Lomovtsev, F. E.

English

Электронная библиотека БГУ

Differential Equations, 17ol. 36, No. 4, 2000, pp. 605-612. Translated from Differentsial'nye Uravneniya, Vol. 36, No. 4, 2000, pp. 542-548. 
Original Russian Text Copyright (~ 2000 by Lomovtsev. 
PARTIAL  
D IFFERENTIAL  EQUATIONS 
The Cauchy Problem for Complete Second-Order 
Hyperbolic Differential Equations 
with Variable Domains of Operator Coefficients 
F. E. Lomovtsev  
Be laruss ian  State  Univers i ty ,  M insk ,  Be la rus  
Received February 10, 1998 
Complete second-order hyperbolic differential equations with constant domains of operator co- 
efficients were investigated in [1, 2]. In the case of variable domains of operator coefficients, 
second-order hyperbolic differential equations with a two-term leading part were analyzed in [3-5]. 
In the present paper, we investigate second-order hyperbolic differential equations with a three-term 
leading part in the case of variable domains of operator coefficients. 
1. STATEMENT OF THE PROBLEM 
Let H be a Hilbert space with inner product (., .) and norm I" I. On a bounded interval ]0, T[ 
of the real axis, we consider the Cauchy problem for the differential equation 
Lu =- d2u/dt 2+ B(t)du/dt + A(t)u = f, t E ]0,T[, (1) 
with the homogeneous initial conditions 
ult=0 =0, duldtlt=o =0,  (2) 
where u and f are functions of the variable t ranging in H, A(t) and B(t), t E O, are linear 
unbounded operators in H with domains D(A(t)) and D(B(t)), respectively, depending on t, and 
O is some set of full measure on [0, T]. 
We assume that the operator A(t), t E O, satisfy the following conditions. 
Condi t ion  A1. The operators A(t), t E O, are self-adjoint in H and satisfy the inequalities 
(A(t)u,u) >_ cl(t)]u] 2 Vu e D(A(t)), (3) 
where cl (t) > 0 are constants independent of u. 
Condi t ion  A2. The inverse operators A-l(t), t E O, have the strong regular derivative 
dA-l(t)/dt e L~(]O,T[,.~(H)) on ]0, T[, which satisfies the inequalities 
dA-l(t)g, ) _  
dt g < c2 (A-l(t)g,g) Vg e H, (4) 
where c2 >_ 0 is a constant independent of g and t. 
The notion of strong regular derivative was introduced in [6] for operators defined almost every- 
where. 
We assume that the operators B(t), t E O, satisfy the following conditions. 
Condi t ion  B1. D(A(t)) C D(B(t)), t e O; B(t)A-1/2(t) e L~(]O,T[,.~(H)), where A-U2(t) 
are the inverses of the square roots A1/2(t) of A(t), t E O, and 
-Re(B(t)u,u) c31ul VueD(A(t)), (5) 
where c3 > 0 is a constant independent of u and t. 
0012-2661/00/3604-0605525.00 (~) 2 00 MAIK "Nauka/Interperiodica" 
606 LOMOVTSEV 
Condit ion B2. A1/2(t)B(t) (dA-l(t)/dt) 9 L~(]O,T[,.~(H)), and 
-Re(B(t)u,A(t)u) <_ C4 A1/2(t)u 2 Vu e D(A(t)), (6) 
where c4 _> 0 is a constant independent of u and t. 
Let us prove the strong solvability of the Cauchy problem (1), (2). We also present an example 
of a mixed problem for a complete second-order hyperbolic partial differential equation whose 
coefficients contain nonintegrable singularities with respect o the variable t. 
