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Semi-linear evolution equations in interpolation spaces
NonlinearAnalysis, Theory, Method.~& Applications,Vol. 1 No, 3, pp. 249 261. PergamonPress, 1977. Printedin Great Britain.
SEMI-LINEAR EVOLUTION EQUATIONS IN INTERPOLATION SPACES EUGENIO SINESTRARI* a n d PAOLA VERNOLE* Istituto Guido Castelnuovo, Universita degli studi--Roma, 00100 Roma, Italy (Received 27 July 1976) Key words: Interpolation spaces, nonlinear evolution equations
INTRODUCTION
Tim OBJECTof this paper is the study of the following semilinear evolution equation: u'(t) = Au(t) + q>(t,u(t))
(0.1)
with initial condition u(0) = x. A: D~, _ E --) E is the infinitesimal generator of an analytic semigroup in the Banach space E. The equation (0.1) has been studied by several authors (see [1]-[4] and references indicated there) under the assumption that q~(t, x) is defined from R+ x DAo tO E where DAO is the domain of 0-th power of - A. We assume that tp(t, x) is defined from R+ x DA(0, ]9) t o E where DA(0, /9) is the real interpolation space between DA and E (with parameters 0e ]0, 1[ and p ~ [1, + ~]), obtained by the Lions-Peetre's method (see [5]). The use of interpolation spaces enables us to give an exact description of DA(O, p) in important applications (for example, when A is an elliptic differential operator with homogeneous Dirichelet conditions, D^(0, p) is a subspace of Sobolev or Besov spaces : see [6] and the example in the last section of this paper) and to give easy conditions to the mild solution in order to be "strong" or classical by using the theory of Da Prato-Grisvard
[7]. We study our problem under rather weak assumptions i.e. when tp is locally Lipschitz continuous (and not necessarily Lipschitz continuous on bounded sets) in the variable x; in this case we establish (see Section 2) a local existence theorem for a mild solution of (0.1) i.e. for a continuous function u: [0, T] ~ DA(O,p) satisfying the integral equation: u(t)
en'x + f'o ea"-s)g°(s'u(s)) ds
0 <~ t <~ T
(0.2)
Section 3 is concerned with the dependence of the solution on all the data (i.e. A, ~a and x); under reasonable assumptions it is shown that the maximal mild solution can be uniformly approximated by the solutions of the approximating problems. In Section 4 we study the regularity properties of the mild solution: in particular we show that the mild solution is classical when q~ is continuous with values into another interpolation space; we also study the problem of the global existence of the solution which is proved under suitable conditions. * Work supported by G.N.A.F.A. of C.N.R. 249
250
E. SINESTRARIAND P. VERNOLE
In Section 5 a simple application is given, The results of this paper can be easily extended to the case when A is dependent on t. The generalization to the case of integro-differential equations with delay will be considered in another paper. I. A N A L Y T I C
SEMIGROUPS
AND INTERPOLATION
SPACES
Let E be a Banach space with norm I[" II"Let A(E) denote the set of the infinitesimal generators of analytic semigroups on E. Assume that the linear operator A: D^ c E ~ E has the following property: (Lt) A ~ A(E) and if {eat} is the semigroup generated by A, we have:*
}[e*'l]~,~, ~< M1,
(t >~ 0),
IlAeat}l~tn)~< M2/t,
t > 0
