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MSC 35A01, 35G20 DOI: 10.14529/mmpl80101
THE CAUCHY PROBLEM FOR THE SOBOLEV TYPE EQUATION OF HIGHER ORDER
A.A. Zamyshlyaeva, E.V. Bychkov
South Ural State University, Chelyabinsk, Russian Federation
E-mail: zamyshliaevaaa@susu.ru, bychkovev@susu.ru
Of concern is the semilinear mathematical model of ion-acoustic waves in plasma. It is studied via the solvability of the Cauchy problem for an abstract complete semilinear Sobolev type equation of higher order. The theory of relatively polynomially bounded operator pencils, the theory of differentiable Banach manifolds, and the phase space method are used. Projectors splitting spaces into direct sums and an equation into a system of two equivalent equations are constructed. One of the equations determines the phase space of the initial equation, and its solution is a function with values from the eigenspace of the operator at the highest time derivative. The solution of the second equation is the function with values from the image of the projector. Thus, the sufficient conditions were obtained for the solvability of the problem under study. As an application, we consider the fourth-order equation with a singular operator at the highest time derivative, which is in the base of mathematical model of ion-acoustic waves in plasma. Reducing the model problem to an abstract one, we obtain sufficient conditions for the existence of a unique solution.
Keywords: semilinear Sobolev type equation of higher order; Cauchy condition; relatively polynomially bounded operator pencils; phase space method.
Introduction
Let Q = (0, a) x (0,6) x (0, c) С R3. In a cylinder Q x R consider equation which arose in a theory of ion-acoustic waves in plasma [1]
д 2u
(A - \)utttt + (A - X)utt + = A(u3) (!)
with the Cauchy-Dirichlet conditions
u(x, 0) = u0(x), ut(x, 0) = u1(x), Utt(x, 0) = u2(x), uttt(x, 0) = u3(x), x e Q, (2)
u(x, t) = 0, (x,t) e дQ x R.
In suitable Banach spaces U and F mathematical model (1), (2) can be reduced to the Cauchy problem
u(k)(0) = uk, k = 0, l,...,n - 1, (3)
for a semilinear Sobolev type equation of higher order
Au(n) = Bn-1u(n-1) + B„-2u(n-2) + ... + Bou + N (u), (4)
where u(k is the time derivative of order k, the operators A, Bn-1, Bn-2,..., B0 £ L(U; F),N £ C^(U; F)- By Sobolev type equations we mean those equations that are not solvable with respect to the highest time derivative in particular when the operator A is not invertible. Such situations often arise when ker A = {0}. Mathematical models representable in form (3), (4) will be called Sobolev type mathematical models of higher order.
It is known that the Cauchy problem (3) for Sobolev type equations is unsolvable in principle for arbitrary initial data u0,ui,... ,un-i. In our opinion, the most fruitful approach to the study of such equations is the phase space method developed by G.A. Sviridyuk and T.G. Sukacheva for the study of semilinear Sobolev type equations of the first order [2]. The essence of this method consists in reducing the singular equation (2) to a regular one, defined, however, not on the entire space, but on some subset containing admissible initial values, understood as the phase space of the original equation.
A theory of complete linear Sobolev type equations of higher order is presented in [3]. Semilinear Sobolev type equations of the first order were studied in [4,5]. Initial-boundary value problems for Sobolev type equations of the first and higher order find application in mathematical modelling [6,7].
The aim of the work is to develop a method of analytical investigation of Sobolev type mathematical models of higher order. In addition to the phase space method, the methods of the theory of relatively polynomially bounded operator pencils [8] are also used. In this article we also rely on the theory of differentiable Banach manifolds [9].
1. Theory of Relatively Polynomially Bounded Operator Pencils
Let U, F be Banach spaces and operators A, B0, B1,..., Bn-1 £ L(U; F)- By B denote the pencil formed by operators Bn-i,Bi, Bo. The sets p (S) = {l £ C : (|nA -lin-iBn-i - ... - B - Bo)-1 £ L(F; U)} and aA(S) = C \ pA(B) are called an A-resolvent set and an A-spectrum of the pencil S respectively. The operator-function of a complex variable RA(S) = (lnA — in-iBn-l — ... — iBi — B0)-1 with the domain pA(S) is called an A-r^olvent of the pencil S.
Definition 1. The operator pencil B is called polynomially bounded with respect to an operator A (or polynomially A-bounded) if 3a £ R+ £ C (|| > a) ^ (RA(B) £ L(F; U)).
