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ИНТЕГРО-ДИФФЕРЕНЦИАЛЬНЫЕ УРАВНЕНИЯ И ФУНКЦИОНАЛЬНЫЙ АНАЛИЗ
INTEGRO-DIFFERENTIAL EQUATIONS AND FUNCTIONAL ANALYSIS
sËmmÊII
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Серия «Математика»
2019. T. 29. С. 120-137
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Онлайн-доступ к журналу: http: / / mathizv.isu.ru
УДК 517.946 MSG 35К70, 35М20
DOI https://doi.org/10.26516/1997-7670.2019.29.120
Ultraparabolic Equations with Operator Coefficients at the Time Derivatives *
A. I. Kozhanov
Sobolev Institute of Mathematics of SB RAS, Novosibirsk, Russian Federation
Abstract. The article is devoted to the study of the solvability of boundary value problems for third-order Sobolev-type differential equations of the third order with two time variables (such equations are also called composite-type equations or equations not solved for the derivative). The peculiarities of the equations under study are, firstly, that the differential operators acting at the time derivatives are not assumed inverse, and, secondly, that the statements of boundary value problems for them are determined by the coefficients of these differential operators. For the problems proposed in the article, we prove existence and uniqueness theorems for regular solutions (solutions having all weak derivatives in the sense of Sobolev involved in the equation). The technique of proving the existence theorems is based on a special regularization of the equations under study, a priori estimates, and passage to the limit.
Keywords: ultraparabolic equations, irreversible operator coefficients, boundary problems, regular solutions, existence, uniqueness.
* The author was supported by the Russian Foundation for Basic Research (Grant 1851-41009).
ULTRAPARABOLIC EQUATIONS WITH OPERATOR COEFFICIENTS 1. Introduction
The equations studied in the article can be called Sobolev-type equations, or composite-type equations. Among the numerous works on the theory of Sobolev-type equations (see [1-7; 10; 11; 13]), distinguish the works devoted to equations with degenerate (noninvertible) operator at the time derivative — see [2; 10; 13]. In the present paper, we also consider equations with noninvertible operators at the higher part but, firstly, the nature of the noninvertibility of the operator coefficients is different than in the works of the predecessors, and, secondly, in contrast to the numerous earlier works, we study equations with two time variables.
2. Statement of the Problem
Suppose that Q is a bounded domain in with smooth boundary r (of class C2), T and A are given positive numbers, Q is the cylinder Q x (o, T) x (0, A) of variables x, t, a, S = r x (0, T) x (0, A) is the lateral boundary of Q, ao(t,a), a\(t,a), (3o(t,a), /?i(t, a), (x,t, a), ml(x,t,a), i,j = 1 ,...,n, rrio(x,t,a), f(x,t,a) are given functions, defined for x € Q, t € [0,T], a € [0, A]. Furthermore, let La, Lg, and M be differential operators whose action at a given function v(x,t, a) is defined by the equalities
Lav = a0 (t, a)v + ai(t, a)Av, Lg v = ,0o(t, a)v + ^i(t, a)Av,
Mv = mj (x, t, a)vXiXj + ml(x, t, a)vxi + m0(x, t, a)v
(A is the Laplace operator with respect to the variables x1, ..., xn, Here and below, repeating indices imply summation from 1 to n).
The aim of the article is the study of the solvability of boundary value problems for the equations
Laut + LgUa - Mu = f (x,t,a). (1)
Introduce the notations
Y1 = {(x, t, a) : x € Q, t = 0, a € (0, A)},
72 = {(x, t, a) : x € Q, t = T, a € (0, A)},
Y3 = {(x, t, a) : x € Q, t € (0,T), a = 0},
74 = {(x, t, a) : x € Q, t € (0,T), a = A}.
As we will show below, in well-posed boundary value problems for equations (1), on each of the sets 7», ¿ = 1,4, boundary conditions can be given or not given.
Let k, i = 1,4, be numbers equal to 0 or 1. Refer as the Pi^i^-condition for equation (1) to the condition that the value of the solution u(x,t,a) is given on 7» ifk = 1, and, respectively, the value u(x,t,a) is not given if k = 0.
The Boundary Value Problem ii1z2z3z4: Find a function u(x,t,a) that is a solution to equation (1) in the cylinder Q and satisfies the Pi^i^-condition and also the condition
u(x, t,a)\s = 0.
