ON PERIODIC SOLUTIONS OF LINEAR PARABOLIC PROBLEMS WITH NONLOCAL BOUNDARY CONDITIONS Текст научной статьи по специальности «Математика»

UDC: 517.9

MSC2010: 35K10, 35B10, 35R10, 35K99

DOI: https://doi.org/10.37279/1729-3901-2021-20-2-7-11

ON PERIODIC SOLUTIONS OF LINEAR PARABOLIC PROBLEMS WITH NONLOCAL BOUNDARY CONDITIONS

© O. V. Solonukha

Federal Research Center "Informatics and Control", Russian Academy of Science, Vavilov str. 40, 119333, Moscow, Russia; RUDN University, Miklukho-Maklaya str. 6, 117198, Moscow, Russia e-mail: [email protected]

On periodic solutions of linear parabolic problems with nonlocal boundary conditions.

Solonukha O. V.

Abstract. A linear parabolic equation with nonlocal boundary conditions of the Bitsadze-Samarsky type is considered. The existence and uniqueness theorem of the periodic solution is proved.

Keywords : nonlocal problem, parabolic equation, monotone operator.

Nonlocal elliptic boundary value problems have been considered since the 30s of the XXth century in the work of T. Carleman. In the 50-60s of the XXth century, the abstract nonlocal elliptic problems were studied by M. I. Vishik, F. Browder, etc. Nonlocal parabolic problems in a bounded cylinder were considered mainly in the cases of parabolic delay differential equations, parabolic integro-differential equations, and parabolic operator- differential equations. In this paper, we consider a parabolic equation with nonlocal boundary conditions of the Bitsadze-Samarsky type, cf. [1]. The peculiarity of these nonlocal conditions is that they are set using shifts in spatial variables in a bounded domain. A method for studying elliptic boundary value problems with such nonlocal conditions was developed in the 80-90s, see [2-4]. In this paper, the time-periodic solutions of a linear parabolic equation with nonlocal boundary conditions are investigated. The proofs are given for a model example. However the method is suitable for the general case of nonlocal boundary conditions of this type.

In the rectangular parallelepiped QT = (0, T) x (0, 2) x (0,1) we consider the parabolic equation

dtw(t,x) di (Aij(t,x)djw(t,x)) = f (t,x) ((t,x) E ), (1)

i,j=1,2

with nonlocal boundary conditions

w(t, xi, 0) = w(t, xi, 1) = 0 (0 < t < T; 0 < x1 < 2), w(t, x) |xi=o = 7iw(t,x)|x1=i, (0 <t<T; 0 < x2 < 1), (2)

w(t,x)|x1=2 = 72w(t,x)|zi=i (0 < t < T; 0 < x2 < 1).

Here f 6 L2(OT), the functions Aj 6 C^(M3) are 1-periodic in xi and T-periodic in t. Moreover, Aj(t,x) = Ajj(t,x) (i,j = 1, 2), and there exists ci > 0 such that

^ Ajj(t,x)6£; > ci ^ |&|2 V(t,x) 6 OT. (3)

=i ,2 i=i,2

Time-periodic solution of (1)-(2) must satisfy the condition

w(0,x) = w(T,x) (x = (xi,x2) 6 Q = (0,2) x (0,1)). (4)

We consider our problem in Sobolev space L2(0, T; W2 (Q)), this is the set of functions u 6 L2 (Ot) such that c^u 6 L2 (OT). Let

L2(0,T; W2i,7(Q)) := {w 6 L2(0,T; W^Q)) : w satisfies (2)}. (5)

In this paper we consider the spaces of real-valued functions. We define the operator A : L2(0,T; W21,y(Q)) ^ L2(0,T; W2-1(Q)) by the formula

(Aw, v) = ^ Aij (t,x)djw(t,x) div(t,x) dxdt Vv 6 L2(0,T; W^Q)).

1<i,j<n Jt

We introduce the unbounded operator dt : L2(OT) D D(dt) ^ L2(OT) with the domain D(dt) := {w 6 L2(0,T; W^Q)) : dtw 6 L2(Ot), w(0,x) = w(T,x)}. (6)

Definition 1. The function w 6 WY is called the generalized solution of problem (1), (2), (4) if it satisfies the operator equation

dtw + Aw = f, w 6 W7, (7)

where W7 := D(dt) n L2(0,T; W21,Y(Q)).

Note that nonlocal conditions bind the values of the unknown function on some parts of boundary with its values on shifts of these parts into domain OT. The above shifts are generated by a certain difference operator. Properties of such difference operators in the spaces L2(Q) and W2(Q) were studied earlier, see [3, 4]. In this paper we use the above results to formulate the properties of difference operators acting in different function spaces RQ : L2(OT) ^ L2(OT) and RQ : L2(0,T; W^Q)) ^ L2(0,T; W21,7(Q)).

We consider the difference operator

Ru(t, x) = u(t, x) + a1u(t, xi + 1, x2) + a-1u(t, xi — 1, x2).

This operator corresponds to boundary conditions (2). We define the operator Rq given by Rq = PqRIq : Lp(Qt) ^ L2). Here Iq : L2) ^ L2((0,T) x Rn) is the operator of extension of functions from L2(QT) by zero in (0,T) x (Rn \ Q), the operator Pq : L2((0,T) x Rn) ^ L2(QT) is the operator of restriction of functions from L2((0,T) x Rn) to QT. Then for any u e L2(0,T; Wl(Q)) and w = RQu we have

w|xi=0 = Rqu\x1=0 = aiu|xi=1, w|x1=2 = Rqu\x1=2 = a_iu|x1=1, w|xi = 1 = Rqu|xi=1 = u|xi = 1.