Let us recall, without proof, some results about the real interpolation spaces; we use the notations of [6] and [8] (where the proofs can be found). Definition 1.1. If 0 ~]0, 1[ and p e [1, + oo] we denote by (DA, E)o., the real interpolation space between D a and E (with parameters 0 and p), obtained by the method of Lions-Peetre [5]. For equivalent definitions of (D^.E).. p see appendix of [7]. We set for 0 < 0 < 1 OA(O,p) = (DA, E)t_o, p
if
1 ~< p < +
DA(O, o0) = the completion of DA with the norm of (OA, E)1-0, ® so that DA is dense in DA(O, p) for each p e [1, + oo] and 0 e l 0 , 1[. Let us denote by II " I[0,p the norm Of DA(O,p). The spaces DA(O, p) have the following interpolation property: let F be another Banach space and F E A(F) : if ~fTe £P(D^, Dr) with IITII~,OA,or, <~ a o
and
T~.~(E,F)
with
IITII , , ,
~ al
then T e .~°(DA(0, p), Dr(O, p))and 11Tl[z,o^,o,.,.o^,0.9,, ~< Caoa 0 ~1 - o where c is a constant independent of T. The proof is in [6] for the case 1 ~< p < + ~ ; when p = + ~ , it is a consequence of the fact that (D^, E)1 -0. ® and (Dr, F)t _ 0, ~ohave the interpolation property and if T ~ La(DA,Dr) n ~e(E, F) then T maps D^(0, ~ ) in Dr(0, co). We will now prove some results which play an important role in the sequel: PROPOSITION 1.1. (i) {eAt} is a strongly continuous semigroup on D^(0, p) and IIe^'ll ~e(o^(0, p)) ~< M
(t > 0)
(ii) HeA'11~,~, o^ ,o,,)) ~< M(1 + 1/t) °
(t > O)
where
M = max (M 1, M2)
(iii) I f t o > 0 and x ~E, then lim ]]e^'x - e^'°xH0.. = 0. t--'to
* If E and F are two Banach spaces, C~(E, F) is the Banach space of continuous linear operators from E to F, with the usual norm. We set &e(E, E) = .o~'(E). t DA is regarded as a Banach space with the graph norm.
S e m i - l i n e a r e v o l u t i o n e q u a t i o n s in i n t e r p o l a t i o n s p a c e s
251
Proof. (i) It is easy to see that IleA'l]~,.~> ~< M,: hence by interpolation IleA'l[~(Oa(0,~), ~< e M, ~< M As D A is a dense subset of DA(O, p), for any x E DA(O, p) and ~. > 0, there exists x~ ~ D A such that II~ - ~,11,,~ < ~,: thus for t > 0 we have: lieA'x - xil0, p ~< lieA'x - er'tx~llo, p +
IleA'x~- X~II0,~ + Ilx,- xllo, ~
~< IleA'x~- X~II0,p+ (M + 1)c:
lim IIeA'x~- X~[IOA = O,
as
l--,O +
lim [leA'x -- XlI0,p = 0: hence {eA'} is strongly continuous on DA(0, p). t~0 ~
(ii) It can be checked that
[leA'll~(~,OZ)~
M, +
M2/t when
t > 0. Hence by interpolation
]le A~ ])~ ~5 > 0 l)eA' x - eA'°xllo,
-
)le A('-o'e
- e A"°-''e A xllo .
AS eA'~x~ D^(O, p), we get the thesis from (i). 2. LOCAL EXISTENCE AND U N I Q U E N E S S OF THE MILD SOLUTION
Definition 2.1. Let X r = C(0, T; E) be the Banach space of continuous functions u: [0, T ] -~ E, with the usual n o r m ; similarly we set Xo, p, r = C(O, T; DA(O, p)). Let q0: R ÷ x DA(O, p) ~ E, (t, x) ~ tp(t, x) be a given function, satisfying the assumption: (F 0 F o r any x ~ DA(O, p), t ~ ¢p(t, x) is a continuous function from R+ to E and there exist r~ and K~ > 0 such that
lifo(t, x,) - ~o(t, x~)ll ~< g
llx, - x~ll0,~
for each t ~ R+ and x~, x 2 ~ P(x, r.~)*. It is easy to see that ¢p is continuous from R+ × DA(0, p) to E.
Definition 2.2. Set for each u ~ X0, p, r and t ~ [0, T] f (u)(t) = ~o(t, u(t)); then f(u) ~ X r. Given x ~ DA(0, p) we shall study the p r o b l e m of finding a function u from R+ to DA(O, p) satisfying:
u'(t) = Au(t) + ¢p(t, u(t)) u(0) = x
t >t 0 (P*)
We call a mild solution of the p r o b l e m (P*) in [0, T ] a function u ~ Xo, p, r satisfying the integral * Let B be a Banach space, x ~ B and r > 0: we set P(x, r) = {y e B, IIY - xll < r} and/5(x, r) = {y ~ B, IlY - xll < r}.