2. THE UNIQUENESS THEOREM FOR STRONG SOLUTIONS 
First, we introduce spaces and give the definition of strong solutions of the Cauchy problem 
(1), (2). As the space of strong solutions, we choose the Hilbert space E that is the completion of 
the set 
D(L) = {u E L2(]O,T[,H) : u(t) e D(A(t)), t e O; d2u/dt 2, A1/2(t)du/dt, 
(A(t)u,u)lt=o=O, I(A(t)u,u)l t=T < +OO; u(O)=du(O)/dt=O} 
in the norm 
IluIiE = dt dt + A1/~(t)u :dt 
0 
As the space of right-hand sides f of Eq. (1), we choose the Banach space F that is the completion 
of L~(]0, T[, H) in the norm 
II/11 = { (f, 9~(t)v)dt /llvllo } 
Here E0 is the set of all functions v e L2(]O, T[,H) such that x/rc~v e L2(]O, T[, H), the derivative 
dr~dr E L~(]0, T[, H) exists, and v satisfies the condition v(0) = 0, equipped with the Hermitian 
norm 
ilvll0 = dv 2 cl(t)H ~dt , 
o o 
and ~(t)v = i T - t )dv/dt .  Note that F is the so-called negative space corresponding to the positive 
space Be equipped with the norm [Iwll~ = H~-l(t)w[10, where ~-l(t)  is the inverse of ~(t). 
The Cauchy problem (1), (2) corresponds to a linear unbounded operator L : E D D(L) ~ F 
with domain D(L). Let this operator L satisfy the closability criterion for linear operators in 
Banach spaces: if un ~ 0 in E and Lun ---* f in F as n ~ co, then f = 0. By L we denote the 
closure of L, and by D (L) we denote its domain. The solutions of the operator equation Lu = f, 
f E F, are referred to as strong solutions of the Cauchy problem (1), (2). 
Now we derive an a priori estimate for strong solutions of the Cauchy problem (1), (2). 
Theorem 1. Let the set D(L) be dense in L2(]0,T[, H), and let the operator L admit the clo- 
sure L. If Conditions A1, A2, and B1 are satisfied, then there exists a constant Co > 0 independent 
of u and such that IluIIE < Co I]LU]IF for all u e D (L). 
Proof. In H, we consider the abstract smoothing operators A['(t)  = [I+eA(t)]-',  e > 0, whose 
ranges are contained in D(A(t)) and which satisfy the following conditions. 
T 
A1. If ~ --* 0, then S IA[ l ( t )v -  vl 2 dt ~ 0 for all v e L2(]O,T[,H). 
0 
A2. The operators All(t)  are strongly and regularly differentiable with respect o t in H, and 
their derivatives satisfy dA[ 1 (t)/dt e Lo~ (]0, T[, .~(g)) .  
These properties were proved in [3]. 
DIFFERENTIAL EQUATIONS Vol. 36 No. 4 2000 
THE CAUCHY PROBLEM FOR COMPLETE SECOND-ORDER HYPERBOLIC ... 607 
Integrating by parts once more, we obtain the identity 
T T 
~, dr'' A;1 dt (A(t)u, A;l(t)u) dt 
o o 
T T 
/B  t du -1 du : 2Re  i ec(T-t) (Lu, A:l(t)~(t)?J)dr-2Re i.c(T-t)(T -t) t ()-~,Ae (t)-~)dt (7) 
o o 
T 
+ f e'(T-t)(T -- t)O,(u, u)dt, 
o 
where ~(u,u)  = ((d(A(t)A[l(t)) /dt)u, u)-c(A(t)A[ l ( t )u,  u). Using the formula [3] 
d(A(t)A;l(t)) /dt = -A(t)A[l(t) (dA-l(t)/dt) A(t)A[l(t), 
inequalities (4), and the relation IA[1/2(t)vl < Ivl in the operator-differential form ~(u,u),  
we obtain 
(~(u, u) <_ (c2 - c) A1/2(t)A-[1/2(t)u 2. (8) 
If in (7) we use the estimate (8) and let e tend to zero in the resulting inequality with regard 
for Property A1, then we obtain the inequality 
T T 
2Re/e'(T-t)(d'u\ dt" ~(t)u)dt + /e.(T-t)(A(t)u'u)dt 
o o 
T T 
/B  t du <2Re/ec(T-t)(Lu,~(t)u)dt-2Re/ec(T-t)(T - t) ~ ( )-~, ~t ) dt- S1q- S2 
o o 
for c _> c2. Integrating by parts once more, we arrive at the inequality 
T T T 
.I ec(T_t) du 2 
dt  dt + f e'(T-t)(A(t)u, u)dt <_ S 1 "J- S 2 -- C f ec (T - t ) (T  -- t)  du  2 dt dr, 
o o o 
whose right-hand side does not exceed $1 for c >_ max {c2, 2Ca} by virtue of (5). This, together with 
simple estimates, yields IlullE <__ c011LullF for all u e D(L). Performing the passage to the limit, 
we can generalize this inequality to all strong solutions of the Cauchy problem (1), (2). 
