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Incomplete iterated Cauchy problems
JOURNAL
OF MATHEMATICAL
ANALYSIS
incomplete
AND
APPLICATIONS
Iterated
552-579 (1992)
168,
Cauchy
Problems
RALPH DELAUBENFELS* Department
of Mathematics, Ohio University, Athens, Ohio 45701 Submitted by C. Foias Received September 18, 1990
We treat numerous higher order incomplete abstract Cauchy problems by writing them in the form (1)
FL,
(dldt-A,)u(t)=O
(t>O),
(21 u“~“(O)=x,(l
u“-‘l(t)bounded(l
The abstract Cauchy problem (l), (2), and (3a) is wellposed, in a certain sense, when A,, .... A, generate bounded C-semigroups and (-A,+ ,), .... ( -A,) generate uniformly stable C-semigroups. For (l), (Z), and (3b), we need A,, _._,A, and 0 1992 Academic Press, Inc. ( -A, + ,), .... (-A,) to generate stable C-semigroups.
I. INTRODUCTION The relationship between strongly abstract Cauchy problem
continuous
semigroups
u(0, x) = x,
~u(t,x)=G(u(t,x))(t~O)
and the
(1.1)
is well known, see, for example, [6, 16, 20, 26, 30, 341. Perhaps the most unified treatment of higher order complete abstract Cauchy problems appears in [31] (see also [21,22]), where the iterated abstract Cauchy problem, u(t)=0
u +l)(o)=xi
(t&O),
(1
(1.2)
is introduced, and is shown to be wellposed, in a certain sense, if and only if, for each k, A, generates a strongly continuous semigroup. Writing a higher order problem in the iterated form (1.2) decomposes it into first order problems. * Research supported by an Ohio University
Research Grant
552 0022-241X/92 $5.00 Copyright 0 1992 by Academic Press, Inc. All rights of reproduction m any form reserved.
INCOMPLETE
ITERATED CAUCHY
553
PROBLEMS
In this paper, we give a unified treatment of incomplete higher order problems via the incomplete iterated abstract Cauchy problems fJ(-$-Ak)u(r)=O
(t>,O), &‘)bounded
tP1)(0)=x,
(l
(1
(1.3)
and fi, (:-A,).(t)=0
zP’)(0)=xi
(t>O), ]im u(i-l) (t)=O t-02
(1
(l
(1.4)
where A,, A,, .... A, are commuting linear operators. For (1.3) to be wellposed, it is suflicient that A,, .... Aj generate bounded strongly continuous semigroups and ( - Ai+ 1), .... ( - A,) generate uniformly stable strongly continuous semigroups. For (1.4) to be wellposed, it is sufficient that A,, .... Aj and (- Aj+ 1), .... (-A,,) generate stable strongly continuous semigroups. Special cases,after an appropriate factorization of (1.3) or (1.4), are u’“‘(t) = B(u(t))
(t b O),
u(‘-“bounded
zP1)(0)=xi
(1
(1 di
(1.5)
and u’“‘(t) = B(u(t))
(t 3 O),
u(‘-‘)(0)=xi
,‘t; u(‘-‘j(t)=0
(1 Giij), (l
(1.6)
and general second order abstract Cauchy problems, u”(t) + 2B(u’(t)) + A(u(t)) =O,
u(O) = x, u bounded,
(1.7)
u(O) =x, lim u(t) = 0. ,+CC
(1.8)
and u”(t) + 2B(u’(t)) + A(u(t)) = 0,
Factoring higher order problems into the iterated form gives them a sort of spectral intuition. The intuition, in (1.2), is that each A, must lie, in some sense, in a left half-plane. The intuition in (1.3) is that A,, ..,, Ai should lie in the left half-plane Re(z) < 0 and Ai+, , .... A, should lie in a closed subset of Re(z) > 0. In (1.4), both A,, .... Ai and (- Aj+ ,), .... (-A,) should lie in the open half-plane Re(z) < 0. These are exactly the results one would obtain if A,, .... A, were complex numbers.
5.54
RALPH DELAUBENFELS
We also generalize our results by weakening the hypotheses on Ak or (- Ak), requiring merely that they generate what are called C-semigroups (Definition 2.3) rather than strongly continuous semigroups. This is a generalization of strongly continuous semigroups that has received much attention recently [S, 7, l&13, 22225, 28, 29, 32, 331). Among other things, this allows us to consider A, B, A, whose resolvents are polynomially bounded, rather than 0(1/w), as with generators of strongly continuous semigroups. Equation (1.5), with n=2 and j= 1, was considered in [l, 151; for the same n andj, (1.6) is in [15, 91. In [17, 181, (1.5) is treated (see Section IV for generalizations of the results in [ 181). Some examples, including incomplete versions of the wave equation and first order perturbations of Laplace’s equation, may be found in 2.15, 2.18, 5.5, 5.9, and 5.10. Here is a rough table of contents. In Section II, we give necessary and sufficient conditions for (1.5) and (1.6), with n = 2 and j= 1, to have a unique solution, for all x E { Cx 1x E the domain of A}. Section III contains the basic results about (1.3) and (1.4). Section IV treats (1.5) and (1.6), and Section V deals with (1.7) and (1.8). All operators are linear, on a Banach space, X. We will write L(X) for the space of all bounded operators from X into itself. We will write D(A) for the domain of the operator A, Im(A) for the image, p(A) for the resolvent set, sp(A) for the spectrum. By a solution of (1.2), (1.3), or (1.4) we mean u such that
and u satisfies (l-2), (1.3), or (1.4). A solution of (1.5) or (1.6) is in C”(R+,X)nC(R+,D(A)) and a solution of (1.7) or (1.8) is in C(R+, D(A)) n C’(R+, D(B)) n C*(R+, X). We will write x for (XI 1“‘9 x,). II. SECOND ORDER INCOMPLETE CAUCHY PROBLEMS AND C-SEMIGROUPS We will give necessary and sufficient conditions for the following special casesof (1.5) through (1.8) to have a unique solution, for all x in C(D(A)). u”(t)=A(u(t))
(t20),
u”(t)=A(u(t))
lim u(t)=O= t-cc
u(0) =x, (t>O),
u bounded.
(2.1)
u(0) = x,
lim u’(t)= lim u”(t). I-K r-m
(2.2)
INCOMPLETE ITERATED CAUCHY PROBLEMS
555
Note that (2.2) may be considered a boundary value problem, with boundary conditions at zero and infinity. DEFINITION 2.3. Suppose C is a bounded, injective operator. The strongly continuous family of bounded operators { W(t)},,o is a C-semigroup if W(0) = C, and W(t) W(s) = C W(t + s), for all s, t B 0. The generator of { W(t)},, 0 is defined by
Ax=C-’
(
!eOf(W(t)x-x)
1
,
D(A) f (x 1limit exists and is in Im(C)} In this paper, C will always be a bounded, injective operator. Basic properties of C-semigroups may be found in [lo]. The generator of a C-semigroup is closed and its domain is left invariant by the C-semigroup. Even when the C-semigroup is exponentially bounded, its generator may have empty resolvent set. However, its C-resoluent set will contain a halfplane. 2.4. The complex number r is in p,(A), the C-resoluent set is injective, and Im(C)c Im(r-A). When A generates an exponentially bounded C-semigroup, { W(t)}lGo, then (r-A)-‘C exists, for Re(r) sufficiently large and equals the Laplace transform of { W(t)},,o (see [28]). DEFINITION
of A, if (r-A)
DEFINITION 2.5. We will call the C-semigroup stable if lim,, m W(t)x = 0, for all x E X. We will call it uniformly stable if lim, _ oD11 W(t)ll = 0. PROPOSITION 2.6 (from [lo] ). SupposeA generatesa C-semigroup. Then (1.1) has a unique solution, for all x E C(L)(A)). If p(A) is nonempty, then the converseis also true. There then exists finite A4 such that IIu(t) I(< M I(C- ‘x/l, for all t 2 0,
x E C(D(A)). DEFINITION
2.7. We will say that the bounded C-semigroup { W(t)},>o
is a bounded nowhere reversible C-semigroup if there exists no nonconstant
bounded continuous v: R -+ X such that W(t) u(s) = Cv(t + s), for all nonnegative t, real s. PROPOSITION 2.8. Suppose { W(t)},ao is a bounded C-semigroup generated by B. Then ( W(t)},>,, is nowhere reversible if and only if there exist no nonconstant solutions of (l.l), with G E (-B).
556
RALPHDELAUBENFELS
THEOREM 2.9. Suppose A has a square root that generates a bounded nowhere reversible C-semigroup. Then (2.1) has a unique solution, for all x E C(D(A)). If p(A) is nonempty, then the converse is also true. There then exists finite M such that jlu(t)(l < M 11 C ‘x,11,for all t > 0, x E C(D(A)).
Exactly the same arguments as in [9] show the following. THEOREM 2.10. Suppose A has a square root that generates a stable C-semigroup. Then (2.2) has a unique solution, for all x E C(D(A)). If p(A) is nonempty, then the converse is also true. We then have the same continuous dependence on initial data as in Theorem 2.9. COROLLARY 2.11. Suppose A = B2, where B generates a uniformly stable C-semigroup. Then (2.1) and (2.2) have unique solutions, for all x E:C( D(A)).
Remark 2.12. The proof of Corollary 2.11 shows that a uniformly
stable C-semigroup is nowhere reversible. A stable C-semigroup may not be nowhere reversible. If W(t)f(x)=f’(x+ t), on L’([O, oo)), then is a stable strongly continuous semigroup that is not nowhere { wm>, reversible (see Example 2.15 and Remark 2.17). The following corollary of Theorem 2.9, when C = Z, appeared in [ 151. Suppose A = B2, where B generates a C-semigroup such that BW(t)EL(X), for t large, and lim,,,I(BW(t)(l =O. { WOl,,o Then (2.1) has a unique solution, for all XE C(D(A)). COROLLARY 2.13.
Remark 2.14. As pointed out in [15],.the results of [l] regarding (2.1) are contained in Corollary 2.13, with C = I. More generally, Corollary 2.13 includes the case when ( W(t)),,o is a bounded holomorphic C-semigroup (see [ 143). EXAMPLE 2.15. Suppose 1 < p < co and g is a nondecreasing nonnegative function on R. Consider the following incomplete wave equation
(-$,t,.,=(&kx)
40, x) =f(xh Let B= dldx, (W(t))f(x)
=f(x
(tbO,xER), sup
lu(x, t)l P g(x) dxl t a 0 < ~0.
(2.16)
I
the generator of the strongly continuous semigroup + t), on X= LP(R, g dx).
557
INCOMPLETE ITERATED CAUCHY PROBLEMS
Equation (2.16) has a unique solution, for all and only if lim, _ m g(x) = co. PROPOSITION.
f ED(B)
if
Proof. If v as in Definition 2.7 existed, then it is not hard to see that it would have the form u(s) = f (s + x), for somef E A’. Thus, by Theorem 2.9, (2.16) has a unique solution, Vf E D(B) if and only if v(s), as defined above, is unbounded, for all f E X. Since
and g is nondecreasing, lim,, loo /o(s)jjp exists, and equals
Cxlim - 03&)I
llfll p.
Thus IIu(s)ll is bounded if and only if lim,, m g(x) = cc. 1 Remark 2.17. In [9], we showed that the stable version of (2.16) has a unique solution, for all f E D(B) if and only if lim, _ _ cug(x) = 0. This would seem to imply some type of symmetry between (2.1) and (2.2)
Even when A has a square root, B, that generates a strongly continuous semigroup { e’B}, a 0, it is sometimes necessary to pass to the C-semigroup iCe*B),a05 for appropriate C, to apply Theorem 2.9 or 2.10. EXAMPLE
2.18. Consider the Laplace equation on an infinite cylinder, Au(x, y) =o
((x,~)~to,
a)xD)
4% Y) = 0
((x,Y)ECO,
a)xaD)
40, Y)=f(Y)
(Y ED)
lim 24(x,y) = 0 x-m
(2.19)
(Y EDI,
where D is an open subset of R” with smooth boundary aD, f EL’(D) and A is the Laplacian on R”+ I. Let A equal the Laplacian on L’(D), D(A)= W’,‘(D)n W;‘(D). Then it is well known that A generates a bounded strongly continuous holomorphic semigroup, hence -A has a square root, B, such that -B generates a bounded strongly continuous holomorphic semigroup, This semigroup is not stable, because Im(B) is not dense, thus w’BLo. we cannot apply Theorem 2.10 here. However, since { emrB}1a,, is holomorphic, (B(I- B)ple-‘B),aO is a stable B(Z- B)-‘-semigroup, and Theorem 2.10 implies the following, after writing du(x, y) = (a/ax)‘(x, y) + Au(x, Y).