(2)
Obviously, there are 16 different problems of the given form. But it is also obvious that, among these problems, there are those similar to each other, or those reducible to one another by the change t' = T — t or a' = A — a (for example, the problems Pnoo and Poon are in essence identical, the problems Piooo and Poioo are reduced to each other by the change t' = T — t etc.). An easy analysis makes it possible to distinguish six basic problems among all the problems Pz1z2z3z4 — the problems Pim, Pmo, -Piioo, Pioio, -Piooo, and Poooo- It is for these problems that we will prove existence theorems for regular solutions below.
Denote by Vq the linear space of functions v(x,t,a) belonging to ¿2(Q) and such that their weak derivatives vt(x,t,a), va(x,t,a), vXi(x,t,a), Vxit tj i VXia(yX,t, a), VXixj(jE)t) VXiXjt(x,t,a), VXiXja(x,t, fit), i,j = l,...,n, exist and also belong to L2(Q). Normalize this space:
Ml Vo =
/
\Q
V2 + V2t + V2a + Y^ (vli + vlit + V2Xia) +
1/2
i= 1
I ^ ^ ^^X^Xj ^ ^X^Xjt ^ ^XiXjQj^j
dx dt da
Obviously, endowed with this norm, Vo becomes a Banach space.
It is in the Banach space Vo that we will establish the solvability of the boundary value problems under study.
3. Solvability of the Boundary Value Problems Рцц and Poooo
The boundary value problems Рцц and Poooo can be called dual to each other — in the first of them, boundary conditions are given on all the sets 71, 72, 73, and 74 (and hence, with account taken of condition (2), the boundary value problem Рцц becomes the Dirichlet problem with defining the boundary data on the whole boundary of the cylinder Q), and in the second, — on the contrary, all the sets 71, 72, 73, and 74 are free from boundary
data, and hence no boundary conditions are given with respect to the time variables t and a.
o
Let w(x) be a function of the space VK • ^ave inequality
in which the number do is defined only by the domain Q [12, Chapter I, § 9; 4, Chapter II, § 2].
Theorem 1. Suppose the fulfillment of the following conditions:
I) mij(x, t, a) € C2(Q), mi(x, t, a) € C^_(Q), _
i,j = 1 mo(x,t,a) € Cl(Q), oo(t,a) € Cl(D), a\(t,a) €
Cl(D), Po(t, a) € Cl(D), fat, a) € Cl{D);
II) ml:>(x, t, a) = t, a), i,j = l,...,n, (x, t, a) € Q,
mij(x,t,a)CiCj > AtolCI2, no > 0, (x,t,a) € Q, £ € Rn;
III) \m%x.(x,t,a) — \mxixj(x,t, a) — mo(x,t,a) > ni > 0, (x, t, a) € Q;
IV) no - \au{t, a) - \l3ia(t, a) > 0, ni -\aot{t, a) - \Pia{t, a) > 0, (t, a) €
[Hi + \aot{t, a) - \Poa{t, a)] + [a0a(t, a) +J30t(t, a)} £0r?0 +
[Hi - \aot{t, a) + \Poa{t, a)} rfi > 0, [t, a) € D, € R, rj0
[Ho - \au{t, a) + \Pia{t, a)] |£|2 - [ala(t, a) + (3lt(t, a)} ^ + ^
[lio + t, a) - ?Pia(t, a)] \r]\2 > ¡12 (|£|2 + M2), № > 0, (t, a) € D,
£ € ]Rn, ?7 <E Rra;
V) oi(0,a) > 0, o0(0,a) = o0i(a) + 0:02(0), «01(a) < 0, 0:02(0) > 0, a\(T,a) < 0, o0(T,a) = 003(a) + 004(a), 003(a) > 0, 004(a) < 0, Oi(T, a) — (¿0004(a) <0, a£ [0, A];
VI) /?i(i,0) >0, Mt,0) = l3oi(t) + /3o2(t), /3oi(t) <0, p02(t) >0, Pi(t,0)-dofMt) > 0, Pi(t,A) < 0, p0(t,A) = po;i(t) + Mt), Pos(t) > 0, Mt) < 0, Pi(t, A) - doPo4 (t) < 0 ,t£ [0, T].
Then, for any function f(x,t,a) such that f(x,t,a) € ^(Q), ft(x,t,a) € L2(Q), fa(x, t, a) € L2(Q), the boundary value problem Poooo has a unique solution in Vq.
Proof. Make use of the regularization method. Let e be a positive number and let L£ be the differential operator whose action at a given function v(x,t,a) is defined by the equality
(3)
L£v = e(Avtt + Avaa - Av) + Lavt + Lfjva - Mv.
