CC BY
12
УДК 517.956.6
A Nonlocal Boundary Value Problem with Constant Coefficients for the Multidimensional Second Order Equation of Mixed Type of the Second Kind
Sirojiddin Z. Dzhamalov*
Institute of Mathematics Uzbekistan Academy of Sciences M. Ulugbek, Tashkent, 100170 Uzbekistan
Received 06.11.2017, received in revised form 28.12.2017, accepted 15.02.2018 Multidimensional second order equation of the mixed type of the second kind is considered in the paper. Unique solvability and smoothness of the solution of a nonlocal boundary value problem with constant coefficients in Sobolev spaces are proved under some conditions on coefficients.
Keywords: multidimensional equations, solvability, generalized solution. DOI: 10.17516/1997-1397-2018-11-4-472-481.
1. Introduction and formulation of the problem
n
Let Q = (ai,Pi), be n-dimensional parallelepiped in the Euclidean space Rn of points
i=i _
(xi,..., xn), 0 < ai < ¡ii < yi = 1, n.
In domain Q = Q x (0, T) we consider a second order differential equation
Lu = K(x,t) uu - (aij(x) uXi) + a (x,t) ut + c (x,t) u = f (x,t). (1)
Here and below repeating indexes mean summation from 1 to n. We assume that all functions below are real-valued and smooth enough.
Let K (x, 0) < 0 < K (x, T) at x e Q. Then equation (1) is an equation of the mixed type of the second kind since function K(x,t) can change sign in the domain Q [1-4].
1.1. The nonlocal boundary value problem
We are to find a generalized solution of equation (1) from Sobolev space W2(Q), (2 < I is a natural number) that satisfies nonlocal boundary conditions
Y • u (x, 0) = u (x, T), (2)
ViDl u\Xi=ai = Dl u\Xi=Pi (3)
dPu _
when p = 0,1, where DX u = , DX u = u, y and = 1,n are some constants which are
Xi dxP Xi
not equal to zero. They will be defined below.
* [email protected] © Siberian Federal University. All rights reserved
Nonlocal boundary value problems for the mixed type second order equation both first and second kinds were considered [2,4-8,12,14,15]. Nonlocal boundary value problems (2), (3) for the mixed type equation of the first kind were studied for the first time by one of the authors of the paper [9].
Here equation (1) is considered in the case K(x, 0) < 0 < K(x,T). Unique solvability and smoothness of the generalized solution of one nonlocal boundary value problem with constant coefficients (2), (3) in Sobolev spaces W%(Q) (2 < 1 e N) are studied for the first time.
__2 n
Let us assume that aij(x)= aji(x); aij(ak) = aji(^k), Vk = 1,n end VCe Rn, |C| Ci ■
i=i
Let us also assume that one of the following conditions holds:
(a) aijCiCj > a0|C|2, where a0 is const > 0,
(b) aijCiCj < a1|C|2, where a1 is const < 0.
Further we assume that ^^ > 1, | Y | > 1 in the case of condition (a), | j | < 1 in the case of condition (b).
W l2(Q) (2 < /-natural number ) is the Sobolev space with the scalar product (, )l and the norm W °(Q ) = L2( Q ) is the space of square integrable functions.
Let v = (vt, vxi ,■■■, vXn) be a unit vector of an exterior normal to the boundary dQ, where vt = cos(v, t), vXi = cos(v, xi), Vi = 1, n.
Further, the Young inequality is often used
ap up vq 1 1
Vu,v > 0, Va > 0, p > 1, u ■ v <--1--, - + - = 1.
p qaq p q
If p = q = 2 then we come to the Cauchy inequality with a [10].
First, we consider the case / = 2, that is, u e W22 (Q) and assume that coefficients of equation (1) are smooth enough functions.
2. Uniqueness of the solution of the problem
Theorem 2.1. Let us assume that above mentioned conditions on coefficients of equation (1)
2
are fulfilled and 2a — Kt + XK ^ 6i > 0; Xc — ct ^ 52 > 0; where X = t In |j| > 0 if |j| > 1 in
2T
the case of condition (a) and X = t In |j| < 0 if |j| < 1 in the case of condition (b), |r/i| ^ 1,
Vi = 1,n, c(x, 0) ^ c(x,T). If a generalized solution of problem (1)-(3) from the space W^KQ) exists for any function f e L2(Q) then the solution is unique and the following inequality holds:
Hi < mf ||o-From this point on m is positive constant.