558
RALPHDELAUBENFELS
PROPOSITION.
Equation (2.19) has a unique solution, VIE B(D(B3)).
LEMMA 2.20. Suppose B generates a bounded C-semigroup and A = B2. Then all solutions of (2.1) are unique if and only if ( 1.1), with G = -B, has no nonconstant bounded solutions.
be the C-semigroup generated by B. Suppose Proof: Let { w(f)},,o there existsf, a nonconstant bounded solution of (1.1) with G = -B. Since is bounded, there exists r E p,(B) (see [28]). Define, for s 3 0, i W%o v(s)=(B-r)
‘C(W(s)f(O)-Cf(s)).
Clearly v(0) = 0. To see that v is a solution of (2.1), note that U’(S)= W(s) B(B- r)-‘Cf(0) + B(B- r)-‘Cf(s), hence is continuously differentiable, since f(s) E D(B), for all s > 0, so that v is twice continuously differentiable, with U”(S) = B2 W(s)(B- r)-‘Cf(0) - B2(B- r)-‘Cf(s) = B2Ms)).
The function v is nontrivial for the following reason. If Cf(s) equalled W(s)f(O), Vs>O, then (d/ds) Cf(s) would equal B(Cf(s)), so that, since (d/ds) Cf(s) equals - B(Cf(s)), by hypothesis, it follows that Cf, and
hence, since C is injective, f, would be constant. Thus v is a nontrivial solution of (2.1), with v(0) = 0, so that the solutions of (2.1) are not unique. Conversely, suppose v is a nontrivial solution of (2.1), with x = 0. For ~20, define
Then Vs>O, f’(s)=C(B-r)-‘Cv”(s)-CB(B-r)P’Cv’(s)=C(B-r)-’ CB2u(s) - C(B-
r)-‘Cv’(s)
= -B(f(s)).
see that f is bounded, note that, if g(s)= C(B-r)-‘Cv’(s) + CB(B - r) ~ ‘Cv(s), then a similar calculation shows that g’(s) = B( g(s)), V/s> 0; this implies that g(s) = W(s) C-‘g(O), hence is bounded. Since CB(B-r)-‘C is bounded, C(B-r)-‘Cv’(s)=g(s)-CB(B-r)-‘Cv(s) is bounded, which implies that f is bounded. All that remains is to verify that f is not constant. If it were, then Vs 2 0, we would have To
( > -f-B
[C(B-r)-‘Cv(s)]=f(s)=f(O)=C(B-r)P’Cu’(0);
by variation of parameters (see [22]), this implies that C(B-r)~LCv(t)=~fW(t-s)(B-r)-‘Cv’(0)ds 0
(t 2 0).
INCOMPLETE
ITERATED CAUCHY
PROBLEMS
559
A calculation shows that (d/ds)( W(s)f(s)) = 0, Vs 3 0, thus C(B-r)-‘CW(t)u’(O) = W(t)f(O) = W(O)f(r) = Cf(0) = C(B-r)-’ C2u’(0), so that, since C and (B- r)-‘C are injective, W(t) u’(0) = Co’(O), Vt > 0. This means that C(B-r)-‘Cu(r)=~‘C(B-r)~‘Cu’(O)ds=tC(B-r)~’Cu’(O), 0
so that u(t) = W’(O). Since u is bounded, this implies that u’(O) = 0, so that v is trivial, which is a contradiction. Thus f is the desired solution of (1.1), concluding our proof. 1 Proof of Proposition 2.8. Suppose there exists f, a solution of (l.l),
as
in Lemma 2.19. Define S>O s < 0. For s,t>O, (d/dt)(W(t)u(s)-Cv(t+s))=B(W(t)v(s)-Cv(t+s)), thus W(t) V(S)-Cu(t+s)= W(t) C’[ W(0) v(s)-Cv(s)] =O, so that { W(t)),>o fails to be nowhere reversible. Conversely, suppose there exists v as in Definition 2.7. Since { W(t)}( ao is bounded, there exists r E p,(B) (see [28]). Let f(s) - (B- r)-lC2v( -s). The following calculation shows that f is the desired solution of (1.1).
= ’ B(B50
=5,$
r)-‘W(y)
v( -s) dy
W-r)-‘CW.4
=(B-r)p’CW(s)
d-s)1 4
u(-s)-(B-r)-‘CW(O)v(-s)
=(B-r)p1C2v(0)-f(s).
This concludes the proof, since f is bounded and nonconstant because u is. 1 Proof of Theorem 2.9. Suppose B generates a bounded nowhere reversible C-semigroup { W(t)}, 3 o and B2=A. Since u(t)= W(t) C-‘x is clearly a solution of (2.1), with the desired growth conditions, the first part of the theorem follows immediately from Lemma 2.19 and Proposition 2.8. For the converse, when p(A) is nonempty, standard arguments (see [lo]) show that A has a square root that generates a bounded C-semigroup. The nowhere reversibility follows from Lemma 2.19 and Proposition 2.8. 1
560
RALPH DELAUBENFELS
Proof of Corollary 2.11. Let { W(t) >,,0 be the C-semigroup generated by B. To show that { IV(Z))~~~ is nowhere reversible, let u be as in Definition 2.7. For fixed w, 01(w) = W(f) u(w - t), Vt > 0, so that I/IV(t) IJ(W- t)li = 0, since u is bounded. Thus Co(w) = 0, IIWw)ll =lim,,, so that, since C is injective, u(w) = 0, VW> 0. The corollary now follows from Theorem 2.9. 1 Proof of Corollary 2.13. To show that { IV(t)} ,>,0 is nowhere reversible, suppose f is a solution of (1.1), as in Proposition 2.8. For t >, 0, w 3 t, (d/dt)(W(t)f(w-t))=O, thus BCf(w)=BW(t)f(w-t). As in the proof of Corollary2.11, f’(w)= -Bf(w)=O, Vw30, which implies that f is constant. By Proposition 2.8, { W(t) }, a 0 is nowhere reversible, thus the corollary follows from Theorem 2.9. 1 III.
MAIN THEOREMS
Note that (2.1) and (2.2) may be written as
where A, = B, A2 = (-B), BZ= A. Theorems 2.9 and 2.10 state that C-wellposedness is equivalent to certain conditions on A, and A,; in particular, for Theorem 2.10, both A, and (-A,) must generate stable C-semigroups. Thus our main theorems, Theorems 3.4 and 3.5, may be thought of as generalizations of Theorems 2.9 and 2.10. The exact nature of the continuous dependence on initial data needs to be specified. Motivated by Lemma 3.7 (from [22]), we will make the following definitions (see Remark 3.3). DEFINITION 3.1. Suppose c is an injective operator on Xi. We will say that (1.3) or (1.4) is nicely solvable with respect to 2; if there exists a unique solution for all x E c(g{,) and there exists finite M such that
Ilu(’
< MIIA’Zi-‘xl1
for 0 < 16 n, f >, 0, where A;-j
A=
l
fi A,, /= 1
Al L 1 ..
0
0
.Aj’
INCOMPLETE DEFINITION
ITERATED CAUCHY
PROBLEMS
561
3.2. Define 4, on X’, by
Remark 3.3. It is appropriate to state here the role of the sets gj,” and the operators 4. We will write our solutions of (1.3) and (1.4) explicitly in the “C-semigroup d’Alembert form” (see [22])
u(t)= i
Wk(f)c-‘Yk,
k=l
is the C-semigroup generated by A,. C(9j,t,,) is the set where { Wk(t))raO in which y,, .... yj must lie, to guarantee that u is a solution of any permutation
where cr is a permutation of { 1, .... n}, of (1.3) or (1.4). The operator LZ$ performs the change of variables from y to x : x = ~$y. THEOREM
3.4. Suppose,for i = 1,2, Ci is a bounded, injective operator,
with (1) (2) (3) group. (4)
For 1
Then (1.3) is nicely solvable with respect to C, 4. THEOREM
3.5. Suppose,for i = 1, 2, Ci is bounded and injective, with
For 1 < k < j, A, generates a stable C,-semigroup. (2) For 1
Then (1.4) is nicely solvable with respect to C, 4.
562
RALPHDELAUBENFELS
Note that hypothesis (2) guarantees that 4. is injective. We will use the following two “C-semigroup D’Alembert formulas,” from
c221. LEMMA 3.6.
Suppose
( 1) For 1 d k d n, Ak generates a C-semigroup, { Wk(t) } f $ 0. (2) For 1O, 1
for all i, then there exists { Y,~};= 1 c Im( C) such that
u(t)= i
W,(t) c-‘Yj
j=l
LEMMA 3.7. Suppose { Ak}i=, , { Wk(t)); = 1, are as in Lemma 3.6, and nI=, p,(A,) is nonempty. Then, VXE CS$(~:,,), there exists a unique solution of (1.2), given by
u(t)=
i
Wdt) c-'Y,,
k=l
where y = d;
lx. There exists finite M such that
Ilu(’ for 0616n,
tB0,
GM(suP{lI
11 ~k~n}HA’(C4,-1~II,
wk(t)il
where
A=
Al [ 1 ..
0
0
IIZII= i
‘A,
LEMMA 3.8. Suppose f: [0, 00) -+X is n C’( [0, co), X), ( -A) generates
kzt))i) t
lIzill.
i=l
continuous, UE C([O, co), a bounded C-semigroup
f3o and
Vt,,O.
INCOMPLETE
ITERATED CAUCHY
Then for all s > 0, lim, _ o. j& W(r) either
f (r + s) dr
563
PROBLEMS
exists and equals - Cu(s), zf
(a) u is bounded and { W(I)}(>~ is uniformly stable, or (b) lim,,, u(t)=O. Proof
For s,t>O, (d/dt)(W(t)u(t+s))=
W(t)f(t+s),
W(r)tc(f+s)=Cu(s)+~’
thus
W(r)f(r+s)dr, 0
and either (a) or (b) implies that lim, _ m W(t) u(t + s) = 0. Proof
Theorem 3.4. Since all C,-semigroups are bounded, is nonempty (see [28]). Thus, by Lemma 3.7,
of
(7:= 1p,(A,)
u(t)= i:
Wk(f) c;‘y,,
k-1
where y = &,:‘x, and { Wk(t)},,O is the C,-semigroup generated by Ak, is a solution of (1.3) (see Remark 3.3), with the desired growth conditions. For uniqueness, suppose u is a solution of (1.3), with xi = 0, for 1 < i
n
(Ai-A,)”
i
((r-Ak)-1C,CZ)“2+1,
S= fi [(r-Ai)-lC,]i. i= I
For 1
1, define w,(t) = (C, C,)‘RS
fi k=j+i
We will show, by induction on i, that, for 1 6 i < n - j+ 1, there exists { Yk,i}jk= 1E gj,j such that
For i = 1, (*) follows from Lemma 3.6 applied to S- ‘w,( t) and the fact that n!=, (d/da-A,) w,(t)=O, since Im(S)c9j,j. Now suppose, for the sake of induction, that (*) holds, for i= 1. For be the C,-semigroup generated by -A,. Since u is a k>.i, let {&(t)),,~
564
RALPHDELAUBENFELS
solution of (1.3), our construction of R implies that w,, r(t) is bounded. By (*I,
thus, by Lemma 3.8(a),
Gw,+,(s)=
-y 0 S,+,(r) CIC, i
W,(r+s) y,,,dr,
k=l
Vs3 0, so that w/+,(s)
=
-
i k=l
w,(s)
,-;
Sj+,(r)
wk(dpc,~+
completing the induction, and establishing (*); note that hypothesis (4) implies that { y k,l+l)ik=1G9j,j. Letting i= n - j+ 1, (*) implies that there exists { yk}i=, s gj,, such that (CICz)“-j+
‘Ru(t) = f:
Wk(t) y,.
k=l
This is a solution of (l-2), with n=j, xi=O, for 1, 0, as desired. 1 Proof of Theorem 3.5. This is the same as the proof of Theorem 3.4,
except that Lemma 3.8(b) replaces Lemma 3.8(a). 1
IV. A HIGHER ORDER PROBLEM The spectral intuition of Theorems 3.4 and 3.5 may be applied in a straightforward way to u(n)= Bu. This factors into
where ~1”= 1. One chooses hypotheses on B so that the spectrum of B”” is contained in a sector, S, such that j rotations of S, akS, are contained in the left half-plane Re(z) < 0 and (n - j) rotations are contained in the right half-plane Re(z) 2 0. As a corollary of Theorems 3.4 and 3.5, we will prove a theorem very
INCOMPLETE ITERATED CAUCHY PROBLEMS
565
similar to Theorem 3.4 of [ 181, for (1.5) and (1.6) (Theorem 4.1), and then generalize it, by removing the requirement that 0 be in p(B) (Theorems 4.2 and 4.3). THEOREM 4.1. Under either of the following sets of hypotheses, (1.5) and (1.6) have unique solutions, for x E D(B) x D(B@ ‘jii) x . . . x II and
there exists finite M such thar
IIUWII 6 M i
(pPX,II,
i=l
for OdlQn. (a) n even, j=n/2, D(B) dense, [O, co) ~p(( - l)‘B), sup{(l + r) Il(r-(-l)jB)-‘JIIr>Oj
Ilu(’
< A4 i i=
for 061
II(rB-’ -I) B1’“xill, 1
rEp(B).