'aa
Consider the boundary value problem: Find a function u(x, t, a) that is a solution in the cylinder Q to the equation
Leu = f(x, t, a) (4)
and satisfies (2) and also the conditions
ut(x, 0, a) = ut(x, T, a) = 0, x € Q,, a£(0,A), (5)
ua{x, t, 0) = ua(x, t, A) = 0, xeil, ie(0,T). (6)
Define the linear space Vo as the set of functions in Vo whose weak derivatives Vtt(x, t, a), Voa(x,t,a), vXitt(x,t,a), vXiaa(x,t,a), vXiXjtt(x,t,a), vXiXjaa(x,t,a), i,j = 1 ,...,n, exist and belong to L2(Q). Normalize the
space Vo:
□
We prove that, for fixed e, under conditions I-VI, the boundary value problem (4), (2), (5), (6) is solvable in Vq for any function f(x,t,a) from Li2(Q). Use the method of continuation in a parameter ([see 14, Chapter III,
§ 14])-
Let A be a number in [0,1]. Consider the problem: Find a function u(x, t, a) that is a solution in the cylinder Q to the equation
LEt\u = eA(uu + uaa - Au) + A [Laut + Lgua - Mu] = f(x, t, a) (4A)
and satisfying conditions (2), (5), and (6). Denote by A the set of those numbers A in [0,1] for which this boundary value problem is solvable in Vo for fixed e and under conditions I-VI for any function f(x,t,a) in L2(Q). If it turns out that this set is nonempty, open, and closed simultaneously, then it will coincide with the whole interval [0,1] (see [14, Chapter III,
§ 14])-
It is obvious that the set A is not empty — it contains 0.
The openness and closedness of A will follow from the a priori estimate
\\u\\yo < N0\\f\\L2(Q) (7)
for all possible solutions u(x,t,a) to the boundary value problem (4a), (2), (5), (6) in Vo, which is uniform over A (again see [14, Chapter III, § 14]). Show that the desired estimate indeed holds.
ULTRAPARABOLIC EQUATIONS WITH OPERATOR COEFFICIENTS 125 Consider the equality
J Le \uu dx dtda = J fu dx dt da. Q Q
(8a)
Integrating by parts and using the boundary conditions, it is not hard to pass from this equality to the following:
n
y^ / {u1it + uxia + uxi ) dxdtda+ i=1Q
+A
mvuXiuXj + -(au + ßia)
i= 1
+
Q
A
1 » 1 a 1 ifl '
- - m0 - 2«ot - 2^0«
A
dx dt da+
u2 dx dt da—
y^ J J cki(0, a)u2.(x, 0, a) dx da—^ ^ J J a\(T, a)u2.(x,T, a)dxda+ 1=1 on t=1 o n
A
A
J J aoi(a)u2(x, 0, a) dx da + ^ J J ao3(a)u2(x,T, a) dx da+
0 n
T
0 n
T
£iM)<(M,0) dxdt-\"£ IJ ßi (t,A)u2Xi(x,t,A) dxdt 1=1 on 1=1 0 n
T
T
J J ßoi(t)u2(x,t, 0) dx dt + ^ J J ßo3(t)u2(x,t, A) dx dt\ =
0 n
A
0 n
A
±JJ ao2(a)u2(x,0,a) dxda — ^j J ao4(a)u2(x,T, a) dxda+
o n
o n
T
T
jj ßo2(t)u2(x, t, 0) dx dt—^ jj ßo4 (t)u2(x,t, A) dx dt+j fudx dt da.
(9)
o n
0 n
<3
Estimating the first four summands on the right-hand side of (9) with the use of (3) and applying Young's inequality to the last summand on the right-hand side of (9) and also inequality (3), we conclude that the conditions of the theorem imply the a priori estimate
n
e ^ / (u2.t + u2.a + u2.) dx dt da+
+xi
Q
i=1Q
n
u2 + ^ Ux.