E. SINESTRARI AND P. VERNOLE
252
equation: u(t)
eA'x + Jo eA"-s)q°(s'u(s))ds
(P)
In this section we prove the local existence and uniqueness of the mild solution under the assumptions (L,) and (Ft). W e must study the integralon the right hand side of (P) when the integrand is considered as a DA(O , p)-valued function. Definition 2.3. If u ~ U(0, T; E)* with 1/(1 - 0) < r ~ + oo we define
(Qu)(t)=f'oeA"-s)u(s)ds
O<~t<~T
Q has the following p r o p e r t y : PROPOSITION 2.1. Q u e Xo, p, T a n d if u e X T then IIQ II.~(x~, xo . . . . ) ,N< fl(T) where fl(T) =
M ( T + 1)°T 1 - ° 1 - 0
Proof. F o r a given t e ]0, T], let us set : v: [0, t[ -+ DA(O, p), v(s) = e A"-')u(s); it is easy to see that v is B o c h n e r - m e a s u r a b l e ; if u • X T we can also p r o v e that v is c o n t i n u o u s : in fact, let 0 ~< s, s + h ~< t - 6 < t: there exists M6 such that IleA~ll ~et~, o, (o, p)) <~ M v Vs >1 6 then
llv(s + h) - v(s)Ila,,, <~ {leA('-s-h~u(s + h) - eA(t-s-h)u(s)llo,p + {leA(t-~-h)U(S) -- eA('-~)U(S)IIO,p <~ M,
Ilu(s +
h) - u(s)l} + [leA"-s-h)u(s) -- eA('-%(S)[10, /
as h -+ 0 the thesis follows using (iii) of P r o p o s i t i o n 1.1. Let us p r o v e t h a t s --+ is integrable on [0, t]. F o r 0 ~< s < t we h a v e a.e.
II*)llo,,
Ile'*-"u(s)ll0, ~ <. M(1 +
1/t - s)°llu(s)l]:
hence, using H f l d e r ' s inequality, v is integrable (owing to 1/1 - 0 < r). Let us p r o v e that Qu • X o p T: we can s u p p o s e 1/1 -- 0 < r < + oo. Let us put iT(t) = u(t) for t e [0, T ] ; a(t) = 0 for t e R/[O, T]. If 0 < t, t + h ~< T we h a v e :
II(eu)(t+ h)-
= IIf2 e'[a(t + h - s)- a(,- s)]dsll0.
~< M
(1 + 1/s) ° ' / ' - * ds
IlEt(t + h - s) - Et(t - s)l I d s )
As r < + oo we get the c o n t i n u i t y o f Qu. Letu•X randO
II(Qu)(t)[Io,~ < Ilullx.
;
M(1 + 1/t - s)° ds < IlUllx~M(t + I) ° o
;o
s - ° ds
< II.ll~ ( T + l) ° T ' - ° 1 --
0
M = HU]IxTfl(T)
* U(0, T; E) is the Banach space of{classes of) functions u: [0, T] -+ E, Bochner-measurable and such that I1~(')11~ I:(0, T) with u s u a l n o r m .
Semi-linear evolution equations in interpolation spaces
253
hence
IIo,II o.p.T -<
l (r)IlUllx .
Definition 2.4. F o r each u ~ Xo, p, T we set O(U)(t) = J t ° eA(t-S)~o(s, u(s))ds
0 <<. t <<. T;
from P r o p o s i t i o n 2.1 we get ~(u) e Xo.p, r. R e m a r k 2.1. • has the following p r o p e r t y : F o r each x o ~ DA(O, p) and T > 0, there exist r o and K o (depending on Xo) such that
ll*(.,)(t) - *(.2)(t)L.
ko (r)[I.1 - "2[Ixe,.,T
for each u, ~ Xo. ,, r, Ilui(t) - xoIIo, p ~< ro (i = 1, 2) and t e [0, T ] In fact from (FI) setting K o = Kxo and r o = rxo, we get:
II
;o
eA(t-s)Fcp(s , lgl(S))
-
-
~0(S, //2(S))] dsl[0, p ~< g o M
i
(1 + 1/t - s)°Hul(s) - u2(s)Ho, p dS o
THEOREM 2.1. (Local existence and uniqueness of the mild solution). Let 0 ~ ] 0 , 1[, p c [1, + ~ ] , A verify (L1) and q~ verify (F 0. F o r each x o ~ DA(O, p), there are positive n u m b e r s 6 o and TO (depending on Xo) such that: if x ~ DA(O, p) and Hx - Xollo ' p <~ 6 o there exists a unique u ~ Xo, p, r mild solution of (P*) (with initial d a t u m x) P r o o f Given x o ~ DA(O , p) let r o and K o be the constants of the preceding remark. Let us choose 6 o such that 0<6
(1)
o < ro/M
Let us define for every T >~ 0 the following functions
o(r) = too(T)=
sup Ile*'xo- Xoll,..
