CC BY
15
MSC 45K05, 47D06, 34K30
DOI: 10.14529/ mmp 170104
REGULARITY RESULTS AND SOLUTION SEMIGROUPS FOR RETARDED FUNCTIONAL DIFFERENTIAL EQUATIONS
A. Favini, Department of Mathematics, University of Bologna, Bologna, Italy, favini@dm.unibo.it,
H. Tanabe, Takarazuka, Japan, bacbx403@jttk.zaq.ne.jp
We show that the solutions of the retarded functional differential equations in a Banach space, whose existence and uniqueness are established in paper of A. Favini and H. Tanabe, have some further regularity properties if the initial data and the inhomogeneous term satisfy some smootheness assumptions. Some results on the solution semigroups analogous to the one of G. Di Blasio, K. Kunisch and E. Sinestrari and to the one of E. Sinestrari are also obtained.
Keywords: retarded functional differential equation; regularity of solutions; analytic semigroup; solution semigroup; C0-semigroup; infinitesimal generator.
Introduction
We consider the following retarded functional differential equation in a complex Banach space X:
d f0
—u(t) = Au(t) + Aiu(t - r)+ a(s)A2u(t + s)ds + f (t), 0 < t < T, dt J-r (u.l)
u(0) = p0, u(s) = pi(s) a.e. s G (-r, 0). A
semigroup etA,t > 0, in X. Suppose 0 G p(A) for simplicity. A1 and A2 are closed linear operators in X such that D(A1) D D(A), D(A2) D D(A), and a is a complex valued function defined in the interval [-r, 0] such that a G L1(-r, 0; C).
The following theorems which are improvements of the results by G. Di Blasio and A. Lorenzi [1] are established in A. Favini and H. Tanabe [2]:
Theorem A Suppose 0 < 9 < 1/p. If the following assumption is satisfied:
(I) ^o G (X,D(A))e+i-iMp, pi G Wd'P(-r, 0; D(A)), f G Wd'p(0,T; X),
u
u G Wd'p(0, T; D(A)) n C([0, T]; (X, D(A))e+i-i/PtP), (0.2)
du/dt G We'p(0,T; X). (0.3)
Theorem B Suppose 1/p <9 < 1. If the following assumption is satisfied: ' po G D(A), pi G Wd'P(-r, 0; D(A)), pi(0) = po,
(n) f G Wd ,p(0,T; X) nC ([0,T];(X,D(A))e-i/p ,p), S f0
Apo + Aipi(-r) + a(s)A2Pi(s)ds G (X, D(A))o-i/ptp,
-r
then, problem (1.1) admits a unique solution u such that
u e We'p(0,T; D(A)), (0.4)
du/dt e We'p(0,T; X) n C([0,T];(X,D(A))o-i/p). (0.5)
Note that since A2pl e C([—r, 0]; X), the integral a(s)A2pl(s)ds is well defined.
In this paper we prove further regularity of solutions u when pl and f satisfy more regularity assumptions. Using this result we prove some results on solution semigroups analogous to the one of G. Di Blasio, K. Kunisch and E. Sinestrari [3] when 9 < 1/p, and to the one of E. Sinestrari [4] when 9 > I/rp. hi case 9 < 1/p it is shown that
the map ^ ^ ^ ( ^ ) )' wkere u is the solution of (0.1) with f (t) = 0 and
ut(s) = u(t + s), —r < s < 0, is a Co-semigroup in (X,D(A))e+l-l/pp x Wep(—r, 0; D(A)), and the characterization of its infinitesimal generater is given. This is nothing but a simple extention of a result of [3] in case where X is a Hilbert space and 9 = 1, p = 2. However, in case 9 > 1/p the situation is a little more complicated. This is caused by the following fact. In this case there appears the space (X, D(A))d+l-l/p,p which is a subset of D(A). If u belongs to this space, Au, i = 1, 2, is defined, but may not belong to (X,D(A))e-l/p>p. Therefore we assume the additional condition AiA-1 e L((X, D(A))ff-l/ptp), i = 1,2. A comment on this assumption will be given in section 6. Under these hypotheses it will be shown that the map pl M- ut is a C0-semigroup in We'p(—r, 0; D(A)) n C([—r, 0]; (X, D(A))e+l-l/pp) with the
u
with f (t) = 0 and ut(s) = u(t + s), —r < s < 0.
For a Banach space y we use the following norm of We'p(0, T; Y):
f-T n t \ 1/p
Ne,p,y, (o,T)(u)=[ / \\u(t) - u(s)\\pY (t - s)-1-epdsdt) ,
oo
||u||^e,P(o ,T;Y) = Ne,pY (0,T)(u) + T- \\u\\lp(0,T;Y)-
1. Regularity of Solutions: Case 6 < 1/p
First consider the case 0 < 9 < 1/p. Assume that:
' pl e Wl+d'p(—r, 0; D(A)), po = Pl(0) (e D(A)), f e W,p(0,T; X) n C([0,T]; (X, D(A))e+l-l/pp),
Apo + AiPi(—r) + a(s)A2pl(s)ds e (X, D(A))e+l-lMp.
(0.6)
(1-1)
Theorem 1. Suppose 0 < 9 < 1/p. Then, under assumption (1-1) the solution of problem (0.1) satisfies
u e W1+9'p(0,T; D(A)) П W2+'p(0, T; X) П C 1([0,T]] (X,D(A))e+i-i/p,p). (1.1)
Since (1-1) is stronger than (I), in view of Theorem A a solution u of (0.1) exists and satisfies (0.2) and (0.3). Set
Ф0 = Афо + Am(-r)+ j a(s)A2^i(s)ds + f(0). (1.2)
J —r
r
Then (1-1) implies
pO E (X, D(A))e+i-i/PiP, p[ E W0,p(-r, 0; D(A)), f E W0,p(0,T; X).
Namely, (I) is satisfied by p'0, p1, f instead of p0, pi, f. Therefore there exists a unique solution v of the following problem:
v'(t) = Av(t) + A1v(t - r)+ ! a(s)A2v(t + s)ds + f (t), 0 < t < T,
J—r
v(0) = p'0, v(s) = p1(s) a.e. s E (-r, 0) satisfying
!
v E W0'p(0,T; D(A)) n C([0,T]; (X,D(A))0+i—i/PtP), dv/dt E W0p(0,T; X).
(1.3)
(1.4)
Set w(t) = p0 + f0 v(t)dr. Then in view of (1.4)
w' E W0 ,p(0,T; D(A)) n C([0,T]; (X,D(A))0+i—i/p>p), w" E W0'p(0,T; X).
