .

Theorem 1. Scope definition maximum compression semigroup is a closed convex

set that is not contained in any closed hyperplane.

Proof. On the opposite, let be its own subset of convex closed shell .

Define dissipative operator tightly defined in as ,

where and – filter defined earlier.

The set is the union of sets where – operator defined earlier.

Having used the equality , we can conclude that

.

Consider the Cauchy problem:

it can be approached by the sequence of Cauchy problems appearing as:

,

where is part of , and is a mapping of: , , .

Let be the projection, thus and

. Define the sequence by placing the induction ,

.

Then , and

.

As a result of Lipchitz condition we have , so there

.

Resulting from theorem given earlier, satisfies the equation in the approximate

Cauchy problem and tends to function evenly at on and function

satisfies the equation in the initial Cauchy problem.

,n kJ T T

k kT T

2 2 kT

, ,: :

m n n k m n kJ J

, , , ,m n n k m n n kJ J T J J T T

k k kT T T

2 2 2 nT

m

T

, , , ,, ( )

m n n k m k n k nJ J J D J ( )

k kA T I

2 kT

tT

( )t

D T

A { ( ) : , ( ) }n

n nAq q f q f q 2

( ) lim( )n

n kq f A f

12

A nA

nA

( )n n

D A

1

2

( ) , { }n n

D A n N

1

2 0

( ) ( ),

( ) , ( ) ( \ ( )) ( )o t

du t Au t

dt

u t u q D T D A

0

( ) ( ),

( ) , ( ) , lim( )

n n

n

n n o k

du t A u t

dt

u t u q q q A f f

0 0 1

0 0 0 00 2

f0

q Aq0 0 n

An

f f f

1 1

2 ( )f D A f Af1

: ( , )p l l

P L R d x ( , )p l l

Pf q L R d x

|| || inf || ||q

q f q f

1

1

( )m

nu t ( )

n

nu t q q

0 0

0 02

1

0

0

( ) ( ( ) )

t

m m

n n nu t P q A u s ds

( ) ( , )m p l l

nu t L R d x

( ) ( ) ( ) ( )

t t

m m m m

n n n n n nu t u t P q A u s ds P q A u s ds

1

0 0

0 0

|| ( ) ( ) ||

t

m m

n n n nA u s A u s ds

1

0

nA || ( ) ( ) ||

m m

n nu t u t

1

( ) lim ( )m

n nm

u t u t

( )n

u t

( )n n

u t

0( )

nu t t [ , ]t

00 ( )

nu t

Page 5: Sequences of semigroups of nonlinear operators and their ...elartu.tntu.edu.ua/bitstream/lib/20714/2/TNTUB... · maximal operators, sequence of semigroups. Received 14.11.2016 Introduction

Mikola Yaremenko

ISSN 1727-7108. Вісник ТНТУ, № 4 (84), 2016 …………………………………………………………………………………… 153

Show that is really a compression semigroup.

Because of dissipativity of and

we get at thus

at , .

Let , . Define descriptive semigroups: ,

, at .

Because and is the compression,

is the compression semigroup.

As is dissipative, we have ,

We show that for

at , , . Really

,

.

Since

and then from ,

, follows that with

.

Denote , , where – arbitrary fixed number with , and

.

Because ,

and , for , we get

, ( ),( )

( ), ( ), ,

t t

t

T f при f D TT f

u t s при f u s s

0

A || || || ||k k

U f U q f q ,f q k kU f T f

2

( )t

f D T || ( ) ( ) || || ( ) ( ) ||u s t u s t u s u s 1 1

, ,s s t 1

0

|| ( ) || || ( ) ||t

u s t T f u s f ( )t

f D T ,s t 0

, , [ , ]s t s t r 0 r R

,

( )n k n

k kT A

1

22 2

,nT

0

, , ,n n n

t s t sT T T

,

k kt J s J

12 2

( )n

kA

1

2k

k kA T

22

,{ : , , ,...}

n k

tT t J J

2 0 1

, ,( ) ( )

