BOUNDARY VALUE PROBLEMS FOR FOURTH-ORDER SOBOLEV TYPE EQUATIONS Текст научной статьи по специальности «Математика»

DOI: 10.17516/1997-1397-2021-14-4-425-432 УДК 517.9

Boundary Value Problems for Fourth-Order Sobolev Type Equations

Alexander I. Kozhanov*

Sobolev Institute of Mathematics Novosibirsk, Russian Federation

Received 10.03.2021, received in revised form 05.04.2021, accepted 20.05.2021

Abstract. The goal of the article is the study of solvability in the Sobolev spaces of boundary value problems for some classes of Sobolev-type fourth-order linear equations. We will prove that an initial boundary value problems well problems with data both at the initial time moment and the final time moments can be well-posed for the equations under study.

Keywords: Sobolev-type fourth-order differential equation, boundary value problem, existence, uniqueness.

Citation: A. I. Kozhanov, Boundary Value Problems for Fourth-Order Sobolev Type Equations, J. Sib. Fed. Univ. Math. Phys., 2021, 14(4), 425-432. DOI: 10.17516/1997-1397-2021-14-4-425-432.

Introduction

The article is devoted to the study of the solvability of boundary value problems for differential equations

3 / дк _\

DAtu + £ AkDku = f (x,t) Dк = дк, к = 0, 4

k=0 ^ '

W

with operators Ak of the form

Ak = dX iaaij'k (x) dj + a0,k (x)

(here and below, summation over repeated indices from 1 to n is carried out).

The differential equations (*) are recently attributed to the class of Sobolev-type equations. Various aspects of the theory of Sobolev-type equations are reflected in monographs [1-7] and also in numerous journal articles (it is impossible to mention even a small part of such articles just because they are numerous).

For Sobolev-type differential equations, best studied is the solvability of the Cauchy problem and initial boundary value problems. At the same time, as is shown in [3,8], in some case, for Sobolev-type equations, simultaneously with initial boundary value problems, other problems can also be well-posed; these include problems with data both at the initial and final time moments. In the present article, for equations (*), we study the solvability both of initial boundary value problems and problems with data at different time moments.

Clarify that the goal of the present article is to prove the solvability of some problem for equations (*) in the classes of regular solutions, i.e., solutions having all weak derivatives in the sense of Sobolev [9-11] occurring in the equation.

Formally, equation (*) with the above operators is a fifth-order equation. The use of the term "fourth-order Sobolev equation" in the title and the article means that the equations under study

* kozhanov@math.nsc.ru © Siberian Federal University. All rights reserved

are fourth-order equations with respect to the time (distinguished) variable, which is the leading variable and defines the statements of the problems.

One more remark: Equations (*) have model and the simplest form. We will speak of some more general equations and of generalizations of the results at the end of the article.

1. Statements of the Problems

Suppose that Q is a bounded domain in 1" with smooth (for simplicity, infinitely differen-tiable) boundary r, Q is the cylinder Q x (0, T) of finite height T, and S = r x (0,T) is the lateral boundary of Q. Furthermore, let aij'k(x), a0,k(x), i,j = 1,... ,n, k = 0,..., 3, f (x,t) be given functions defined for x e Q and t e [0, T] and let Ak and L be the differential operators whose action at a given function v(x, t) is defined by the equalities

d

Akv = dx (atJ,k(x)vxj) + ao,k(x)v, 3

Lv = Dkv + AkDkv.

k=0

Boundary Value Problem I: Find a function u(x,t) that is a solution to the equation

Lu = f (x,t) (1)

in the cylinder Q such that

u(x,t)\s =0, (2)

Dku(x,t)\t=0 xen = 0, k = 0,1, 2, 3. (3)

Boundary Value Problem II: Find a function u(x,t) that is a solution to equation (1) in Q and satisfies conditions (2) and also the condition

D'ku(x,t)\t=0txen = 0, k = 0,12 D3u(x,t)\t=T,xen = 0 (4)

Boundary Value Problem III: Find a function u(x,t) that is a solution to equation (1) in Q that satisfies conditions (2) and also the condition

u(x,t)\t=0,xen = Dtu(x,t)\t=0tXen = DMx,t)\t=o,xen = Dt u(x,t)\t=0tXen = (5)

Boundary Value Problem I is a usual initial boundary value problem for nonstationary differential equations of the fourth order (with respect to time). Boundary Value Problem II is a modified V. N. Vragov's problem (see [12-14]) for fourth-order quasihyperbolic equations. Finally, Boundary Value Problem III is in fact an elliptic boundary value problem.