Since p0 E D(A), v E W0p(0,T; D(A)) C Lp(0,T; D(A)), one has
(1.5)
wO = po + v(r)dr E C([0,T]; D(A)) C Lp(0,T; D(A)). (1.6)
Jo
In view of (1.5) w' E W0,p(0,T; D(A)) C Lp(0,T; D(A)). By virtue of this and (1.6)
w E W1 'p(0, T; D(A)) C W0'p(0, T; D(A)). (1.7)
It follows from (1.7) and (1.5) that
w e W1+0'p(0,T; D(A)) n W2+0'p(0,T; X). D(A) C (X,D(A))0+i—i/p,p, one has
w E C([0, T]; D(A)) C C([0,T]; (X, D(A))0+—i/p„). From this and (1.5) it follows
w E Ci([0,T];(X,D(A))0+i—i/pp). (1.8)
We are going to show
w(t) = u(t), 0 < t < T. (1.9)
If this is proved, then (1.1) follows from (1.7) and (1.8).
Set v(t) = j p^) -<< <T0 Then' v E W0'p(-r,T; D(A)). Problem (1.3) is transformed to the following integral equation:
o
a(
o o —r
ft
v(t) = etAP0 + / e{t—s)AAiv(s - r)ds + / e(t—s)A a(a)A2v(s + a)dads
+ / e(t—s)Af'(s)ds. (1.10)
o
This implies
et rt rt гт
/ v(r)dr = eTApodr + / / e(T-s)AÂiv{s - r)dsdr Jo Jo Jo Jo
rt Î'T Î'O rt rT
+ / / e(T-s)A a{a)Â2v{s + a)dadsdr +/ / e(T-s)A f (s)dsdr. (1.11) Jo Jo J-r Jo Jo
(i) Case T < r. In view of the définition (1.2) of p'o one observes
[teTAp'odr =[etA - I]A-lp'o Jo
= etApo - Po + [etA - I]A-1(^Am(-r) + J a(s)A2p1(s)ds + f (0^ . (1.12) With the aid of a change of the order of intégration and an intégration by parts
ft I-T rt r-r
r-s)A л „V/ \ il__/ / JT-s
/ e(T-s)AÂ1v(s - r)dsdT = / e(T-s)AÂ1pl1(s - r)dsdr Jo Jo Jo
[\e(t-s)A - I]A-1Aip[(s - r)ds Jo
-[etA - I]A-1A1 'M-r)+ f e(t-s)AA1 'Ms - r)ds. (1.13)
o
Again changing the order of integration and integrating by parts one obtains
t T o t t o
/ / e(T-s)A a(a)A2v(s + a)dadsdr = / e(T-s)A a(a)A2v(s + a)dadrds 'o Jo J-r Jo J s J-r
rt ro ft ft fo
= a(
0 -r s 0 -r
Г t Г-s
,(t-s)A _ T] A-1
i(a) e(T-s)AdrA2v{s + a)dads = / a(a)[eSt-s)A - I]A-1A2v(s + a)dads
j [e(t-s'A - I]A-1 J a(a)A2p[(s + a)dads
t0 1
+ / [e(t-s)A - I]A-1 a(a)A2v(s + a)dads = h + h, (1.14)
0 -s
where
I1 = Î [e(t-s)A - I]A-1 f a(a)A2p[(s + a)dads,
Ю J-r
ft {• o
-1
I2 ~.........
I2 = [e(t-s)A - I]A-1 a(a)A2v(s + a)dads.
Ю
s
Changing the order of integration and integrating by parts yield
r-t r t
Ii = a(a) [e(t—s)A - I]A-1A2p[(s + a)dsda
1—14 j
Ii
J —r J0
/0 i' — a
a(a) [e(t—s)A - I]A—1A2pi(s + a)dsda t Jo
-—t { ft
tA
(a) j-[etA - I]A—1A2pi(a) + J^ e{t—s)AA2Vl(s + a)d^ da + / a(a) {[e(t+a)A - I]A—1 A2-1 (0) - [etA - I]A—1 A2-1(a)} da
—r 0
^^---------------1—1 ' - (O) ^ ^ 1
'—t
+ J a(a) j J^ e(t—s)AA2-1(s + a)ds^ da
/0 p—t ft
a(a)[etA - I]A—1A2-1(a)da + / a(a) e(t—s)AA2-1(s + a)dsda
r —r 0
/0 p0 p—a
a(a)[e(t+a)A - I]A—1A2-1(0)da + / a(a) e(t—s)AA2-1(s + a)dsda. t —t 0
The sum of the second and fourth terms of the last side of the above equalities is equal to
t —t t —s
/ e(t—s)A a(a)A2-1(s + a)dads +/ e(t—s)A a(a)A2-1(s + a)dads
0 —r 0 —t
= J e(t—s)A J a(a)A2-1(s + a)dads. Therefore
/0 p 0
a(a)[etA - I]A—1A2-1(a)da + J a(a)[e(t+a)A - I]A—1 A2-i(0)da
+ [ e(t—s)A f a(a)A2-1(s + a)dads. (1.15)
0—r
We can show without difficulty
I2 = ( a(a) ( [e(t—s)A - I]A—1A2v(s + a)dsda
J—t J—a
/0 p t d C s+a
a(a) J [e(t—s)A - I]A—1 J A2v(r)drdsda
/0 pt ps+a
a(a) e(t—s)A A2v(t )drdsda
t J—a J0
r0 pt / p s+a \ p0 pt
= I a(a) I et—s)AA2 ' - ' ' V(T)JT\ dsda I a(a) I ^S
l—t J—a \ J0 / J—t J—a
p0 r t
a(a) l e(t—s>'a2
l—t J—a 0
+ J a(a)[I - e(t+a)A]A—1 A2-0da. (1.16)
/t / ps+a \ p0 pt
e(t—s)AA2 -0 + v(r)dr dsda - / a(a) e(t—s)AA2-0dsda a ( J0 ) J—t J—a
(a) J e(t—s)AA2 ^-0 + + v(r)dr^J dsda
From (1.14) - (1.16) and p0 = pl(0) it follows that
nt i'T i'0
/ / e(T—s)A a(a)A2v(s + a)dadsdr = h + h J0 J0 J—r
/0 pt p—s
a(a)[etA — I]A—lA2pl(u)du +/ e(t—s)A a(a)A2pl(s + a)dads
r J0 J—r
/0 i-1 / r- s+a \
a(a)j e(t—s)AAj p0 + J v(t )dr\ dsda. (1.17)
As is easily seen
t T t t t
/ / e(T—s)Af'(s)dsdT = / e(T—s)AdTf'(s)ds = [e(t—s)A — I]A—lf(s)ds
J0 J0 J0 J s J0
= —[etA — I ]A—lf (0) + /\(t—s)Af (s)ds. (1.18)
0
From (1.11) - (1-13), (1-17), (1.18) the following equality follows easily:
t t —s
w(t) = etAp0 +/ e(t—s)AAlPl(s — r)ds +/ e(t—s)A a(a)A2dads
0 0 —r
/0 nt n t
a(a) e(t—s)AA2w(s + a)dsda + / e(t—s)Af(s)ds. (1.19)
t J—a J0
Set w(t) = | W((t) 0 << << 0 (l-^) is rewritten as
n t n t n 0
w(t) = etAp0 + / e(t—s)AAlW(s — r)ds + / e(t—s)A a(o)A2w(s + a)dads
0 0 —r
+ /* e(t—s)Af (s)ds. 0
Consequently (1.9) is obtained.