It follows from Theorem 1 that if there exists a strong solution of the Cauchy problem (1), (2), 
then it is unique and continuous with respect o f. 
3. THE EXISTENCE THEOREM FOR STRONG SOLUTIONS 
The following statement establishes the strong solvability of the Cauchy problem (1), (2). 
Theorem 2. Let the set D(L) be dense in L2(]0, T[, H), and let the operator L admit the clo- 
sure L. If Conditions A1, A2, B1, and B2 are satisfied and the operator dA-l(t)/dt has the 
2 1 2 1/2 ~ 1 z strong regular derivatived A- (t)/dt such that A (t) (d A- (t)/dt ) E Loo(]O,T[,.9~(H)), i = 1,2, 
then for each f E E, the strong solution u E E of the Cauchy problem (1), (2) exists, is unique, 
and satisfies the inequality HallE <_ Coil filE. 
Proof. By virtue of a corollary of the Hahn-Banach theorem on the continuation of linear 
continuous functionals, to prove the strong solvability of the Cauchy problem (1), (2), it suffices to 
show that the range R(L) of the operator L is dense in F, i.e., if 
T 
(Lu, = e D(L) (9) 91( t)v)dt 0 Yu 
o 
for some v E E0, then v = 0. 
DIFFERENTIAL  EQUATIONS Vol. 36 No. 4 2000 
608 LOMOVTSEV 
In (9), we set u = A-l(t)h, where h 9 L~(]O,T[,H) is such that dhldt, d2hldt ~ 9 L2(]O,T[,H) 
and condition (2) is satisfied; then we obtain 
T 
( d2h A-i(t)e~(T-t)~(t)w) dt 
S t  t dt 2' 
0 
T T 
o o 
T 
dh e<(T_t)~(t)w) dr, - i (B(t) dA~(t) h + B(t)A-~(t)-~, 
0 
(10) 
where 
(I)(h, w)= \(d2A-l(t)h-~ +2 d (t) dhdt,_~ 
and w is found from the Canchy problem C~(t)w = eC(t-T)Ct(t)v, w(0) = 0. We can readily see 
that w 9 E0. 
It follows from (10) that the function A-l(t)~(t)w has the derivative in n2(]0, T[, H) and vanishes 
for t = T. In (10), we perform integration by parts, extend the resulting relation to all functions 
h 9 L2(]O, T[, H), dhldt 9 L2(]O, T[, H), h(0) = 0, with the use of the passage to the limit, 
set h -- w, and take the doubled real part; then we obtain 
T T i ( ;,d [A-'(t)e'(T-t)(T-t)~t ]) dt-f-2Re Se'(T-O(T -- t) (w, ~t ) dt 
0 o 
T 
= -2Re i er -- t)~(w, w)dt 
0 
T 
- dw t dw -2Re  i eC(V-t)(T- t)(B(t) dAd:(t) w + B(t)A-I( )-~ , -~ ) dt. 
0 
(11) 
The following statement is used for integration by parts in the first integral occurring on the 
left-hand side of (11). 