Remark 1. If there exists an operator A-1 £ L(F; U) then the pencil B is A-bounded.
In [10] A.A. Zamyshlyaeva received the necessary condition for the construction of projectors
J Ik rA(B )di = O, k = 0,1,...,n - 2, (5)
Y
where the circuit 7 = {1 £ C : ||| = r > a}.
Lemma 1. [10] Let the operator pencil B be polynomially A-bounded and condition (5) be fulfilled. Then the operators
P = 2-j RA(B)in-1 Adi, q = 2- i vn-lAR£(BW
are projectors in spaces Я and F respectively.
Denote Я0 = ker P, F0 = ker Q, Я1 = im P, F1 = im Q. According to lemma 1 Я = Я0 ф Я1, F = F0 Ф F1- By Ak (Bk) denote restriction of operators A (B) on U, k = 0,1; l = 0, l,---,n - 1-
Theorem 1. [10] Let the operator pencil B be polynomially A-bounded and condition (5) be fulfilled. Then
(i) Ak e L(Uk; Fk), k = 0,1;
(ii) Bk e L(Uk; Fk), k = 0,1, l = 0,1,---,n - 1;
(Hi) operator (A1)-1 e L(Fl; Я1) exists; (iv) operator (B0)-1 e L(F°; Я0) exists.
Using theorem 1 construct operators H0 = (B0)-1Ao e £(Я°), H1 = (B0)-1B0 e L(U0),---, Hn-1 = (Bo0)-1Bn-1 e £(Я°) and So = (A1)-1B0 e ¿(Я1), S1 = (A1)-1B1 e С(Я1), - - -, Sn-1 = (A1)-1B1n-1 eL(Я1).
Definition 2. Define the family of operators {K, K2, - - -, K'^} as follows:
Kf = O, s = n, Kn = I, K1 = Ho, K2 = -H1,---,Kf = -Hs-1,---,Kn = Hn-1, K = K-Ho, K2 = k- - Kn-1H1, ---,Kf = к- - Kqn-Hs-1, ---, Kf = K- - K-Hn-u q = 1, 2, - - - -
A
те
(|nA - fin-1Bn-1 - --- - B - Bo)-1 = - J] iq Kn(Bo0)-1(I - Q) +
q=0
те
+ J] i-q (vn-1Sn-1 + • • • + + So)q L-1Q. q=1
A
resolvent of the operator pencil B. Definition 3. The point то is called
• a removable singularit у of an A-resolvent of the pencil B, if Kf = O, s = 1, 2, - - - ,n;
• a pole of order p e N of an A-resolvent of the pencil B, if3p such that Kf ф O, s = 1, 2,---,n, but Ksp+l = O,s = 1, 2,---,n;
• an essential singularit у of an A-resolvent of the pencil B, if K ф O for a 11 q e N.
Further a removable singularity of an A-resolvent of the pencil B will be called a pole of order 0 for brevity. If the operator pencil B is polynomially A-bounded and the point
to is a pole of order p £ {0} U N of an A-resolvent of the pencil B then the operator pencil B is called polynomially (A,p)-bounded.
Theorem 2. [3] Let A, Bn-1, ..., B1} B0 £ L(U, F) and A be a Fredholm operator. Then the following statements are equivalent:
BB A
by number (p + n - 1) £ {0} U N and the chain of length (p + n - 1) exists.
BB (A, p)
2. Banach Manifolds
Let M be a Ck-manifold modelled by a Banach space U. By TM denote a tangent bundle of the manifold M and by TnM denote a tangent bundle of order n. The set TM has the structure of a smooth Ck-1-manifold, modelled by Banach space U by construction, and tangent bundle TnM is a manifold of class Ck-n. Further we assume that k > n.
By nl denote a canonical projection from a tangent bundle of order l to a tangent bundle of order l - 1 where l = 1, 2,...,n and by denote projection from tangent bundle of order l to a manifold M, i.e. = nln2 .. .nl.
Consider a curve a : J l M of class Cs, (s < k) where J is some interval containing zero. By canonical lifting of the curve a we call a curve a1 in TM a1 : J l TM such that nlal = a. Similarly, % the lifting of order l of curve a in TlM we call a curve al : J l TlM such that nl a1 = a. Therefore lifting of order l of the curve is a mapping of class s - l > 1.
On the basis of the definition of a second-order differential equation [9] introduce
Definition 4. A differential equation of order n on a manifold M is a vector field £ of class Ck-n on the tangent bundle Tn-1M such that for all v £ Tn-1M the equality
nn£(v) = v
holds.