n k n n

k k k kA T A A

1

22 2

, , ,|| || || ||

n n n

k t kA T f A f

( , ).p l l

f L R d x, ,

|| ||n m

t n t mT u T u

,n

n k ku q q

12 ( ) ,

k kq A f

1

0

,k k k k

q q f A q 1

0 ,n m n0

( , )k

0

kt J r

0 2

, ,|| || || ||k k

n m p p

n m n mj jT u T u u u

2 2

, , , ,

( ) ( ) ( ) ( )|| || || ||k k k k

jn m p n m p

n m n mi i i ii

T u T u T u T u

1

1 2 1 2 1 2 1 21

, , , , , , , ,,k k k k k k

j pk n n m m n m n m

k n k m n m n mj j j j j ji

A T u A T u T u T u T u T u

1 2

1

2 2 2 2 2 20

2

, , , ,|| ||k k

jk n n m m p

k n k mj ji

A T u A T u

1

1

2 20

2

, , , , , ,

, ,

, ( ) ( )

( ) ( )

k k k k

k k

n n m m n n m m

n k m k n k mi i i i

pn n m m

k n k mi i

A T u A T u A T u A T u

A T u A T u

1 1

2 2 2 2

21 1

2 2

2 2

2 2 0

, , , ,|| ( ) || || ||k k k

n n n n n n

n k n ni i iT u A T u A T u

1

2 2 22 2 , , ,

|| || || ||n n n

k t kA T f A f

( , )p l l

f L R d x, ,

|| || || || ( )k k

n m p p n m k

n m ki iT u T u r q

1

2 24 2 2 2

kj

2 2

k

k kt j

2 [ ],

k

kj t t r 2 0 n N

,|| ( ) ||k

n k

j n niT u u j

22

, , , ,

( )

k

k k k

n n n n

n n ni i iT u T u A T u ds

2

1 2 2 2

0

( ) ( ) ( )

k

k k k

n n n nu j u j A u j s ds

2

0

1 2 2 2

,lim ( ) ( )

n

k n n nA u t A u t

( , )k

, ,|| ( ) || ( )

k

k

k n n k

j n n n niu j T u A A u j s ds

2

1 2

0

2 2, , ,

|| ( ) ||

k

k

n k n n

n niA u j s A T u ds

2

2

0

2

Page 6: Sequences of semigroups of nonlinear operators and their ...elartu.tntu.edu.ua/bitstream/lib/20714/2/TNTUB... · maximal operators, sequence of semigroups. Received 14.11.2016 Introduction

Sequences of semigroups of nonlinear operators and their applications to study the cauchy problem for parabolic equations

154 …………………………………………………….… ISSN 1727-7108. Scientific Journal of the TNTU, No 4 (84), 2016

is performed for inequality

.

Then, using induction, we get:

when , therefore

we have .

As when , for fixed , we

have , so – is a relatively compact set in .

When we get

since

Because of the uniform continuity and indeed there is an inequality

with , . In other words is a

contraction semigroup, but this leads to conflict with maximality of semigroup , so we get

a contradiction. We show that the region is not contained in some closed hyperplane in

.

Suppose for some and 1|||| e .

Let at , then is also a compression semigroup and

the definitional domain of is the set , which has an empty intersection with ,

so

is an extension of , but is a compression semigroup,

ie there is a conflict with maximality . Theorem 1 is proved.

Theorem 2. The closure of the set , where is the maximum dissipative operator

is a set which is convex in .

Proof. We will use contradiction method. Let and

when . Suppose that , then put with . Using

statement 8, we get ,

, just as , since

, we have that and

, thus we get that , but this contradicts the

( || ||) ( )k k n k n k k

j j jq

0 1

02 2 2 2 1 2 2

|| ||n k

q 0

02 2

(|| || || ||)n n n n

r r n r r

j k ke re q q q q e re

12 2 0 2 2

0 0 02 2 2 k

j r

0 2

,lim ( )

n

t n nT u u t

lim ( )n

k k kn

A A q A q

12 ( ) ( , )

p l l

kq D A L R d x

,lim || ||k k

n

n ni inT u T u

2 20 { }

n nu

0 ( , )p l l

L R d x

, , ( ) [ , ]k k k

s i t j s t i j r

2 2 2 0

,|| ( ) || || ( ) ( ) || || ( ) ||

n

t n n t s nu s t T f u s t u s t u s t T u

,|| || || || || || || ( ) ||

n

t s n t s n t s n t s nT u T u T u T f T u f u s f

3 6

, ,|| ( ) || || ( ) ( ) || || ( ) || || || .