Lemma [3]. Let El, F1, and G1 be Banach spaces, TI : E1 -+ F1 be a linear bounded operator, 
and $1 : F1 --+ G1 be a linear closed operator with dense domain. If the domain of the product 
$1 o T1 of the operators T1 and $1 is dense in El, then the adjoint operator ($1 o T1)* is equal to 
the weak closure of the product T~ o S~ of the adjoint operators T~ and S~, respectively. 
Applying this lemma in E1 = F1 = L2(]O, T[,H) and G1 = L2(]O, T[, H) • H to TI = 
A-I(t)ec(T-O(T - t) and Slg = {dg/dt, g(O)) with domain 
D(S1)-- {g e L2(]O,T[,H) : dg/dt 9 L2(]O,T[,H),g(T) =O} 
in the first two terms of the expression 
T 
0 
_•2 
t=O 
+Te cT A-1/2(t) 
(12) 
DIFFERENTIAL EQUATIONS Vol. 36 No. 4 2000 
THE CAUCHY PROBLEM FOR COMPLETE SECOND-ORDER HYPERBOLIC ... 609 
we find that it is equal to 
T 
0 
9- Te "T A_l/2(t)~t t 2 t=0" 
T 
dw ~tt ) dr- / ( (d  [A_l(t).c(T_t)(T_t)] ) dWdt ' --~) dt 
o (13) 
Note that TI* = T1, St({p,p(O)}) = -dp/dt ,  D (S~) = {{p,p(0)} e G1 : dp/dt e L2(]0, T[, H)}, 
and T~ o S~dw/dt = -d  [A-l(t)ec(T-t)(T - t)dw/dt] /dt 9- (d [A-l(t)e c(T-t) x (T - t ) ] /dt )  dw/dt. 
The fact that dw/dt E D (($1 o T~)*) follows from (10), since this relation can be rewritten as 
follows: 
T 
dw / ( d [A-l(t)'c(T-t)(T-t)] dhdt' ~)dt 
o T 
~- : ((d [A-l(t).c(T-t)(T-t)]) dh 
dt ' 0 
T 
dw 
o 
T T 
-/ec(T-t)(T-t)O(h,w)dt - / (B(t) dA~t(t) h+ B(t)A-l(t)~t , ec(T-t)-,~(t)w) dt. 
o o 
Therefore, integrating the terms occurring on the left-hand side of (11) by parts with the use 
of (12) and (13), we obtain 
T 
f cc(T-t) A-1/2(t)~ t 2dr 9- Yc cT A-1/2(t] dw ,, 
o dt t=0 
T 
= f e~(T-t)(T -- t)Ol(w, w)dt, 
o 
T 
9- f e c(T-t) [w[2dt 
o (14) 
where 
--~ \ dt 2 , -~  - 3 dt ' 
- -2Re  (U(t)dAd:(t)w , ~tt ) -2Re (B(t)A-l(t)~t , d-~t ) . 
dw 
In the operator-differential form r w), we use inequalities (4) and (6) and some elementary 
inequalities. Then we find that r w) does not exceed 
- (c - 3c2 - 2c4 - 2) A-1/2(t)dw/dt 2 - (c - c5 - c6)Iwl 2, (15) 
where c5 = [IA1/2(t)d2A-l(t)/dt2[12.(H) and c6 = [IA1/2(t)B(t)dA-l(t)/dtll~(H). 
The quantity (15) with c _> c7 = max {3c2 + 2c4 + 2, c5 + c6} is nonpositive. Therefore, if we 
T 
estimate the left-hand side of (14) with c _> c~ below by the quantity f [wl2dt and the right-hand 
0 
side by zero, then we obtain w -- 0 and hence v = 0. The proof of Theorem 2 is complete. 