In the present article, we propose sufficient conditions on the coefficients of (1) new compared to the previous works that guarantee the existence and uniqueness of regular solutions to boundary value problems I, II, or III.

2. Solvability of boundary value Problems I-III

Theorem 1. Suppose the fulfillment of the conditions

aij'k(x) e CX(Q), i,j = 1,..., n, a0k(x) e C(Q), k = 0,1, 2; (6)

aij'3(x) e C2(Q), aij'3(x) = aji'3(x), i,j = 1,...,n, a0,3(x) e C(Q), (7)

-aij'3(x)£ij > m0\t\2, m0 > 0, x e Q, £ e 1". (8)

Then, for every function f (x,t) in L2(Q), Boundary Value Problem I has a solution u(x,t) such that Dku(x,t) e L2(0,T; W%(Q) n W 1(Q)), k = 0,1, 2, 3, Dfu(x,t) e L2(Q).

Proof. Make use of the method of continuation in a parameter. Let A G [0,1]. Consider the following problem: Find a function u(x,t) that is a solution to the equation

Dfu + A3D3u + Ak Dk u = f (x, t)

(9)

k=0

and Q that satisfies conditions (2) and (3). Note that, for A = 0, this problem has a solution u(x,t) belonging to the desired class; this follows from the fact that, for A = 0, equation (9) is a usual parabolic equation with respect to uttt(x,t). Furthermore, by the theorem on the method of extension in a parameter (see [15, Chapter III, Sec. 14], the boundary value problem (9), (2), (3) has a regular solution u(x,t) if f (x,t) G L2(Q) and problem (9), (2), (3) is solvable in the class of regular solutions for A = 0 if all derivatives occurring in (9) are uniformly bounded in L2(Q).

For proving the desired boundedness, let us first consider the equality

0 Jn

D4u + A3 D3u + A Y, Ak Dku

k=0

D3 u dx dr

fDTu dx dr.

(10)

0n

Integrating by parts, applying Young's inequality and the inequality

/ w2(x,t) dx ^ T / w2 (x,T) dx dr

Jn J0 Jn

(11)

which is valid for functions w(x,t) vanishing for t = 0, and using conditions (6)-(8) and Gron-wall's lemma, it is not hard to obtain from (10) the estimate

i DTu(x,t)]2 dx + VÎ i [d3uxi)2 dxdT < C\ [ f

Jn ,_-, Jo Jn * Jo

(12)

where the constant C1 is defined only by the functions aij'k(x), i,j = 1,...,n, a0,k(x), k = 0,1, 2, 3, and the number T. Now, consider the equality

0n

D4u + AtDTu + A^2AkDku\ AtDTudxdT = - / fA3DTudxdT.

k=0

0n

Integrating by parts once again, applying Young's inequality, inequality (11), estimate (12), conditions (6)-(8), and also the second main inequality for elliptic operators (see [10, Chapter III, Stc. 8], and Gronwall's lemma, we conclude that solutions u(x,t) to the boundary value problem (9), (2), (3) satisfy the second a priori estimate

• Op n pt n n

V / DTuxi(x,t)]2 dx / DTuxiXj)2 dxdT < C2 f2 dxdt,

„•-1 Jn , „•_-, Jo Jn Jo

(13)

i,j=1

where the constant C2 is defined only by the functions aij'k(x), a0,k(x), i,j k = 0,1, 2, 3, the domain Q, and the number T.

