On one Sobolev type mathematical model in quasi-Banach spaces Текст научной статьи по специальности «Математика»

MSC 46A16, 35G05

DOI: 10.14529/mmpl50112

ON ONE SOBOLEV TYPE MATHEMATICAL MODEL IN QUASI-BANACH SPACES

A.A. Zamyshlyaeva, South Ural State University, Chelyabinsk, Russian Federation [email protected],

H.M. Al Helli, South Ural State University, Chelyabinsk, Russian Federation

The theory of Sobolev type equations experiences an epoch of blossoming. In this article the theory of higher order Sobolev type equations with relatively spectrally bounded operator pencils, previously developed in Banach spaces, is transferred to quasi-Banach spaces. We use already well proved for solving Sobolev type equations phase space method, consisting in reduction of singular equation to regular one defined on some subspace of initial space. The propagators and the phase space of complete higher order Sobolev type equations are constructed. Abstract results are illustrated by specific examples. The Boussinesq-Love equation in quasi-Banach space is considered as application.

Keywords: Sobolev type equations; quasi-Banach spaces; propagators; phase space.

Introduction. Let U, $ be Banach spaces, operators L,M E L(U; F) (i-e. linear, continuous, defined on U and acting into F)- Consider a Sobolev type equation (the term was introduced by R.E. Showalter [1])

Sobolev type equations have been already well researched. The first monograph devoted to such equations was published in 2003 [2]. Here degenerate analytical (semi)group, and degenerate Co-semigroups used in the study of equations of the first order were constructed. Linear Sobolev type equations of higher order in Banach spaces were studied in [3]. The results of Sobolev type equations theory are used in the theory of dynamical measurements [4], optimal control theory [5, 6], in the study of dichotomies of equations of the form (1) [7, 8]. In addition, the theory of degenerate groups and semigroups of operators was transferred into a locally convex spaces.

Equations that are not solved with respect to the highest derivative in time were studied for the first time by A. Poincare, however, a systematic study of them was started in the middle of the last century after the fundamental work of S.L. Sobolev. Now the Sobolev type equations theory is actively studied area of nonclassical equations of mathematical physics, and a number of monographs, completely devoted to them, or in part, is growing like an avalanche.

In this paper Sobolev type equations of the second order of the form (1) are considered in quasi-Banach spaces. As it is well known [9, p. 3.10], a quasi-Banach space is not a normed one, but it can be made metrizable. One of the examples of a quasi-Banach space is a space of sequences lq, q E (0,1). In [10] there was constructed a quasi-Banach space l™, q E (0,1) mE R, l0 = lq which was called a quasi-Sobolev space. These spaces we will be used to illustrate the abstract results of the paper.

The authors consider it their pleasant duty to express there sincere gratitude to Professor G. A. Sviridyuk for the statement of the problem and fruitful discussions.

U

a field R is called a quasi-normed, if there is a function • I : U ^ R with the following properties:

(i) U.IMI > 0 for all u E U, and kIMI = 0 exactly when u = 0, where 0 is a zero in U

Ли" = Б1и' + B0u, ker A = {0}.

(1)

(ii) u||au|| = Mu||u|| f°r all n G U a G Rj

(iii) uHu + v|| — C(u||u|| +u IMI) f°r u,v G U, where the constant C > 1.

A function uH ' H with properties (i) - (iii) is called a quasi-norm. In particular, if C = 1 a quasi-norm ull ' H is called a norm, and lineal U with the norm ull ' H is a normed one. A quasi-normed lineal (u; ull ' H) is metrizable (see [9, Lemma 3.10.1]), so we have a conception of the fundamental (or Cauchy) sequence {uk} C H: u||uk — uiH — 0 f°r k,l — to. Define the quasi-Banach space as a full quasi-normed lineal. Example 1. Let {Ak} C R+ be a monotonic sequence such that lim Ak = +to, and

q G R+. Put

C = ju = {uk} C R : £ lukl) " < • Lineal im for a 11 m G R, q G R+ with a quasi-norm of element u = {uk} G im

i/q

_n \ q \

huh = \>: a

/ <x \

(E (a? luk i)9J

is a quasi-Banach space (for q G [1, +to) it is a Banach space). Note that if q G (0,1), then constant C = 2 /q in (iii). The spaces im are called quasi-Sobolev in [10].