DIFFERENTIAL EQUATIONS Vol. 36 No. 4 2000 
610 LOMOVTSEV 
Remark.  We can generalize the assertion of Theorem 1 in the above-described way (possi- 
bly, with a larger value of the constant Co) and the assertion of Theorem 2 by the Schauder- 
Ladyzhenskaya method of continuation with respect o the parameter to the case of an equation 
with a leading part, namely, 
d2u/dt 2+ B(t)du/dt + A(t)u + [t(t)du/dt + A(t)u = f, 
where [t(t),.4(t)A-1/2(t) 9 Loo(]O,T[,-~(H)). 
4. EXAMPLE 
In the bounded omain G = ]0, T[ • ]0,/[ of the variables t and x, we consider the equation 
02u(t,x)/Ot ~- t-ZO2u(t,x)/OxOt - t-"O2u(t,x)/Ox 2 = f(t ,x),  (16) 
where a >_ 4 and fl _< a/2 - 1, with the boundary conditions 
Ou(t,O)/Ox = u(t,l) = 0 (17) 
and the homogeneous initial conditions 
u(O,x) = O, Ou(O,x)/Ot = 0. (18) 
In this example, as the Hilbert space g~ of strong solutions of problem (16)-(18), we choose the 
completion of the set 
D(L) = {u e L2(G) : u(t,x) 9 W~(O, 1), Ou(t,O)/Ox = u(t , l )= 0 Vt 9 ]0, T[; 
02u(t,x)/Ot 2, t-"/202u(t,x)/OxOt, t -~ x 02u(t,x)/Ox 2 9 L~(G); 
l l 
t -~flOu(t,x)/Oxl2dx -- O, t -~flOu(t,x)/Oxl2dx T < +oo; 
0 t=O 0 = 
u(O,x) = au(O,x)/Ot = 0} 
% 
in the Hermitian orm 
llu(t,.)ll.= JOu(t,x)/OtJ'dxdt+ t-" leu(t,x)/ex]'dxdt 
0 0 0 0 
As the Banach space (I) of right-hand sides f(t, x) of Eq. (16), we choose the completion of L2(G) 
in the norm 
I I f(t,x)l lr  -sUP, ex0 { 
T l 
/ / S(t,x)(T - t)(Ov(t,x)/Ot)dxdt 
0 0 
/,v,,., 0} 
where ~0 is the set of all functions v E L~(G) such that t-"/~v E L2(G), the derivative Ov/Ot E L2(G) 
exists, and v(0, x) = 0; this set is equipped with the norm 
(y; J J ) Hv(t,x)llo-- ]Ov(t,x)/Otl2dxdt+ t-" Iv(t,x)]:dxdt 
0 0 0 0 
i/2 
The following theorem claims that the mixed problem (16)-(18) is well posed in the sense of 
strong solvability. 
DIFFERENTIAL EQUATIONS Vol. 36 No. 4 2000 
THE CAUCHY PROBLEM FOR COMPLETE SECOND-ORDER HYPERBOLIC ... 611 
Theorem 3. For each f 9 r the mixed problem (16)-(18) has a unique strong solution u 9 
satisfying the inequality Ilull~ < Collfll~. 
Proof. It suffices to show that all assumptions of the abstract Theorem 2 are valid. First, 
we show that the operators A(t), t E ]0, T[, induced by the differential expression A(t)u(t, x) = 
-t-"O2u(t,x)/Ox 2, t E ]0, T[, and the boundary conditions (17) satisfy all assumptions of the 
abstract Theorem 2 in the Hilbert space H = L2(0, l). The operators A(t), t E ]0, T[, with domains 
D(A(t)) = {u 9 L2(G) : u(t,x) 9 W~(0,/) Vt 9 ]0, T[; t-"O2u(t,x)/Ox 2 9 L2(G); 
Ou(t ,o ) /ox  = u(t,1) = 0} 
are self-adjoint in L2(0, l), since they are obviously symmetric in L2(0, l) and have bounded inverses 
A-l(t)g = t ~ f_ g(t,T)dT ds 
0 
on L2(0, I). Their boundedness follows from the inequalities IIA-l(t)gll~,a < (1/3)14T2~llgll~, ~ for 
all g 9 L2(O, 1), where H" H0,a is the norm in the nilbert space L2(O,l). The operators A(t) satisfy 
inequalities (3) with o( t )= 21-2t -~. 
