ON THE QUASIUNIQUENESS OF SOLUTIONS OF DEGENEFIATE EQUATIONS IN HILBERT SPACE

In this paper, we study the quasiuniqueness (i.e., f1≐f2 if f1−f2 is flat, the function f(t) being called flat if, for any K>0, t−kf(t)→0 as t→0) for ordinary differential equations in Hilbert space. The case of inequalities is studied, too.

In this paper we study the quasiuniqueness of solutions of the abstract equation of the form t-B(t)u t [O,T], 0 T +(R) (0.I) Here B(t) is p unbounded non-symmetric operator in Hilbert space.The quasiuni- queness of solutions of a somewhat more general problem of the form du llt[-B(t)u(t)]I < t(t)IIu(t)l!(o.2) is stuoied too.Here B(t) is ,( the same tyl;(-as in (0.!), anG ,(t) is a contin- uous non-regatie fuF, ction ir the interval !.
Recall that by quasiuniqueness we mea, ur, iqueness ii the class of functions that iffer by flat functicns.We sa) that the fl'F, ction u(t) IS flat furctio, if Vk 0 t-ku(t) Ii Section 1, ve study the simplest model, which is furthe developed in Section 2. In Section, 3 the nair} theorems are ubtained: Theorem 3 for the problem (0.2) and Theorem 4 for the problem (0.1).Our conditions of quasiuniqueness generalize the corresponding conditio,s cf [1] (we do not present She analog of Theore I-] of [1] since it is trivial).Theorem 2 of section 2 corresponds to Theorem 1-3 of [1] and generalizes Theorem 1-2 of the same paper.Our Theorems 3, 4 of Section 3 are a further generalization of Theorem 2, section 2 as well as of Theorems 1-2, I-3 of [1].Section 4 is devoted to reeF,arks about previous sections.We point out that n the paper we used methods different from those of AliF, hac-Baouendi in [I].Problems (0.1), (0.2) and these which can be reduced to them were recently studied by a number of authors (see [1][2][3][4][5][6]). Thus in [4] an example of a particular equation which could be reduced to the fov (0.1), where B(t) B(O) is self- adjoint, was considered, and the quasiunique,ess was proved for it.Further in [2] and [3], equation (0.1) was studied for B(t) B(O) + tBl(t) with 6(0) bounded (Fuchs-type equation).In the paper [5], the quasiunique,ess was proved for a certain class of elliptic operators with a degeneration in a single point.Condi- tiens which are difficult te verify were imposed, but a simple class of elliptic operators satisfying them was indicated.In our paper [6] elliptic equations with a possible degeneration on a hyperplane or in a single point are studied.In [6], the quasiuniqueness was proved for (0.1)-(0.2) with self-adjoint operator B(t).
Methods employed here were first used by Agmon and Nirenberg ( [7], [8]) for studying the Cauchy problem in the non-degenerate case. 1.

MODEL CASE.
Let H be a Hilbert space with scalar product (.,.) and norm 'II, is the interval [O,T] with 0 < T < +, (t) a continuous non-negative function on I, u(t) (CI(I,H), A, a linear operator in H, with domain D A and A A 1 + A 2, A A the self-adjoin part of A, and A -A 2 the anti-self-adjoint part of A. We shall assume that u(t) D A and that Au(t) (C(I,H).Set D tt-THEOREM 1.Let u(t) be a solution of the inequality IIDu(t) Au(t)ll t(t)u(t) (I.I) We suppose that all the conditions introduced above hold and that commutator [AI,A2] O. Let u(t) be a flat function (i.e., V k > O, t-ku(t) t/O 0).Then u(t) 0 in I.
f(s) > 0, and for t < t o T < t o We have s.Assume that the flat-function u(t) satisfying (1.1) is not identically zero.From (1.17) we have that u(t) is not a flat function.This is a contradiction.Therefore Theorem 1 is proved.
We assume that the function B(t)x is differentiable for 0 t T for all x DB(t), and set B(t)x B(t)x (2.1) Let u(t) be a solution of Du(t) (2.3a) where y(t), (t) are non-negative continuous functions in the interval I.If u(t) is a flat-function, then u(t) 0 in I.