(ii) Case r < T < 2r. By virtue of the result established in the previous case 0 < T < r we already know that w(t) = u(t) for 0 < t < r. Hence
r t t t
w(t) = p0 + v(t )dT + v(t )dT = w(r) + v(t )dT = u(r) + v(t )dT.
Since
Au(r) + Aiu(0) + f a(s)A2u(r + s)ds = u'(r) - f (r) e (X, D(A))e+i—i/p,p,
the following facts hold:
' Що,г] e Wi+9p(0,r; D(A)), f e W^ 'p(r,T; X) П C([r,T]; (X,D(A))e+i—i/pp),
Au[o'r](r) + Aiu[o'r](0) + / a(s)A2u[o,r](r + s)ds e (X,D(A))e+i—i/pp.
—r
О
r
Hence (1-1) is satisfied with [-r, replaced by [0,r], M[o,r] respectively. Therefore, by
the method of the previous case we can show w(t) = u(t) for r < t < T.
We can proceed to show (1.9) in the general case, and the proof of Theorem 1 is complete.
In case 1/p < 9 < 1 we assume
' ^ G w1+0p(-r, 0; D(A)) ( pi(0) G D(A)),
f e W1+0'p(0,T; X) n C 1[0,T]; (X,D(A))0-i/p>p),
A^(0) + Am(-r) + / a(s)A2^i(s)ds G (X, D(A))0-i/p>p,
(II-l)
r _r
(• o
Ap[(0) + A1^l1(-r) + a(s)A2^1(s)ds G (X,D(A))0-1/PtP,
J —r
( D—M0)) = Ayi(0) + Am(-r) + — a(a)A2^i(a)da + f(0).
Theorem 2. Suppose 1/p < 9 < 1. If assumption (II-l) is satisfied, the solution u of (0.1) satisfies
u g W1+0'p(0, T; D(A)) n W2+0'P(0,T; X) n C2([0, T]; (X, D(A))0—i/p#). (1.20)
If hypothesis (II-l) holds, then (II) is satisfied by and f' in place of ^^d f respectively. Therefore according to Theorem B there exists a unique solution v of the following problem
d f0
dj~v(t) = Av(t) + A1 v(t - r)+ a(s)A2v(t + s)ds + f'(t), (1.21)
dt J_r
v(s) = p[(s), -r < s < 0 (1.22)
satisfying
Set
v G W0'P(0,T; D(A)), (1.23)
dv/dt G W0P(0,T; X) n C([O,T];(X,D(A))0—1/p). (1.24)
w(t) = po + / v(t)dr. (1.25)
In view of (1.23), (1-24) one has
w' G W0'p(0,T; D(A)), (1.26)
w" G W0'p(0, T; X) n C([0, T]; (X, D(A))0-i/p„). (1.27)
Since 9 > 1/p, W0'p(0,T; D(A)) C C([0,T]; D(A)). Hence v G C([0,T]; D(A)). From this and ^o G D(A) it follows that w G C 1([0,T]; D(A)). This implies w G W1 'p(0,T; D(A)) C W0'p(0,T; D(A)). Hence with the aid of (1.26), (1.27) we deduce
w g W1+0'p(0,T; D(A)) n W2+0'p(0,T; X). (1.28)
o
Since D(A) c (X,D(A))g -i/p,p, one also has w e C 1([0,T]; (X, D(A))0-i/p,p). From this and (1.27) it follows that
w e C2([O,T];(X,D(A))0-i/p,p). (1.29)
If it is shown that w(t) = u(t), 0 < t < T, then in view of (1.28) and (1.29) the proof of Theorem 2 is complete. This part of the proof is almost the same as that of Theorem 1, and so it is omitted.
2. Solution Semigroup: Case 0 < 1/p
Suppose assumption (I) is satisfied. Set
Z =(X,D(A))e+i-i/pp x W0 ,p(-r, 0; D(A)).
Following G. Di Blasio, K. Kunisch and E. Sinestrari [3] the solution semigroup for (0.1) is defined as follows:
S «( £) = ( t>) < * 0, *»(e^
where u is the solution of problem (0.1) with f (t) = 0, and u(t) = | P(t) 0 << < <°0
ut(s) = u(t + s) for -r < s < 0 Since u e W0p(0, o; D(A)) n C([0, o);(X,D(A))e+i-i/pp), where u e W0p(0, o; D(A)) means u e W0p(0,T; D(A)) for any 0 <T < o u(t) e (X, D (A)) 0+i-i/p,p for t > 0, u e W0 'p(-r, o; D(A)), and hence ut e W0'p(-r, 0; D(A)) for 0 < t < o. Therefore S(t) : Z ^ Z and S(0) = I. It can be shown without difficulty that S(t) is a Co-semigroup in Z.
S(t)
D(A) = {( ^ ) ; Pi e Wi+0 ,p(-r, 0; D(A)), pi (0) = po ,
Apo + Aipi(-r) + J a(s)A2Pi(s)ds e (X, D(A))0+i-iMp J , Po\ = ( Apo + Aipi(-r) + f-r a(s)A2Pi(s)ds \
I Pi J I pi )'
This theorem can be establised by showing the following statements following G. Di Blasio, K. Kunisch and E. Sinestrari [31:
(i) S(t)D(A) c D(A),
(ii) D(A) is dense in Z,
(iii) A c infinitesimal generator of {S(t)}
(iv) A : D(A) c Z ^ Z is closed.
Problem (0.1) is rewritten as
d ( u(t) \ , I u dt \ ut u(0) uu0
(2.1)
])+{7 0") ■0 * t* r'
=( ф)■
(2.2)
The mild solution of (2.2) is expressed as
(t >) = * (i)( £) +i s (t - <f 0") (
If ^ ^ ^ G D (A) and f G C1([0,T]; (X, D(A))e+1-1/Pp)^,n ^ ^ ^ is a strict solution, and
u G C 1([0,T];(X,D(A))e+i-i/pp),
u. G C 1([0,T]; W0>p(-r, 0; D(A))), (2.3)
m ') = S + S w( f 0>)+1' s (t - ,( f 0ds.