te[O, Yi
sup Hq~(t,Xo)H re[0, TI
7o(T) = ~o(T) + mo(T)[3(T) + r o K o B ( T )
It is easy to see (using (i) of P r o p o s i t i o n 1.1) that 7o: R+ ~ R+ is continuous, strictly increasing and 7o(0) = 0, 7o(+ oo) = + oo. Let us choose n o w To > 0 such that : 7o(To) ~< r o - M6 o
(2)
from this follows K o f l ( T o ) < 1. Let us take X ~ D A ( O , p ) such that Ilx -Xo[Io, p <. 6 o and set uo(t) = x o, v(t) = cA'x, Vt ~ [0, To]. By virtue of R e m a r k 2.1, ~ : P(u o, to) ~_ Xo, p, ro ~ Xo.p, To is a contraction m a p with Lipschitz constant a ~< Kofl(To) < 1. The mild solutions of (P*) in [0, To] are the solutions of the equation u - @(u) = v
in
Xo, v. to"
254
E. SINESTRARIAND P. VERNOLE
It is well known, [see also L e m m a 3.1 and R e m a r k 2.2] that we have a unique solution if
(3)
I1~ - no + OtUo)ll~o,, ~o ~ rot1 - ~)holds. The inequality (3) is verified since }Iv - Uollxo,.,~o = sup
IleA'x - Xollo, v -%< sup
t~[O, To]
t~[O, To]
IleA'(x - Xo)ll0,, +
sup
re[0, To]
I1:%-
Xok,
<~ M 6 o + % ( T o)
and IIO(Uo)Lo ,.To ~
~(To)mo(To)
so that IIv - u o + (1)(Uo)llxs,~,ro ~< M 6 o + % ( T o) + mo(To)fl(To) = M 6 o + ?o(To) - roKofl(To)
~< rol-1 - K o f l ( T o ) ] <~ ro(1 - cO. PROPOSITION 2.2. Let x 1, x 2 ~P(xo, 30). If u I and u 2 ~ X o , p. ro are the mild solutions of (P*) corresponding to the initial values x 1 and x 2 we have
where M CO
l -- Kofl (To)
In fact, if we set v i = u i - O(ui) we have
1
1 sup leAtx1 -- eAtx2}l I1~, -~211xo,~,~o ~< i - K o f l ( T o ) ,~t0,ro]
Ilu~ - uzllxo,~,To "%<~
REMARK 2.2. Let u ~ Xo. p, r be a solution of (P) in [0, T ] and v e Xo, v, rl verify
eA,
/)(t)
+ fl °
+
, ts))ds
0. t
.<
/4)
If we set fu(t) w(t) = ( v ( t - T)
O <.%t <<. T T <~ t <~ T +
T1
(5)
it can be checked that w e X0,p, r + r , and verifies (P) in [13, T + 7~]. Conversely ifw is a solution of (P) in [0, T + TI] and if we define u and v by (5), it is easy to see that v verifies (4). where u(T) = w(T). The uniqueness on the large of solutions to (P) now follows. 3. D E P E N D E N C E
F o r every n e N let An:
D A --~ E --~
OF SOLUTION
ON THE DATA
E be a linear o p e r a t o r verifying the condition (L1) with the
Semi-linear evolution equations in interpolation spaces
255
same constants M 1 a n d M 2. Let us also suppose that the following condition holds: (A) lim IleA'X --
e^'xll0., = 0 uniformly in t e I, a c o m p a c t interval of R+, and x ~ K, a com-
n .-~ oo