i= 1
dxdtda<N\ J f2dxdtda, (Юл)
Q
in which the number is defined by the functions rrf^x, t, a), m%(x, t, a), i,j = 1 ,...,n, rrio(x,t,a), ao(t,a), a\(t,a), (3o(t,a), /?i(i,a), and also by the numbers do, T, A, and e. Consider the equality
— J Le>\u{uu + uaa) dxdtda = — J f{uu + uaa)dxdtda. (Ha)
Q Q
Integrating by parts once again and using the boundary conditions (2), (5), and (6), applying conditions I-VI and inequality (3), we conclude that this equality implies the estimate
n „
E J «« + <ta + <aa + <t + <a) dt da+
e
i= 1
Q
+4
Q
Ut + + (uXit + UXia)
i= 1
dxdtda <N2 J f2 dxdtda, (12д)
Q
where the number N2 is defined by the functions (x,t,a), m%(x,t,a), i,j = 1 ,...,n, mo{x,t,a), ao(t,a), ot\(t,a), (3o(t,a), (5\{t,a), and also by the numbers do, T, A, and e. Consider the equality
— J Le\u/S.u dxdtda = — J fAu dxdtda. (13л)
Q Q
Integrating by parts, using the second main inequality for elliptic operators (see [8], [9, Chapter III, § 8], estimate (12л), and Young's inequality, we conclude that (13л) implies the estimate
e J [(Aut)2 + (Aua)2 + (Au)2] dxdtda+
Q
ultraparabolic equations with operator coefficients 127
+A E J uxixj dx dtda < N3 J f2dxdtda, (14a)
Q
in which the constant N3 is defined by the functions rrf^x, t, a), m%(x, t, a), ij, = 1,... ,n, m o(x, t, a), ao(t, a), ot\(t, a), (3o(t, a), (5\{t, a), the domain Q, and also by the numbers T, A, and e. Finally, consider the equality
— J Le>\u(Auu + Auaa) dxdtda = — J f(Autt + Auaa) dx dt da. (15a)
Q Q
Integrating by parts again, using estimates (12a) and (14a), the second main inequality for elliptic operators, and Young's inequality, we conclude that solutions u(x,t,a) to the boundary value problem (4a), (2), (5), (6) satisfy the estimate
e J [(Autt)2 + (Auaa)2] dxdtda+
Q
+A E J(uxixjt + uxixja) dx dtda < N4 J f2 dxdtda, (16a)
l'j=lQ Q
with the constant N4 defined by the functions ml:>(x, t, a), ml(x, t, a), i, j = 1,... ,n, rrio(x,t, a), ao(t, a), ot\(t, a), fio(t, a), (5\{t, a), the domain Q, and also the numbers T, A, and e.
From (12a), (14a), and (16a), we have the estimate
e J [(Autt)2 + (Auaa)2 + (Au)2] dx dt da < N5 J f2 dxdtda,
Q Q
in which the constant N5 is defined by the functions rrf^x, t, a), m%(x, t, a), i,j = l,...,n, mo(x,t,a), ao(t,a), ot\(t,a), (3o(t,a), (5\{t,a), the domain Q, and also the numbers T, A, and e. This estimate implies the desired estimate (7).
As we already said above, estimate (7) implies that the boundary value problem (4a), (2), (5), (6) is solvable in Vo for fixed e, under conditions I-VI, and for all A € [0,1].
Thus, the boundary value problem (4i), (2), (5), (6) has a solution u{x,t,a) belonging to Vq. Let us demonstrate that, under conditions I-VI, for f(x,t,a) such that f(x,t,a) € L2(Q), ft(x,t,a) € L2(Q), fa(x,t,a) € Li2(Q), this solution satisfies a priori estimates uniform over e.
Observe first of all that, under conditions I-VI, equality (81) implies the estimate
У^ J (u2.t + u2.a + u2.) dxdtda + J
i=l Q Q
U
+ £
U ry
i= 1
dx dt da <
< N\ J f2 dx dt da,
Q
(17)
with the constant N\ defined by the functions mt:>(x, t, a), ml(x, t, a), i, j = 1 ,...,n, mo(x,t,a), ao(t,a), a\(t,a), (3o(t,a), and (5\(t,a), and also the number do-
Further, it is not hard to transform equality (111) to the form
— J LEiiu(utt + Uaa) dx dt da = J(ftUt + faUa) dx dt da.
Q
Q
Transforming the left-hand side of this equality as it was done in proving estimate (12a), using the conditions of the theorem and estimate (17), and applying Young's inequality, we conclude that solutions u(x, t, a) to the boundary value problem (4i), (2), (5), (6) satisfy the estimate
n „
/ (uxitt + uxita + uxiaa) dx dt da+
+ 1
Q
V* +Ul + Y] (и1л + и2)
i= 1
dx dt da < N2 j (/2 + f2 + f2) dx dt da,
Q
with the constant N2 defined by the functions mtJ(a;, t, a), ml(x, t, a), i, j = 1 ,...,n, mo(x,t,a), ao(t,a), a\(t,a), (3o(t,a), and (5\(t,a), and the number do-
Equality (13i), estimates (17) and (18) and also the conditions of the theorem and the second main inequality for elliptic equations imply the third a priori estimate
e J [(Aut)2 + (Aua)2] dxdtda+
Q
+ Y1 j uliXj dx dt da < N3 J (/2 + f2 + f2) dx dt da, (19)
Q
in which the constant N3 is defined by the functions ml:>(x, t, a), ml(x, t, a), i,j = 1 ,...,n, rrio(x,t,a), ao(t,a), ot\(t,a), (3o(t,a), (5\{t,a), and also by the domain Q.
