On some properties of solutions to one class of evolution Sobolev type mathematical models in quasi-Sobolev spaces

Introduction. Let U be a Banach space, L (U) be a space of linear and bounded operators. Mapping U E C (R; L ( U )) Is called a. semigroup of operators If for all s,t E R₊

U ^sU t=U^s + t.                                      (1)

Usually a semigroup is identified with its graph {Ut : t E R₊ } . A semigroup {Ut : t E R₊ } is called holomorphic if it can be analitically continued to some sector of complex plane containing half axis R₊ preserving property (1). A holomorphic semigroup is called degenerate if its unit P = s— t lim Ut is a projector in U.

Firstly holomorphic degenerate semigroups appeared in [1, 2] as solving semigroups for evolution Sobolev type equation

Lu=Mu,                               (2)

where operator L E L (U; F), and оperator M E Cl(U; F), F is a Banach space. Explicit theory of such semigroups can be found in [3].

Interest in Sobolev type equations has recently increased significantly [4-6], moreover, there arose a necessity for their consideration in quasi-Banach spaces. The need is dictated

ИЗ not so much by the desire to fill up the theory but by the aspiration to comprehend non-classical models of mathematical physics [7] in quasi-Banach spaces [8].

Since the Cauchy problem for the Sobolev type equation is not solvable for arbitrary initial data it is necessary to construct the phase space of equation as the set of admissible initial values containing all solutions of equation [3]. The phase spaces of evolution and dynamical Sobolev type equations were constructed earlier in Banach spaces [3]. Moreover there were found conditions when the phase space splits into direct sum of invariant with respect to equation spaces and the solutions have exponential dichotomies [9]. By now these problems are completely solved in Banach spaces [6]. Our goal is to spread these ideas to one class of evolution Sobolev type equations in quasi-Banach spaces of sequences.

• 1.    Holomorphic Degenerate Semigroups of Operators. Let U be a lineal over R. An ordered pair ( U, _U ||. ||) is called a quasi-norm cd space. If the function _U . :U ^ R satisfies the following conditions:

• 1-    _U l| u| >  0 for all u E U. moreover _U |u| = 0 1 ff u = 0. where 0 Is a zero element In U:

• 2.    _U ||au| | = |a|_U |u| for all u E U. a E R;

• 3.    _U Iu + v| =C( _U |u| + _U || v| |) for all u, v E U. where the constant C >  1.

The function _U ||u| | with properties (i)-(iii) is called a quasi-norm. Obviously, in case C = 1 this function is a norm.

The metrizable complete quasi-normed space is called a quasi-Banach space. The spaces of sequences lq, q E (0 , 1) are well known quasi-Banacli spaces (for q E [1 , + to ) the spaces lq are Banach spaces).

Let henceforth {Xk} C R+ be a monotone sequence such that quasi-Banach space lim Xk = + to. The k→∞

∞m     q im = |u={uk}: X1 (X2 |uk0 <+to j

^ mm q 1/q with a, quasl-norm m ||u|| = Ik=1 ( Xk |uк|) )   , m E R Is called /

Obviously, for q E [1 , + to ) the spaces lm are Banach spaces: dense and continuous embedding ln in to lm for n > m and q E R₊.

a

quasi-Sobolev space.

iq = lq , and there is a

Example 1. Let U = lm+2ⁿ. F = lm: Qn ( X ) Ire a. polynornial of power n. Consider operator Qn (Л) u = {Qn ( Xk ) uk}, n E N, where {uk} E U. It is easy to see that operator Qn (Л) E L (U; F), moreover Qn (Л) : lm+2ⁿ ^ lm is a toplinear isomorphism.

Let U and F Ire quasi-Banach spaces, operators L E L (U; F) arid M E Cl (U; F). following [1], [2], take into consideration L-resolvent set p^L ( M ) = {p E C : ( pL—M ) ~ ¹ E L (F; U) } arid L-spectrum a^L ( M ) =C \p^L ( M ) of operator M. It Is easy to show that the set p^L ( M ) Is always opened. therefore the L -spectrum of operator M Is always closed.