pact set of E. F o r every n ~ N, let tpn: R+ x D^(O,p) ~ E be a continuous function, verifying the condition (F 0 with the same constants r x and K~. M o r e o v e r let the following conditions hold : (B) lim II~o.(t,x) - ~(t, x)ll -- o uniformly in t ~ I, a c o m p a c t interval of R+, and x e K, a comtl--~ oO
pact set OfDA(0, p). The following t h e o r e m describes the dependence of the solution to (P) from the data A, ¢p and x. THEOREM 3.1. G i v e n x o • DA(0, p), let 60, TO be the positive constants of T h e o r e m 2.1. If we choose x. D^(O.p). IIx. - xollo., ~< 6o and T0 ~ ]0, To[,, then for each n t> n o (depending on x o and To) there exists a unique u. e X0, p, rb such that
u.(t)=eA"'xn+f'oe^""-S)q~.(s,u.(s))ds
O<~t<~To
(P)n
If in addition lim x. = x
DA(O,p)
in
(3.1)
n---b o o
and u ~ Xo, p, r6 is the mild solution to (P) with initial value x, then we have lim u. = u
in
Xo, p, T6
M--* o o
Proof. Let ro, Ko, 60, To be the constants of T h e o r e m 2.1. As in the p r o o f of T h e o r e m 2.1 let us define for each n e N and T > / 0 0t(o")(T) =
sup
Ile^-'Xo - xoll0.,
sup
![¢pn(t,Xo)[[
le[O, To]
m(on~(T)=
te[O, 7"]
T(o")(T) = a(on)(T) +
m(o")(T)~(T)+ roKofl(T)
F r o m (A) a n d (B) it follows that lira 7(o")(T) = ?o(T) for each T >i 0 and x o a
DA(O,p). If we choose
R~oo
T Oe
]0, To[- we have lira ~,(o")(To)-- yo(T0) < yo(To) ~< r o - M6 o
hence there exists n o (depending on x o and T0) such that ),to")(T0) <~ r o - M 6 o
for
n ~> n o
(3.2)
Let us suppose n >/no and x. eD (0, p) such that [Ix. - Xoll0.p ~< 6 o. Set v.(t) = eA'x., o each t e [0, To]; the solutions to the equation (P). are the solutions u. ~ X0,p. r~ of the equation
Uo(t) = x o for
u. - q).(u.) = v.
(3.3)
256
E.
SINESTRARI A N D
P.
VERNOLE
where ~ . : P(uo, ro) ~- Xo, p, To ~ Xo. o. T6
¢.(u)(t)=f'oeA""-s'~,(s,u(s))ds
is a contraction map with Lipschitz constant a. ~< Kofl(Td) < 1 because of(3.2). The problem (3.3) admits a unique solution if we have ]l~.
-
Uo + *.(Uo)[]xo, p,~.o <~ ro (1 - ~.)
As in the proof of the T h e o r e m 2.1 we get
I[~, - Uollxop,r; < M 6 o + a(o"'(To) ]l(I).(Uo)]lxo,.,.,; ~< B(Vo)m(o~'(T'o) hence because of(3.2)
[J~. - Uo + *.(Uo)]lxo,.,T, < to[1 - KoB(T~)] < r o 0 - a.) T o prove the last part of the theorem we need a simple lemma: LEMMA 3.1. Let a be a Banach space with n o r m II" 11and for each n ~ N let * , : P(xo, r) ~_ B ~ B be an application such that: (1) HtI).(xl) - (I).(x2) H ~< allx 1 - x2{ [ Vxl, x 2 e P ( x o , r ) , n e N w i t h O (2) there exists for any x e P(x o, r), lira q).(x) = q)(x).
< a < 1.
n~ct3
(3) y. e P ( x o - ~.(Xo), r(1 - a)) and there exists lira y. = y.. Then there exist unique x, x. e P(x o, r) such that "~ ~ x. - (l).(x.) = y.