(a) n even, j= n/2, D(B) dense, (0, co) E p(( - l)‘B), B injectiue, sup{s~~(s-(-1)‘B)-‘J~~s>0}
Ilu”‘(t))ll GM i i=
for O
IJ(rB-’ -I)
B”“xJl,
1
rEp(B).
(a) n even, j=n/2, L)(B) and Im(B) dense, (0, co)~p(( sup{s~((s-(-1)‘B)-‘~~~s>0}<00. 409/168/2-19
-l)jB),
566
RALPH DELAUBENFELS
(b) n odd, j = (n + 1)/2, Im( B) dense, ( - 1)j’ ‘B generates a bounded holomorphic strongly continuous semigroup. We will need the following lemma, which may be established by numerical computation, where S, = {z 1larg(z)l < 0). LEMMA 4.4. Suppose n EN, a = e2nrln, j= n/2, when n is even, and j = (n + 1)/2, when n is odd. Define
{akS,,2n Ik=1,2, ....n}, Fn,jr
i akeniinSni2,,I k = 1, 2, .... n}, {c~~S,,~(k = 1, 2, .... n}, { txkeni’“Snln)k = 1, 2, .... n},
tf n is odd and J' is even tfn is odd and j is odd tf n is even and j is odd if n is even and j is even.
Then j members of &, j are contained in the open left half-plane Re(z) < 0, while (n - j) members are contained in the open right half-plane Re(z) > 0. Proof of Theorems 4.1, 4.2, and 4.3. The following definition will make the intuition more clear. If 0 d f3< rr, we will say that the operator A is of type 13if A is densely defined, sp(A) G VB= {z 1larg(z)l 6 0} and V$ > 0, there exists finite M, such that Vz$ Vti, I\(z-A))~II 0 if and only if A is of type (7r/2- 4). If B is as in Theorem 4.2(a), then there exists 8
WI 1. Hence, in Theorem 3.4 and/or 3.5, we may choose C, = I, C2 = B(r - B))‘. It is not hard to seethat D(B) G 9;,:,,, and that, if G = B’“, A computation then D(B) x G(D(B)) x ... x Gjpl (D(B)) s Aj(D(B)j). shows that B(s - B)) ‘,“4,:’ is bounded, for any s E p( B), so that applying Theorem 3.4 and/or 3.5 gives the desired results. 1
INCOMPLETE ITERATED CAUCHY PROBLEMS
567
V. MORE SECOND ORDER PROBLEMS We will use Section II and III to consider (1.7) and (1.8), under very general hypotheses on A and B, when A and B commute, by factoring, d 2 z +zB$+A=($-A,)($A2),
0
where A,- -B-(B*-AA)“‘, A,= -B+(B’-A)‘j2. The equation (1.8) may be considered a boundary value problem, with boundary conditions at 0 and co. By letting A equal A, the Laplacian, Theorems 5.2 and 5.8 may be applied to perturbations of the Laplace equation. THEOREM 5.1. Supposethat both A and - (B) are injective and generate bounded holomorphic strongly continuous semigroups of angle greater than 7~14that commute. Then (1.7) has a unique solution, Vx E D(B3) n D(BA). There exists finite M such that
Ilu(t
G Mllxll,
Vt > 0, x E D(B3) n D(BA).
THEOREM 5.2. SupposeA and B are as in Theorem 5.1, and either A or B has dense range. Then (1.8) has a unique solution, QXED(B3)r\ D(BA), with the wellposednessof Theorem 5.1.
Remark 5.3. Theorems 5.1 and 5.2 are best possible in the following sense. If 4 < 7~14,then there exist A, B such that Oep(A) n p(B) and A( -B) generates a uniformly bounded holomorphic strongly continuous semigroup of angle 4(7r/2) that commutes, but the solutions of (1.7) and (1.8) are not unique. (Note that an operator, G, generates a bounded holomorphic strongly continuous semigroup of angle $ if and only if, Vb < tj, G generates a uniformly bounded holomorphic strongly continuous semigroup of angle 4.) We will prove this after the proof of Theorems 5.1 and 5.2. The hypotheses on A and B, in Theorems 5.1 and 5.2 imply that the resolvents (w-A)-’ and (0-B))’ have 0(1//o/) growth rate. This may be replaced by polynomial growth. THEOREM 5.4. Suppose m, n E N, both sp( -A) and sp(B) are contained in {reUIr>O, [dl
Ilu(t GMl/B”A”xlI,
‘dt>O, XED(B’“+~A”“+~).
568
RALPH DELAUBENFELS
EXAMPLE 5.5. Theorem 5.2 may be applied to first order perturbations of the Laplace equation, with boundary conditions at 0 and co,
du(t, x)+2B
(
%(I, x) =o 1
((6 X)E co, aJ)xD)
u(t, x)=0
((1, X)E [O, co) x aD)
40, xl =f(x)
(XED)
lim u(t,x)=O I--rK
(XED),
where D is a region in R” with smooth boundary aD, CELL (1
of (1.7) and (1.8). u”(t) - 2bBu’( t) - B2u( t) = 0 (t b 0),
u(0) = x, u bounded.
(5.6)
u”(t)-2bBu’(t)-B2u(t)=O(t>0),
u(O) =x, lim u(t) = 0. (5.7) I-W
THEOREM 5.8. Suppose B generates a stable (unzformly stable) C-semigroup. Then (5.7)( (5.6)) has a unique solution, Vx E C(D(B’)). There exists finite M such that
Vxc C(D(B’)),
Ilu(t GwC-‘xll>
t 20.
When B is a positive self-adjoint operator and b > 0, the following was proposed in [2] as a mathematical model for elastic systems with structural damping (see also [3,4, 12, 271). u”(r)+2bBu’(r)+B2u(t)=O(t>0),
u(0) =x* .) u’(0) =x2.
(5.9)
An incomplete version of this is u”(t)+2bBu’(t)+
B’u(t)=O
(t>O),
14(0)=x, lim u(t)=O. 1-m
(5.10)
In Theorems 5.11 and 5.12, the value of b has a qualitative effect, taking us from the complete case (5.9) to the incomplete (5.10). THEOREM
5.11. Suppose 0 d b < 1. Let 0 s arccos(b), and suppose that
either
(a) both e-‘#B and -eieB generate stable C-semigroups, or (b) both --e- “B and eieB generate stable C-semigroups.
INCOMPLETE ITERATED CAUCHY PROBLEMS
569
Then (5.10) has a unique solution, for all x E C(D( B2)), and there exists jikite M such that Ilu(~ THEOREM
QWC-‘XII,
VXEC(D(B*)),
t20.
5.12. Suppose that B is injective and either
(a) b > 1 and - B generates a C-semigroup, or (b) 0 < b < 1 and both -e”B and -ePiOB generate C-semigroups, where 0 = arccos(b). Then (5.9) has a unique solution, for all (x,, X~)E C(D(B*)) X BC(D(B’)), of the form u(t) = W(tb,) C’y,
1)1’2,b, = b-(b*Y, Y E C(D(B*)), b, = b+(b’is the C-semigroup generated by - B.
wherex = I& ,:,I and { Wt)l,ao
+ W(tb,) C-‘y,,
1)1’2,
For 0 6 b Q 1, the spectral picture of Theorems 5.11 and 5.12 is shown in Fig. 1. In Theorem 5.11, the C-spectrum is contained in either the upper ((b)) or lower ((a)) diagonally shaded areas. In Theorem 5.12(b), when is bounded, the C-spectrum is contained in the horizontally s!hz!i ::a. 5.13. Suppose ~20, r~~,~~(B)~S,,,, {II~2-m~~--B)-‘IIIw4SIL,~} LEMMA
FIG. 1. Graph of two regions, a diagonally shaded one and a horizontally shaded one. The two lines make an angle of 0 with the y-axis. The dotted lines are the x and y axes.
570
RALPH DELAUBENFELS
is bounded, f is holomorphic on S,,, x SEJ,E,and bounded and continuous on -s*,, x s3.E3 and, if F = 0, f(z, w)(w - B)-‘(z - B)-’ is bounded as (z, w) approaches (0,O) in Se. Then
(a)
Vk>m,
(B-rJY=&Jas if
(W-N-~ t3.i
&,
E>O.
(b)
rfm=
1 and E=O, then Vk>2, 13(B-r)-i=lj;lj as3
w(w-W’
dw (W-r)k’
Cd)vxEXja,,r,ftw, w)(w-B)P’x(dw/(w-r)m+l)~D(B), with
BJ"aso.zf(
w, w)(w-B)-‘x(dw/(w-r)“+‘)
=J
axe
Prooj (a) t > 0, define
wf(w, w)(w-B)P1x(dw/(w-r)m+l).
Let f=i?(S,u
(z[ IzI < [2rl}),
g(w)=(l3rl
- B)-’dw, T(t)=&Jre-‘t&“)(w By Cauchy’s theorem, 2niT(t) = J eprgc”)(w - B)-’ as,.,
dw.
+ w)“*.
For
INCOMPLETE
ITERATED CAUCHY
571
PROBLEMS
Thus, for x in D(B), 27ciT(t)x = j
e-‘g’“‘(w-E+B-r)(w-B)~‘x 8.k
dw (w--T)
by a calculus of residues argument; continuing this argument k times gives 27ciT(t)(B-r)-‘=j
e~‘g(“‘(w-B)~‘&.
-e.c
By dominated convergence, lim, _ OT( t)( B - r) -k exists and equals jas,,, (w -B)-‘(dw/(w-r)k). Hence, it is sufficient to show vx E D( I?“).
lim T(t)x = x, r-0
The calculation follows, for x E D(P). 2ni( T(t)x - x) e -“‘“‘(w-B)-‘x-~x]dw
= ,[r
= j e p’8’.‘l(vr+B-B)(w-B)p1x-x]& F
=
Ir
e -“‘W’)(w-r+B-B)(w-B)-l(B-r)x&
= [j I-
e -Q?(w)&]
(B- r)x
+ j e~“‘“.‘(w--B)~‘(B--r)‘x~ r
(*I
572
RALPH DELAUBENFELS
Each term in the sum is divisible by t, hence goes to zero, as t does. The integral converges to zero, as t does, by dominated convergence. This establishes (*) and (a). (b) Let f be as in the proof of (a). By Cauchy’s theorem,
5
=e
dw --(W-r)k-
w(w--B)-
srw+-B)-’
= Jl s
dw (W-r)k
+B(w-S))l)&
=B .rr(w-W’
dw (W-rr)k,
by calculus of residues and the fact that B is closed,
by the same argument as in (a). (c) Let rl =as,,,, r+as,,,, where tj < $ < 0. We calculate as follows, using the residue theorem and the resolvent equation
sWh s fkW-s)Lo-B)-‘&& *Se.&
= SIr1
r2
1 AZ3 w, -----((w-B)-*-(z-B)-‘) (z - ‘.$I) dw
X(W
as desired.
dz (Z-r-y
INCOMPLETE
ITERATED CAUCHY
573
PROBLEMS
(d) For w E as,,, B(w B)-lx = (B w + w)(w B)-lx = w(w-B)‘x-x, so that Il(l/(w-r)“+‘)(B-r)(w-B)-‘xl] is O(Jwl-*), for Iw] large. Since B is closed, this implies that jas,, f(w, w)(w - B)-’ x(dw/(w - r)mfl) E D(B), with B
saye f(w w)W-‘~(~-~)~+,
dw
=
f(w, w)Cw(w-W’x-xl sa&?.,
=
sah wf(w~W)(w-B)~‘~(~_~)_+~,
cw-d;m+, dw
by the residue theorem. 1 Proof of Theorem 5.1. Let O,( 0,) be the angle of the bounded holomorphic strongly continuous semigroup generated by A( -B). ( - B*) generates a bounded holomorphic strongly continuous semigroup of angle 20, - 7c/2 (see [8] or [19]). Let (A -B)* be the generator of IA -m* i LO9 and let (B* - A)“* be the square root of (B* - A) such that :&4fi* generates a bounded holomorphic strongly continuous semigroup. Finally, let A, be the generator of {e~‘Be-“B2~A)“Z),t0. We will construct A, by defining, for t > 0, 1 _ B)- 1 Wt) = (271i)* i s e’(W-(W2+Z)‘~2)w(w asI as+ xz(z+A)-’
dw ~~ (1 +w)2
dz (1
where S, is as in Lemma5.13, n/2-@,<~
+z)2’
n/2-@,
is a uniformly stable A( 1 - A)-*B( 1 + B)-*-semi(1) 1wt)L,o group. (2) If C,EA(~-A)-~B(~ +B)-4, and -A, is defined to be the generator of ( W(t)},aO, then Im(C,) E D(A, A*), D(B) and D(A), with (&I)(
;-A,).,.,t)=((;r+2B-$+A)
c,u(t),
whenever U: [O, co) H X is twice continuously differentiable, t 2 0. (3) If x E D(B3) n D(BA), then u(t) 5 efA1xE C”([O, co), X) A
C’W, NY CWUl)n
NO, 4
CWH).