n n

s n n n s n s n s nu s T u u s u s u s T u T u T u

3

tT f ( )u t

|| ( ) || || ( ) ||t

u s t T f u s f ( )t

f D T , , [ , ]s t s t r 0 tT

tT

( )t

D T

( , )p l l

L R d x

( ) { ( , ) : , }p l l

tD T f L R d x f e M ( , )

p l le L R d x

( )t t

S f e T f e ( )t

f D T tS

tS ( )

te D T ( )

tD T

( )

( )

t t

t

t t

T f при f D TT f

S f при f e D T

tT tT

tT

( )D A A

( , )p l l

L R d x

, ( )q D A ( )f q 1

0 1 [ ( )]f D A q q q 1

q Aq1

( ) ( ) ( )A f q A f A q f q

1 1 1

lim || || || ||f q f q

0

lim ( ) || ||A f f

1

0

|| || lim ( ) ( )q q A f A f

1 1

0

|| || || || || ||q f f q lim( ) ( ) , [ , ]A f q

1

01 0 1

lim( ) || ||A f q f q

1

0lim( )A f f

1

0

Page 7: Sequences of semigroups of nonlinear operators and their ...elartu.tntu.edu.ua/bitstream/lib/20714/2/TNTUB... · maximal operators, sequence of semigroups. Received 14.11.2016 Introduction

Mikola Yaremenko

ISSN 1727-7108. Вісник ТНТУ, № 4 (84), 2016 …………………………………………………………………………………… 155

assumption that . Theorem 2 is proved.

Theorem 3. 1) The maximum compression semigroup has tightly defined

generator and it has been generated by a maximum dissipative operator. 2) If the maximum

dissipative operator is single valued, then the semigroup generated by this operator is a maximal

contraction semigroup.

Proof. Proposition 1) is a consequence of previous theories and assertions. Indeed local

generator is consistently expressed in , maximum dissipative extension

operator generates a compression semi-group , then semigroup is an extension of

semigroup , but out of maximality we get = . 1) has been proven.

Proposition 2) prove by contradiction. Let the operator generates semigroups

and – the maximum extension . Assume the opposite and

, use 1) generator let be consistently defined in .

Because of closure there is an element and

using the maximum dissipativity A get that exist and

. Since , weakly differentiable and on , and

and thus ,

since .

That is, we have , directing to zero, we

obtain a contradiction

and . Theorem is

proved.

Sequences of nonlinear semigroups in spaces established that: Cauchy

problem , , with each has

only a weak solution if the operator is maximal dissipative

operator; Let be consistently defined generator of compression semigroup , while its

maximum dissipative expansion generates the same compression semi-group ; In addition,

it was found that the generator of nonlinear compression semigroup is consistently defined in

[ ( )]f D A

tT

tT A0

( )t

D T A

A0

tS tS

tT tT tS tT

A tT

tS tT ( ) ( )t t

D S D T

( ) ( )t t

D S D T tSA ( )

tD S

( )t

D T ( )f D A ( ) ( , )p l l

tf D T L R d x

( ) ,q A f

10

lim [ ( )] ( , )p l l

q q D A L R d x

0

( )q D A tT q t

limh

hA q Aq

0

q fAq

lim , || ||

p p

hh

T q q q f q f q fh

2

0

, ,

, || || , ,

p p

h h h

p pp

h h

S f f q f q f S f T q q f q f

T q q q f q f q f T q q q f q f

2 2

2 2

|| || || || || ||h h h h

T q S f S q S f q f

,lim

pp

h

h

S f f q f q f q f

h

2

0

lim ,

, ,

ph

h

p p

S f fq f q f

h

Af q f q f Af q f q f

2

0

2 2 || ||p

q f

0

( , ).p l l

L R d x

( ) ( ), ( ) ( , ),p l ld

u t u t u t L R d xdt

0

[0, ],t t0

(0)u u0

( )u D A

: ( , ) ( , )p l l p l l

L R d x L R d x

0

tT

tT

( , ).p l l

L R d x

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Sequences of semigroups of nonlinear operators and their applications to study the cauchy problem for parabolic equations

156 …………………………………………………….… ISSN 1727-7108. Scientific Journal of the TNTU, No 4 (84), 2016

Theorem 4. Let be a sequence of nonlinear semigroups, satisfying

condition: and the sequence of operators is a generator

of nonlinear semigroups sequence and there is a sequence of numbers

, so that . Denote the border of elements from

with as a border in -norm.