G

im iq

Let u uH ' II) and (F; fII ' H) quasi-Banach sp aces, a linear operator L : u — F with

the domain domL = u is called continuous if lim Luk = L ( lim uk 1 for any the sequence

{uk} C u which is convergent in u. Note that in this case the linear operator L : u — F is continuous precisely when it is bounded (i.e., displays bonded sets to bounded sets). Denote by L(u; F) a lineal (over the field R) of linear bounded operators. It is a quasi-Banach space with quasi-norm

L(UF) HLH = suP FHLuIl uM=1

Now let the operators A,B1,B0 G L(u;F)- Following [3], introduce the sets pA(B) = {^ G C : (^2A - B — Bo)-1 G L(F; u)^d aA(B) = C\pA(B), which are called an A-resolvent set and an A-spectrum of the pencil B, respectively.

The operator-function RA(B) = (^2A — ¡j,B1 — B0)-1 with domain pA(B) is called an A BB

BB

AA

3a G R+ G C (H > a) ^ RA(B) G L(F;u))-

Fix y = {^ G C : /^l = r > a} which is a contour bounding a disk containing aA(B). We require additional condition

[ra (B)d^ = 0- (A)

Jj

This condition was introduced in [3] and is very important in considering of the Sobolev type equations of higher order. Note that if there exists an operator A-1 G L(F; u) or operator B1 = O (equation is incomplete), then condition (A) holds; and if operator A = O and there exists an operator B-1 G L(F;u), then condition (A) is not fulfilled.

SHORT NOTES

Let the pencil B be polynomially A bounded, and (A) be fulfilled. Then there exist the following integrals of analytic operator functions:

P Y Q Y vARA(B)d».

Lemma 1. Let the pencil B be polynomially A-bounded and (A) be fulfilled. Then the operators P E L(U) and Q E L(F) are projectors.

Denote U0 = ker P, F0 = ker Q, U1 = im P, F1 = im ^^mma U = U0 © U1, F = F0 © F1- By Ak (Bk) denote restriction of operator A (Bi) rato Uk, k,l = 0,1.

Theorem 1. Let the pencil B be polynomially A bounded, and (A) be fulfilled. Then actions of operators split:

(i) Ak E L(Uk; Fk), k = 0, 1;

(ii) Bk E L(Uk; Fk), k,l = 0,1;

(Hi) there exists an operator (A1)-1 E L(Fl;U1); (iv) there exists an operator (B0)-1 E ^(F0;U0).

Example 2. Using formula An = [Xkuk}, u E im we introduce a Laplace quasi-operator.

It is easy to show [10] that operator Л : l^2 ^ is a toplinear isomorphism for all m E R q E R+. The inverse operator Л-1и = {A-1u]} is called a Green quasi-operator. Next, construct the operators A = A — Л and B1 = а(Л — A') B0 = в (Л — A")

а, в E R+.

Lemma 2. Let

(i) A E {Ak} or (ii) (A E {Ak})A(A = A'). Then the pencil B = (B1,B0) is polynomially A-bounded, moreover, ж is a removable singularity of A-resolvent of B.

(Hi) (A E {A]}) A (A = A') A (A = A"). Then the pencil B = (B1,B0) is polynomially A-bounded moreover, ж is a pole of order 1 of A-resolvent of B.

Remark 1. In case (i) A-spectrum of pencil B aA(B) = {/I'2 : к E N}, where щ1]2 are the roots of equation

(A — A] )/2 + а (A' — Ak )/ + в (AAk) = 0. (2)

In case (ii) aA(B) = {/]'2 : к E N, A] = A} U {/ : l E N, Ai = A}, where /л]'2 are the roots of (2) when A = Ak, and щ is the root of (2) when A = A^ In case (iii) °A(B) = {/k2 : к E N, A] = A}.

Remark 2. It is easy to see that if (A E {Ak}) A (A = A' = A") then the pe ncil B is not A

Now let's check the condition (A) In case (i) there exists an operator A-1 E L(F;U), (A)

f sr^_< фк, • > фк d/_= у- <фк, • > = о

2пг J (A — A])/2 + а (A' — A])/ + в (AAk) = ¿^ a(A' — A])фк = '

(A)

(A)

2. The Phase Space and Propagators. Now consider the Sobolev type equation of the second order (1) with the initial condition

u(m)(t) = u(m, m = 0,1- (3)

Definition 2. The operator-function U* G C^(R; L(U)) is called a propagator of equation (1) if for any u G U vector- function u(t) = Utu is a solution of this equation.