It follows from the definition that £ is a differential equation of order n iff every integral curve ¡3 for £ is a lifting of order n - 1 of the curve nn-1ft- In other words
(<-1P )n-1 = 3.
MU
Tn-1M, the main part of differential equation
f : Tn-1M l Un
has n components f = (f1, f2,..., fn) each of which maps Tn- 1M into U.
f Ck-n
n
f (91,g2,... ,9n) = (92,g3,... ,9n, fn(91,92,..., 9n)). Following [9] we formulate and prove
M Ck £ n
class Ck-n. Then for any point (u0,u1 ,...,un-1) £ Tn-1M there exists a unique curve u £ C1^(—t, t); M), t = t(u0,u1,... ,un-1) > 0 l > n, lying in M, passing through the point (u0,u1,... ,un-1) such that
u(n) = fn(u,u ,u,...,u(n-1)), ()
u(k) (0)= uk, k = 0,1, . . . , n - 1. { >
Proof. Since Tn-1M is a Ck-n+1-manifold and £ is a vector field of class Cl on Tn-1M, then for any point (u0,u1,... ,un-1) £ Tn-1M, there exists a unique integral curve <(t),t £ (—t, t), passing through the point (u0,u1,..., un-1) (<(0) = (u0,u1,..., un-1)).
n
<(t) = (u(t),u1(t),... ,un-1(t)) £ M x Un-1. f£
<= (u(t),U1(t),...,lin(t)) = f (u(t),u1(t),. . . ,un-1(t)) = = (u1(t),.. . ,un-1(t),fn(u(t),u1(t), . . .,un-1(t))). Therefore, the differential equation can be rewritten in more convenient form
u(t) = u1 (t) ,
u1(t) = u2(t),
un-1(t) = fn(u(t),u1(t), . . . ,un-1(t))
or u(n)(t) = fn(u(t), u1(t),... ,un-1(t)). Making the reverse substitution, we obtain
u(n) = fn(u,u,u,...,u(n-1)). Thus, the curve (n*<)(t) = u(t),t £ (—t,t), lies in M and satisfies (6).
□
3. The Cauchy Problem
Turn to problem (3), (4) and give definition of its solution.
Definition 5. If a vector-function u £ C^((—t,t); U^ t £ R+ satisfies equation (4) then it is called a solution of this equation. If the vector-function satisfies in addition condition (3) then it is called a solution of (3), (4).
Definition 6. The set P is called a phase space of (4), if
(i) for all (u0,u1,... ,un-1) £ Tn-1P there exists a unique solution of (3), (4);
(ii) a solution u = u(t) of (4) lies in P as a trajectory, i.e. u(t) £ P for all t £ (—t, t).
If ker A = {0} then equation (2) can be reduced to an equivalent equation
u(n) = F (u,u,...,u(n-1)),
where F(u,u,... ,u(n-l)) = A-l(Bn-lu(n-l) + Bn-2u(n-2) +... + B0u + N(u)) is a mapping of class C^ by construction. The existence of a unique solution u of (3), (4) for all (u0,u:,... ,un-l) follows from theorem 3.
Let ker A = {0} and operator pencil B be (A, 0)-bounded, then by theorem 1 equation (4) can be reduced to an equivalent system of equations
0 = (I - Q)(Bo + N)(u0 + ul), dn dn-l dn-2 (7)
—ul = A-lQ(Bn-l -nl + Bn-2 + ... + B0 + N )(u0 + ul), 1 j
where ul = Pu, u0 = (I — P)u.
Now consider a set M = {u E U : (I — Q)(B0u + N(u)) = 0} Let the set M be not empty, i.e. there is a point u0 E M. Denote u0l = Pu E Ul.
The set M is called a Banach Ck-manifold at point u0 if there exist neighborhoods
0 C ^md Ol C Ul ^^^ints u0 and ul respectively and a Ck-diffeomorphism 8 : Ol ^ O
such that 8^ is a of projector P on O. ^te set M is called a Banach Ck-
manifold modelled by the space Ul if it is a Banach Ck-manifold at any point. Connected Ck
Let the following condition be fulfilled
(I — Q)(B0 + N'Uo) : U0 ^ F0 is a toplinear isomorfism. (8)
According to the implicit function theorem [11] there exist neighborhoods O0 C U0 and
01 C Ul of points u0 = (I — P)u0, ul = Pu0 and the operator IB E C(X(Ol; O0) such that u0 = B(u0^. Construct an operator 8 = I + B : Ol ^ M, 8(u0) = u0. Then the operator ^^^^^^^^ with the set Ol a map of M and is a restriction of P on 8[Ol] = O C M. Thus we prove
Lemma 3. The set M = {u E U : (I — Q)(B0u + N(u)) = 0} under condition (8) is a C™-manifold at point u0.