At the last step, consider equality (15i). Transforming the right-hand side of this equality by integration by parts, using the conditions of the theorem, the second main inequality for elliptic operators, and estimates (17)-(19), we conclude that solutions u(x,t,a) to the boundary value problem (4i), (2), (5), (6) satisfy the fourth a priori estimate
e J [(Autt)2 + (Auaa)2] dxdtda+ ^ J u2.Xjtdxdtda+
Q l'j=lQ
+ È / u*i*j* dx dt J (/2 + f2 + f2) dx dt da, (20)
l'j=lQ Q
in which the constant N4 is defined by the functions ml:>(x, t, a), ml(x, t, a), i,j = 1 ,...,n, mo(x,t,a), ao(t,a), ot\(t,a), (3o(t,a), (5\{t,a), and also the domain Q.
Estimates (17)-(20) are desired estimates of solutions u(x, t, a) to the boundary value problem (4i), (2), (5), (6) uniform over e. These estimates and the reflexivity of a Hilbert space imply that there exists a sequence {£m}m= 1 °f positive numbers {um(x, t, a)}^= 1 of solutions to the boundary value problem (4i), (2), (5), (6) and a function u(x,t,a) such that, as m —> 00 and for i, j = 1,..., m, we have the convergences
^m ^ 0,
um(x, t, a) —> u(x, t, a) weakly in W^iQ),
umXiXj(x,t,a) uXiXj(x,t,a) weakly inL2(Q),
umXiXjt(x, t, a) uXiXjt{x, t, a) weakly in L2(Q),
umXiXja(x, t, a) uXiXja(x, t, a) weakly in L2(Q),
emAum(x,t,a) —> 0 weakly in L2(Q),
emAumtt(x,t,a) 0 weakly in L2(Q),
emAumaa(x,t,a) 0 weakly in L2(Q).
Obviously, the limit function u(x, t, a) belongs to Vq and is a desired solution to the boundary value problem Poooo-
Uniqueness in Vq of solutions to the boundary value problem Poooo obviously follows from the equality
J (Laut + Laua — Mu)udxdtda = 0. (21)
Q
The theorem is proved.
Turn to investigating the solvability of the boundary value problem
Pun.
The solvability of the boundary value problem Pim in Vq will be proved again by the regularization method, we will again use equation (4) but no additional boundary conditions will be given. Define some conditions to be used below:
VIIi. ai(0,a) < 0, ao(0,a) > 0, ai(T,a) > 0, a0(T,a) < 0, a € [0, A]-, VIIZ. ai(0,a) < 0, ao(0,a) > 0, ai(T,a) > 0, a0(T,a) < 0, a € [0, A]-, VII3. ai(0,a) < 0, ao(0,a) > 0, ai(T,a) > 0, a0(T,a) < 0, a € [0, A]-, VI4. ai(0,a) < 0, ao(0,a) > 0, ai(T,a) > 0, a0(T,a) < 0, a € [0, A]-, vilh. AM) < 0, Po(t,0) > 0, fh (t,A) > 0, /3o(t,A) < 0 ,te [0, T]; VIII2. /3i(t, 0) < 0, /3o(t, 0) > 0, /?!(t,A) > 0, /3o(t,A) < 0, t € [0,T]; VIIIs. Pi(t,0) < 0, ¡3o(t, 0) > 0, fi\(t, A) > 0, /3o(t,A) < 0, t € [0, T]; VIII4. f3\(t, 0) < 0, po(t,0) > 0, fait, A) > 0, /3o(t,A) <0,te [0,T],
Theorem 2. Suppose the fulfillment of conditions I-IV, of one of conditions VII 1-VII4, and of one of conditions Villi- VIII4. Then, for any function f(x,t,a) such that f(x,t,a) € L2(Q), ft(x,t,a) € L2(Q), fa(x,t,a) <E L2(Q), fix, 0, a) = f\x,T,a) = 0 for (x,a) € Q x (0 ,A), fix,t, 0) = f(x,t,A) = 0 for ix,t) € Q, x (0,T), the boundary value problem Pini has a unique solution in Vq.