574
RALPH DELAUBENFELS
By Lemma5.13(b), W(0)=A(1-A)-2B(1+B))2. Applying Lemma 5.13(c) twice shows that W(s) W(t) = W(0) W(t+s), Vs, t 20. The continuity and uniform stability of { W(t)},ao follows from dominated convergence and the fact that (W- (w’ + z)“‘) is in the open left half-plane, when (z, w) E S,,, x S,,,. This proves (1). For (2), define, for t B 0, 1
-l(d + zpw)( w _ B) - 1
T(t) = (27ci)2 s8.Qcas, e dw xz(z+A)-1(1+W)4(1+z)3’
dz
where 4 and $ are as in the definition of W(t). By Lemma 5.13(b), T(0) T C,. Because of the rate of decay of the integrand, r(t) is twice continuously differentiable; by Lemma 5.13(d), ~(~)xED(B*)~D(A), VXEX, with
0
f
2 T(l)= (B2-A 1 T(t)>
vlt > 0.
is bounded, and - (B2 - ,4)‘j2 generates a bounded Since { T(f)},,o strongly continuous holomorphic semigroup, Corollary 2.13 (see Remark 2.14) implies that T(t) = e ~ f(B2~ A)“2C1)
Vt>O.
Thus,
1 (w2+z)l’2w(w-B)-’ =(27ci)2 sas, sasi ~~ dw
xz(z+A)r’
(I+
w)4
dz (1
+z)3’
so that, by Lemma 5.13(d), A,C, =-
1
asp as (-w-(W2+z)“2)W(W-13-1 I *
(27ci)2 s
dw xz(z+A)-’
(1+
w)4
dz (1
+z)3’
INCOMPLETE
ITERATED CAUCHY
575
PROBLEMS
Applying Lemma5.13(c) shows that, for t30, exists and equals
(-d/ds) W(s) T(t)l,,,
dw dz ____~ xM.(W--B)~1Z(Z+A)I(1+W)4(1+z)3
1 3
SO that Im(T(t)) G D(A,), with A,T(t) equal to the integral inside the brackets above. Arguing with {AZT(t)},t,,, as we did with { r(t)},ao, we find that, VXEX, A,C~X=A,T(O)XED(A,), with
A,A,C,x=-
1 iismis~(-w-(w2+z)1’2)(-w+(w2+z)1’2) (27ci)’ s I dw dz xw(w-~)~1z(z+A)-1X(1+W)q(1+z)3.
Thus we may establish (2) with the following calculation, for t 2 0.
=C,u”(t)-(A,+A,)C,u’(t)+A,A,C,u(t) =-
1 w(w-B)-‘z(z+A)-’ (27ri)’ I as,siis* x [u”(t)-(W-(W2+z)1’2-w+(W2+Z)“2)U’(t)
+(-w-(W2+z)1’2)(-W+(W2+z)1’2)~(t)l =-
ds
1 w(w-B)-‘z(z+A)-’ (2ni)’ sas,sas*
uf’(t)+2wu’(t)--zu(t)l =
dw
(1+w)4(1+z)3
K
$ 2+2B;+A >
dw dz ___~ (1 + w)” (I +z)3
1
C,u(t).
For (3), suppose x E D(B3) n D(BA). Since x E D(B(B’ -A)), it follows that x E D(B(B’ - A)‘12), so that x E D(A) n D(B2) A D(B(B2 - A)‘12) E D(Af). This implies that u(t) = .P’x is twice continuously differentiable. Since x E D(A), u(t) E D(A), Vt 3 0, and Au(t) = @IAx is continuous. Since
576 D(B3)
RALPH
n D(BA)
G D(B*)
DELAUBENFELS
n D(B(B*-A))
z D(B2)
n D(B(B2
-A)‘12)
c
D(BA , ), it follows that u’(t) = A, @Ix E D(B), V’t>/ 0, and Bu’( t) = kAIBA I x is continuous. This concludes the proofs of assertions (1 t(3). For x E D(B3) n D(M), let u(t) = eIA1x.By assertions (2) and (3) and Theorem 3.4, for t 3 0,
Since C, is injective, this implies that u(t) is a solution of (1.7). For uniqueness, suppose u is a solution of (1.7), with x = 0. By assertion (2), C, o(t) is a solution of (1.3), so by Theorem 3.4, C, v(t) = 0, so that, since C, is injective, u(t) = 0, as desired. The wellposednessfollows from the fact that {erAI} is a bounded strongly continuous semigroup. 1 Proof of Theorem 5.2. This is the same as the proof of Theorem 5.1, except that we must verify that erA’ is stable, and use Theorem 3.5 instead of 3.4. A bounded holomorphic strongly continuous semigroup {e’G},aO is stable if and only if Im(G) is dense (see [26, Chap. A-IV, Corollary 1.41). Either Im(A) or Im(B2) is dense, thus {erAe-rB2},a0 is stable, so that Im(B2 - A), and hence Im( ( B2 - A)“‘), is dense. This means that {efAIJra,,= {e-‘Be~“B2-‘A”Z}r~0 is stable, as desired. 1 ~ Proof of Remark 5.3. Let X-~*(Sn12-),, x S+i) (see Lemma 5.13 for terminology), A and - B defined as the generators of the following strongly continuous semigroups.
(efAf)(z, w) = eC’;f(z, w),
(e-IBf)(z, w) = eC’“f(z, w).
Choose E such that 0 < E< (7r/4- 4). Then 3R such that
whenever Iwl >R, 7c/4a(arg(w)l >(rr/4-s/2), and n/2-4>$>~/4+~. This implies that, if Es { (eeti, w)lIwl>R, n/4~larg(w)l>71/4--E/2, x/2-4>$>71/4+~}, then V(Z,W)EE, Z(W+(W~+Z)“‘), and hence ( - w + ( W* + z)“~) = ]zI ‘/F( w + (w* + z)i’*), is in the left half-plane. Clearly (- w - (w2 + z)“‘) is also in the left half-plane, V(z, w) E E. Thus, l,(z, W)-ee”~“.+‘“.*+=)‘l*) (u(t))(z , w) E ,1(-11’-(11~*+=)‘4 1AZ, w) is a nontrivial solution of (1.7) and (1.8) such that u(O)=O.
1
INCOMPLETE
ITERATED
CAUCHY
577
PROBLEMS
Proof of Theorem 5.4. There exists E> 0 such that {z 1IzI 6 E} 5 p(A) n p(B). For i= 1,2. t > 0, define, with the terminology of Lemma 5.13,
where fi(w, z) E w + (w’ + z)“~, f2(w, z) = --w + (w’ + z)“*. is a uniformly stable As in the proof of Theorem 5.1, { Wi( t)},,, B-“A -m-semigroup, for i = 1,2. Let A, be the generator of ( W,(t)}, a0 and (-A*) be the generator of { Wz(t)},,,. As in the proof of Theorem 5.1, it may be shown that D(B”+2A” + ‘) c D(A, A*), D(BA,) and D(AT), with ($-
A,)($= [(
f
~~))B-(n+*)~-(m+l~~(t)
)
*+2&$+/t
1
B-(“+2)A-(m+‘)U(t),
(*)
Vt > 0, whenever u: [0, co) I-+ X is twice continuously differentiable.
For XE D(B2n+2A2m+2), u(t)= W,(t) B”A”x is a solution of (1.7) and (1.8) by (*), as in the proof of Theorem 5.1. Uniqueness also follows from (*), as in Theorem 5.1. The continuous dependence on initial data follows from the fact that { W,(t)},ao is a bounded B-“A-“‘-semigroup. 1 Proof of Theorem 5.8. It is clear that, if Ai= b,B, where 6, = (b + (b2 + 1)“2) and b, = (b - (b* + l)“‘), then, Vt z 0, (-$A,)($-A,)u(t)=z/(t)-2bBu’(t)-B’u(t), whenever u E C( [0, co), [D(B*)]) n C’( [0, co), [D(B)]) n C’( [0, a~), X). Hence, since D(A:)nD(A,A2)=D(B2), and the solutions of (d/dt- A,) (d/dt-A,)u(t)=O are given by u(t)= W(tb2)Cp’x, where {(W(t)},2o is the C-semigroup generated by B, the result follows from Theorems 3.4 and 3.5. 1 Proof of Theorems 5.11 and 5.12. Let Ai-b,B, where b, s -b+ (b* - l)“*, b, c -b - (b* - l)“*. Then, for u as in the proof of Theorem 5.8, (-$-A,)(;-A,).(t)=.“(t)+2bB..(t)+B’.(r),
Vt>O.
A computation shows that, under hypothesis (a) of Theorem 5.11, both (-A,) and A, generate stable C-semigroups, under hypothesis (b) of
578
RALPH DELAUBENFELS
Theorem 5.1I, both A, and ( --A*) generate stable C-semigroups, while in Theorem 5.12, both A, and A, generate C-semigroups. Hence, Theorem 5.11 follows from Theorem 3.5, and Theorem 5.12 follows from Lemma 3.7. 1
REFERENCES
I. A. V. BALAKRISHNAN,Fractional powers of closed operators and the semigroups generated by them, Pacific J. Math. 10 (1960), 419437. 2. G. CHEN AND D. L. RUSSELL,A mathematical model for linear elastic systems with structural damping. Quart. Appl. Muk (1982), 433454. 3. S. CHENAND R. TRIGGIANI,Proof of extensions of two conjectures on structural damping for elastic systems, Pacific J. Math. 136 (1989), 15-55. 4. S. CHEN AND R. TRIGGIANI,Differentiable semigroups arising from elastic systems with gentle dissipation: The case 0
INCOMPLETE ITERATED CAUCHY PROBLEMS
579
AND N. TANAKA, Exponentially bounded C-semigroups and integrated semigroups, Tokyo J. Math. 12, No. 1 (1989), 99-115. 26. R. NAGEL (Ed.), “One-Parameter Semigroups of Positive Operators,” Lecture Notes in Math., Vol. 1184, Springer-Verlag, New York/Berlin, 1986. 27. F. NEUBRANDER, Integrated semigroups and their application to the abstract Cauchy problem, Pacific J. Mafh. 135 (1988) 11I-155. 28. F. NEUBRANDER AND R. DELAUBENFELS, Laplace transform and regularity of C-semigroups, in preparation. 29. M. M. PANG, Resolvent estimates for Schrodinger operators in LP(R”) and the theory of exponentially bounded C-groups, preprint, 1989. 30. A. PAZY, “Semigroups of Linear Operators and Applications to Partial Differential Equations,” Springer, New York, 1983. 31. J. T. SANDEFUR, Higher order abstract Cauchy problems, .I. Mafh. Anal. Appl. 60 (1977),
25. 1. MIYADERA
728-742. 32. B. STRAUB,
“Uber Generatoren von C-Halbgruppen und Cauchyprobleme zweiter Ordnung,” Diploma&it, Tiibingen, 1989. 33. N. TANAKA, On the exponentially bounded C-semigroups, Tokyo J. Math. 10, No. 1 (1987) 107-117. 34. J. A. VAN CASTEREN,Generators of strongly continuous semigroups, in “Research Notes in Math.,” Vol. 115, Pitman, New York, 1985.