Starting from
u. G C([0,T]; Lp(-r, 0; D(A))) ^ u G Lp(-r,T; D(A))
one can show that (2.3) is equivalent to i G Wl+e'p(-r,T; D(A)). Thus the following assertion holds:
Theorem 4. If the following assumptions are satisfied:
V1 G W1+0 ,p(-r, 0; D(A)), Vo = ?1(0), f G C 1([0,T]; (X, D(A))0+1-1/p,p), Avo + Am(-r) + j-r a(s)A2V1(s)ds G (X,D(A))0+1-1/pp, then a solution of (0.1) satisfying
u g W1+0 ,p(0,T; D(A)) n C 1([0,T]; (X,D(A))0+1-1/pp) exists and is unique.
3. Regularity of Solutions: Case 6 > 1/p
In this section we suppose that the following assumptions are satisfied:
(II-2) A1A-1,A2A-1 e L((X,D(A))0-1/pp, (X,D(A))0-1/pp),
( V1 G W0 ,p(-r, 0; D(A)) n C([-r, 0]; (X, D(A))0+-1/p,p), (H-3) <
\f e W0,p(0,T; X) n C([0,T];(X,D(A))0-1/pp).
Remark 1. Set v0 = V1(0) Then it follows from (II-3) that v0 G (X, D(A))0+1-1/pp. Hence Av0 G (X, D(A))0-1/pv1 G C([-r, 0]; (X, D(A))0+1-1/pp) it follows that
AV1 G C([-r, 0]; (X,D(A))0-1/pp), i a(s)Av1(s)ds e (X,D(A))0-1/p,p.
-r
Hence by (II-2)
A1V1 G C([-r, 0];(X,D(A))0-1/pp), i a(s)A2V1(s)ds e (X,D(A))0-1/pp. (3.1)
-r
Hence the final condition of (II) is satisfied. Therefore (II-2) and (II-3) imply (II). Remark 2. (11-2) is equivalent to
A, E L((X, D(A))e+i-i/PiP, (X,D(A))e-1/pД г = 1, 2■
A comment on assumption (II-2) will be given in the final section.
Theorem 5. Suppose 9 > 1/p, and assumptions (II-2) and (II-3) are satisfied. Then the solution и of problem (0.1) satisfies
u E W0 'p(0,T; D(A)) П C([0,T]](X,D(A))e+1-1/pp), (3.2)
du/dt E W0'p(0,T; X) П C([0,T];(X,D(A))e-i/p(3.3)
Suppose first T < r. Let u0 be the function defined by
uo(t) = etA[pi(0) + A-1f(0)] + /4e(t-s)A l(s)ds - A-1 f(0), (3.4)
Jo
where o
f(s) = Am(s - r) + f(s) + a(a)A2pi(a)da, ~ ~ ~ —r
f*(s) = f(s) - f(0) = Am(s - r) + f(s) - Aipi(-r) - f (0)■ It follows from (II-3) and (3.1) that
7E W0 ,p(0,T; X) П C([0,T]; (X,D(A))e—i/p>p), (3.5)
f E W0'p(0, T; X) П C([0, T]; (X, D(A))g—i/p„), (3.6)
Фо + A—if(0) E (X, D(A))e+i—i/pp. (3.7)
The solution of (0.1) is obtained as the solution of the following integral equation
u(t) = uo(t) + f e(t—s)A i a(a)[A2u(s + a) - A2^i(a)]dads, (3.8)
J0 J—r
where u(s) = | ф^^) s< 0' ^^^ eclua^on sovled by successive approximation: un+i(t) = uo(t)+ f e(t—s)A i a(a)[A2un(s + a) - A2^i(a)]dads, n =1, 2, 3,■■■■ (3.9)
0 —r
It is shown in A. Favini and H. Tanabe [2] that un E W0'p(0,T; D(A)), un(0) = ф0, n = 0,1, 2, ■ ■ ■. From (3.9) it follows that
un+i(t) - un(t) = f e(t—s)A[ a(a)[A2Un(s + a) - A2Un—i(s + a)]dads■ (3.10)
о —r
If -r < a < -T, then s + a < t - T < 0. Hence
A2un(s + a) - A2un—i(s + a) = pi(s + a) - pi(s + a) = 0.
Therefore (3.10) is rewritten as
un+i(t) - un(t) = f e(t-s)A i a(a)[Ä2Un(s + a) - Ä2Un-1(s + a)]dads. (3.11)
J0 J-T
It is proved in [2] that
\\Un+1 - Un\\weP(0,T;D(A)) < ^hW^L^—Tfl) [(Qp)-1/PC'o + l] \\un - Un-1 \\weP(0,T;D(A))
for some constants C0, C2 independent of T and k2 = \\Ä2Ä-1 \\. Therefore if T is so small that
C'2k2\\a\\ L1(—T, 0) [(dp)-1/pC'o + ^ < l, (3.12)
then
те
У^ \\Un+1 - un\\we>p(0,T;D(A)) < (3.13)
n=1
Set
Wl(0, T; X) = {u E Wep(0, T; X); u(0) = 0}.
The following lemma is due to G. Di Blasio [5] (Theorem 10 if в < l/p and Theorem 8 if 9 > l/p). Also c.f. Lemma 1 of G. Di Blasio and A. Lorenzi [1].
Lemma 1. Suppose в = l/p. If x E (X, D(Ä))l+1-1/PtP, then
e-Ax E Wl ,p(0,T; D(Ä)) П C([0,T]-,(X,D(Ä))e+1-1/p>p). The following lemma is Theorem 24 of G. Di Blasio [5].
Lemma 2. Suppose в > l/p. Then, if f E Wl'p(0,T; X), the function V(t) = f0 e(t-s)A f (s)ds satisfies
V E Wl'p(0,T; D(Ä)),
dV/dt = ÄV + f E C([0,T]; (X,D(Ä))e-1/pp), (3.14)
and the following inequality holds with a constant C2 independent ofT:
\\V\\w0p(o ,T;D(A)) < C2\\f \\w0P(O ,T;X)- (3.15)
In A. Favini and H. Tanabe [2] it is shown that the constant C2 above can be chosen independent of T if we choose (0.6) as the norm of Wl'p(0, T; D(Ä)). Vf
\\V '\w 9.p(0 ,T ;X) = \\ÄV + f \\we,p(0, T ;X) < \\V\\w0,p(O , T ;D(A)) + \\f \\we,P(0, T ;X)
< (C2 + l)\\f \\w в'Р(0, T ;X) ■ (3.16)
From Lemma 2 the following lemma follows:
Lemma 3. Suppose в > l/p. Let f E Wlp(0,T; X) П C([0,T]; (X,D(Ä))e-1/p,p)- Then, for V(t) = /0 e(t-sS)Af (s)ds one has
V E W1 'p(0, T; D(Ä)) П C([0,T]; (X,D(Ä))e+1-1/p,p).