(3.4)
x - ¢(x) = y
(3.5)
M o r e o v e r we have lim x. = x n---b oO
P r o o f The first part of the lemma follows from the contraction mapping principle, applied to the applications of P(x o, r) into itself: x ~ y. + ~.(x)
and
x ~ y + ~(x)
F r o m (3.4) and (3.5) we get
I1~. - ~ll = II0.(~.) - ¢(x) + y. - yll < IfO.(x.) - ¢.(x)H + II¢.(x) - ¢(x)rJ + [ly. - y[f ~< allx,, - xll + ]l(1),,(x) - (I)(x)ll + Ily. - yll; hence we have 1
11~. - xll < -i-~_ ~ (ll¢.(x) from this the thesis follows.
-
¢(x)l[ + []y.
-
Y[I);
Semi-linear evolution equations in interpolation spaces
257
Let us complete the p r o o f of t h e o r e m 3.1 by using the preceding lemma. Let t e [0, To]; we have
Ile^-,x. - e ,x O,p <~ IleA"tXn -- e^°,x O,p + Ile^-'x < Mllx.-
xllo,.
+
O,p
Ile "'x - eat. II
from (A) we get lim v. = v in Xo, p, r6 Let us p r o v e that
lira n~oo •
f:
;o
e ^"~'- ~)q).(s, u(s)) ds =
e ^<'- ~)~o(s, u(s))ds
uniformly for t e [0, To]. As
f'oe^"('-~)q~.(s,u(s))ds-f'oeA"-~)q~(s,u(s))ds =
u s)) -
u(s))] ds +
-
s
u(t -
s)) d s -
-
s
u(t -
s))
s
T h e first integral can be majorized by
fl(To) m a x ]lq~.(t, u(t)) - ~o(t, u(t))ll re[O, Tb]
hence it goes to zero for n ~ ~ (because of (B)) ; the last difference converges to zero uniformly on [0, To] for the a s s u m p t i o n (A). T h e next t h e o r e m shows that a solution of (P) can be uniformly a p p r o x i m a t e d by solutions of (P). in every c o m p a c t interval [0, T]. THEOREM 3.2. F o r each n e N, let A., q~. and x. verify the assumptions of T h e o r e m 3.1. Let u e Xo,v, + be a mild solution to (P*) in [0, T]. F o r sufficiently large n there exists u. e Xo, p, t, a solution of (P). in [0, T], and we have lira u. = u in Xo, p, ~. n--+ co
Proof We proved that given y e DA(O,p) there exist 6 r, Tr and ny such that if x, x . ~ P(y, 6y) and n >~ n r there exist u and u. in Xo,., ry, solutions of (P) and (P). respectively and !irn~ u. = u in Xo, p,r.; we can suppose that if we change ~o with (t, x) ~ q~(t + h, x) and ~o. with (t, x) ~ % (t + h, x) we can use the same constants 6~. Tr and n~ for each h e [0, ~ ] " it is sufficient to set in the proofs of T h e o r e m s 2.1 and 3.1, t o o ( T ) = sup [Iq~ (t, Xo)ll and m(o")(T)= sup IIq~. t E [ 0 , ~ + T]
t ~ [ 0 . ~ + T]
(t, Xo)II. Let F be the range of u: [0, ~ ] ~ DA(O, p) and H a finite covering of F m a d e by 1~ = P(y, 6y), y ~ DA(O, p). Let x o be such that u(0) ~ Ixo ~ H ; there exists n o such that if n >~ n o then x. ~ lxo and (P). has solution u. in [0, Txo] a n d t i m u. = u in Xo, p, rxO" If Txo < ~ let x 1 be such that u(T~) I~, ~ I1 ; there exist nl such that if n ~> n o + n 1 then u.(T~o) e Ix, and (P.) has solution u. in [0, T~o + T~,] and lira u. = u in Xo,.,r~ +r~ (see R e m a r k 2.2). After k times the solutions of (P). are defined (for sufficiently large n) on an interval of amplitude 1 >t zk where r = min {Ty, Ix ~ H} : so that when k >t T/T the thesis follows.