OF MATHEMATICAL
ANALYSIS
incomplete
AND
APPLICATIONS
Iterated
552-579 (1992)
168,
Cauchy
Problems
RALPH DELAUBENFELS* Department
of Mathematics, Ohio University, Athens, Ohio 45701 Submitted by C. Foias Received September 18, 1990
We treat numerous higher order incomplete abstract Cauchy problems by writing them in the form (1)
FL,
(dldt-A,)u(t)=O
(t>O),
(21 u“~“(O)=x,(l
u“-‘l(t)bounded(l
The abstract Cauchy problem (l), (2), and (3a) is wellposed, in a certain sense, when A,, .... A, generate bounded C-semigroups and (-A,+ ,), .... ( -A,) generate uniformly stable C-semigroups. For (l), (Z), and (3b), we need A,, _._,A, and 0 1992 Academic Press, Inc. ( -A, + ,), .... (-A,) to generate stable C-semigroups.
I. INTRODUCTION The relationship between strongly abstract Cauchy problem
continuous
semigroups
u(0, x) = x,
~u(t,x)=G(u(t,x))(t~O)
and the
(1.1)
is well known, see, for example, [6, 16, 20, 26, 30, 341. Perhaps the most unified treatment of higher order complete abstract Cauchy problems appears in [31] (see also [21,22]), where the iterated abstract Cauchy problem, u(t)=0
u +l)(o)=xi
(t&O),
(1
(1.2)
is introduced, and is shown to be wellposed, in a certain sense, if and only if, for each k, A, generates a strongly continuous semigroup. Writing a higher order problem in the iterated form (1.2) decomposes it into first order problems. * Research supported by an Ohio University
Research Grant
552 0022-241X/92 $5.00 Copyright 0 1992 by Academic Press, Inc. All rights of reproduction m any form reserved.
INCOMPLETE
ITERATED CAUCHY
553
PROBLEMS
In this paper, we give a unified treatment of incomplete higher order problems via the incomplete iterated abstract Cauchy problems fJ(-$-Ak)u(r)=O
(t>,O), &‘)bounded
tP1)(0)=x,
(l
(1
(1.3)
and fi, (:-A,).(t)=0
zP’)(0)=xi
(t>O), ]im u(i-l) (t)=O t-02
(1
(l
(1.4)
where A,, A,, .... A, are commuting linear operators. For (1.3) to be wellposed, it is suflicient that A,, .... Aj generate bounded strongly continuous semigroups and ( - Ai+ 1), .... ( - A,) generate uniformly stable strongly continuous semigroups. For (1.4) to be wellposed, it is sufficient that A,, .... Aj and (- Aj+ 1), .... (-A,,) generate stable strongly continuous semigroups. Special cases,after an appropriate factorization of (1.3) or (1.4), are u’“‘(t) = B(u(t))
(t b O),
u(‘-“bounded
zP1)(0)=xi
(1
(1 di
(1.5)
and u’“‘(t) = B(u(t))
(t 3 O),
u(‘-‘)(0)=xi
,‘t; u(‘-‘j(t)=0
(1 Giij), (l
(1.6)
and general second order abstract Cauchy problems, u”(t) + 2B(u’(t)) + A(u(t)) =O,
u(O) = x, u bounded,
(1.7)
u(O) =x, lim u(t) = 0. ,+CC
(1.8)
and u”(t) + 2B(u’(t)) + A(u(t)) = 0,
Factoring higher order problems into the iterated form gives them a sort of spectral intuition. The intuition, in (1.2), is that each A, must lie, in some sense, in a left half-plane. The intuition in (1.3) is that A,, ..,, Ai should lie in the left half-plane Re(z) < 0 and Ai+, , .... A, should lie in a closed subset of Re(z) > 0. In (1.4), both A,, .... Ai and (- Aj+ ,), .... (-A,) should lie in the open half-plane Re(z) < 0. These are exactly the results one would obtain if A,, .... A, were complex numbers.
5.54
RALPH DELAUBENFELS
We also generalize our results by weakening the hypotheses on Ak or (- Ak), requiring merely that they generate what are called C-semigroups (Definition 2.3) rather than strongly continuous semigroups. This is a generalization of strongly continuous semigroups that has received much attention recently [S, 7, l&13, 22225, 28, 29, 32, 331). Among other things, this allows us to consider A, B, A, whose resolvents are polynomially bounded, rather than 0(1/w), as with generators of strongly continuous semigroups. Equation (1.5), with n=2 and j= 1, was considered in [l, 151; for the same n andj, (1.6) is in [15, 91. In [17, 181, (1.5) is treated (see Section IV for generalizations of the results in [ 181). Some examples, including incomplete versions of the wave equation and first order perturbations of Laplace’s equation, may be found in 2.15, 2.18, 5.5, 5.9, and 5.10. Here is a rough table of contents. In Section II, we give necessary and sufficient conditions for (1.5) and (1.6), with n = 2 and j= 1, to have a unique solution, for all x E { Cx 1x E the domain of A}. Section III contains the basic results about (1.3) and (1.4). Section IV treats (1.5) and (1.6), and Section V deals with (1.7) and (1.8). All operators are linear, on a Banach space, X. We will write L(X) for the space of all bounded operators from X into itself. We will write D(A) for the domain of the operator A, Im(A) for the image, p(A) for the resolvent set, sp(A) for the spectrum. By a solution of (1.2), (1.3), or (1.4) we mean u such that
and u satisfies (l-2), (1.3), or (1.4). A solution of (1.5) or (1.6) is in C”(R+,X)nC(R+,D(A)) and a solution of (1.7) or (1.8) is in C(R+, D(A)) n C’(R+, D(B)) n C*(R+, X). We will write x for (XI 1“‘9 x,). II. SECOND ORDER INCOMPLETE CAUCHY PROBLEMS AND C-SEMIGROUPS We will give necessary and sufficient conditions for the following special casesof (1.5) through (1.8) to have a unique solution, for all x in C(D(A)). u”(t)=A(u(t))
(t20),
u”(t)=A(u(t))
lim u(t)=O= t-cc
u(0) =x, (t>O),
u bounded.
(2.1)
u(0) = x,
lim u’(t)= lim u”(t). I-K r-m
(2.2)
INCOMPLETE ITERATED CAUCHY PROBLEMS
555
Note that (2.2) may be considered a boundary value problem, with boundary conditions at zero and infinity. DEFINITION 2.3. Suppose C is a bounded, injective operator. The strongly continuous family of bounded operators { W(t)},,o is a C-semigroup if W(0) = C, and W(t) W(s) = C W(t + s), for all s, t B 0. The generator of { W(t)},, 0 is defined by
Ax=C-’
(
!eOf(W(t)x-x)
1
,
D(A) f (x 1limit exists and is in Im(C)} In this paper, C will always be a bounded, injective operator. Basic properties of C-semigroups may be found in [lo]. The generator of a C-semigroup is closed and its domain is left invariant by the C-semigroup. Even when the C-semigroup is exponentially bounded, its generator may have empty resolvent set. However, its C-resoluent set will contain a halfplane. 2.4. The complex number r is in p,(A), the C-resoluent set is injective, and Im(C)c Im(r-A). When A generates an exponentially bounded C-semigroup, { W(t)}lGo, then (r-A)-‘C exists, for Re(r) sufficiently large and equals the Laplace transform of { W(t)},,o (see [28]). DEFINITION
of A, if (r-A)
DEFINITION 2.5. We will call the C-semigroup stable if lim,, m W(t)x = 0, for all x E X. We will call it uniformly stable if lim, _ oD11 W(t)ll = 0. PROPOSITION 2.6 (from [lo] ). SupposeA generatesa C-semigroup. Then (1.1) has a unique solution, for all x E C(L)(A)). If p(A) is nonempty, then the converseis also true. There then exists finite A4 such that IIu(t) I(< M I(C- ‘x/l, for all t 2 0,
x E C(D(A)). DEFINITION
2.7. We will say that the bounded C-semigroup { W(t)},>o
is a bounded nowhere reversible C-semigroup if there exists no nonconstant
bounded continuous v: R -+ X such that W(t) u(s) = Cv(t + s), for all nonnegative t, real s. PROPOSITION 2.8. Suppose { W(t)},ao is a bounded C-semigroup generated by B. Then ( W(t)},>,, is nowhere reversible if and only if there exist no nonconstant solutions of (l.l), with G E (-B).
556
RALPHDELAUBENFELS
THEOREM 2.9. Suppose A has a square root that generates a bounded nowhere reversible C-semigroup. Then (2.1) has a unique solution, for all x E C(D(A)). If p(A) is nonempty, then the converse is also true. There then exists finite M such that jlu(t)(l < M 11 C ‘x,11,for all t > 0, x E C(D(A)).
Exactly the same arguments as in [9] show the following. THEOREM 2.10. Suppose A has a square root that generates a stable C-semigroup. Then (2.2) has a unique solution, for all x E C(D(A)). If p(A) is nonempty, then the converse is also true. We then have the same continuous dependence on initial data as in Theorem 2.9. COROLLARY 2.11. Suppose A = B2, where B generates a uniformly stable C-semigroup. Then (2.1) and (2.2) have unique solutions, for all x E:C( D(A)).
Remark 2.12. The proof of Corollary 2.11 shows that a uniformly
stable C-semigroup is nowhere reversible. A stable C-semigroup may not be nowhere reversible. If W(t)f(x)=f’(x+ t), on L’([O, oo)), then is a stable strongly continuous semigroup that is not nowhere { wm>, reversible (see Example 2.15 and Remark 2.17). The following corollary of Theorem 2.9, when C = Z, appeared in [ 151. Suppose A = B2, where B generates a C-semigroup such that BW(t)EL(X), for t large, and lim,,,I(BW(t)(l =O. { WOl,,o Then (2.1) has a unique solution, for all XE C(D(A)). COROLLARY 2.13.
Remark 2.14. As pointed out in [15],.the results of [l] regarding (2.1) are contained in Corollary 2.13, with C = I. More generally, Corollary 2.13 includes the case when ( W(t)),,o is a bounded holomorphic C-semigroup (see [ 143). EXAMPLE 2.15. Suppose 1 < p < co and g is a nondecreasing nonnegative function on R. Consider the following incomplete wave equation
(-$,t,.,=(&kx)
40, x) =f(xh Let B= dldx, (W(t))f(x)
=f(x
(tbO,xER), sup
lu(x, t)l P g(x) dxl t a 0 < ~0.
(2.16)
I
the generator of the strongly continuous semigroup + t), on X= LP(R, g dx).
557
INCOMPLETE ITERATED CAUCHY PROBLEMS
Equation (2.16) has a unique solution, for all and only if lim, _ m g(x) = co. PROPOSITION.
f ED(B)
if
Proof. If v as in Definition 2.7 existed, then it is not hard to see that it would have the form u(s) = f (s + x), for somef E A’. Thus, by Theorem 2.9, (2.16) has a unique solution, Vf E D(B) if and only if v(s), as defined above, is unbounded, for all f E X. Since
and g is nondecreasing, lim,, loo /o(s)jjp exists, and equals
Cxlim - 03&)I
llfll p.
Thus IIu(s)ll is bounded if and only if lim,, m g(x) = cc. 1 Remark 2.17. In [9], we showed that the stable version of (2.16) has a unique solution, for all f E D(B) if and only if lim, _ _ cug(x) = 0. This would seem to imply some type of symmetry between (2.1) and (2.2)
Even when A has a square root, B, that generates a strongly continuous semigroup { e’B}, a 0, it is sometimes necessary to pass to the C-semigroup iCe*B),a05 for appropriate C, to apply Theorem 2.9 or 2.10. EXAMPLE
2.18. Consider the Laplace equation on an infinite cylinder, Au(x, y) =o
((x,~)~to,
a)xD)
4% Y) = 0
((x,Y)ECO,
a)xaD)
40, Y)=f(Y)
(Y ED)
lim 24(x,y) = 0 x-m
(2.19)
(Y EDI,
where D is an open subset of R” with smooth boundary aD, f EL’(D) and A is the Laplacian on R”+ I. Let A equal the Laplacian on L’(D), D(A)= W’,‘(D)n W;‘(D). Then it is well known that A generates a bounded strongly continuous holomorphic semigroup, hence -A has a square root, B, such that -B generates a bounded strongly continuous holomorphic semigroup, This semigroup is not stable, because Im(B) is not dense, thus w’BLo. we cannot apply Theorem 2.10 here. However, since { emrB}1a,, is holomorphic, (B(I- B)ple-‘B),aO is a stable B(Z- B)-‘-semigroup, and Theorem 2.10 implies the following, after writing du(x, y) = (a/ax)‘(x, y) + Au(x, Y).