In view of Lemma 1, 3 and (3.6), (3.7) one observes
e~A[<pi(0) + A—if(0)] e C([0,T]](X,D(A))g+1-1/Pp), fo e-(-s)AK(s)ds e C([0,T]; (X,D(A))g+1-1/Pp).
Moreover, A—if(0) is a constant function with a value in (X, D(A))g+i—i/p>p. Consequently
uo e C([0,T]-,(X,D(A))g+i-iMp). (3.17)
Suppose for some n = 1, 2,...
un e C([0,T];(X,D(A))g+i-i/pp). (3.18)
Then «n e C([-r,T]-, (X, D(A))g+i-i/pp). Rence A2un e C([-r,T]; (X, D(A))g-ipp) in view of Remark 2. Therefore it is easy to show
i a(a)[A2tint + a) - A2^(a)]da e C([0,T]; (X,D(A))g-ipp). (3.19)
J —r
Since un e Wg p(0,T; D(A)), one h as un e Wg 'p(-r,T; D(A)), and he nee A2un e Wg'p(-r,T; X). The following lemma is proved in A. Favini and H. Tanabe [2, Lemma 2.5]:
Lemma 4. Suppose v e Wg 'p(-r,T; X), 0 <9 < 1,9 = 1/p. Then f—r a(a)v(• + a)da e Wg'p(0,T; X), and
r-0
a(a)v(■ + a)da
< \\а\\ьц-т,0) [Ng ,p,(-r,T) (v) + T \\v\\Lp(-r,T;X)] ■
W e,P(0, T ;X)
Applying Lemma 4 to A2un one observes J° a(a)A2un(• + a)da e Wg'p(0,T; X). Therefore, noting (3.1) one deduces
/ a(a)[A2iin( + a) - A2<pi(a)]da
—r
/0 n o
a(a)A2Un^ + a)da - a(a)A2^i(a)da e Wgp(0,T; X). (3.20)
r —r
By virtue of (3.19), (3.20) and Lemma 3 one obtains
J e('—s^AJ° a(a)[A2Un(s + a) - A2^i(a)]dads e C([0, T]; (X, D(A))g+i—ipp). (3.21)
From (3.9), (3.17) and (3.21) it follows that (3.18) holds with n + 1 in place of n. Next, we estimate the following norm:
\\V'\\LP(0,T;(X, D(A))sp) = (iV'fLP(o,T;X) + £ VWopd^ , (3-22)
r
where \ • \o,p is the semi norm defined by
/ [• ™ \ 1/p
\u\ep =( J \\tl-dAetAu\\pt-1dt) .
The following inequality was shown in the proof of Lemma 2 of G. Di Blasio [5]:
\V'\lp(0,t;X) < ((p - 1)/9p)(p-1)/p MiTeNop,(0,T)(f) + Mo\\f\lp(0,T;X), (3.23)
where M0, M1 are constants such that \\etA\\ < M0, \\(d/dt)etA\\ < M]_/t. In the proof of Theorem 26 of [5, p. 81] it was shown that
^\V'(t)\pdpdt < 24p-2MlP(9-p + (1 - 9)-p) (Npdp((o,t)(f) + (^p)-1 [t-0p\\f(t)\\pdt)
i'T
p-1
+2p-1Mp t-0p\\f (t)\\p dt (s + 1)-p s-1+p-p0 ds. (3.24)
00
It is easy to show the following inequality holds for f E W° 'p(0,T; X) with a const ant c T
0 t-0p\\f (t)\\pdt < c\\f \\p^ p(oTX). (3.25)
From (3.22) - (3.24) and (3.25) it follows that the following inequality holds with a constant C1 independent of T:
1 \V'(t)\Pppdt < C\\\p^ p(otx). (3-26)
By virtue of (3.22), (3.23) and (3.26) the following inequality holds for f E W°'p(0,T; X) with a constant C2 independent of T:
\\V' \ \ LP(0 , T;(X, D(A))e ,p) < C2(T<° + 1)\\f \\we ■ P(0, T ;X )■ (3.27)
The following lemma is also due to Lemma 11 of G. di Blasio [5]. Lemma 5. Suppose 9 > p. Then
Wo,p(0, T; X) n Lp(0, T; (X, D(A))op) C C([0, T]; (X, D(A))o-1/pp),
and the following inequality holds for u E Wo'p(0,T; X) n Lp(0,T; (X, D(A))0p) with a constant C3 independent of T:
\\u\\c([0,T];(X,D(A))e_1/pp) < C3 (Td-l/p\\u\\w»P(0,T;X) + T-l^\\u\\LP(0,T;(X,D(A))giP)
Inequality (3.28) follows from the one in case T = 1 and considering a function u(Tt) , 0 < t < 1
In view of Lemma 5
\\V'\\c([0,TUX,D(A))e-i/p, p) < ^
< C3 (T0-1/p\\V'\\weP(0,T;X) + T-1/p\\V'\\LP(0,T;(X,D(A))gp)) ■
(¡() Bulletin of the South Ural State University. Ser. Mathematical Modelling, Programming
& Computer Software (Bulletin SUSU MMCS), 2017, vol. 10, no. 1, pp. 48-69
Inequalities (3.16), (3.27) and (3.29) yield
\\У' Wc([0,T\,(X,D(A))e-i/Pp) < CT\\f \\w 0 ,p(0,T ;X), where CT = C3 ((C2 + C2 + 1)Te + C2) T-i/p. Hence
(3.30)
\\V\\c([0 ,T];(X,D{A))e+1-1/p, p) \\AV\\C([0,T];(X,D(A))e-1/p, p)
= \\V' — f \\c([0,T];(X,D(A))e-i/p pp) < CT\\f \\w<> p(0,T;X) + \\f \\c ([0,T];(X,D(A))e-i/p „)• (3-31)
We apply (3.31) to f°T a(a)[A2Un(• + a) — A2Un-i( + a)]da. Then V = Un+i — un (c.f. T
C'2k2\\a\\Li(-T,0)[(ep)-l/pC0 + 1] < i.