Definition 1. Operator M is called strongly ( L,p)-sectorial, p E { 0 } U N, if (i) there exist constants a E R and 9 E ( п/ 2; п ) such that the sector

SL,_e ( M ) = {p E C: | a rg ( p—a ) | <9, p > a} C p^L ( M );

• (ii)    there exists a constant K E R₊ such that

  ≤^K

  П ^p =0 ^|pk^—a|,

  
  

max L^ ( u ) || RL^ )⁽ M )^ ,L (₅) || LL^ )⁽ M ) ^

for all p о ,p 1 ,-.;Цр ^E SLg ^{( M} )• Here R L_p,p )⁽ M ) = П к =o RLk ^{( M} ) ^{is the} fo^ and LL^p )( M )= П к =₀ L^k ( M ) is the left ( L,p )-resolvent оf operator M, and RL ( M ) = ( pL—M ) ^{- 1} L and LL ( M ) = L ( pL—M ) ^{- 1} are the right and the left L -resolvents of operator M respectively.

(hi) there exists a dense in F lineal F0 such that

,,                           .                      ,,              const.

F || M ( XL - M ) - ¹ LL„ )( M ) f | <              for all f E F⁰ ,

|X| k=0 \pk | where const = const(f): for all X,pk E SL(M). k = 0,... ,p.

L №я)||(XL M) L tp)(M )|| < |X| П p =0 Ipk | for arbitrary X, pk E Sg (M). k = 0,... ,p and some const E R+.

Example 2. Let Я= Im ⁺² n. F= Im- m E R. q E R₊. Qn ( X ) = £ П =0 aX Rs ( X ) = £ s =0 djX^j be polynomials of powers n and s respectively ( n < s ) with real coefficients CSf <  0), without common roots. Construct operators L = Qn (Л), M = Rs (Л) as in example 1. It Is easy to show that Rs (Л) E Cl (Я; F)- dom Rs (Л) = Im ^{+2 s} . tlre L -spectrum a^L ( M ) of operator M consists of points pk = Rs ( X_k )( Qn ( X_k )) ^- 1. k E N: X_k Is not the root of the polynomial Qn ( X ).

Lemma 1. [10] Operator M defined in exam,pie 2 is strongly ( L, 0)-sectorial.

Theorem 1. [10] Let operators M and L be defined as in example 2. Then

(if operators L and M generate. holomoephie semigroups {Ut : t E R₊ } and {Ft : t E R₊ } on stKiees Я and F respectively given by

Ut

2 ni

IRL ( M ) e^tdp г

EL (Я)

Ft

-P [ LL ( m ) e“dp e l (F) 2 ni     ^

г

for t E R₊. where, thc. contour Г C p^L ( M ) is sueh that \argp\ ^ 9 при p ^ to. p E Г.

• (ii)    there exist semigroup’s units which are the projectors P E L (Я) a nd Q E L (F) given by

• 2. Invariant Spaces and Exponential Dichotomies of Solutions. Let U and F be quasi-Sobolev spaces of sequences, operators L E L (U; F) and M E Cl(U; F) be constructed in example 2. Consider linear evolution Sobolev type equation

{ I ,                          If X_k Is not the root of Qn ( X ) for all k E N;

I —   ^ < ., ek > ek, If there exist I E N : Xl is the root of Qn(X), keN: к=I

(the. projector Q has the. same. form), splitting the. quasi-Banach spaces Я and F into direct sums

Я=Я⁰ ® Я¹ ,   F=F0 ® F¹;

(Hi) there is splitting of operator actions L_k E L (Я ^k ; F ^k ), M_k E Cl (Я ^k ; F ^k ), k = 0 , 1, and existence of operators M- ¹ E L (F⁰; Я⁰), L- ¹ E L (F¹;Я¹);

(iv) operators H = M0 ¹ L 0 (G = L 0 M0 1) arc. nilpotent and operators S = L 1¹ M1 : dom MП U¹ ^ U¹ anid T=MiL- 1: M [dom M ] П F¹ ^ F¹ are. sectorial.

Lu=Mu.

Vector-function u E C 1(R+;U), satisfying (4) pointwise is called (a classical solution of this equation. The solution u = u ( t ) of (4) is called a solution to the weakened Cauchy problem (in sense of S.G. Krein), if in addition for u 0 E U

lim u (t) = u0. t→0+ holds.

Definition 2. The set P C U is called a phase space of equation (4), if

• (1)    any solution u = u ( t ) of (4) lies In P pointwIse. i.e. u ( t ) E P for all t E R+:

• (ii)    for all u 0 E P there exists a unique solution to (4), (5).