Proof Consider the boundary value problem: Find a function u(x, t, a) that is a solution to equation (4) in the cylinder Q and satisfies the Pim-condition and also condition (2). The proof of the solvability of this problem in Vq for fixed e and f(x,t,a) € L2(Q) is again carried out by the method of continuation in a parameter and a priori estimates. In view of the analogy of the procedure of applying of the method of continuation in a parameter with the corresponding procedure used in the proof of Theorem 1, we just show that solutions to the above-proposed boundary value problem satisfy estimates in Vo valid if f(x,t,a) € L2(Q), and then — that, under additional conditions on the function f(x,t,a), there are estimates uniform over e.
Consider equality (81). Using the Pim-condition, condition (2), conditions I-IV, we infer that solutions u(x, t, a) to the boundary value problem for equation (4) with the Pim-condition and condition (2) satisfy estimate (10i).
Now, consider equality (Hi). After integration by parts with the use of the boundary condition, this equality takes the form
n „
e ^^ / (-u2,tt + 2u2x.ta + u2.aa + u2x.t + u2x.a) dx dt da+
+(ßot + a0a)uaut +
1 i 1 ii 1 1o
2 mxi - 2 ~ m° + 2aoi ~ 2
1 i 1 a 10 1
- 2m%ixj ~mo + 2l3°a~ 2aot
u2t +
ua } dx dt da+
mvuXi
/1 1 \ tUXjt + ( -ßla ~ 2 «Ii ) J2Ulit + ^ ' i= 1
/1 1 \ ""
+ ( g ait ~ 2^la ) ^ ~ + aia^UxitUxi ^ ' ¿=1
dx dt da+
+
m/uXiuXjt + m%uXiuXja + (mj t - m\)uXiut+
Q
+(mxia ~ mla)uXiua - motuut - m0auua
dx dtday +
A
A
\ J J ao(0,a)u2(x,0,a) dxda~ J J ai (®>a)uxit(x>Q>a) dxda+
on 1=1 0 n
A n A
\J J ao(T, a)u2(x,T, a) cfeda + ^E J J ai(T, a)ux.t(x,T, a)dxda+
on i=l 0 n
T T
J J ßo(t,0)ul(x,t,0) dxdt — ^^^J J ßi(t,0)ulia(x,t,0) dxdt-
o n
T
o n
T
—i J J /3o(t, A)u2(x,t, A) dx dt — J J Pi(t,A)ux.a(x,t,A) dxdt =
on 1=1 o n
= — J f(utt + Uaa) dx dt da. Q
Under the fulfillment of one of conditions VIIi, Vila, VII3, or VII4, one of conditions Villi, VIII2, VIII3, or VIII4, the sum of the last four sum-mands on the left-hand side of this equality is nonnegative. Using this fact and conditions I-IV, we infer that a solution u(x, t, a) to the boundary value problem for equation (4) with the Pim-condition and condition (2) satisfies estimate (12i).
Further analysis of equalities (13i) and (15i) gives the fulfillment of estimates (14i) and (16i) for a solution u(x, t, a) to the boundary value problem for equation (4) with the Pim-condition and condition (2). The sum of estimates (10i), (12i), (14i), and (16i) gives estimate (7); this estimate implies the existence in Vq of a function u(x, t, a) that is a solution to equation (4) in the cylinder Q and satisfies the Pim-condition and also condition (2).
The presence of a priori estimates uniform over e for solutions u(x, t, a) to the boundary value problem for equation (4) with the Pim-condition and condition (2) is proved by an additional integration by parts in (Hi) and (15i). The possibility of choosing a sequence converging to a solution to the boundary value problem Pim stems from the obtained estimates uniform over e and the reflexivity of a Hilbert space.
The uniqueness of solutions to the boundary value problem Pim in Vq is easy to show with the use of equality (21) and conditions I-IV.
The theorem is completely proved. □
4. Solvability of the Boundary Value Problems Pino, Pioio,
Pi ioo, and Piooo •
The boundary value problems Pmo, Pioio, Pnoo, and Piooo are problems intermediate between the problems Poooo and Pmi — in them, part of the sets 71, 72, 73, and 74, are free of boundary value conditions, whereas the value of the solution is given on the remaining part. The solvability of these problems in the space Vq is studied by a combination of the methods of investigating the problems Poooo and Pmi- More exactly, we once again use the regularization method, where equation (4) again serves as the regularizing equation. Only the boundary conditions change — for each of the problems Pmo, P1010, P1100, and Piooo, on the sets on which the values of the solution are not given, in the regularizing problem, the values of the derivative ut or ua are given (for example, in the problem regularizing the problem Pi no, the additional condition
ua(x,t,a)=0 for i eO, T
is given). The technique of the necessary a priori estimates as for fixed e as of estimates uniform over e completely corresponds to the technique used in the proof of Theorems 1 and 2. The convergent sequence is chosen in a standard manner: with the use of the estimates obtained and the reflexivity property of a Hilbert space.