Proof. Let us assume that a generalized solution of problem (1)-(3) exists in the space W 2 (Q). Taking into account conditions of Theorem 1 and the Cauchy inequality with a from problem (1)-(3), it is easy to obtain the following inequality
2 Lu ■ exp I — Xt — H-ix-i) ■ ut dxdt ^ exp I — Xt — NJ fax.i) {(2a — Kt + XK) ■ +
Jq V i=i ) JQ \ i=1 )
+ Xaij uxiuxj + (Xc — ct) ■ u2} dxdt — a ■ HuxH2 — /J2a-1 ■ ||ut||0 +
+ exp I — Xt — ji xA {Ku2tvt — 2aij uxH utVxH + aij uxH ux^ vt + c ■ u2vt} ds, (4) Jsq V 7=i /
2
where 0 < ji = — ln \ni\, 0 <Oi = (Pi — ai), a and a-1 are coefficients of the Cauchy inequality Oi
with a. Conditions of Theorem 1 provide non-negativity of the integral over the domain Q and on the boundary dQ. Because u G W22(Q) satisfies boundary conditions (2), (3) and j2 = e-XT = ev
i2 = e^i 6i then
/ exp I - Xt - V] Hi xA {Ku "j vt - 2 aij uXi ut vXi + aj uXi uXj vt + cu2 vt} ds =
JdQ V i=i J
= f 'exp [ - ^ Hi ^ {[K(x, T) e-XTy2 - K(x, 0)] uj (x, 0)+
Jai \ i=i J
+ [e-XtY2 - 1]uXi(x, 0) + [c(x,T)e-XTy2 - c(x, 0)]u2(x, 0)}dx-
f t
- 2[exp(-Hifii Wi - exp( -Hiai)] exp(-Xt) uXi(-ai,t)ut(ai,t)dt >
Jo
> exp [ - ^ Hi x<) {[K (x,T) e-XTY2 - K (x, 0)]uj(x, 0)+ J a \ i=i J
+ [c(x, T)e-XtY2 - c(x, 0)]u2 (x, 0^dx ^ 0. (5)
Omitting positive boundary integrals, we obtain from (5) the following inequality
n n
2 Lu • exp(-Xt - Hi x-i) • utdxdt ^ exp(-Xt - NJhi x-i,) {(2a - Kt + XK) • u'2+
Jq i=i Jq i=i
+ XaT uXi + (X c - ct) • u2} dx dt - a \\uXi ||0 - H • a- • ||ut ^2 , (6)
where aT = a0 in the case of condition (a), aT = ai in the case of condition (b). Setting coefficients XaT - a > X0 > 0, Si - H2a-:L > S0 > 0, we obtain from inequality (6) the first a priori estimate
\\u\\i < m\\f\\0.
Uniqueness of the generalized solution of problem (1)-(3) in W^ (Q) follows from this estimate.
□
3. The equations of composite type
To prove the existence of the solution of problem (1)-(3) in W2;(Q) we use the method of "e-regularisation" together with Galerkin method [1,3,8,13].
Let us consider a nonlocal problem for composite type equation
d
LEuE = -e— &uE + Lue = f (x, t), (7)
YDqt uE\t=0 = DquE\t=T, q = 0,1, 2, (8)
ViDXi uE\x=^ = DXi , p = 0,1, (9)
d2u n d2u dpu
where Au = d^ + = dx2 is the Laplace operator, D Xiu = ^~ , D 0Xi u = u, p = 0,1,
d q u
dtï _
that \ y \ > 1 in the case of condition (a), \ y \ < 1 in the case of condition (b), \ni\ > 1, Vi = 1, n.
D qu = —— , q = 1, 2; D t u = u, e is a small enough positive number, n, Y = const = 0, such
In what follows we use composite type equation (7) as the e-regularization equation for equation (1) [1,8].
Let us denote a class of functions such that uE(x,t) G W 2(Q) and £ G L2(Q) satisfying conditions (8),(9) by W.
Definition. Function ue (x,t) G W satisfying equation (7) is denoted the regular solution of problem (7)-(9).