Then the closure of in norm, which we denote with generates

nonlinear semigroup , which can be defined as -uniform border:

on any finite interval .

In addition, the semigroup is the only one in the class of semigroups which

satisfies the following conditions:

1) for any element function is strongly absolutely continuous on any finite

interval;

2) for any element for all and and is

continuous by the norm at by .

3) for any element , there is a strong continuous derivative except

perhaps countable number of points.

Proof. The proof methods used are similar to those that were used above. For

convenience and to avoid confusion code sequence is set in brackets, i.e. semigroup

generator will continue to be marked as and respectively.

Let the elements

therefore the operator

is a dissipative operator. Since , then for and every

the equality is true, so is the maximum

dissipative operator. Then fix sequence index, one that is in parentheses, and use previous

results, which is always possible when then there is for this operator the

evaluation is true for .

{ : , }t

nT t n N 0

n n t

t tT f T g e f g

{ : }

nn N

{ : , }t

nT t n N 0

,n

10

,n p l l

nR L R d x

,p l l

L R d x limn

nf

f ,

p l lL R d x

,p l l

L R d x

{ : }t

T t 0 ,p l l

L R d x

limn

t tn

T f T f

,t t0

0

{ : }t

T t 0

( )f D tT f

( )f D t 0 ( )t

T f D 0t tD T f T f

0 tT f

( , )p l l

L R d x t 0 t

( )f D 0

t t

dT f T f

dt

n

tT

n

( )n

tT ( )n

, ( ) ( , )p l l

f g D L R d x

( ) ( )

( ) ( )

,

lim , ,

pn n

n np p

f g f g f g

T f f T g gf g f g f g

2

2

0 ( )n

( )( ) ,

n p l l

nR L R d x n n

n

1

1

n ( ),

n p l l

nR L R d x ( )n

m

( )n

m

1

, ,p l l

f g L R d x

( ) ( )n n

f g f gm m m

1 1 1

1

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Mikola Yaremenko

ISSN 1727-7108. Вісник ТНТУ, № 4 (84), 2016 …………………………………………………………………………………… 157

Let us put by definition . So nonlinear operators are

generators of nonlinear semigroups and for these semigroups for

assessment is true. For elements

a border exists.

Define the Cauchy difference for semigroups sequences using index

: , and mark it , that is , and because

the way acts is important only for large , it can be assumed that , thus the

"tail" of sequence is investigated.

Using estimates obtained and given the already introduced symbols, we get for :

,

,

then if we denote

,

we get similar assessment of the norm in the segment :

.

Then we prove that the convergence in the limit , of any

number is uniform regarding n.

Proof. Fix an arbitrary number and element . Since

, there is a natural number and the number that at ,

( )

( )

n

n

mm

m

1

( )n

m

( ){ : , }

n

mtT t n N 0

, ,p l l

f g L R d x( ) ( )

t

n n mmt mt

T f T g e f g

11

( )[ ]

nf D

( ) ( )lim ,

n n

t mtm

T f T f t

0

( ){ : , }

n

mtT t n N 0

m( ) ( )

,n n

mt ktT f T f t 0

( )n

mktP

( ) ( ) ( ),

n n n

mkt mt ktP f T f T f t 0

( )n

mktP ,k m ,k m p

( )n

mktP

0

( ) ( )n p n

mkP f pe f

2

( ) ( )