One has
\\Un+i — un\\c([0,T]-,(X,D(A))e+i-i/p>p)
< Ct
a(a)[A2Un( + a) — A2u,n-i (• + a)]da
T
+
a(a)[A2Un( + a) — A2Un-i( + a)]da
T
W 0 >p(0,T; X) c([0,T ];(X,D(A))o-i/p ,p)
It is shown in [2] that the following inequality holds:
a(a)[A2Un(• + a) — A2Un-i( + a)]da
T
W 0 >p(0,T; X)
< k2\\a\\Li(-T,0) [(Op)-l/pC0 + 1] \\Un — Un-l\\w0 ,p(0,T; D(A)) •
As is easily seen
a(a)[A2Un( + a) — A2Un-i( + a)]da
T
sup
s€[0,T ]
c([0,T ];(X,D(A))o-i/plp)
T
a(a)[A2Un(s + a) — A2Un-i(s + a)]da
(X,D(A))o-i/p л
< / \a(a)\da sup \AUn(r) — A2Un-i(r)\\(x,d(A)) J-T r£[-T,T]
< k2\\a\\Li(-T,0) sup \\AU,n(r) — AU,n-i(r)\\(x,d(A))
0-i/p ,p
0-1/p , p
T ф,т ]
= k2\\aWL1(-T,0)\\Un — Un-i\\c([0,T]; (X,D(A))e+i-i/p,p). The following inequality follows from (3.32) - (3.34):
\\Un+i — Un\\c([0,T] ; [X,D(A))e+i-i/p „)
< Ctk2\\a\\Li(-T,0) [(Op)-i/PC0 + 1] \\Un — Un-i\\w0>p(0,T;D(A)) + k2\\aWL1(-T,0)\\Un — Un-i\\c ([0,T ]; (X,D(A))o+i-i/p ,p).
(3.32)
(3.33)
(3.34)
0
0
0
0
Summing both sides from n = 1 to m one gets
m
W«n+i - un\\c([0,T];(X,D(A))e+1_1/p, p)
n=i
m
< CTk2\\a\\Li(—T,0) [(9P) — i/pC0 + ^ y \\un - un—i\\ws• p(0,t;D(A))
n=i
m
+ k2\\a\\L1(—T,0)^2 \\un - un—i\c([0,T]iX,D(A))e+1-1/p , p)- (3-35)
n=i
Substituting
m
Y \\un - un—i\\c(l0,TUX,D(A))e+i_i/PpP) n=i
m
W^n+i - un\\c([0,T];(X,D(A))e+i_i/p„) + \\ui - u0\\c([0,T];(X,D(A))e+i_i/p„)
n=i
in the last side of (3.35) one gets
m
Y \\un+i - un\\c([0,T];(X,D(A))e+i_i/p, p) n=i
m
< CTk2\\a\\Li(—T,0) [(9P) — i/pC0 + ^ Y \\un - un—i\\we,p(0,T;D(A))
n=i
m
+ k2\\a\\Li(—T,0)YJ \\un+i - un\\c([0,T] ; (X,D(A))g+i_i/pp)
n=i
+ k2\\a\\Li(—T,0)\\ui - u0\\c([0,T];(X,D(A))g+i_i/pp)-
This implies
m
(1 - hhWL^—T^Yl \\un+i - un\\c([0 ,T ]; (X>D(A))e+i-i/p , p)
n=i
m
< CTk2\\a\\Li(—T,0) [(9P) — i/pC0 + ^ \\un - un—i\\we,p(0,T;D(A))
n=i
+ k2\\a\\Li(—T,0)\\ui - u0\\c([0,T];;(X,D(A))e+l_l/ppp)■ (3.36)
Letting m —y to in (3.36) one obtains in view of (3.13)
<x
(1 - hhWL^—Tfl))^ \\un+i - un\\c([0 ,T ];(X,D(A))e+i_i/p, p)
n=i
<x
< Ctk2\\a\\Li(—T,0) [(9p) — i/pC0 + 1] ^ \\un - un—i\\w°.p(0,t;d(A))
n=i
+ k2\\a\\Li(—T,0)\\ui - u0\\c([0,T];;(X,D(A))e+i-i/pp) < (3-37)
Let T satisfy k2|H\Li(-T,0) < 1 besides T < r and (3.12). Then by virtue of (3.37) and (3.13) one obtains
<x
\\un+1 - un\\c([0,T];(X,D(A))e+i_i/p„) <
n=1
Hence {un} is convergent in C([0, T]; (X, D(A))o+1-1/p,p). Since un ^ u in Wo'p(0,T; D(A)), one concludes u E C([0,T]; (X, D(A))o+1—1/pp) and
un ^ u in Wo,p(0,T; D(A)) n C([0,T]; (X, D(A))o+1-1/pp).
T0
0 <To < r, C2k2\\a\\Li(-To,o) [(9p)-1/pC0 + 1] < 1, k2\\a\\Li(-To,o) < 1■
Then, by the result just proved one has u E Wo'p(0,T; D(A)) n C([0,To]; (X,D(A))o+1-1/p,p). Hence
u E Wo,p(-r,T; D(A)) n C([-r,To]; (X,D(A))o+1-1/p,p).
T0 < T
uu \[To—r,To] E Wo,p(To - r, To; D(A)) n C([To - r,To]; (X,D(A))o+1-1/p,p),
u
d f0
-ru(t) = Au(t) + A1u(t - r)+ a(s)A2u(t + s)ds + f (t), T0 < t < T,
dt -r
u(s) = u\ [To—r,To] (s), To - r < s < To. Therefore, by virtue of the result already proved
u E Wo,p(0,T; D(A)) n C ([0, min{2To,T}]; (X, D(A))o+1-1/pp).
Continuing this process we can complete the proof of Theorem 5.
Next, we consider the case where the following assumption is satisfied: E W 1+o'p(-r, 0; D(A)) n C 1([-r, 0]; (X, D(A))o+1-1/pp), (II-4) { f E Wo+1p(0,T; X) n C ([0,T]; (X, D(A))o+1-1/pp) n C 1([0,T]; (X,D(A))o-1/pp), V1(0) = A?1(0) + Am(-r) + — a(a)A2^1(a)da + f (0).
u
satisfies
u e W 1+o'p(0,T; D(A)) n W2+o'p(0,T; X) n C 1([0,T]; (X,D(A))o+1-1/p,p), , ,
U
G C([0,T]-(X,D(A))e-i/p).
Proof. If hypotheses (II-2) and (II-4) are satisfied, then (II-l) and (II-3) are also satisfied. In view of Theorem 2 and Theorem 5 it suffices to show
u' G C([0,T];(X,D(A))e+i-i/p).
Since (II-4) is satisfied, (II-3) is satisfied by <p'Vl f in place of f. Therefore in view of
v
d f0
—v(t) = Av(t) + Aiv(t - r)+ a(s)A2v(t + s)ds + f (t), dt J—r
v(s) = ^i(s), -r < s < 0
v e Wg,p(0, T; D(A)) n C([0, T]; (X, D(A))g+i—ipp), dv/dt e Wg'p(0, T; X) n C([0, T]; (X, D(A))g—i/p).