Theorem 3.1. Let us assume that above mentioned coefficient conditions for equation (1) are
2
fulfilled and 2a — \Kt\ + X > ^ > 0, X c — ct > S2 > 0, where X = ^ ln M > 0 if M > 1 in
2
the case of condition (a) and X = ^ In |y| < 0 if |y| < 1 in the case of condition (b), \ni\ ^ 1,
c(x, 0) = c(x,T), a(x, 0) = a(x,T), a(ai,t) = a(pi,t), K(ai,t) = K(pi,t), Vi = 1,n. Then for any function f,ft G L2(Q), such that y • f (x, 0) = f (x,T) there is a unique regular solution of problem (7)-(9), and the following inequalities are true:
I )
II )
£( \\ue tt\\l + \W tx\\l) + \\ue II? < m
dAuF
dt
+ \K\\? <
+ \\ft\\l
Proof. The proof of Theorem 2 is carried out using Galerkin method with special basis functions. [8,10].
2
2
m
0
3.1. Proof of the first a priori estimate I)
Consider the following spectral problems. Let (x,t) be eigenfunction of the following prob-
lem
^=^+dx=-v? *,
(10)
DP hj\t=o = Dpt hj\t=T , P = 0,1, (11)
DX hj\x=o = DX hj\x=v (12)
It follows from the general theory of linear self-adjoint elliptic operators that all { hj (x,t) } are eigenfunctions of problem (10)-(12). They form fundamental system in W22(Q), and they are orthonormal in L2(Q) [10,11]. Then we construct the solution of an auxiliary problem using these functions:
exp
^(yXt + ]C Mixij
Wjt = ,
Y • Wj (x, 0) = Wj (x, T),
(13)
(14)
where, y = const = 0, such that \ y \ > 1 in the case of condition (a), \ y \ < 1 in the case of
2 _
condition (b), 0 < = — ln \ni\, \ni\ ^ 1,Vi = 1,n. Obviously, problem (13), (14) is uniquely Vi
solvable and its solution has the from
exp
mi • xi\ [
M • I
( Xt j 1
exp[YJ dr + —
exP ( y) dt
(15)
W
j
0
N
It is clear that functions wj (x,t) are linearly independent. Indeed, if cjwj = 0 for some
j=i
N
set of functions wi,w2,... ,wN then acting on this sum by the operator I, we have cjIwj =
j=i
N _
= 2 cj4>j = 0. Then we obtain that cj = 0 for any j = 1,N. It follows from the construction
j=i
of function fy (x,t) that functions wj (x,t) satisfy the following conditions
YDt w\t=0 = DtWi\t=T, q = 0,1, 2 (16)
^ Wi\Xi=ai = DPi Wi\Xi=^, P = 0,1. (17)
N
We take the approximate solution of (7)-(9) in the from w = uN = cjwj where coefficients
j=i
cj are defined for any j = 1, N as solutions of the linear algebraic system
LEuN • e 2 fyj dxdt = f • e 2 fyj dxdt. (18)
We prove the unique solvability of algebraic system (18). Multiplying every equation of (18) by 2cj and summing up with respect to j from 1 to N and taking into account (12), (13), (18), we obtain
f -(xt+ E tHXi) r -(xt+ E ^i)
/ LEw • e i=1 • wtdxdt = / f • e i=1 • wtdxdt. (19)
Upon integrating identity (19), by virtue of theorem 2 we obtain for the approximate solution of problem (7)-(9) the estimates I), i.e.
e(\Kt 112 + IKxWI ) + \K ||2 < m \\f. (20)
This implies the solvability of algebraic system (18). In particular, from estimate (20) we obtain a weak solution of problem (7)-(9) [3,10].
3.2. Proof of the second a priori estimate II.)
Taking into account problem (10)-(14), from identity (18) we obtain
--T LEw e t Aiw- dxdt =--^ f e t Aiw- dxdt, (21)
vj2 Q vj2 Q
where,
n
-(Xt +J2 Hixi)
A lu* = exp
A j - X ujtt - Hj ujxx +---j , Auj = ujtt + u3
xx
X2 + h2
jtt - Hj Wj XX +
Multiplying each equation of (21) by 2v2cj and summing up with respect to j from 1 to N and considering (15), (16), (21), we have the following identity
-2 LEw • e-^- • AIwdxdt = -2 f • e-^-• Alw dxdt. (22)
Integrating (22) and taking into account conditions of Theorem 2.1 and boundary conditions (15), (16), we obtain the following inequality
ft\\0 + \\f ||0
d Aw
> E dt
2 C -(X-t+ y iliXi)
+ e {(2a -\Kt\ + XK ) w 2tt+
0 Jq
2 2 2 } i ßiXi) 2
+ (2a — \Kt\ + XK )w tx. + Xw x.x. +Xw tx,j dxdt + I e i=1 [(K wtt — 2awt wtt+ * * * * jsq
+ w2xixi + 2wx*xi wtt — w2xit + Kw2xit + 2cw (wtt + wxx)vt+ + (2Kwtt wx*t — 2wtt wx*t + 2awt wx*t) Vx*]ds — a ( \\wxx\\0 + \\wxt\\0 ) —
2
12
— ^2a-1 \utt\0 — m ( \\f\0)^ Ji, (23)
i=i
where, J is the integral over the domain, J2 is the integral over the boundary.