Let us give the exact results for each of the problems.
Theorem 3. Suppose the fulfillment of conditions I-IV, of one of conditions VII 1, VII2, VII3, or VII4, of the condition
Pi(t, A) < 0, fa(t, A) = p03(j;) + Putt), Pos(t) > 0, PoS) < 0,
,1 (t,A) - do,004 (t) < 0, t e [0,T] ;
and also of one of the conditions
0i(t, 0) < 0, ,o (t, 0) > 0, e [0, T];
or
,i(t, 0) < 0, ,o (t, 0) > 0, e [0,T].
Then, for any function f (x,t,a) such that f (x,t,a) e L2(Q), ft(x,t,a) e L2(Q), fa(x, t, a) e L2(Q), f (x, 0, a) = f (x,T,a) =0 for (x,a) e fix(0,A), f (x,t, 0) = 0 for (x,t) e fi x (0, T), the boundary value problem P1110 has a unique solution in V0.
Theorem 4. Suppose the fulfillment of conditions I-IV, of one of conditions VIIt, VII2, VII3, or VII4, and also of the conditions
,1 (t, 0) > 0, 0o(t, 0) = ,01 (t) + ,02(t), ,01 (t) < 0, ,02(t) > 0,
0i(t, 0) - d0,02(t) > 0, t e [0, T];
,1(t, A) < 0, ,0(t, A) = ,03(t) + ,04(t), ,03 (t) > 0, ,04(t) < 0,
,1 (t, A) - d0,04(t) < 0, t e [0,T].
Then, for any function f (x,t,a) such that f (x,t,a) e L2(Q), ft(x,t,a) e L2(Q), fa(x, t, a) e L2(Q), f (x, 0, a) = f(x,T,a) = 0 for (x,a) e fi x (0, A), f (x, t, 0) = f (x,T, a) = 0 for (x,t) e fi x (0,T), the boundary value problem P1100 has a unique solution in V0.
Theorem 5. Suppose the fulfillment of conditions I-IV, of one of the conditions
a1(0,a) < 0, «0(0, a) > 0, a e [0,A];
or
«1(0, a) < 0, «0(0, a) > 0, a e [0,A]; of one of the conditions
,1(t, 0) < 0, ,0(t, 0) > 0, e [0, T];
or
,1(t, 0) < 0, ,0(t, 0) > 0, e [0,T],
and also of the conditions
a1(T, a) < 0, a0(T, a) = a03(a) + a04(a), a03(a) > 0, a04(a) < 0,
a1(T,a) — d0a04(a) < 0, a e [0, A]; ,1(t, A) < 0, ,0(t, A) = ,03(t) + ,04 (t), ,03(t) > 0, ,04(t) < 0,
Pi(t,A)-doPo4(t)<0, t € [0,T],
Then, for any function f(x,t,a) such that f(x,t,a) € L2(Q), ft(x,t,a) € L2(Q), fa(x, t, a) € L2(Q), f(x, 0, a) = 0 for (x, a) e fix (0, A), f{x, t, 0) = 0 for (x,t) € Q, x (0,T), the boundary value problem Pioio has a unique solution in Vq.
Theorem 6. Suppose the fulfillment of conditions I-IV, of one of the conditions
ai(0, a) < 0, cko(0, a) > 0, a € [0, A]-,
or
cki(0, a) < 0, cko(0, a) > 0, a € [0, A]-, and also of the condition
a\(T,a) < 0, ao(T,a) = aQS(a) + ao4(a), ao3(a) > 0, 0:04(0) < 0, a\(T, a) — doao4(a) <0, a € [0, A]-, Pi(t, 0) > 0, ,0(t, 0) = ,01 (i) + ,02 (i), ,01 (t) < 0, Po2(t) > 0,
Pi(t, 0) — doPo2(t) > 0, ie[0,T];
Pi(t, A) < 0, p0(t, A) = ,03(t) + ,04(t), ,03(t) > 0, ,04(i) < 0,
Pi{t,A) -d0p0i{t) < 0, t € [0,T],
Then, for any function f(x,t,a) such that f(x,t,a) € L2(Q), ft{x,t,a) € L2{Q), fa(x, t, a) € L2(Q), f(x, 0, a) = 0 for (x, a)ei!x (0, A), the boundary value problem P1000 is uniquely solvable in Vq.
5. Conclusion
We have studied the solvability of boundary value problems for third-order Sobolev-type equations with two time variables not solved for the derivatives. Existence and uniqueness theorems are prove for regular solutions.