258
E. S[NESTRAmAND P. VERNOLE 4. C L A S S I C A L
SOLUTIONS
AND
A PRIORI
ESTIMATES
In the present section we use some results of [7] for the solutions in X r to the problem: u'(t) = Au(t) + 9(0
0 <~ t <~ T (4.1)
u(0) = 0
where 9 E X r. PROPOSITION 4.1. Let u e Xo, p, r be a mild solution to (P*). Set g(t) = q~(t, u(t)) t ~ [0, T ] ; there exist u. ~ Xo, p, r, continuously differentiable from ]0, T ] to E such that lim u. = u in X r and n ' - * O0
g. E X r such that lim 0. = 0 in X r and n.--~ oo
u'.(t) = Au.(t) + O.(t)
0 < t <~ T
(4.1n)
u.(0) = x
Proof We can write u = u o + u I where Uo(t) = eAtX and ul is a strong solution (according to [7]) of the problem (4.1) where O(t) = q~(t, u(t)) t ~ [0, T]. The conclusion follows from the theorem 4.5of[7]. PROPOSITION 4.2. Let ~0: R+ x DA(O, 19)--~ DA(O', p), O' E l 0 , 1[, p'~ [1, + oo] be a continuous function and u e Xo.v.r a mild solution to (P*): then u is a classical solution to (P*), i.e. for each t > O, u(t) ~ O^, u is continuously differentiable from ]0, T] to E, and (P*) holds.
Proof. As u E Xe, p, r, we have g(. ) = ~0(., u(. )) e Xe; v', r - C(0, T; (DA, E ) I _ 0', oo) thus setting
ugt)=foe^"-"O(s)ds from theorem 4.7 of [7] we know that u x e C(0, T, D^) n C~(0, T; E) and verifies (4.1) in [0, T]. Suppose now that A and q~verify the assumptions of Theorem 2.1: it is easy to see that for each x ~ D^(O, p) there exists a unique maximal mild solution u*: [0, T * [ - * DA(O, p) where T* ~< + oo. F r o m theorem 3.2 it follows easily: PROPOSITION 4.3. F o r each n ~ N, let A., ~o. and x. verify the assumptions of Theorem 3.1. Let us choose T < T*: for sufficiently large n, there exist u. ~ Xe, p, r, solutions to (P). such that lim u. = u* in Xo, p, r. n---* oo
PROPOSITION 4.4. Let ~o verify the following condition for each T > 0: if u: [0, T [ --* DA(O, p) is continuous and bounded then q~ (-, u(.)) e E(0, T; E) with r > 1/(1 - 0). Then if u*: [0, T*[ is a maximal mild solution to (P*) and T* < + oo, we have sup IIu*(t)IIo, p t ~ [o. T*[
=+00.
Proof. Suppose that u* ~ U°(O, T*, DA(O, p)); from (F2) it follows that (o(., u*(')) ~ E(O, T*, E). Hence, by proposition 2.1, Qu* E Xo, p,r.- Since u* can be extended to a solution of(P) in [0, T*] we get a contradiction by applying Theorem 2.1.
Semi-linear evolution equations in interpolation spaces
259
THEOREM 4.1. Let ~p verify the following condition:
(F3)
D+llxll~(t,x) <. 0
x ~ D^(O,p),
t >10.
Let {eAt} be a contraction semigroup i.e.?
(t~)
Ile^'ll~,,~, .< 1.
Let u: [0, T [ ~ DA(O, p) (0 < T ~< + oo) be a solution to (P)in [0, T[; then Ilu(t)ll <. [lu(O)l[ for each t e [0, 7~[.
Proof. We need the following lemma: (see [9]). LEMMA 4.1. Let v e Cl(a, b; E): then t ~ IIv(t)ll is Lipschitz continuous and we have
IIv(t)ll --V(a)II + ~' DUv(s)ll,~'(s)ds
t~[a,b].
We give now the proof of the theorem. If we choose T e ]0, ~[, u is the mild solution to (P*) in [0, T]: let u n and fin be the functions of the Proposition 4.1. Let 0 < ~, < t ~< T: if we apply the Lemma 4.1 to the functions u, e C1(~, T; L0 from (4.1n) of Proposition 4.1 we get
Ilu~(t)ll- II..(*)ll =
ollu#( ))lu~(s)ds
<- f' D +llu,,(s)llAu,,(s)d~ + ~' D +li,,,,,(s)li,,,(s)ds <~
ds
by virtue of (L2). Using Theorem 11.3 of [10] and taking max lim we get n-¢oo
II~t)l[- I~011 -< ~<
;
max lim n~oo
max
lim D+ Ilu.(s)llo.(~)ds < n -,-~ oo
f" O+llu.(s)llg~(s)d~
;o+
Ilu(s)llq,(s, ~(s))ds .< 0.