Thus, if a1 = y1, a-1 = y2, and u e L2(0,T; W2(Q)), then the function w = Rqu e L2(0,T; W2 (Q)) satisfies the nonlocal boundary conditions (2), i.e. Rq(L2(0,T; W21(Q))) C L2(0,T; W21,7(Q)), where 7 = {71,72}. Conversely, if 7172 = 1, it can be proved that L2(0,T; W21,Y(Q)) C RQ(L2(0,T; W1(Q))), see [3, 4]. Therefore, Rq maps continuously and bijectively L2(0,T; W2(Q)) onto L2(0,T; W2y (Q)). Hence there exists a bounded inverse operator R-1 : L2 (0,T; W2 Y (Q)) ^ L2(0,T; W2(Q)). Consequently, instead of equation (7), we can consider the equation

dtRQu + ARqu = f, u e W, (8)

where W := D(6t) n L2(0,T; WV^(Q)).

Definition 2. A linear operator A : L2(fiT) D D(A) ^ L2(fiT) is monotone if

(Au,u) /M-u^ > 0 Vu e D(A).

A linear densely defined monotone operator A is maximally monotone if there is no linear monotone operator that is a strict extension of A.

As is known, in reflexive strictly convex with its conjugate spaces, the maximum monotonicity of the operator is equivalent to the condition:

(Au,u}> 0 Vu e D(A), (A*u,u}> 0 Vu e D(A*), (9)

see Lemme 1.1 [5, Chapter 3]. It is also known that dt with domain (6) is maximally monotone and dt* = —dt, see [5, Chapter 3, sec.2.2].

Lemma 1. Let y1y2 = 1. Then dtRQ : L2(QT) D W ^ L2(QT) is maximally monotone.

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Proof. Let RQ = 2(Rq + RQ) and RQs = 1(Rq - RQ). Note that

(RQu(t),u(i))i2(0) = (u(i),RQu(i))i2(Q) = (RQu(i),u(i))i2(Q) = (rQu(i),u(i))i2(0). Moreover,

dt (RQu(t),u(t))i2(Q) = (diRQu(t),u(t))i2(Q) + (RQu(t) diu(t))i2(Q)

= (RQ5t«(t),«(i))i2(o) + (u(i),RQdtu(i))i2(Q) = 2 (dtRQu(t), u^))^. Since (dtRQu, u) = 0, we obtain

T

(dtRqu,u) = (dtRQu,u) = J (dtRQu(T),u(r))i2(0) dr = 1 (RQu(t),u^)^ |T=o = 0

0

due to u(T, x) = u(0,x). On the other hand,

((dt R0)*u,u) = (RQdfu, u) = (dfRQu,u) = -(dtRQu,u) = -(dtRQu,u) = 0.

The condition (9) is fulfilled. □

Lemma 2. Let RQ > 0, functions A^- e C^(M3), A^-(t,x) = Aji(i,x) (i, j = 1, 2) are l-periodic in xi and T-periodic in t, and inequality (3) holds. Then

(ARQu,u) > c2 IIuy L2(0,T;WW21(Q)), (10)

where c2 > 0 does not depend on u.

For the operator (ARg)(t, ■), similar estimate is proved in [3, 4]. Note that operator ARQ : L2(0,T; W1(Q)) ^ L2((0,T; W2-i(Q)) satisfying (10) is monotone and coercive in terms of [5].

Theorem 1. Let |y1 + y2| < 2, functions A^- e C~(R3), A^-(t,x) = Aji(t,x) (i, j = 1,2) are 1-periodic in x1 and T-periodic in t, and inequality (3) holds. Then for any f e L2(^t) there exists a unique generalized solution of problem (1), (2), (4).

Proof. As stated above, the generalized solution of (1), (2), (4) is the function w = Rqu, where u is the solution of equation (8). Note that if |y1 + Y2| < 2, then y1Y2 = 1 and RQ > 0, see Examples in [3, 4]. According to Lemma 1, the operator dtR^ is maximally monotone, and according to Lemma 2, the operator ARq is monotone and coercive. The conditions of Theorem 1.1 [5, Chapter III, §1] hold. Thus, the solution of equation (8) exists. The uniqueness of this solution follows from (10). □

This work is supported by the Ministry of Science and Higher Education of the Russian Federation: agreement no. 075-03-2020-223/3 (FSSF-2020-0018).

References

1. BITSADZE A. V. AND SAMARSKII A. A. (1969) On some simple generalized linear elliptic problems. Dokl. Akad. Nauk SSSR. 185:4. p. 739-740.

2. SKUBACHEVSKII A. L. (1986) The first boundary value problem for strongly elliptic differential-difference equations. J. of Differential Equations. 63. p. 332-361.

3. SKUBACHEVSKII A. L. (1997) Elliptic Functional Differential Equations and Applications. Birkhauser, Basel-Boston-Berlin.

4. SKUBACHEVSKII A. L. (2016) Boundary-Value Problems for Elliptic Functional-Differential Equationa and its Applications. Russ.Math.Surv. 71:5. p. 801-906.

5. LIONS J.-L. (1969) Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Paris.

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