The equations under study have model form. It is not hard to obtain analogous theorems on the solvability of the corresponding boundary value problems also for general equations. For example, the Laplace operator can be replaced by an arbitrary second-order elliptic operator whose coefficients ao, «1, ,0, and ,1 can depend also on the variables x\, ..., xn, and the number of time variables can be arbitrary. The calculations and conditions for such general equations will ne substantially more cumbersome but the essence of the results on solvability will not change.
References
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Alexandr Kozhanov, Doctor of Sciences (Physics and Mathematics), Professor, Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, 4 Koptyug Ave., Novosibirsk, 630090, Russian Federation, tel.: (8-383)3297683 8-9139276052 (e-mail: [email protected])
Received 11.06.19
Ультрапараболические уравнения с операторными коэффициентами при временных производных
А. И. Кожанов
Институт математики им. С.Л. Соболева СО РАН, Новосибирск, Российская Федерация
Аннотация. Работа посвящена исследованию разрешимости краевых задач для дифференциальных уравнений соболевского типа третьего порядка с двумя временными переменными (подобные уравнения называются также уравнениями составного типа, или уравнениями, неразрешенными относительно производной). Отличительными особенностями изучаемых уравнений являются, во-первых, то, что дифференциальные операторы, действующие на временные производные, не предполагаются обратными, во-вторых, то, что постановки краевых задач для них определяются коэффициентами этих дифференциальных операторов. Для предложенных задач в работе доказываются теоремы существования и единственности регулярных решений (решений, имеющих все обобщенные по С. Л. Соболеву производные, входящие в уравнение). Техника доказательств теорем существования основана на специальной регуляризации изучаемых уравнений, априорных оценках и предельном переходе.
Ключевые слова: ультрапараболические уравнения, необратимые операторные коэффициенты, краевые задачи, регулярные решения, существование, единственность.
Список литературы
1. Demidenko G. V., Uspenskii S. V. Partial Differential Equations and Systems not Solvable with Respect to Highest Order Derivatives. New-York : Marsel Dekker Inc., 2003. https://doi.org/10.1201/9780203911433
2. Favini A., Yagi A. Degenerate Differential Equations in Banach Spaces. New-York : Marsel Dekker Inc., 1999. https://doi.org/10.1201/9781482276022
3. Hayashi N, Kaikina E. I., Naumkin P. I., Shismarev I. A. Asymptotics for Dissipative Nonlinear Equation. Berlin : Springer-Verlaq, 2006.
4. Копачевский H. Д. Интегродифференциальные уравнения Вольтерра в гильбертовом пространстве. Симферополь : Таврич. нац. ун-т, 2012.
5. Корпусов М. О. Разрушение в неклассических нелокальных уравнениях М. : Либроком, 2011.
6. Kozhanov A. I. Composite Type Equations and Inverse Problems. Utrecht : VSP, 1999.
7. Kozlov V., Maz'ya V. Differential Equation with Operator Coefficients with Applications to Boundary Value Problems for Partial Differential Equations. Berlin : Springer-Verlaq, 1999.
8. Ладыженская О. А. Об интегральных оценках сходимости приближенных методов и решениях в функционалах для линейных эллиптических операторов // Вести. ЛГУ. 1958. № 7. Сер. математики, механики, астрономии. Вып. 2. С. 60-69.
9. Ладыженская О. А., Уральцева Н. Н. Линейные и квазилинейные уравнения эллиптического типа. М. : Наука, 1973.
10. Pyatkov S. G. Operator Theory Nonclassical Problems. Utrecht : VSP, 2003. https://doi.org/10.1515/9783110900163
11. Линейные и нелинейные уравнения соболевского типа / А. Г. Свешников, А. Б. Алынин, М. О. Корпусов, Ю. Д. Плетнер. М. : Физматлит, 2007.
12. Соболев С. Л. Некоторые применения функционального анализа в математической физике. М. : Наука, 1988.
13. Sviridyuk G. A., Fedorov V. Е. Linear Sobolev Type Equations and Degenerate Semigroups of Operators. Utrecht : VSP, 2003.
14. Треногин В. А. Функциональный анализ. M. : Наука, 1980.
15. Умаров X. Г. Полугруппы операторов и точные решения задач анизотропной фильтрации. М. : Физматлит, 2009.
Александр Иванович Кожанов, доктор физико-математических наук, профессор, Институт математики им. С. JI. Соболева СО РАН, Российская Федерация, 630090, г. Новосибирск, пр. Коптюга, 4 тел.: (383)3297683 (e-mail: [email protected])
Поступила в редакцию 11.06.19