Estimates (12) and (13) imply the obvious third estimate

i ( (D4u)2 dxdT < C3 i f2 dxdt, 0 n o

1

n

of solutions u(x,t) to the boundary value problem (9), (2), (3); the constant C3 in this estimate is again defined only by the functions aij'k(x), a0k(x), i,j = 1,... ,n, k = 0,1, 2, 3, the domain l, and the number T.

Estimates (12)-(14) give the desired uniform boundedness over A in L2(Q) of all derivatives occurring in (9). As we already said above, this boundedness and the solvability of the boundary value problem (9), (2), (3) for A = 0 give the solvability of this problem in the desired class also for A = 1. This exactly means the validity of the theorem.

The theorem is proved. □

Before proving the following theorem on the solvability of Problem I in the class of regular solutions, we formulate an auxiliary assertion on the nonnegativity of the scalar product of a pair of second-order differential operators.

Let A and B be differential operators whose action is defined by the equality

d ■■

Av = —— (a13 (x)vx.) + ao(x)v, dxi 1

d

Bv = dhi ^(x)vxj) + bo(x)v. Proposition 1. Suppose the fulfillment of the conditions

aij(x) G C2(l), bij(x) G C2(l), aij(x) = aji(x), bij(x) = bji(x), x G l, i,j = 1,...,n; a0(x) G C 1(l), b0(x) G C 1(l), a0(x) < -a0 < 0, b0(x) < -b0 < 0, x G l; 3 a (x) : ai(x) G C (l), ai (x) > 0, x G l, i = 1,...,n, ai(x)£2 < aij (x)££ < M0ai(x)g, x G l, £ G Rn;

laxk (x)| < M\\J ai (x), x G l, i,j,k = 1, ... ,n;

aij (x)vivj = 0 for x G T; bij(x)££ > m0|£|2, m0 > 0, x G l, £ G Rn;

a0(x)bij(x) + b0(x)aij(x) + 1 (aj (x)bkl(x))xi + 2 (bj (x)akl(x))xi -

- (a%k (x)bik (x))] £i£j < 0, x G l, £ G Rn;

a0(x)b0(x) + 2 (a0xi (x)bij (x))x, + 2 (b0xH (x)aij (x))x, > 0, x G l.

◦

Then every function v(x) G W2(l) n W 2(l) satisfies the inequality

/ AvBv dx > 0 JQ

This assertion is proved in [16].

We say that operators A and B of the above form satisfy the (A, B)-condition if the coefficients of these operators satisfy all conditions of Proposition 1.

Theorem 2. Suppose the fulfillment of the (-A3,-A2)-condition and also of the condition

aij'k(x) G C*(!), i,j = 1,...,n, a0,k(x) G C(l), k = 0,1. (15)

Then, for every function f (x,t) such that f (x,t) G L2(Q), ft(x,t) G L2(Q), f (x, 0) = 0 for x G ll, Boundary Value Problem I has a solution u(x,t) such that D£u(x,t) G L^(0, T; W^d n

W l(l)), k = 0,1, 2, 3, Dfu(x,t) G L(0,T; L2(l)).

Proof. Observe first of all that the (—A3, —A2)-condition in particular means that —A3 is an elliptic-parabolic operator in Q and -A2 is an elliptic operator. Let e be a positive number. Define operators A3,e and Le:

A3,e = A3 + eA2, le = L + eA2D3.

Consider the following boundary value problem: Find a function u(x, t) that is a solution to the equation Leu = f in Q that satisfies conditions (2) and (3). Obviously, this boundary value problem is Boundary Value Problem I and that it satisfies all conditions of Theorem 1. Moreover, due to the condition f(x,t) G L2(Q), ft(x,t) G L2(Q), a solution u(x,t) to this problem satisfies the memberships

Dku(x,t) G LTO(0,T; W22(Q) n WW 1(Q)), k = 0,1, 2, 3,4, D5tu(x,t) G L2(Q) (16)

(this fact stems from its validity for the "shortened" equation Dfu + Az^D^u = f(x,t) and the corresponding a priori estimates).