BB A (A)

Then there exist propagators of equation (1):

U{ = — i RA(B)Ae^d^,t G R, U0 = — I RA(B)(v(A — B^d^t G R-2n% . Y ^ 2n%

Further, if the pencil B is polynomially A-bounded, condition (A) holds and to is a pole of order p G {0} (J N of A-resolvent of B, then the pencil Bis called (A,p)-bounded.

Definition 3. The Set P C U is called a phase space of equation (1) if

(i) for any uj G P, j = 0,1, there exists a unique solution of (1), (3);

(ii) any solution u = u(t) of equation (1) lies in P as a trajectory (i.e., u(t) G P for tGR

Theorem 2. Let the operator M be (L,p)-bounded, p G {0} U N. Then the subspace U1 is a phase space of equation (1). Moreover, for any uj G U1, j = 0,1, there exists a unique solution of the Cauchy problem (3) for equation (1) which can be represented as: u(t) = U0uo + Ut^u1.

Example 3. Let U = im+2, F = i^> m G R, q G (0,1) where i™ is a quasi-Sobolev space defined in example 1, and A,B1,B0 are the operators constructed in example 2.

Consider the Boussinesq-Love equation as one of the most well-known non-classical equations of mathematical physics of the second order in time [11]

(A — A)u = a (A — A')u + /3 (A — A")u, u(t) G U (4)

Theorem 3. Let A G {Ak} or (A G {Ak}) A (A = A') A (A = A"). Then for any sequence

u= iu(ja Gk1 = i u if A = Ak,k G N; 7 = 01 uj = {u(jk} G u = j {u G u : ui = 0, A = Ai}, j = 0 1

there exists a unique solution of (3), (4), which also has the form u(t) = £ '

k k k

e^kt — e^lt

vk(A — Ak) + a(A' — Ak) t + vk(A — Ak) + a(A' — Ak) e^kt (A — Ak )(vk— vk) (A — Ak )(vk— vk) .

uokek+

+

/ 1 2\ u1kek, (vk— vk)

k

A = Ak. Here nl'2 are the roots of (2), the vectors ek = (0, 0, • • •, 0,1, 0, • • •), where unit stands on the k-th place. The set U1 is a phase space of equation (4).

Y

References

1. Showalter R.E. The Sobolev Type Equations. I (II). Appl. Anal, 1975, vol. 5, no. 1 (2), pp. 15-22 (pp. 81-99).

2. Sviridyuk G.A., Fedorov V.E. Linear Sobolev Type Equations and Degenerate Semigroups of Operators. Utrecht, Boston, Köln, VSP, 2003. 216 p. DOI: 10.1515/9783110915501

3. Zamyshlyaeva A.A. Lineynye uravneniya sobolevskogo tipa vysokogo poryadka [Linear Sobolev Type Equations of High Order]. Chelyabinsk, Publ. Center of the South Ural State University, 2012. 107 p.

4. Shestakov A.L., Keller A.V., Nazarova E.I. Numerical Solution of the Optimal Measurement Problem. Automation and Remote Control, 2012, vol. 73, no. 1, pp. 97-104. DOI: 10.1134/S0005117912010079

5. Manakova N.A., Dyl'kov A.G. Optimal Control of the Solutions of the Initial-finish Problem for the Linear Hoff Model. Mathematical Notes, 2013, vol. 94, no. 1-2, pp. 220-230. DOI: 10.1134/S0001434613070225

6. Zamyshlyaeva A. A., Tsyplenkova O.N. Optimal Control of Solutions for Showalter-Sidorov-Dirichlet Problem for the Boussinesq-Love equation. Differential equations, 2013, vol. 49, no. 11, pp. 1390-1398. DOI: 10.1134/S0012266113110049

7. Sviridyuk G.A., Keller A.V. Invariant Spaces and Dichotomies of Solutions of a Class of Linear Sobolev Type Equations. Russian Mathematics (Izvestiya VUZ. Matematika), 1997, vol. 41, no. 5, pp. 57-65.

8. Sagadeeva M.A. Dikhotomii resheniy lineynykh uravneniy sobolevskogo tipa [Dichotomy of Solutions of Linear Sobolev Type Equations]. Chelyabinsk, Publ. Center of the South Ural State University, 2012.