558
RALPHDELAUBENFELS
PROPOSITION.
Equation (2.19) has a unique solution, VIE B(D(B3)).
LEMMA 2.20. Suppose B generates a bounded C-semigroup and A = B2. Then all solutions of (2.1) are unique if and only if ( 1.1), with G = -B, has no nonconstant bounded solutions.
be the C-semigroup generated by B. Suppose Proof: Let { w(f)},,o there existsf, a nonconstant bounded solution of (1.1) with G = -B. Since is bounded, there exists r E p,(B) (see [28]). Define, for s 3 0, i W%o v(s)=(B-r)
‘C(W(s)f(O)-Cf(s)).
Clearly v(0) = 0. To see that v is a solution of (2.1), note that U’(S)= W(s) B(B- r)-‘Cf(0) + B(B- r)-‘Cf(s), hence is continuously differentiable, since f(s) E D(B), for all s > 0, so that v is twice continuously differentiable, with U”(S) = B2 W(s)(B- r)-‘Cf(0) - B2(B- r)-‘Cf(s) = B2Ms)).
The function v is nontrivial for the following reason. If Cf(s) equalled W(s)f(O), Vs>O, then (d/ds) Cf(s) would equal B(Cf(s)), so that, since (d/ds) Cf(s) equals - B(Cf(s)), by hypothesis, it follows that Cf, and
hence, since C is injective, f, would be constant. Thus v is a nontrivial solution of (2.1), with v(0) = 0, so that the solutions of (2.1) are not unique. Conversely, suppose v is a nontrivial solution of (2.1), with x = 0. For ~20, define
Then Vs>O, f’(s)=C(B-r)-‘Cv”(s)-CB(B-r)P’Cv’(s)=C(B-r)-’ CB2u(s) - C(B-
r)-‘Cv’(s)
= -B(f(s)).
see that f is bounded, note that, if g(s)= C(B-r)-‘Cv’(s) + CB(B - r) ~ ‘Cv(s), then a similar calculation shows that g’(s) = B( g(s)), V/s> 0; this implies that g(s) = W(s) C-‘g(O), hence is bounded. Since CB(B-r)-‘C is bounded, C(B-r)-‘Cv’(s)=g(s)-CB(B-r)-‘Cv(s) is bounded, which implies that f is bounded. All that remains is to verify that f is not constant. If it were, then Vs 2 0, we would have To
( > -f-B
[C(B-r)-‘Cv(s)]=f(s)=f(O)=C(B-r)P’Cu’(0);
by variation of parameters (see [22]), this implies that C(B-r)~LCv(t)=~fW(t-s)(B-r)-‘Cv’(0)ds 0
(t 2 0).
INCOMPLETE
ITERATED CAUCHY
PROBLEMS
559
A calculation shows that (d/ds)( W(s)f(s)) = 0, Vs 3 0, thus C(B-r)-‘CW(t)u’(O) = W(t)f(O) = W(O)f(r) = Cf(0) = C(B-r)-’ C2u’(0), so that, since C and (B- r)-‘C are injective, W(t) u’(0) = Co’(O), Vt > 0. This means that C(B-r)-‘Cu(r)=~‘C(B-r)~‘Cu’(O)ds=tC(B-r)~’Cu’(O), 0
so that u(t) = W’(O). Since u is bounded, this implies that u’(O) = 0, so that v is trivial, which is a contradiction. Thus f is the desired solution of (1.1), concluding our proof. 1 Proof of Proposition 2.8. Suppose there exists f, a solution of (l.l),
as
in Lemma 2.19. Define S>O s < 0. For s,t>O, (d/dt)(W(t)u(s)-Cv(t+s))=B(W(t)v(s)-Cv(t+s)), thus W(t) V(S)-Cu(t+s)= W(t) C’[ W(0) v(s)-Cv(s)] =O, so that { W(t)),>o fails to be nowhere reversible. Conversely, suppose there exists v as in Definition 2.7. Since { W(t)}( ao is bounded, there exists r E p,(B) (see [28]). Let f(s) - (B- r)-lC2v( -s). The following calculation shows that f is the desired solution of (1.1).
= ’ B(B50
=5,$
r)-‘W(y)
v( -s) dy
W-r)-‘CW.4
=(B-r)p’CW(s)
d-s)1 4
u(-s)-(B-r)-‘CW(O)v(-s)
=(B-r)p1C2v(0)-f(s).
This concludes the proof, since f is bounded and nonconstant because u is. 1 Proof of Theorem 2.9. Suppose B generates a bounded nowhere reversible C-semigroup { W(t)}, 3 o and B2=A. Since u(t)= W(t) C-‘x is clearly a solution of (2.1), with the desired growth conditions, the first part of the theorem follows immediately from Lemma 2.19 and Proposition 2.8. For the converse, when p(A) is nonempty, standard arguments (see [lo]) show that A has a square root that generates a bounded C-semigroup. The nowhere reversibility follows from Lemma 2.19 and Proposition 2.8. 1
560
RALPH DELAUBENFELS
Proof of Corollary 2.11. Let { W(t) >,,0 be the C-semigroup generated by B. To show that { IV(Z))~~~ is nowhere reversible, let u be as in Definition 2.7. For fixed w, 01(w) = W(f) u(w - t), Vt > 0, so that I/IV(t) IJ(W- t)li = 0, since u is bounded. Thus Co(w) = 0, IIWw)ll =lim,,, so that, since C is injective, u(w) = 0, VW> 0. The corollary now follows from Theorem 2.9. 1 Proof of Corollary 2.13. To show that { IV(t)} ,>,0 is nowhere reversible, suppose f is a solution of (1.1), as in Proposition 2.8. For t >, 0, w 3 t, (d/dt)(W(t)f(w-t))=O, thus BCf(w)=BW(t)f(w-t). As in the proof of Corollary2.11, f’(w)= -Bf(w)=O, Vw30, which implies that f is constant. By Proposition 2.8, { W(t) }, a 0 is nowhere reversible, thus the corollary follows from Theorem 2.9. 1 III.
MAIN THEOREMS
Note that (2.1) and (2.2) may be written as
where A, = B, A2 = (-B), BZ= A. Theorems 2.9 and 2.10 state that C-wellposedness is equivalent to certain conditions on A, and A,; in particular, for Theorem 2.10, both A, and (-A,) must generate stable C-semigroups. Thus our main theorems, Theorems 3.4 and 3.5, may be thought of as generalizations of Theorems 2.9 and 2.10. The exact nature of the continuous dependence on initial data needs to be specified. Motivated by Lemma 3.7 (from [22]), we will make the following definitions (see Remark 3.3). DEFINITION 3.1. Suppose c is an injective operator on Xi. We will say that (1.3) or (1.4) is nicely solvable with respect to 2; if there exists a unique solution for all x E c(g{,) and there exists finite M such that
Ilu(’
< MIIA’Zi-‘xl1
for 0 < 16 n, f >, 0, where A;-j
A=
l
fi A,, /= 1
Al L 1 ..
0
0
.Aj’
INCOMPLETE DEFINITION
ITERATED CAUCHY
PROBLEMS
561
3.2. Define 4, on X’, by
Remark 3.3. It is appropriate to state here the role of the sets gj,” and the operators 4. We will write our solutions of (1.3) and (1.4) explicitly in the “C-semigroup d’Alembert form” (see [22])
u(t)= i
Wk(f)c-‘Yk,
k=l
is the C-semigroup generated by A,. C(9j,t,,) is the set where { Wk(t))raO in which y,, .... yj must lie, to guarantee that u is a solution of any permutation
where cr is a permutation of { 1, .... n}, of (1.3) or (1.4). The operator LZ$ performs the change of variables from y to x : x = ~$y. THEOREM
3.4. Suppose,for i = 1,2, Ci is a bounded, injective operator,
with (1) (2) (3) group. (4)
For 1
Then (1.3) is nicely solvable with respect to C, 4. THEOREM
3.5. Suppose,for i = 1, 2, Ci is bounded and injective, with
For 1 < k < j, A, generates a stable C,-semigroup. (2) For 1
Then (1.4) is nicely solvable with respect to C, 4.
562
RALPHDELAUBENFELS
Note that hypothesis (2) guarantees that 4. is injective. We will use the following two “C-semigroup D’Alembert formulas,” from
c221. LEMMA 3.6.
Suppose
( 1) For 1 d k d n, Ak generates a C-semigroup, { Wk(t) } f $ 0. (2) For 1
for all i, then there exists { Y,~};= 1 c Im( C) such that
u(t)= i
W,(t) c-‘Yj
j=l
LEMMA 3.7. Suppose { Ak}i=, , { Wk(t)); = 1, are as in Lemma 3.6, and nI=, p,(A,) is nonempty. Then, VXE CS$(~:,,), there exists a unique solution of (1.2), given by
u(t)=
i
Wdt) c-'Y,,
k=l
where y = d;
lx. There exists finite M such that
Ilu(’ for 0616n,
tB0,
GM(suP{lI
11 ~k~n}HA’(C4,-1~II,
wk(t)il
where
A=
Al [ 1 ..
0
0
IIZII= i
‘A,
LEMMA 3.8. Suppose f: [0, 00) -+X is n C’( [0, co), X), ( -A) generates
kzt))i) t
lIzill.
i=l
continuous, UE C([O, co), a bounded C-semigroup
f3o and
Vt,,O.
INCOMPLETE
ITERATED CAUCHY
Then for all s > 0, lim, _ o. j& W(r) either
f (r + s) dr
563
PROBLEMS
exists and equals - Cu(s), zf
(a) u is bounded and { W(I)}(>~ is uniformly stable, or (b) lim,,, u(t)=O. Proof
For s,t>O, (d/dt)(W(t)u(t+s))=
W(t)f(t+s),
W(r)tc(f+s)=Cu(s)+~’
thus
W(r)f(r+s)dr, 0
and either (a) or (b) implies that lim, _ m W(t) u(t + s) = 0. Proof
Theorem 3.4. Since all C,-semigroups are bounded, is nonempty (see [28]). Thus, by Lemma 3.7,
of
(7:= 1p,(A,)
u(t)= i:
Wk(f) c;‘y,,
k-1
where y = &,:‘x, and { Wk(t)},,O is the C,-semigroup generated by Ak, is a solution of (1.3) (see Remark 3.3), with the desired growth conditions. For uniqueness, suppose u is a solution of (1.3), with xi = 0, for 1 < i
n
(Ai-A,)”
i
((r-Ak)-1C,CZ)“2+1,
S= fi [(r-Ai)-lC,]i. i= I
For 1
1, define w,(t) = (C, C,)‘RS
fi k=j+i
We will show, by induction on i, that, for 1 6 i < n - j+ 1, there exists { Yk,i}jk= 1E gj,j such that
For i = 1, (*) follows from Lemma 3.6 applied to S- ‘w,( t) and the fact that n!=, (d/da-A,) w,(t)=O, since Im(S)c9j,j. Now suppose, for the sake of induction, that (*) holds, for i= 1. For be the C,-semigroup generated by -A,. Since u is a k>.i, let {&(t)),,~
564
RALPHDELAUBENFELS
solution of (1.3), our construction of R implies that w,, r(t) is bounded. By (*I,
thus, by Lemma 3.8(a),
Gw,+,(s)=
-y 0 S,+,(r) CIC, i
W,(r+s) y,,,dr,
k=l
Vs3 0, so that w/+,(s)
=
-
i k=l
w,(s)
,-;
Sj+,(r)
wk(dpc,~+
completing the induction, and establishing (*); note that hypothesis (4) implies that { y k,l+l)ik=1G9j,j. Letting i= n - j+ 1, (*) implies that there exists { yk}i=, s gj,, such that (CICz)“-j+
‘Ru(t) = f:
Wk(t) y,.
k=l
This is a solution of (l-2), with n=j, xi=O, for 1
except that Lemma 3.8(b) replaces Lemma 3.8(a). 1
IV. A HIGHER ORDER PROBLEM The spectral intuition of Theorems 3.4 and 3.5 may be applied in a straightforward way to u(n)= Bu. This factors into
where ~1”= 1. One chooses hypotheses on B so that the spectrum of B”” is contained in a sector, S, such that j rotations of S, akS, are contained in the left half-plane Re(z) < 0 and (n - j) rotations are contained in the right half-plane Re(z) 2 0. As a corollary of Theorems 3.4 and 3.5, we will prove a theorem very
INCOMPLETE ITERATED CAUCHY PROBLEMS
565
similar to Theorem 3.4 of [ 181, for (1.5) and (1.6) (Theorem 4.1), and then generalize it, by removing the requirement that 0 be in p(B) (Theorems 4.2 and 4.3). THEOREM 4.1. Under either of the following sets of hypotheses, (1.5) and (1.6) have unique solutions, for x E D(B) x D(B@ ‘jii) x . . . x II and
there exists finite M such thar
IIUWII 6 M i
(pPX,II,
i=l
for OdlQn. (a) n even, j=n/2, D(B) dense, [O, co) ~p(( - l)‘B), sup{(l + r) Il(r-(-l)jB)-‘JIIr>Oj
Ilu(’
< A4 i i=
for 061
II(rB-’ -I) B1’“xill, 1
rEp(B).