Differentiate the equation Leu = f (x, t) with respect to t (this is possible due to memberships (16)), multiply it by Dfu(x,t), and integrate it over the cylinder {x G Q, 0 < t <t}. Involving the ellipticity of the operators —A3,£ and — A2, applying Young's inequality, inequality (11), and Gronwall's lemma, we obtain the estimate

e[ ( (A2D4u)2 dxdT + V [ [Djuxi(x,t)]2 dx +i [A2D3u(x,t)]2 dx < C4 [ ftdxdt, (17)

J0 Jn i=i-JQ

where the constant C4 is defined only by the functions aij'k(x), a0,k(x), i,j = 1,... ,n, k = 0,1, and also the number T.

Let {em}m=i be a sequence of positive numbers converging to zero and let {um(x,t)}^=1 be a sequence of solutions to the equation LFmu = f satisfying (2) and (3). Estimate (17), the second main inequality for elliptic operators, and the reflexivity of a Hilbert space mean that there exists a sequence {umi (x,t)}f=1 and a function u(x,t) that satisfy the following weak convergences as l ^ x> in L2(Q):

emi A2D\u(x, t) ^ 0,

Dtumi (x,t) ^ D4u(x, t),

AkDkumi (x, t) ^ AkDku(x, t), k = 0,1,2, 3.

Obviously, the limit function u(x,t) is a solution to Boundary Value Problem I and this solution still satisfies (17). Therefore, the function u(x,t) is the desired solution to the problem under study.

The theorem is proved. □

Turn to investigating the solvability of Boundary Value Problem II.

The main difference of Boundary Value Problem II from Boundary Value Problem I is that, in its study, it is impossible to use Gronwall's lemma. Gronwall's lemma can be replaced by small-ness conditions.

We will give the simplest version of the theorem in the solvability of a Boundary Value Problem II, whose prove involves smallness conditions.

Let operators A0 and A1 be defined with the use of the parameter ¡3 and the operators A0 and A^

~ ~ ~ d ( d \

A0 = ¡3A0, Ai = ¡3Ai, Ak = dxi. \^ij'k(x)+ 7l0i(x), k = 0,1. (18),

Theorem 3. Suppose the fulfillment of the conditions

aij'k (x) € C2(Q), aij'k (x) = aji'k (x), ij = l,...,n, k =2, 3;

aij'k(x) € C 1(Q), i,j = l,...,n, a0,k(x) € C(Q), k = 0,1; aij'k(x)£i£j > m0\£\2, m0 > 0, x € Q, £ € R", k = 2, 3; a0,k(x) € C(Q), k = 0,1, 2, 3, a0k (x) < 0, k = 2, 3. Then there exists a positive number f0 such that for \f\ < f0 and f (x,t) € L2(Q), Boundary Value Problem II has a solution u(x,t) such that D^u(x,t) € L2(0,T; W2(Q) fl W 1(Q)), k = 0,1, 2, 3, Dfu(x,t) € L2(Q).

Proof. For A = 0, Boundary Value Problem II for equation (9) has a solution u(x, t) in the desired class; this follows from the fact that for A = 0 equation (9) is an inverse parabolic equation with respect to Dfu(x,t). Further, consider (10). Integrating by parts and estimating the last two summands on the left-hand side (10) from above with the use of (11), we infer that there exists a positive number f1 such that for \f\ < f0 we have the a priori estimate

V [ D3uXi f dxdt < C5 [ f2 dxdt (19)

i=iJ Q JQ

with the constant C5 defined only by the coefficients of the operators Ak, k = 0,1,2,3. At the next step, consider the equality

D4 u + AsDfu + AkDktu

k=0

A2Dtudxdt = fA2Dtudxdt.

Q

Reckoning with the ellipticity of A2 and A3 and using the second main inequality for a pair of elliptic operators [10, Chapter III, Sec. 8], it is not hard to show that there exists a number f0 such that 0 < f0 < f1, and for \f \ < f0, for solutions u(x,t) to Boundary Value Problem II for equation (9), estimate (13) holds with some constant C6 on the right-hand side that is defined only by the coefficients of the operators Ak, k = 0,1, 2, 3, and the domain Q.