Taking into account conditions of Theorem 2.1 and boundary conditions (14), (15), we obtain for coefficients A — a ^ Ao > 0, 61 — ¡j?a-1 > 50 > 0 that Ji > 0 and J2 ^ 0. Now we have from inequality (23) the second estimate
I -f
I il f II2 ^
+ ||u£ ||2 < m •
\0 + \\ft\0
(24)
Hence, from the well-known theorem on weak compactness [10] the obtained estimations (20), (24) allow one to take the limit N ^ x> and to conclude that a subsequence {u^} converges in L2(Q) together with the first and the second order derivatives to the unique regular solution uE(x,t) of problem (7)-(9) with the properties specified in Theorem 2.1 [3,6,8,10]. By virtue of (24) the following inequality holds for uF(x,t)
9 A dtAU
+ \K\\2 < m
10 + \\ft\\0
(25)
Theorem 2.1 is proved.
m
2
E
0
2
E
0
4. Existence of solution for the problem
4.1. The method of "e-regularization"
Now by means of the method of "e-regularization" we prove solvability of problem (1)-(3).
Theorem 4.1. Let us assume that all conditions of theorem 2.1 are satisfied. Then the generalized solution of problem (1)(3) in space W2(Q) exists and it is unique
Proof. The uniqueness of the solution of problem (1)-(3) in W22 (Q) is proved in Theorem 1.1. Now we prove existence of the generalized solution of problem (1)-(3) in W2(Q). For this purpose, we consider equation (7) in the domain Q with nonlocal boundary conditions (8), (9) at e > 0. Because all conditions of Theorem 2.1 are fulfilled then there exists unique regular solution of problem (7)-(9) at e > 0, and estimates I),II) are true for it.
It follows from the well-known theorem on weak compactness [10] that it is possible to take from the set of functions [uE] , e > 0 weakly converging sub sequence of functions in W such that {uEi} ^ u at ei ^ 0. Let us show that limit function u(x,t) satisfies the equation Lu = f (1).
d Au
sequence
bounded in L2(Q), and operator L is linear, then we have
Indeed, as sequence { uEi} converges weakly in W2(Q), sequence ^ E, (£ > 0) is uniformly
Lu - f = Lu - Lu£i + £i • ~AjU~L = L(u - u£i ) + £i • dAU£i ■ (26)
Taking the limit £i ^ 0, we obtain from (26) the unique solution of problem (1)-(3) in W2(Q) [1,6,8].
Theorem 3.1 is proved. □
5. Smoothness of solution for the problem
Now we prove a more general case l > 3. Further we assume that coefficients of equation (1) are infinitely differentiated in the closed domain Q .
Theorem 5.1. Let us assume that conditions of Theorem 3.1 are fulfilled and 2(a + pKt) -\Kt\ + XK > S> 0,
Dm K\t=0 = Dm K\t=T, Dm aU = Dm a\t=T, Dm c\t=0 = Dm c\t=T. Then for any function f (x,t) such that f G W^(Q),Df+1f G L2(Q), yDm f\t=0 = Dm f\t=T where m = 0,1, 2, 3,... ,p there exists unique generalized solution of problem (1)-(3) in the space W t+2(Q), where p =1, 2, 3,... .
Proof. It follows from smoothness of the solution of problem (10)-(14) that the approximate solution of problem (7)-(9) satisfies conditions w = u^ G CQ);
YDq w\t=0 = Dq w\t=T, q = 0,1, 2,...,
ViDXi w\x.=_oli = DXi w\3U=p., p = 0,1.