Lets act with the Frechet derivative 8(n) i i , of order n on the second equation of 8(ul) = u
Jn
8(n) i i )ul(n) = d- (8(ul))
(ulou1,-K-i) dtn v J
we obtain equation u(n) = F(u, u,..., u(n-l)), where
F (u,u ,...,u(n-l) ) = 8((n)ui ui )A-lQ(Bn-lu(n-l) + Bn-2u(n-2) +
(u0 >ui >■■■ K-i)
+B0u + N(u)) E C~(U).
By virtue of theorem 3, we get
Theorem 4. Let the operator pencil B be (A, 0)-6ounded; N E C^(U; F) and condition (8) be fulfilled. Then for any (u0,ul,... ,un-l) E Tn-lM there exists a unique solution of M
4. Mathematical Model of Ion-Acoustic Waves in Plasma
Turn to the model example (1), (2). In order to reduce (1), (2) to (3), (4) set
U = {u e Wl2+2{Ü) : u(x) = 0,x e дП}, F = Wl2{Ü).
3 _ д2
io = -а ш--
Define operators A = A — X, B2 = (X' — A), B0 = —adX?, B3 = B\ = O. Operators A, B3,B2,B1, Bo are G L(U; F) for all l G {0} U N. 3
Denote the eigenfunctions of the Dirichlet problem (2) for the Laplace operator by <£kmn = {sin ^a1 sin sin }> wkere K m n G N and denote the eigenvalues by
Xkmn = —\j (nkk )2 + (IT )2 + (nn )2 • The spectr um a (A) is negative, discrete, finite and tends only to — ro. Since {pkmn} C C^(Q) we obtain
Ii4A - ¡I3B3 - ß2B2 - ßB1 - B
CO
^—^ 4 2 / nn\2
= [(Xkmn — X)f + (Xkmn — X)f — « (-J ] < ^kmn, • > ^kmn,
k,m,n=l
where < •, • > is the inner product in L2(Q).
Remark 2. In the case when (i) X G a (A) the A-spectrum of pencil B aA(B) = {fjrmn : r,m,n G N, j = 1,..., 4} where fjrmn are the roots of equation.
\ 2
—J = 0. (9)
In the case when (ii) (X G a (A)) A (X = X') the A-spectrum of pencil B aA(B ) = {fj k : k G N}, where fjk are the roots of equation (9) with X = Xi. In the case when (iii) (X G a (A)) A (X = X') the A-spectrum of pencil B aA(B ) = {j k : k G N, k = l}.
Check condition (5) In case (i) there exists A-1 G U1) therefore condition (5) is fulfilled [3]. In case (ii)
1 f ^ r J
1 / V^ f < ^kmn, • > Pkmndj
E
W . ^ , (Xkmn - X)l4 + (Xkmn - X')i2 - a(f)2
Y к, m, n=l c
_ 1 i l1Г < фкти, ' > фkmndЦ _ _
_W k mn, (Xkmn - X')^2 - a(nn)2 _ '
Y к ,m ,n= 1 c
when r _ 1, therefore condition (5) is not fulfilled and this case is excluded from further considerations. In case (iii) (X G а (А)) Л (X _ X' ) condition (5) is fulfilled.
Lemma 4. Let (г) X G а (А)) or (ii) (X G а(А)) Л (X _ X' ). Then реп cil 13 is polynomially (A, 0)-bounded.
Proof. In case (i) ker A _ {0} that is, the operator A has no eigenvectors and, by remark 1 the pencil B is (A, 0)-bounded.
In case (ii) A G a(A) and A = A' construct the chain of B-adjoined vectors of an eigenvector = Y1 akmntykmn G ker A \ {0}. Sinee B3 = B\ = O the first three
^ — ^kmn
B-adjoined vectors can be taken equal to zero. On the fourth we obtain
B0^0 = B0( £ akmntykmn) = —a (~) ^^ akmntykmn G imA,
^ — ^kmn ^—^kmn
since Y, \akmn\ > 0.