( ) ( ) ( )

n n

n n n

mkt m kP f T f T f

m k

1 1

( ) ( ) ( ) ( )n n n n

m m k kT f T f

m k

( )p ne f

m k

1 1

( ) ( )( ) inf{ : }

n p n

mkconst p e g g

( ) ( ) ( ) ( )p

n n n n

m k m kT f T f T f T f

2

0

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

p

n n n n

n n n nm k m k

m k m kT f T f T f T f d

m k m k

21 1 1 1

( )( )

ppp p n

c p e fm k

2

1

1 1

,p l l

L R d x ( )n

mkP , 0

( ) ( )n n ppmk mk

P e

( ) ( )lim ,

n n

t mtm

T f T f t

0

, 0

, 0 f D

( )lim

n

nf f

n

0 M 0 n n0

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Sequences of semigroups of nonlinear operators and their applications to study the cauchy problem for parabolic equations

158 …………………………………………………….… ISSN 1727-7108. Scientific Journal of the TNTU, No 4 (84), 2016

, and , that for large indexes of norm sequence generators are

uniformly bounded on each element.

Obviously, the set:

is limited.

Using the previous inequalities we find out that for every number there is such a

number , that for elements and inequality

is true.

We choose among numbers large enough, so they fit the inequality

, and get at and

thus at we have:

1.

Let us show that for inequality is true.

Indeed we have estimates

.

So, for a valid assessment is . Since

and operator is dissipative as the border of operators

( )nf D

( )nf M

( ) ( )

( ) ( ) ( ) ( ) ( ) ( ){ , :

, , , , }

n np

n n n n n nm k

m k m k m kT f T f T f T f T f T f

m k

n n m p k p

1 1

2

00

0

( ) 0 ,f g ( )f g

pp p

pf f g g const

p M e

2 2

22

,k m k p 0

min ,

( )

p

p

p Me

k c p p

2

20 1

2

, 0

( ) ( )

( ) ( ) ( )

n n

n n n

mkt m kP f T f T f

m k

1 1

pMe

k

0

2n n

0

( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

pn n

mkt mkt

pn n n n

n n n n

m k m k

P f P f

T f T f T f T fm k m k

2

21 1 1 1

p

pconst

p M e

22

n n0

( ).

n p

mk

( )( ) inf{ : }

p nconst p e g g

( ) ( ) ( ) ( )

pn n n n

m k m kT f T f T f T f

2

0

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

p

n n n n

n n n nm k m k

m k m kT f T f T f T f d

m k m k

21 1 1 1

;p p

p

p M e

p M e

2

22

( )( )

pp

p nc p e f

m k

2

1

1 1

( )( )

pp p

p p Mec p

k c p

2

1

0 12

,k m k0

( ) ( )

, ,

supn n p

m kn n

T f T f e

00

( , )p l l

R L R d x 0

Page 11: Sequences of semigroups of nonlinear operators and their ...elartu.tntu.edu.ua/bitstream/lib/20714/2/TNTUB... · maximal operators, sequence of semigroups. Received 14.11.2016 Introduction

Mikola Yaremenko

ISSN 1727-7108. Вісник ТНТУ, № 4 (84), 2016 …………………………………………………………………………………… 159

at , we get the assertion of uniform convergence of semigroups

. To complete the proof we use Lemma 4 [6, 7].

Lemma 4. Let then the operator has a unique extension which is

determined in the entire area and for which in the entire assessment

is correct. In addition and the operator

is a maximal dissipative operator.

Theorem 4. (on the generalized Cauchy problem in ). Generalized Cauchy

problem , where

, each has at every only a single weak solution.

Conclusions. We have constructed operator functions of exponential type, investigated

the link between these operator functions and generalized initial Cauchy problem for equations

of parabolic type. The existence of a solution of the generalized Cauchy problem for equations

of parabolic type has been proven. The results can be generalized to classes of differential

operators of more general type operating in certain functional spaces.

References

1. Medynsky I.P., Ivasyshen S.D. On global solvability of the Cauchy problem for some quasilinear parabolic

equations, Intern. Conf. “Nonlinear Partial Differential Equations” (Kiev, August 21 – 27, 1995): Book of

abstracts, K., 1995, 111 p.

2. Medynsky I.P. The local solvability of the Cauchy problem for the quasilinear parabolic system with

degeneration on the initial hyperplane, Intern. Conf. “Nonlinear Partial Differential Equations» (Kiev,

August 26 – 30, 1997): Book of abstracts, Donetsk, 1997, pp. 129 – 130.

3. Minty G. Monotone (nonlinear) operators in Hilbert space, G. Minty, Duke Math. J, 1962, vol. 29, pp.

341 – 346.