Since u(t) = ^i(0) + I v(t)dr (c.f. Proof of Theorem 2),
0
satisfying
u
v e C([0,T]-,(X,D(A))o+i-i/p,p).
□
4. Solution Semigroup: Case 6 > 1/p
In this section we assume that hypotheses (II-2) and (II-3) are satisfied. Let Z = Wg ,p(—r, 0; D(A)) n C([—r, 0]; (X, D(A))e+i-i/p>p). For p1 e Z let u be the solution of (0.1) with f (t) = 0: d f0
—u(t) = Au(t) + A1 u(t - r)+ a(s)A2u(t + s)ds, 0 < t< ^
dt J-r (4.1)
u(s) = p1(s), —r < s < 0. u
u e Wg,p(0,T; D(A)) n C([0,T]; (X,D(A))0+i-i/p,p), u' e Wg,p(0,T; X) n C([0,T]; (X,D(A))g-ipp).
Therefore u e Wg'p(—r, m; D(A)) n C([—r, m); (X,D(A))g+i-i/p,p^ere u e Wgp(—r, m; D(A)) means u e Wgp(—r,T; D(A)) VT > 0. This implies ut e Z for t > 0. Therefore if we set
S(t)pi = ut, t > 0,
S(t) maps Z to Z.
Let u ^e ^te solution of (4.1), v be the solution of (4.1) with the initial function uT and w(t) = u(t + t) for t > 0, t > 0. Then
d f0
—v(t) = Av(t) + Aiv(t — r)+ a(s)v(t + s)ds, 0 < t< m,
dt -r
v(s) = u(t + s), —r < s < 0, d d
—w(t) = — u(t + t ) = Au(t + t ) + Aiu(t + t — r)+ a(s)u(t + t + s)ds dt dt J-r
= Aw(t) + Aiw(t — r)+ a(s)w(t + s)ds, 0 < t< m,
-r
w(s) = u(t + s), —r < s < 0.
Therefore v = w, and hence vt = wt. On the other hand vt = S (t)uT = S (t)S (t )p1,
Wt(s) = w(t + s) = u(t + s + T) = Ut+T (s) = (S (t + T )^1)(s).
Thus
S (t)S (t )<p1 = S (t + t )V1.
It is easy to see that the mapping [0,T] 9 t M ut E C([-r, 0};;(X,D(A))e+1-1/pp) is continuous. The continuity of [0, T] 3 t M ut E We'p(-r, 0; D(A)) is shown in the following lemma.
Lemma 6. For v E Wd'p(-r,T; D(A)) the mapping [0,T] 3 t M vt E Wd'p(-r, 0; D(A)) is continuous.
Proof. The lemma is proved by the following step:
(i) For w E W 1p(-r,T; D(A)) limT^t \\wt - Wt\\w°,p(-r,o;D(A)) = 0.
(ii) For v E Wd'p(-r, T; D(A)), w E W 1'p(-r,T; D(A)) such that \\v-w\\we,p{-rT;D(A)) < e one has
\\vT - vt\\we,p(-r,o;D(A))
< \\wT - wt\\we,p(-r,o;D(A)) + \\vT - wt\\we,p(-r,o;D(A)) + \\vt - wt\\we,p(-r,o;D(A))
< \\wt - wt \ \we p(-r,o;D(A)) + 2\\v - w\\we p (-r,T;D(A)) < \\wt - wt\\we ,p(-r,o;D(A)) + 2e.
□
Hence the mapping [0,T] 9 t M S(t)p1 = ut E Z is continuous. Thus it has been shown that {S(t), t > 0} is a Co-semigroup.
The following result is an analog to Theorem 4.4 of E. Sinestrari [4]:
S(t)
D(A) = {E Wl+d'p(-r, 0; D(A)) n C1 ([-r, 0];;(X,D(A))o+1-1/p,p); <p1(0) = A^(0) + Am(-r)+ j a(a)A2<p1(v)dv},
r-0
a(a)A?Wi (a)
' —r
Api =
Proof. We show
(i) 5(t)D(A) с D(A).
(ii) D(A) is dense in Z = W0'p(—r, 0; D(A)) n C([-r, 0]; (X, D(A))e+i—i/pp).
(iii) Л с infinitesimal generator of {S(t)}.
(iv) Л : D(Л) с We'p(—r, 0; D(A)) ^ We'p(—r, 0; D(A)) is closed.
(i) Let фi G D^) and u be the solution of (4.1). Then in view of Theorem 6 and its proof u(0) = ^i(0) and u'(0) = <^(0). Hence u G Wl+e,p(—r, ж; D(A)) П Cl([—r, ж]; (X, D(A))e+i—i/pp). Hence
ut G W^p(—r, 0; D(A)) П Cl([—r, 0]; (X, D(A))e+i—i/pp).
For t > 0
(ut)'(0) = Km Tlt(s) - Tlt(0) = hm u(t + s) - Tl(t) = hm u(t + s) - u(t s^0 s S^—0 s S^—0 s
= u'(t) = Au(t) + Aiu(t - r)+ i a(s)A2u(t + s)ds
—r
f 0
= Aut(0) + Aiut(-r) + a(s)A2ut(s)ds.
Hence ut e D(A).
(ii) Let e Z and ut = S(t)pi. Set = /0utdt. p€(s) = /0ut(s)dt = /0u(t + s)dt.
(p'e(s) = ds^ J^ utdt) (s) = d J0 u(t + s)dt = u'(t + s)dt
d_
ds 0 ds 0 0 u(e + s) - u(s) = u(e + s) - ^i(s) = u^s) - (s), -r < s < 0. (4.2)
Therefore
^ = ue - pi e Wg ,p(-r, 0; D(A)) n C([-r, 0]; (X, D(A))g+-i/pp).
Hence
^ e Wg+i,p(-r, 0; D(A)) n C\[-r, 0]; (X, D(A))g+-i/p„). By virtue of (4.2)
Ap,(0) + Aiye(-r)+ j a(a)A2^,(a)da
—r
P€ P€ p0 P€
= A u(t)dt + Ai u(t - r)dt + / a(a)A2 u(t + a)dtda
J0 J0 J—r J0
= J0 (^Au(t) + Aiu(t - r) + J a(a)A2u(t + a)da) dt
= f u'(t)dt = u(t) - u(0) = ue(0) - pi(0) = p'e(0).