^—^kmn
Therefore the eigenvector doesn't have a B-adjoined vector of order four, the length of the chains of B-adjoined vectors of operator A is bounded by three, and the chain of length three exists.
□
Construct projectors. In case (i) P = ^d Q = I. In case (ii)
P 1 ^ ^ < Pkmm • > Pkmni
^ ^kmn
and the projector Q has the same form but it is defined on space F- Construct the set M = {u G U : <a(-j u + A(u3), tykmn > tykmn = 0}.
_ V c J
^—^kmn
By theorem 4 we have
Theorem 5. (i) Let X G 0"(A), (u0, ul,..., un-1) E Un. Then for sorne r = t(u0,ul,... ,un-l) > 0 there exists a unique solution u E Cn((—t,t),H) of problem (1), (2).
(ii) Let (X G 0"(A)) A (X = X'), (u0,u\,... ,un-i) E Tn-1 M and condition (8) be fulfilled. Then for some t = t(u0,ul,... ,un-l) > 0 there exists a unique solution u E Cn((—t, t), M) of problem (1), (2).
Acknowledgment. The work was supported by Act 211 Government of the Russian Federation, contract no. 02.AOS.21.0011.
References
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Received November 10, 2017
УДК 517.9 DOI: 10.14529/mmpl80101
ЗАДАЧА КОШИ ДЛЯ УРАВНЕНИЯ СОБОЛЕВСКОГО ТИПА ВЫСОКОГО ПОРЯДКА
A.A. Замышляева, Е.В. Бычков
Южно-Уральский государственный университет, г. Челябинск,
Российская Федерация
В статье исследована полулинейная математическая модель ионно-звуковых волн в плазме на основе разрешимости задачи Коши для абстрактного полного полулинейного уравнения соболевского типа высокого порядка. Используется теория относительно полиномиально ограниченных пучков операторов, теория дифференцируемых банаховых многообразий и метод фазового пространства. Построены проекторы, расщепляющие пространство в прямую сумму, и уравнение на два эквивалентных уравнения. Одно из уравнений определяет фазовое пространство, и его решением является функция со значениями из собственного подпространства оператора при старшей производной по времени. Решением второго уравнения является функция со значениями из образа проектора. Таким образом, были получены достаточные условия разрешимости изучаемой задачи. В качестве приложения рассмотрено уравнение четвертого порядка с сингулярным оператором при старшей производной по времени, лежащее в основе математической модели ионно-звуковых волн в плазме. Редуцировав модельную задачу к абстрактной, были получены достаточные условия существования единственного решения полулинейной математической модели ионно-звуковых волн в плазме.
Ключевые слова: уравнение соболевского типа высокого порядка; полулинейное уравнение; полиномиальный пучок операторов; метод фазового пространства.
Литература
1. Замышляева, A.A. Вычислительный эксперимент для одной математической модели ионно-звуковых волн / A.A. Замышляева, A.C. Муравьев // Вестник ЮУрГУ. Серия: Математическое моделирование и программирование. - 2015. - Т. 8, № 2. - С. 127-132.
2. Свиридюк, Г.А. Фазовые пространства одного класса операторных полулинейных уравнений типа Соболева / Г.А. Свиридюк, Т.Г. Сукачева // Дифференциальные уравнения. - 1990. - Т. 26, № 2. - С. 250-258.
3. Замышляева, A.A. Линейные уравнения соболевского типа высокого порядка / A.A. Замышляева. - Челябинск: Издательский центр ЮУрГУ, 2012.
4. Свиридюк, Г.А. О галеркинских приближениях сингулярных уравнений типа Соболева / Г.А. Свиридюк, Т.Г. Сукачева // Известия высших учебных заведений. Математика. -1989. - № 10. - С. 44.
5. Манакова, H.A. О решении задачи Коши - Дирихле для уравнения Баренблатта - Гиль-мана / H.A. Манакова, Е.А. Богатырева // Известия Иркутского государственного университета. Серия: Математика. - 2014. - Т. 7. - С. 52-60.
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Алена Александровна Замышляева, доктор физико-математических наук, доцент, кафедра «Прикладная математика и программирование:», ЮжноУральский государственный университет (г. Челябинск, Российская Федерация), zamyshliaevaaa@susu.ru.
Евгении Викторович Б ЫЧКОВ, К^НДИДсХТ физико-математических наук, кафедра «Уравнения математической физики», Южно-Уральский государственный университет (г. Челябинск, Российская Федерация), bychkovev@susu.ru.
Поступила в редакцию 10 ноября 2017 г.