in [I]
Indeed, the condition (1.4)  with y(t), B(t) non-negative continuous functions in the interval I. 3.
M,IN THEOREMS It ca.n be easily seen frem the proof of Theorem 2 for the case (2.3a) that the following is true" THEOREM 3. Let B(t) be a linear operator with domain DE(t), B(t) Bl(t) + B2(t) A;ere BI (t) B(t) is the self-adjoint part of B(t), B2(t) -B(t) is the anti-self-adjeint part of l(l,H), B(t)u(t) CI(I,H).We shall assume tFat u(t) DB(t), that u(t) C Let (t) denote a non-negative continuous function iv the interval I. Let a flat-function u(t) be a solution of where y(t), B(t) are non-negative centinuous functions in the interval I. Then u(t) 0 in I. Now consider, instead of inequality (3.1), the equation t--B(t)u(t) with the same assumptions regarding B(t) as in Theorem 3. The following is true" THEOREM 4. Let. with the assumptions of Theorem 3, u(t) be a flat-function and solution of the equation (3.3).Then, if   (BIU'U) 2 ,B2]u u) + (tllU ) 211BIU (J,u) u + ([B I (3.4) > -y(t)t (BIU,U)I B(t)tllu(t)ll 2 then u(t) _= 0 in I. PROOF.Let Next it follows from (3.7) that (t) is twice diffeentiable, and e( u,0ulq (tlU,U ll(B1ull 2 + Re(BlU,B2u) (BlU,U)2 ( and from (3.7) it follows that D2(t) _> -y(t)tlD(t)I 2tB( [;2(t) + y(t)tID(t) + 2tB(t) > 0 (3.12)From (3.12) an(i Lemma 2.2 it follows imwiately that u(t) 0 in the ipterval I. REMARKS.REMARK 1.Our key step in proving all the therems was to obtain an inequality for (t) of the form D2(t) + t(t)ID(t) + tB(t) _> 0 (4.1)Therefore, it follows from (2.12) in the case of equation (3.3) (problem (0.1), i.e., when f 0), that the following equality holds: ]u,u) IIBI u q ull + tBlU'U) +-( 2In the course of deducing (4.1), one obtains the condition (3.4), (BlU'U) 2 2 211BlU q u + (tBlU,U) + ([Bi,B2]u,u) > -ty(t)IBlU,U tB(t)Iiu(t)l Let us point out that it seems to us that this conditior.,obtained from (4.2), must be close enough to being necessary (for quasiuniqueness).i.e., if u is a flat solution of (0.2) and (4.4) holds, then u 0 in (see also Remark 6).REMARK 3. The method of the proof of the theorems concerning the quasiunique- ness of the solution of (0.1)-(0.2) presented in sctions 2 an 3 allows one to assert, even in cases when there is no quasiuniqueness, that a given solution is trivial if the appropriate conditions are true for this solution.We have in mind the cenditions (2.3), (2.3a), (3.2), (3.4).
It may quite happep that these conditions do not hold for all the solutions of (0.1)-(0.2).On the other hand, if for some specific solution u(t) of (0.1) or (0.2) the appropriate condition does hold, then its triviality follows from the flatness of this specific u(t).Quasiuiqueness of solution of (0.1)-(0.2) follows in the case when these conditiors are satisfied by the whole class of possible solutions.
REMARK 4. It fellows from Theorems 3 and 4 of section 3 that the quasiunique- Here Bl(t) can be replaced by B(t) and C for the problem (0.2) and C 2 for the problem (0.1) correspondingly.
On the other hand, in case (iii) there even exists a classical uniqueness in the case of (0.1).This stems from the following" t-B(t) du (t,u(t)} Re(B(t)u,u) 0 1 d 1.dq -tt u,u) ;dt 0 and if q(O) O, then q(t) 0 for all t.REMARK 5.The conditions in Theorems 2-4 do not seem natural, at any rate not at first sight.The following conditions seem more natural" (lU,U) _> -y(t){l(BlU,U) + liul{ 2} with y(t) a continuous function in I.