Taking into account conditions of Theorem 2.1 at £ > 0, nonlocal conditions at t = 0, t = T and equality
-— T \ ¡t=T / -xt dAue -\t t=T / -xt j,- nn ,t=T
we obtain
[e 2 ■ Leue) |t=0 ={-e ■ e 2 + e 2 ■ Lue) Uo = [e 2 - f[x,t)) lt=o
\\Y • Uettt (x, 0) - ue ttt(x, T )\\0 < const. Hence, function ve(x,t) = ue t(x,t) belongs to W and satisfies the following equation
PeVe = LeVe = ft - a t Uet - Ct U e = Fe. (27)
It follows from theorem 2.1 that the set of functions {Fe} is uniformly bounded in the space L2(Q), i.e.
\\Fe\\o < m [\\f\0 + \\ft\\0_ .
Further, it can be easily obtained from conditions of Theorem 3.1 that coefficients of the operators Pe (e > 0) satisfy conditions of Theorem 4.1. Then on the basis of estimates I), II) for function {ve} we obtain similar estimates
e ( \\Vett\\0 + \\VetxWl ) + \\Ve\\l < m (\\f\\0 + \\ft\\0 ), (28)
dAvF
dt
+ IM2 <
l? + llfttllo
(29)
Function {uF} satisfies parabolic equation with conditions (2), (3)
n uF
U£t
-£ (a
«) „=f+p
dAuF dt
- K(x,t)uFtt - (a - l)uFt - cuF
here G L2(Q). Set of functions {$e} is uniformly bounded in W22(Q), i.e.
№fllO < m
I? + llfttllo
< m I
12 •
(30)
(31)
On the basis of a priory estimates for parabolic equations [1], [10] and inequality (31) we obtain
y uf y 3 < m"£ 1,2
2 '
Further, one can prove in a similar way that ||we||.p+2 ^ m \\f llp+i , where p = 2, 3,... . □
Remark. In the formulation of problem (1)-(3) the sign at the quadratic form does not play an essential role. However, in the case
(a) aij^ ao|C|2; aij = aji, where ao = const > 0, x £ Q, £ £ 1" the class of equations (1) includes parabolic equations and in the case
(b) aij(x)£i£j ^ ai|£ |2; aij = aji, where ai = const < 0, x £ Q
the class of equations (1) includes inverse parabolic equations. Nevertheless, similar results are obtained only with the change in the value of 7 for problem (1)-(3) in the case of conditions (a) and (b).
Therefore, the following question arises: whether or not restrictions on 7 are essential? In this connection we consider the following examples.
Examples. In the rectangle Q = (0,1) x (0, T) we consider the following problem
ni« = ut — uxx = 0, Yu (x, 0) = u (x, T), u(0,t) = u(l,t) = 0.
(32)
(33)
(34) 2nk
Solving problem (32)-(34) by the Fourier method, we find Yk = exp(—XkT) < 1, Xk = , k = 0,1, 2,.... It is easy to verify that all conditions of Theorem 1 are fulfilled but functions uk = Cke-Xk* sin Xkx (where Ck are arbitrary constants) are nontrivial solutions of this boundary value problem.
In the same way, we consider the following problem
n2u = ut + uxx = 0 , Yu (x, 0) = u (x, T), u(0,t) = u(l,t) = 0.
(35)
(36)
(37)
Solving problem (35)-(37) by the Fourier method, we find that functions uk = CkeXkt sin Xkx with any Ck are nontrivial solutions of this boundary value problem. In this case Yk = exp(XkT) > 1.
2
m
o
ij F X
Hence, we see that restrictions on 7 for both conditions (a) and (b) are essential. If these conditions are not satisfied then we do not have the uniqueness of the problem as shown above.
The author would like to thank prof. R. Ashurov and reviewer for useful comments and suggestions.
References
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Об одной нелокальной краевой задаче с постоянным коэффициентом для многомерного уравнения смешанного типа второго рода, второго порядка
Сирожиддин З. Джамалов
Институт математики Академия наук Республики Узбекистан М. Улугбека, Ташкент, 100170 Узбекистан
В данной 'работе при выполнении некоторых условий на коэффициенты многомерного уравнения смешанного типа второго рода в пространстве доказываются однозначная разрешимость и гладкость решения одной нелокальной краевой задачи с постоянным коэффициентом в пространствах С.Л.Соболева.
Ключевые слова: многомерные уравнения, разрешимость, обобщенное решение.