Estimate (14) with the corresponding constant C7 on the right-hand side obviously follows from the previous estimates.

The obtained estimates of solutions to Boundary Value Problem II for equation (9) and the theorem on the method of continuation in a parameter and give the solvability of Boundary Value Problem II for equation (1) in the desired class.

The theorem is proved. □

Theorem 4. Suppose the fulfillment of the conditions aij'k (x) € C2(Q), aij'k(x)= aji'k (x), a0,k(x) € C(Q), i,j = 1,...,n, k = 0,1, 2, 3;(20)

aij'k(x)£i£j > m0\£\2, m0 > 0, x € Q, £ € R", a0,k(x) < 0, k = 2, 3; (21)

-aij'k(x)£i£j > m1 \£\2, m1 > 0, x € Q, £ € R", a0k(x) > 0, k = 0,1; (22)

Ao = @Ao. (23) Then there is a positive number f0 such that, for \f \ < f0 and f (x,t) € L2(Q), Boundary Value

Problem III has a solution u(x,t) such that Dk u(x,t) € L2(0,T; W22(Q) f W 1(Q)), k = 0,1, 2, 3, D4u(x,t) € L2(Q).

Q

Proof. Show that solutions u(x,t) to Boundary Value Problem III of the class mentioned in the statement of the theorem satisfy the desired a priori estimates.

Multiply equation (1) by D\u(x,t). Integrating over Q, applying integration by parts, and using (20)-(22), it is not hard to obtain the first a priori estimate for solutions u(x, t) to Boundary Value Problem III:

'Q

(Dlnf + £ D

dx dt ^ C% f dx dt;

Q

(24)

here the constant C8 is defined only by the coefficients of the operators Ak, k = 0,1,2,3.

At the next step, multiply equation (1) by A2Dfu(x,t) and integrate it over Q. Using conditions (20)-(23), inequality (11), and also the second main inequality for a pair of elliptic operators, we conclude that there exists a number ¡¡0 such that for \fl \ < ¡¡0 we have a second estimate

n

)2

I (Dtuxi xjY dx dt ^ Cg J f dx dt;

(25)

i,j=iJ Q

with the constant C9 defined only by the coefficients of the operators Ak, k = 0,1,2,3, and the domain Q.

The last a priori estimate

2

u

I (Dfu)2 dxdt < C10 I f2 dxdt (26)

Jq Jq

obviously stems of the previous two estimates.

Using estimates (24)-(26) and the method of continuation in a parameter (for example, with the use of the equation

Dfu + A2D2u + X(A3D^u + A1Dtu + A0u) = f (x, t)),

it is not hard to obtain the desired solvability of Boundary Value Problem III.

The theorem is proved. □

3. Conclusion.

Observe first of all that the conditions of Proposition 1 are fulfilled, for instance, if the numbers a0 and b0 are large.

Furthermore, it is not hard to generalize the obtained results to equations more general than (1); for example, to equations with general second-order elliptic operators Ak.

Some of the conditions of the proven theorems can be changed: for example, we can discard the sign-definiteness of the operator A0 from Theorem 4.

Observe finally that conditions (18) and (23) mean that A\ and A0 are fixed operators, whereas the number ¡3 is a parameter (namely, a smallness parameter).

The work of the author was carried out in the framework of the State Contract of the Sobolev Institute of Mathematics (Project 0314-2019-0010).

References

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Краевые задачи для уравнений соболевского типа четвертого порядка

Александр И. Кожанов

Институт математики им. С. Л. Соболева СО РАН Novosibirsk, Российская Федерация

Аннотация. Целью статьи является исследование разрешимости в пространствах Соболева краевых задач для некоторых классов линейных уравнений четвертого порядка соболевского типа. Докажем, что начально-краевые задачи с данными как в начальный момент времени, так и в конечные моменты времени могут быть корректными для исследуемых уравнений.

Ключевые слова: дифференциальное уравнение четвертого порядка соболевского типа, краевая задача, существование, единственность.