Theorem 2. [10] Let operators M and L be defined as in example (2). Then the subspace U¹ is a phase space, of (4).

Consider the following condition:

Let a^L ( M )= aL ( M ) U aL ( M ) ai id aL ( M ) Is not empty.         1

there exists a bounded domain G1 C C with a boundary of class C ¹ , >      (6)

such that G1 D aL ( M ) and G1 Q aL ( M ) Is empty.

If this condition holds then there exist [11] operators given by integrals

P 1 =     [ Rf ( M ) dp and Q 1 = 1- [ Lf ( M ) dp,

2ni J^ 1 и                   2ni J^ 1 и where y 1 = dG1. By constructIon operators P1 E L(U) arid Q 1 E L(F)-

Lemma 2. Let L,M E L (U; F) be defined in example 2 and condition (6) hold then operators P 1 E L (U) a nd Q 1 E L (F) are projectors in corresponding spaces.

Put U¹¹ = im P 1. F¹¹ = im Q 1. U¹⁰ = ker P 1. F¹⁰ = ker Q 1: am 1 by L 11 ( M 11) denote restriction of operator L MI) оnto U¹¹.

Theorem 3. Let conditions of lemma 2 be fulfilled. Then (i) operators L 11 , M 11 E L (U¹¹; F¹¹) -

(ii) there exists an operator L- E L (F¹¹; U¹¹).

Proof. Statement (i) follows from the construction of operators P 1 E L (U) and Q 1 E L (F), since LP 1 = Q 1 L = L_n arid MP 1 = Q 1 M = M_n.

Statement (ii) follows from theorem 1 since operator L 11¹

is equal to restriction of

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operator --- ( pL — M ) - ¹ dp onto sirbspace F¹¹-

2 ni

γ 1

Corollary 1. Let conditions of lemma 2 be fulfilled. Then P 1 = PP 1 = P 1 P and Q 1 = QQ i = Q i Q.

Construct operators P 2 = P — P 1 and Q ₂ = Q — Q 1. Due to corollary 1 these operators are projectors. Put U¹² = im P 2, F¹² = im Q ₂ and by L ₁₂ MI 12) denote restriction of operator L (M) 01 ito U ¹².

Corollary 2. Let conditions of lemma 2 be fulfilled. Then

( i) U = U⁰ ® U¹ , F = F ® F¹ , U¹ = U¹¹ ® U¹² , F¹ = F¹¹ ® F¹² .

(it) operators L ₁₂ ,M ₁₂ G L (U¹²; F¹²),.'

(Hi) there exists an operator L- G L (F¹²; U¹²).

Definition 3. Let P be a phase space of (4). The subset J C P is called an invariant space of equation (4), if for arbitrary u 0 G J the solution u = u ( t ) of (4), (5) lies in J pointwise (i.e. u ( t ) G J for all t G R +).

Theorem 4. Let operators L,M G L (U; F) be defined as in example 2 and condition (6) hold then the image of group

T f R L ( M ) d^t G R, 2 ni Y 1  ^

• V t =

  
  

is an invariant space of (4).

Proof. The statement follows from equality im V• = im P 1 = U¹¹, that follows from theorem 2, corollaries 1 and 2.

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Definition 4. We say that solutions of (4) have exponential dichotomy, if

• (i) the phase space of (4) can be represented as P = J¹ ® J², where J¹⁽²⁾ are invariant spaces of equation (4);

• (n) for arbitrary u 0 G J ¹ ( u 0 G J ²) soli.ition u = u ( t ) of (4). (5) is such that _U fu ( t ) || < C 1 ( u 0 ) e^-“t (_U fu ( t ) || > C ₂( u 0 ) e“t ) for some a >  0 and all t G R ₊ .

Theorem 5. Let operators L,M G L(U; F) be defined as in example 2 and condition aL (M) П iR = 0

hold. Then solutions of (4) have exponential dichotomy.

Proof. The estimates of solutions can be received in dependence of location of components of L -spectrum of operator M. Due to condition (5) we can consider aL ( M ) containing the points of L -spectrum of operator M located in the right halfplane and construct projectors P 1, P 2 and invariant spaces J¹, J² respectively. Obviously, the needed estimates for solutions hold.

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