(a) n even, j= n/2, D(B) dense, (0, co) E p(( - l)‘B), B injectiue, sup{s~~(s-(-1)‘B)-‘J~~s>0}
Ilu”‘(t))ll GM i i=
for O
IJ(rB-’ -I)
B”“xJl,
1
rEp(B).
(a) n even, j=n/2, L)(B) and Im(B) dense, (0, co)~p(( sup{s~((s-(-1)‘B)-‘~~~s>0}<00. 409/168/2-19
-l)jB),
566
RALPH DELAUBENFELS
(b) n odd, j = (n + 1)/2, Im( B) dense, ( - 1)j’ ‘B generates a bounded holomorphic strongly continuous semigroup. We will need the following lemma, which may be established by numerical computation, where S, = {z 1larg(z)l < 0). LEMMA 4.4. Suppose n EN, a = e2nrln, j= n/2, when n is even, and j = (n + 1)/2, when n is odd. Define
{akS,,2n Ik=1,2, ....n}, Fn,jr
i akeniinSni2,,I k = 1, 2, .... n}, {c~~S,,~(k = 1, 2, .... n}, { txkeni’“Snln)k = 1, 2, .... n},
tf n is odd and J' is even tfn is odd and j is odd tf n is even and j is odd if n is even and j is even.
Then j members of &, j are contained in the open left half-plane Re(z) < 0, while (n - j) members are contained in the open right half-plane Re(z) > 0. Proof of Theorems 4.1, 4.2, and 4.3. The following definition will make the intuition more clear. If 0 d f3< rr, we will say that the operator A is of type 13if A is densely defined, sp(A) G VB= {z 1larg(z)l 6 0} and V$ > 0, there exists finite M, such that Vz$ Vti, I\(z-A))~II
WI 1. Hence, in Theorem 3.4 and/or 3.5, we may choose C, = I, C2 = B(r - B))‘. It is not hard to seethat D(B) G 9;,:,,, and that, if G = B’“, A computation then D(B) x G(D(B)) x ... x Gjpl (D(B)) s Aj(D(B)j). shows that B(s - B)) ‘,“4,:’ is bounded, for any s E p( B), so that applying Theorem 3.4 and/or 3.5 gives the desired results. 1
INCOMPLETE ITERATED CAUCHY PROBLEMS
567
V. MORE SECOND ORDER PROBLEMS We will use Section II and III to consider (1.7) and (1.8), under very general hypotheses on A and B, when A and B commute, by factoring, d 2 z +zB$+A=($-A,)($A2),
0
where A,- -B-(B*-AA)“‘, A,= -B+(B’-A)‘j2. The equation (1.8) may be considered a boundary value problem, with boundary conditions at 0 and co. By letting A equal A, the Laplacian, Theorems 5.2 and 5.8 may be applied to perturbations of the Laplace equation. THEOREM 5.1. Supposethat both A and - (B) are injective and generate bounded holomorphic strongly continuous semigroups of angle greater than 7~14that commute. Then (1.7) has a unique solution, Vx E D(B3) n D(BA). There exists finite M such that
Ilu(t
G Mllxll,
Vt > 0, x E D(B3) n D(BA).
THEOREM 5.2. SupposeA and B are as in Theorem 5.1, and either A or B has dense range. Then (1.8) has a unique solution, QXED(B3)r\ D(BA), with the wellposednessof Theorem 5.1.
Remark 5.3. Theorems 5.1 and 5.2 are best possible in the following sense. If 4 < 7~14,then there exist A, B such that Oep(A) n p(B) and A( -B) generates a uniformly bounded holomorphic strongly continuous semigroup of angle 4(7r/2) that commutes, but the solutions of (1.7) and (1.8) are not unique. (Note that an operator, G, generates a bounded holomorphic strongly continuous semigroup of angle $ if and only if, Vb < tj, G generates a uniformly bounded holomorphic strongly continuous semigroup of angle 4.) We will prove this after the proof of Theorems 5.1 and 5.2. The hypotheses on A and B, in Theorems 5.1 and 5.2 imply that the resolvents (w-A)-’ and (0-B))’ have 0(1//o/) growth rate. This may be replaced by polynomial growth. THEOREM 5.4. Suppose m, n E N, both sp( -A) and sp(B) are contained in {reUIr>O, [dl
Ilu(t GMl/B”A”xlI,
‘dt>O, XED(B’“+~A”“+~).
568
RALPH DELAUBENFELS
EXAMPLE 5.5. Theorem 5.2 may be applied to first order perturbations of the Laplace equation, with boundary conditions at 0 and co,
du(t, x)+2B
(
%(I, x) =o 1
((6 X)E co, aJ)xD)
u(t, x)=0
((1, X)E [O, co) x aD)
40, xl =f(x)
(XED)
lim u(t,x)=O I--rK
(XED),
where D is a region in R” with smooth boundary aD, CELL (1
of (1.7) and (1.8). u”(t) - 2bBu’( t) - B2u( t) = 0 (t b 0),
u(0) = x, u bounded.
(5.6)
u”(t)-2bBu’(t)-B2u(t)=O(t>0),
u(O) =x, lim u(t) = 0. (5.7) I-W
THEOREM 5.8. Suppose B generates a stable (unzformly stable) C-semigroup. Then (5.7)( (5.6)) has a unique solution, Vx E C(D(B’)). There exists finite M such that
Vxc C(D(B’)),
Ilu(t GwC-‘xll>
t 20.
When B is a positive self-adjoint operator and b > 0, the following was proposed in [2] as a mathematical model for elastic systems with structural damping (see also [3,4, 12, 271). u”(r)+2bBu’(r)+B2u(t)=O(t>0),
u(0) =x* .) u’(0) =x2.
(5.9)
An incomplete version of this is u”(t)+2bBu’(t)+
B’u(t)=O
(t>O),
14(0)=x, lim u(t)=O. 1-m
(5.10)
In Theorems 5.11 and 5.12, the value of b has a qualitative effect, taking us from the complete case (5.9) to the incomplete (5.10). THEOREM
5.11. Suppose 0 d b < 1. Let 0 s arccos(b), and suppose that
either
(a) both e-‘#B and -eieB generate stable C-semigroups, or (b) both --e- “B and eieB generate stable C-semigroups.
INCOMPLETE ITERATED CAUCHY PROBLEMS
569
Then (5.10) has a unique solution, for all x E C(D( B2)), and there exists jikite M such that Ilu(~ THEOREM
QWC-‘XII,
VXEC(D(B*)),
t20.
5.12. Suppose that B is injective and either
(a) b > 1 and - B generates a C-semigroup, or (b) 0 < b < 1 and both -e”B and -ePiOB generate C-semigroups, where 0 = arccos(b). Then (5.9) has a unique solution, for all (x,, X~)E C(D(B*)) X BC(D(B’)), of the form u(t) = W(tb,) C’y,
1)1’2,b, = b-(b*Y, Y E C(D(B*)), b, = b+(b’is the C-semigroup generated by - B.
wherex = I& ,:,I and { Wt)l,ao
+ W(tb,) C-‘y,,
1)1’2,
For 0 6 b Q 1, the spectral picture of Theorems 5.11 and 5.12 is shown in Fig. 1. In Theorem 5.11, the C-spectrum is contained in either the upper ((b)) or lower ((a)) diagonally shaded areas. In Theorem 5.12(b), when is bounded, the C-spectrum is contained in the horizontally s!hz!i ::a. 5.13. Suppose ~20, r
FIG. 1. Graph of two regions, a diagonally shaded one and a horizontally shaded one. The two lines make an angle of 0 with the y-axis. The dotted lines are the x and y axes.
570
RALPH DELAUBENFELS
is bounded, f is holomorphic on S,,, x SEJ,E,and bounded and continuous on -s*,, x s3.E3 and, if F = 0, f(z, w)(w - B)-‘(z - B)-’ is bounded as (z, w) approaches (0,O) in Se. Then
(a)
Vk>m,
(B-rJY=&Jas if
(W-N-~ t3.i
&,
E>O.
(b)
rfm=
1 and E=O, then Vk>2, 13(B-r)-i=lj;lj as3
w(w-W’
dw (W-r)k’
Cd)vxEXja,,r,ftw, w)(w-B)P’x(dw/(w-r)m+l)~D(B), with
BJ"aso.zf(
w, w)(w-B)-‘x(dw/(w-r)“+‘)
=J
axe
Prooj (a) t > 0, define
wf(w, w)(w-B)P1x(dw/(w-r)m+l).
Let f=i?(S,u
(z[ IzI < [2rl}),
g(w)=(l3rl
- B)-’dw, T(t)=&Jre-‘t&“)(w By Cauchy’s theorem, 2niT(t) = J eprgc”)(w - B)-’ as,.,
dw.
+ w)“*.
For
INCOMPLETE
ITERATED CAUCHY
571
PROBLEMS
Thus, for x in D(B), 27ciT(t)x = j
e-‘g’“‘(w-E+B-r)(w-B)~‘x 8.k
dw (w--T)
by a calculus of residues argument; continuing this argument k times gives 27ciT(t)(B-r)-‘=j
e~‘g(“‘(w-B)~‘&.
-e.c
By dominated convergence, lim, _ OT( t)( B - r) -k exists and equals jas,,, (w -B)-‘(dw/(w-r)k). Hence, it is sufficient to show vx E D( I?“).
lim T(t)x = x, r-0
The calculation follows, for x E D(P). 2ni( T(t)x - x) e -“‘“‘(w-B)-‘x-~x]dw
= ,[r
= j e p’8’.‘l(vr+B-B)(w-B)p1x-x]& F
=
Ir
e -“‘W’)(w-r+B-B)(w-B)-l(B-r)x&
= [j I-
e -Q?(w)&]
(B- r)x
+ j e~“‘“.‘(w--B)~‘(B--r)‘x~ r
(*I
572
RALPH DELAUBENFELS
Each term in the sum is divisible by t, hence goes to zero, as t does. The integral converges to zero, as t does, by dominated convergence. This establishes (*) and (a). (b) Let f be as in the proof of (a). By Cauchy’s theorem,
5
=e
dw --(W-r)k-
w(w--B)-
srw+-B)-’
= Jl s
dw (W-r)k
+B(w-S))l)&
=B .rr(w-W’
dw (W-rr)k,
by calculus of residues and the fact that B is closed,
by the same argument as in (a). (c) Let rl =as,,,, r+as,,,, where tj < $ < 0. We calculate as follows, using the residue theorem and the resolvent equation
sWh s fkW-s)Lo-B)-‘&& *Se.&
= SIr1
r2
1 AZ3 w, -----((w-B)-*-(z-B)-‘) (z - ‘.$I) dw
X(W
as desired.
dz (Z-r-y
INCOMPLETE
ITERATED CAUCHY
573
PROBLEMS
(d) For w E as,,, B(w B)-lx = (B w + w)(w B)-lx = w(w-B)‘x-x, so that Il(l/(w-r)“+‘)(B-r)(w-B)-‘xl] is O(Jwl-*), for Iw] large. Since B is closed, this implies that jas,, f(w, w)(w - B)-’ x(dw/(w - r)mfl) E D(B), with B
saye f(w w)W-‘~(~-~)~+,
dw
=
f(w, w)Cw(w-W’x-xl sa&?.,
=
sah wf(w~W)(w-B)~‘~(~_~)_+~,
cw-d;m+, dw
by the residue theorem. 1 Proof of Theorem 5.1. Let O,( 0,) be the angle of the bounded holomorphic strongly continuous semigroup generated by A( -B). ( - B*) generates a bounded holomorphic strongly continuous semigroup of angle 20, - 7c/2 (see [8] or [19]). Let (A -B)* be the generator of IA -m* i LO9 and let (B* - A)“* be the square root of (B* - A) such that :&4fi* generates a bounded holomorphic strongly continuous semigroup. Finally, let A, be the generator of {e~‘Be-“B2~A)“Z),t0. We will construct A, by defining, for t > 0, 1 _ B)- 1 Wt) = (271i)* i s e’(W-(W2+Z)‘~2)w(w asI as+ xz(z+A)-’
dw ~~ (1 +w)2
dz (1
where S, is as in Lemma5.13, n/2-@,<~
+z)2’
n/2-@,
is a uniformly stable A( 1 - A)-*B( 1 + B)-*-semi(1) 1wt)L,o group. (2) If C,EA(~-A)-~B(~ +B)-4, and -A, is defined to be the generator of ( W(t)},aO, then Im(C,) E D(A, A*), D(B) and D(A), with (&I)(
;-A,).,.,t)=((;r+2B-$+A)
c,u(t),
whenever U: [O, co) H X is twice continuously differentiable, t 2 0. (3) If x E D(B3) n D(BA), then u(t) 5 efA1xE C”([O, co), X) A
C’W, NY CWUl)n
NO, 4
CWH).