The operators A-l(t) have the first strong derivative 
dA-l(t) f dt g= g(t, r)dT ds, 
0 
which is a bounded operator in L2(O, 1), namely, II(dA-l(t)/dt)gll~,a <- (1/3)c 214r "-211gllg,a for all 
g 9 L2(0,1), and satisfies inequalities (4), since 
0,fl 0 0 
where (., ")o,n is the inner product in L2(0, l). The operators dA-l(t)/dt have the strong derivative 
~-~ g=a(a -  1)t ~-2 fg( t ,T)dT as 
0 
such that A1/2(t)(d~A-l(t)/dt i) 9 L~ (]0, T[,.gP (L2(0, l))), i=  1,2, since 
A1/2(t) (dA-l(t)/dt) g i,o 22 a -2  2 L2(O,l), < (1/2)a I T Ilgllo,  Vg 9  
Al/2(t) (d2A-l(t)/dt2) g ,:,a - < (1/2)a2(a -1)122Ta-4 Hgl10,~22 Vg 9 L2(O,I). 
Second, we show that the operators Bit), t E ]0, T[, induced by the differential expression 
B(t)u(t, x) --- -t-~Ou(t, x)/Ox, t E ]0, T[, also satisfy all assumptions of the abstract Theorem 2 in 
the Hilbert space H = L2(O,l). The operators B(t) are subordinate to A1/2(t): IIB(t)ull2o,a < 
T ~-e~ IIA1/2(t)ull~,n for all u E n (Ai/2(t)), where D (A1/2(t)) are the domains of the opera- 
tors A1/2(t), and satisfy inequalities (5), namely, -2Re(B(t)u,u)o,a = -t-~lu(t,O)l e < 0 for all 
u(t,x) E D(A(t)). Since 
2 2 a -2~-2  2 A1/2(t)B(t) (dA-l(t)/dt) g o,~ <- a T Ilgllo,o Vg e L2(0, l), 
DIFFERENTIAL EQUATIONS Vol. 36 No. 4 2000 
612 LOMOVTSEV 
it follows that A1/~(t)B(t)dA-l(t)/dt 9 Lo~ (]0,T[,-~ cz (L~(0,/))), and from the inequalities 
-2Re(B(t)u,A(t)u)o,~ = -t-~-~lOu(t,1)/Ox]2 < 0 Vu(t,x) 9 D(A(t)) 
we find that inequalities (6) hold. 
Finally, it is obvious that the set D(L) is dense in L2(G); in a standard way, we can show that 
the linear operator L : ~ ~ D(L) ~ 9 admits the closure. The proof of Theorem 3 is complete. 
REFERENCES 
1. Yakubov, S.Ya., Dokl. Akad. Nauk SSSR, 1971, vol. 196, no. 3, pp. 545-548. 
2. Lomovtsev, F.E. and Yurchuk, N.I., Differents. Uravn., 1976, vol. 12, no. 12, pp. 2242-2250. 
3. Lomovtsev, F.E., Differents. Uravn., 1992, vol. 28, no. 5, pp. 873-886. 
4. Lomovtsev, F.E., Differents. Uravn., 1994, vol. 30, no. 8, pp. 1412-1425. 
5. Lomovtsev, F.E., Differents. Uravn., 1997, vol. 33, no. 10, pp. 1394-1403. 
6. Lomovtsev, F.E., Differents. Uravn., 1995, vol. 31, no. 7, pp. 1132-1141. 
DIFFERENTIAL EQUATIONS Vol. 36 No. 4 2000 


The Cauchy Problem for Complete Second-Order Hyperbolic Differential Equations with Variable Domains of Operator Coefficients

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The Cauchy Problem for Complete Second-Order Hyperbolic Differential Equations with Variable Domains of Operator Coefficients

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