4. Minty G. On the generalization of a direct method of the calculus of variations, G. Minty, 1967, vol. 73, no.

3, pp. 315 – 321.

5. Nash J. Continuity of solutions of parabolic and elliptic equations, J. Nash, Amer. J. Math, 1958, vol. 80,

pp. 931 – 954.

6. Yaremenko M.I. Semi-linear equations and non linear semi-groups, М.І. Yaremenko, К.: NTUU "КPІ",

2013, 201 p.

7. Yaremenko М.І. Semigrups and its application to solution of quasiliniar equations, М.І. Yaremenko, К.:

NAN Ukraine, 2014, 247 p.

8. Goeleven D. Dynamic hemivariational inequalities and their applications, D. Goeleven, M. Miettinen,

P.D. Panagiotopoulos, J. Optimiz. Theory and Appl., 1999, vol. 103, pp. 567 – 601.

9. Panagiotopoulos P.D. On a type of hyperbolic variational – hemivariational inequalities,

P.D. Panagiotopoulos, G. Pop, J. Applied Anal., 1999, vol. 5, no. 1. pp. 95 – 112.

10. Sova M. Cosine operator functions, M. Sova, Rozprawy Matematyczne, 1966, vol. 47, pp. 3 – 47.

11. Varopoulos N.Th. Analysis on Lie groups, N.Th. Varopoulos, J. Funct. Anal.,1988, vol. 76, pp. 346 – 410.

12. Yaremenko M.I. Second order quisi-linear elliptic equation with matrix of Gilbarg – Serrin in l

R and

nonlinear semi-groups of contraction in p

L , M.I. Yaremenko, Conference materials “12th International

Conference Academician M. Kravchuk, May 15 – 17, 2008, Kyiv”, Kyiv, 2008, 473 p.

13. Yaremenko M.I. Second order quisi-linear elliptic equation with matrix of Gilbarg – Serrin in l

R and

nonlinear semi-groups of contraction in p

L , M.I. Yaremenko, Conference materials “International

Conference on problems of decision making under uncertainties (PDMU-2008), Мay 12 – 17, 2008”, 2008,

43 p.

( )n n( ) ( )

lim ,n n

t mtm

T f T f t

0

m m

1

m

( , )p l l

L R d x ( , )p l l

L R d x

m m

f gf g

m

1

1

m

m

1

1

( , )p l l

L R d x

0( ) ( ), ( ) ( , ), [0, ],

p l ldu t u t u t L R d x t t

dt

0(0)u u

: ( , ) ( , )p l l p l l

L R d x L R d x 0( )u D A

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Sequences of semigroups of nonlinear operators and their applications to study the cauchy problem for parabolic equations

160 …………………………………………………….… ISSN 1727-7108. Scientific Journal of the TNTU, No 4 (84), 2016

УДК 517.9

ПОСЛІДОВНОСТІ НАПІВГРУП НЕЛІНІЙНИХ ОПЕРАТОРІВ ТА ЇХ

ЗАСТОСУВАННЯ ДЛЯ ДОСЛІДЖЕННЯ ЗАДАЧІ КОШІ ДЛЯ

РІВНЯННЯ ПАРАБОЛІЧНОГО ТИПУ

Микола Яременко

Національний технічний університет України «Київський політехнічний

інститут імені Ігоря Сікорського», Київ, Україна

Резюме. Розглянуто операторні функції експоненціального типу, досліджено зв’язок між такими

функціями (напівгрупами) та задачами Коші для диференціального параболічного рівняння. Встановлено

умови, за яких напівгрупа буде асоційованою з задачею Коші, досліджено послідовності напівгруп та їх

збіжність до певної напівгрупи. Розглянуто максимальні дисипативні оператори та максимальні

напівтрупи, а також задачу про існування розв’язку нелінійних диференціальних рівнянь у частинних

похідних параболічного типу з вимірними коефіцієнтами, нелінійний доданок яких задовольняє умови

форм – обмеженості коефіцієнтів.

Ключові слова: квазілінійні диференціальні рівняння, дисипативні оператори, метод форм,

напівгрупа, максимальні оператори, послідовності напівгруп.

Отримано 14.11.2016