9. Berg J., Löfström J. Interpolation Spaces. An Introduction. Berlin, Heidelberg, N.Y., 1976. 222 p.

10. Sviridyuk G.A., AI Delfi D.K. Splitting Theorem in Quasi-Sobolev spaces. Matematicheskie zametki YaGU, 2013, vol. 20, no. 2, pp. 180-185. (in Russian)

11. Love A.E.H. A Treatise on the Mathematical Theory of Elasticity. N.Y., Dover Publications, 1944. 643 p.

Received September 4, 2014

УДК 517.9 DOI: 10.14529/mmpl50112

ОБ ОДНОЙ МАТЕМАТИЧЕСКОЙ МОДЕЛИ СОБОЛЕВСКОГО ТИПА В КВАЗИБАНАХОВЫХ ПРОСТРАНСТВАХ

A.A. Замышляева, Х.М. Ал Хелли

Теория уравнений соболевского типа переживает эпоху бурного расцвета. В данной работе теория уравнений соболевского типа высокого порядка с относительно спектрально ограниченным пучком операторов, развитая в банаховых пространствах, переносится в квазибанаховы пространства. Мы используем уже хорошо зарекомендовавший себя при решении уравнений соболевского типа метод фазового пространства, заключающийся в редукции сингулярного уравнения к регулярному, определенному на некотором подпространстве исходного пространства. Построены пропагаторы и фазовое пространство полного уравнения соболевского типа второго порядка. Абстрактные результаты иллюстрированы конкретными примерами. В качестве приложения рассмотрено уравнение Буссинеска - Лява в квазибанаховом пространстве.

Ключевые слова: уравнения соболевского типа высокого порядка; квазибанаховы пространства; пропагаторы; фазовое пространство.

Литература

1. Showalter, R.E. The Sobolev type equations. I (II) / R.E. Showalter // Appl. Anal. - 1975. -V. 5, № 1 (2). - P. 15-22 (P. 81-99).

2. Sviridyuk, G.A. Linear Sobolev Type Equations and Degenerate Semigroups of Operators / G.A. Sviridyuk, V.E. Fedorov. - Utrecht, Boston; Köln: VSP, 2003. - 216 p.

3. Замышляева, A.A. Линейные уравнения соболевского типа высокого порядка / A.A. За-мышляева. - Челябинск: Издат. центр ЮУрГУ, 2012. - 107 с.

4. Шестаков, А.Л. Численное решение задачи оптимального измерения / А.Л. Шестаков, A.B. Келлер, Е.И. Назарова // Автоматика и телемеханика. - 2012. - № 1. - С. 107-115.

5. Манакова, H.A. Оптимальное управление решениями начально-конечной задачи для линейных уравнений соболевского типа / H.A. Манакова, А.Г. Дыльков // Вестник ЮУрГУ. Серия: Математическое моделирование и программирование. - 2011. - № 17 (234). -С. 113-114.

6. Замышляева, A.A. Оптимальное управление решениями задачи Шоуолтера-Сидорова-Дирихле для уравнения Буссинеска-Лява / A.A. Замышляева, О.Н. Цыпленкова // Дифференциальные уравнения. - 2013. - Т. 49, № 11. - С. 1390-1398.

7. Свиридюк, P.A. Инвариантные пространства и дихотомии решений одного класса уравнений типа Соболева / P.A. Свиридюк, A.B. Келлер // Известия высших учебных заведений. Математика. - 1997. - №5. - С. 60-68.

8. Сагадеева, М.А. Дихотомии решений линейных уравнений соболевского типа / М.А. Сагадеева. - Челябинск: Издат. центр ЮУрГУ, 2012. - 139 с.

9. Берг, И. Интерполяционные пространства. Введение / И. Берг, И. Лефстрем - М.: Мир, 1980. 264 с.

10. Свиридюк, Г.А.Теорема о расщеплении в квазисоболевых пространствах /P.A. Свиридюк, Д.К. Аль-Делфи // Математические заметки ЯГУ. - 2013. - Т. 20, № 2. С. 180-185.

11. Ляв, А. Математическая теория упругости / А. Ляв; пер. с англ. Б.В. Булгаков, В.Я. Натанзон. - Москва; Ленинград: ОНТИ, 1935. 674 с.

Алена Александровна Замышляева, ксшдидсхт физико-математических наук, доцент, кафедра «Уравнения математической физики>, Южно-Уральский государ-ствбнныи университет (г. Челябинск, Российская Федерация), [email protected].

Аль Хелли Хамис Монем Абдулькадум, магистрант, кафедра «Уравнения математической физики>, Южно-Уральский государственный университет (г. Челябинск, Российская Федерация).

Поступила в редакцию 4 сентября 2014 г.