0
Therefore e D(A). Since [0, to) 9 t — ut e Z is continuous,
- I (ut - pi)dt e J0
1 r
<~ \\ut - Pi\\Zdt — 0 as e — 0. z e J0
(iii) Let e D(A^d u be the solution to (4.1). As was noted in the proof of (i) one has u e Wi+g,p(-r, to; D(A)) n Ci([-r, to]; (X, D(A))g+i—i/ptp). Hence
u' e Wg,p(-r, to; D(A)) n C([-r, to]; (X, D(A))g+i—i/p>p).
r
Therefore
и'(to + -)da — и'
W 0 P(-r,0;D(A))
< / \\U'(to- + ■) — U' \\w0P(-r,0;D(A))do- ^ 0,
u' (to + ■)do — u'
C([-r,0];(X,D(A))e+1_1/p p)
< \\u'(to + ■) — u'\\c([-r,0];(X,D(A))e+i_i/pp)do ^ 0-
{u'(to + ■) — u' )do
W 0 P(-r,0;D(A))
Hence
S(t)pi - pi ut - uo u(t + •) - %(•) 1 f1 d -=-= -= - I -—u(ta + •)da
t t t t J0 da v 7
= [ u'(ta + •)da — u'(•) = Api
o
in Z.
(iv) Let pin E D(A), pin — pi, p'in = Apin — ^ in Z. Then pi = ^ E Z. Hence
pi E W 1+d'p(-r, 0; D(A)) n C 1([-r, 0]; (X, D(A))e+i-i/Pp).
Since pin — pi in C([-r, 0];(X,D(A))e+i-i/PtP), one has Apin(0) — Api(0), Apin(-r) — Api(-r), f-r a(a)A2pin(a)da — f-r a(a)A2pi(a)da in (X, D(A))0-i/PtP. One also has p'in(0) — ip(0) = pi(0) in (X, D(A))d+i-i/pp. Therefore, from
it follows that
p'ln(0) = Apln(0) + Aipin(—r) + / a(o)A2pln(o)do
p[(0) = Api(0) + Aipi(-r) + a(o)A2Pi(o)do.
Therefore p1 E Лр1 = гф.
i
i
о
r
о
r
□
5. Remark on Hypothesis (II-2)
Let A be the realization in Lp(Q), 1 < p < to, of a strongly elliptic linear partial differential operator of second order with the Dirichlet boundary condition, where Q is a bounded domain in Rn. Let Ai,i = 1, 2, be linear partial differential operators of second order in Q. Assume that the coefficients of A,Ai,i = 1, 2, and the boundary dQ of Q are sufficiently smooth. Then D(A) = W2'p(Q) n W0'p(Q)- Assume that A has a bounded inverse. Suppose 1/p < 9 < 3/(2p). Then 0 < 29 - 2/p < 1/p. Hence by virtue of the results of E. Seeley [6] (also c.f. H. Triebel [7, Theorem 4.3.3]).
(Lp(Q),D(A))e-i/p,p = (Lp(Q),W2,p(Q) n Wi'p(Q))o-i/P,P = , .
= bZ-2/p(Q) = W2d-2/p,p(Q). [ '
Since
A"1 E L(Lp(Q),W2p(Q) n Wl'p(tt)) n L(W 1p(Q),W3'p(Q) n W1'p(Q)),
one has
A"1 E L(W20"2/p'p(Q),W2+2d~2/p'p(Q) n Wlp(Q)). (5.2)
Let f E (Lp(Q),D(A))d"i/p^ In view of (5.1) and (5.2) A"1f E W2+2°"2Mp(Q) n W01,p(Q).
Hence
AtA"1f E W2d"2/p,p(Q) = (Lp(Q),D(A))d"1/pp, i = 1, 2.
Therefore
AiA"1 E L((Lp(Q),D(A))ff"1/ptp, (Lp(Q), D(A))"1/pp), i =1, 2. (5.3)
Next, consider the case of the Neumann boundary condition. In this case D(A) = {u E W2'p(Q); du/dn = 0 on 3Q} ,
where du/dn is the outer conormal derivative with respect to A. Suppose
1/p < 9 < 3/(2p) + 1/2, 9 = 1/2 + 1/p. Then, again by virtue of the results of R. Seeley [6]
or H. Triebel [7]
(Lp(Q),D(A))d"1/pp = B2p(dp"1/p) (Q) = W2(0"1/p)'p(Q),
and (5.3) follows as in the case of the Dirichlet boundary condition.
References
1. Di Blasio G., Lorenzi A. Identification Problems for Integro-Differential Delay Equations. Differential Integral Equations, 2003, vol. 16, no. 11, pp. 1385-1408.
2. Favini A., Tanabe H. Identification Problems for Integrodifferential Equations with Delay: an Improvement of the Results from G. Di Blasio and A. Lorenzi. Appear in Funkcialaj Ekvacioj.
3. Di Blasio G., Kunisch K., Sinestrari E. L2-regularity for Parabolic Partial Integrodifferential Equations with Delay in the Highest-Order Derivatives. Journal of Mathematical Analysis and Applications, 1984, vol. 102, issue 1, pp. 38-57. DOI: 10.1016/0022-247X(84)90200-2
4. Sinestrari E. On a Class of Retarded Partial Differential Equations. Mathematische Zeitschrift, 1984, vol. 186, pp. 223-246.
5. Di Blasio G. Linear Parabolic Evolution Equations in Lp-Spaces. Annali di Matematica Pura ed Apphcata (IV), 1984, vol. 138, issue 1, pp. 55-104. DOI: 10.1007/BF01762539
6. Seeley R. Interpolation in Lp with Boundary Conditions. Studia Matematica, 1972, vol. 44, pp. 47-60.
7. Triebel H. Interpolation Theory, Function Spaces, Differential Operators. Amsterdam, N.Y., Oxford, North-Holland, 1978.
Received November 28, 2016
УДК 517.9
БО!: 10 Л4529/ттр 170104
РЕЗУЛЬТАТЫ РЕГУЛЯРНОСТИ И РАЗРЕШАЮЩИХ ПОЛУГРУПП ДЛЯ ФУНКЦИОНАЛЬНО-ДИФФЕРЕНЦИАЛЬНЫХ УРАВНЕНИЙ С ЗАПАЗДЫВАНИЕМ
Показано, что решения функционально-дифференциальных уравнений с запаздыванием в банаховом пространстве, существование и единственность которых показана раннее в работах А. Фавини и X. Танабе, обладают дополнительными свойствами регулярности, если исходные данные и неоднородный член удовлетворяют некоторым предположениям о гладкости. Кроме того, получены некоторые результаты о разрешающих полугруппах.
Ключевые слова: функционально-дифференциальное уравнение с запаздыванием; регулярность решений; аналитической полугруппы; полугруппы решения; С0-полугруппы; инфинитезимальный генератор.
Анджело Фавини, кафедра математики, Болонский университет (г. Болонья, Италия ), favini@dm.unibo.it.
Хироки Танабе (г. Такарадзука, Япония), bacbx403@jttk.zaq.ne.jp.
А. Фавини, X. Танабе
Поступила в редакцию 28 ноября 2016 г.