574
RALPH DELAUBENFELS
By Lemma5.13(b), W(0)=A(1-A)-2B(1+B))2. Applying Lemma 5.13(c) twice shows that W(s) W(t) = W(0) W(t+s), Vs, t 20. The continuity and uniform stability of { W(t)},ao follows from dominated convergence and the fact that (W- (w’ + z)“‘) is in the open left half-plane, when (z, w) E S,,, x S,,,. This proves (1). For (2), define, for t B 0, 1
-l(d + zpw)( w _ B) - 1
T(t) = (27ci)2 s8.Qcas, e dw xz(z+A)-1(1+W)4(1+z)3’
dz
where 4 and $ are as in the definition of W(t). By Lemma 5.13(b), T(0) T C,. Because of the rate of decay of the integrand, r(t) is twice continuously differentiable; by Lemma 5.13(d), ~(~)xED(B*)~D(A), VXEX, with
0
f
2 T(l)= (B2-A 1 T(t)>
vlt > 0.
is bounded, and - (B2 - ,4)‘j2 generates a bounded Since { T(f)},,o strongly continuous holomorphic semigroup, Corollary 2.13 (see Remark 2.14) implies that T(t) = e ~ f(B2~ A)“2C1)
Vt>O.
Thus,
1 (w2+z)l’2w(w-B)-’ =(27ci)2 sas, sasi ~~ dw
xz(z+A)r’
(I+
w)4
dz (1
+z)3’
so that, by Lemma 5.13(d), A,C, =-
1
asp as (-w-(W2+z)“2)W(W-13-1 I *
(27ci)2 s
dw xz(z+A)-’
(1+
w)4
dz (1
+z)3’
INCOMPLETE
ITERATED CAUCHY
575
PROBLEMS
Applying Lemma5.13(c) shows that, for t30, exists and equals
(-d/ds) W(s) T(t)l,,,
dw dz ____~ xM.(W--B)~1Z(Z+A)I(1+W)4(1+z)3
1 3
SO that Im(T(t)) G D(A,), with A,T(t) equal to the integral inside the brackets above. Arguing with {AZT(t)},t,,, as we did with { r(t)},ao, we find that, VXEX, A,C~X=A,T(O)XED(A,), with
A,A,C,x=-
1 iismis~(-w-(w2+z)1’2)(-w+(w2+z)1’2) (27ci)’ s I dw dz xw(w-~)~1z(z+A)-1X(1+W)q(1+z)3.
Thus we may establish (2) with the following calculation, for t 2 0.
=C,u”(t)-(A,+A,)C,u’(t)+A,A,C,u(t) =-
1 w(w-B)-‘z(z+A)-’ (27ri)’ I as,siis* x [u”(t)-(W-(W2+z)1’2-w+(W2+Z)“2)U’(t)
+(-w-(W2+z)1’2)(-W+(W2+z)1’2)~(t)l =-
ds
1 w(w-B)-‘z(z+A)-’ (2ni)’ sas,sas*
uf’(t)+2wu’(t)--zu(t)l =
dw
(1+w)4(1+z)3
K
$ 2+2B;+A >
dw dz ___~ (1 + w)” (I +z)3
1
C,u(t).
For (3), suppose x E D(B3) n D(BA). Since x E D(B(B’ -A)), it follows that x E D(B(B’ - A)‘12), so that x E D(A) n D(B2) A D(B(B2 - A)‘12) E D(Af). This implies that u(t) = .P’x is twice continuously differentiable. Since x E D(A), u(t) E D(A), Vt 3 0, and Au(t) = @IAx is continuous. Since
576 D(B3)
RALPH
n D(BA)
G D(B*)
DELAUBENFELS
n D(B(B*-A))
z D(B2)
n D(B(B2
-A)‘12)
c
D(BA , ), it follows that u’(t) = A, @Ix E D(B), V’t>/ 0, and Bu’( t) = kAIBA I x is continuous. This concludes the proofs of assertions (1 t(3). For x E D(B3) n D(M), let u(t) = eIA1x.By assertions (2) and (3) and Theorem 3.4, for t 3 0,
Since C, is injective, this implies that u(t) is a solution of (1.7). For uniqueness, suppose u is a solution of (1.7), with x = 0. By assertion (2), C, o(t) is a solution of (1.3), so by Theorem 3.4, C, v(t) = 0, so that, since C, is injective, u(t) = 0, as desired. The wellposednessfollows from the fact that {erAI} is a bounded strongly continuous semigroup. 1 Proof of Theorem 5.2. This is the same as the proof of Theorem 5.1, except that we must verify that erA’ is stable, and use Theorem 3.5 instead of 3.4. A bounded holomorphic strongly continuous semigroup {e’G},aO is stable if and only if Im(G) is dense (see [26, Chap. A-IV, Corollary 1.41). Either Im(A) or Im(B2) is dense, thus {erAe-rB2},a0 is stable, so that Im(B2 - A), and hence Im( ( B2 - A)“‘), is dense. This means that {efAIJra,,= {e-‘Be~“B2-‘A”Z}r~0 is stable, as desired. 1 ~ Proof of Remark 5.3. Let X-~*(Sn12-),, x S+i) (see Lemma 5.13 for terminology), A and - B defined as the generators of the following strongly continuous semigroups.
(efAf)(z, w) = eC’;f(z, w),
(e-IBf)(z, w) = eC’“f(z, w).
Choose E such that 0 < E< (7r/4- 4). Then 3R such that
whenever Iwl >R, 7c/4a(arg(w)l >(rr/4-s/2), and n/2-4>$>~/4+~. This implies that, if Es { (eeti, w)lIwl>R, n/4~larg(w)l>71/4--E/2, x/2-4>$>71/4+~}, then V(Z,W)EE, Z(W+(W~+Z)“‘), and hence ( - w + ( W* + z)“~) = ]zI ‘/F( w + (w* + z)i’*), is in the left half-plane. Clearly (- w - (w2 + z)“‘) is also in the left half-plane, V(z, w) E E. Thus, l,(z, W)-ee”~“.+‘“.*+=)‘l*) (u(t))(z , w) E ,1(-11’-(11~*+=)‘4 1AZ, w) is a nontrivial solution of (1.7) and (1.8) such that u(O)=O.
1
INCOMPLETE
ITERATED
CAUCHY
577
PROBLEMS
Proof of Theorem 5.4. There exists E> 0 such that {z 1IzI 6 E} 5 p(A) n p(B). For i= 1,2. t > 0, define, with the terminology of Lemma 5.13,
where fi(w, z) E w + (w’ + z)“~, f2(w, z) = --w + (w’ + z)“*. is a uniformly stable As in the proof of Theorem 5.1, { Wi( t)},,, B-“A -m-semigroup, for i = 1,2. Let A, be the generator of ( W,(t)}, a0 and (-A*) be the generator of { Wz(t)},,,. As in the proof of Theorem 5.1, it may be shown that D(B”+2A” + ‘) c D(A, A*), D(BA,) and D(AT), with ($-
A,)($= [(
f
~~))B-(n+*)~-(m+l~~(t)
)
*+2&$+/t
1
B-(“+2)A-(m+‘)U(t),
(*)
Vt > 0, whenever u: [0, co) I-+ X is twice continuously differentiable.
For XE D(B2n+2A2m+2), u(t)= W,(t) B”A”x is a solution of (1.7) and (1.8) by (*), as in the proof of Theorem 5.1. Uniqueness also follows from (*), as in Theorem 5.1. The continuous dependence on initial data follows from the fact that { W,(t)},ao is a bounded B-“A-“‘-semigroup. 1 Proof of Theorem 5.8. It is clear that, if Ai= b,B, where 6, = (b + (b2 + 1)“2) and b, = (b - (b* + l)“‘), then, Vt z 0, (-$A,)($-A,)u(t)=z/(t)-2bBu’(t)-B’u(t), whenever u E C( [0, co), [D(B*)]) n C’( [0, co), [D(B)]) n C’( [0, a~), X). Hence, since D(A:)nD(A,A2)=D(B2), and the solutions of (d/dt- A,) (d/dt-A,)u(t)=O are given by u(t)= W(tb2)Cp’x, where {(W(t)},2o is the C-semigroup generated by B, the result follows from Theorems 3.4 and 3.5. 1 Proof of Theorems 5.11 and 5.12. Let Ai-b,B, where b, s -b+ (b* - l)“*, b, c -b - (b* - l)“*. Then, for u as in the proof of Theorem 5.8, (-$-A,)(;-A,).(t)=.“(t)+2bB..(t)+B’.(r),
Vt>O.
A computation shows that, under hypothesis (a) of Theorem 5.11, both (-A,) and A, generate stable C-semigroups, under hypothesis (b) of
578
RALPH DELAUBENFELS
Theorem 5.1I, both A, and ( --A*) generate stable C-semigroups, while in Theorem 5.12, both A, and A, generate C-semigroups. Hence, Theorem 5.11 follows from Theorem 3.5, and Theorem 5.12 follows from Lemma 3.7. 1
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I. A. V. BALAKRISHNAN,Fractional powers of closed operators and the semigroups generated by them, Pacific J. Math. 10 (1960), 419437. 2. G. CHEN AND D. L. RUSSELL,A mathematical model for linear elastic systems with structural damping. Quart. Appl. Muk (1982), 433454. 3. S. CHENAND R. TRIGGIANI,Proof of extensions of two conjectures on structural damping for elastic systems, Pacific J. Math. 136 (1989), 15-55. 4. S. CHEN AND R. TRIGGIANI,Differentiable semigroups arising from elastic systems with gentle dissipation: The case 0
INCOMPLETE ITERATED CAUCHY PROBLEMS
579
AND N. TANAKA, Exponentially bounded C-semigroups and integrated semigroups, Tokyo J. Math. 12, No. 1 (1989), 99-115. 26. R. NAGEL (Ed.), “One-Parameter Semigroups of Positive Operators,” Lecture Notes in Math., Vol. 1184, Springer-Verlag, New York/Berlin, 1986. 27. F. NEUBRANDER, Integrated semigroups and their application to the abstract Cauchy problem, Pacific J. Mafh. 135 (1988) 11I-155. 28. F. NEUBRANDER AND R. DELAUBENFELS, Laplace transform and regularity of C-semigroups, in preparation. 29. M. M. PANG, Resolvent estimates for Schrodinger operators in LP(R”) and the theory of exponentially bounded C-groups, preprint, 1989. 30. A. PAZY, “Semigroups of Linear Operators and Applications to Partial Differential Equations,” Springer, New York, 1983. 31. J. T. SANDEFUR, Higher order abstract Cauchy problems, .I. Mafh. Anal. Appl. 60 (1977),
25. 1. MIYADERA
728-742. 32. B. STRAUB,
“Uber Generatoren von C-Halbgruppen und Cauchyprobleme zweiter Ordnung,” Diploma&it, Tiibingen, 1989. 33. N. TANAKA, On the exponentially bounded C-semigroups, Tokyo J. Math. 10, No. 1 (1987) 107-117. 34. J. A. VAN CASTEREN,Generators of strongly continuous semigroups, in “Research Notes in Math.,” Vol. 115, Pitman, New York, 1985.