Letting e ~ 0 +, the conclusion follows. THEOREM 4.2. Let (F3) and (L2) hold and 9 verify the following assumption: (see [11] p. 175) (F,). There are continuous functions a(.), b(.) from R÷ to R÷ such that
Ilk(t, x)ll -< a(llxll) + b(llxll)llxlle,~t >1 0, x
~
Do(O, p).
Then if T < + oo and u: [-0, TI- ~ DA(0, p) is a solution to (P) we have sup Ilu(t)lk~ < + ~ .
te[O, T[
f If x, y ~ E, D]]xl]y(D + ]]xHy) denotes the derivative (right derivative) of the function t -* Ux + ty]l at zero.
260
E. SINESTRARIANDP. VERNOLE
Proof From Theorem 4.1 it follows that for any t ~ [0, T[Hu(t)H <<. I]u(0)]l and from (P) we get ]Ju(t)lJ°'P "< Mllxl]°'P+
f'o M(1
+
1/t- s)°(a(llu(s)]l)+ b(lJu(s)llu(s)Jlo.) ds
As a(.)and b(.)are continuous they are bounded on the interval [0,
Ilu(t)J]o,. <. Co + cl
(t
s)-°llu(s)llo,
-
Ilxll], hence we have
ds
From a lemma (similar to Gromwall's Lemma) proved in [4] we get
Ilu(t)ll0, .<
c s e r'
hence the conclusion follows. F r o m the Proposition 4.4 and from Theorem 4.2 we get the following theorem about the global existence of the mild solution to (P*). THEOREM 4.3. If A verifies the assumptions (L1), (L2) and ~p verifies the assumptions (Ft), (F2), (F3), (F4); then for any x ~ D^(O, p) there exists a unique mild solution to the problem (P*) in
[0, + 5. APPLICATION We give now a very simple application of the preceding theory. Let f~ be a bounded domain in R* with smooth boundary df~. Let 0 ~ ] 1/2, 1[ and p > n/20 - 1. We put E = LP(~). We consider the following problem
c~u(t, x) ~ c~2u(t,x) = ~ ~x-----~i + Z., (u(t, x))"°(D,ju(t, x)) nj ~t i=1 j=l u(O, x) = h(x) u(t,x) = 0
x ~ x ~R~t
~ [0, T]
where no, ni are some non-negative integers, h is an assigned function in W 20"p (f~). It is known that the operator A defined by D A = W2'/'(~) n WI'P(~)
Au = Au verifies (L1), (L2). Moreover it can be proved that DA(O, p) = I/Vo2O'P(~'))~ W 1' P(~')) (see [8]). Setting
~p(u) = ~ un°(Dx~u) ~j j=l
and using Sobolev's lemma, we can verify the assumption (F1). Conditions (F2), (Fa) , (F, 0 hold for instance when p is even and n o = nj = 1.
Acknowledaements The authors would like to thank ProfessorsG. Da Prato and P. Grisvard for their helpful advice.
Semi-linear evolution equations in interpolation spaces
261
REFERENCES 1. SOBOLEVSKIP. E., Equations of Parabolic Type in a Banach Space Trudy Mosk. mat. Obshch. 10, 297--350 (1961). 2. FRIV.DMANA., "Remarks on Nonlinear Parabolic Equations" Proc. Syrup. appl. Math. 17, 5-23 (1965). 3. MARUO K., Integral Equation Associated with Some Non-Linear Evolution Equation. J. math. Soc. Japan, 26, 433~39 (1974). 4. WEB8 G. F. Exponential Representation of Solutions to an Abstract Semi-Linear Differential Equation (to appear). 5. LioNs J. L. & PEETRE J., Sur ane Classe d'Espaces d'Interpolation Publications Math~matiques de I'I.S.H.E. 19 5-68 (1964). 6. GRISVARDP., Spazi di Tracce ed Applicazioni. Rend. Mat. 5 (4), 657-729 (1972). 7. DA PRATOG. & GRISVARDP., Sommes d'Op6rateurs Lin6aires et Equations Diff6rentielles Operationnelles. J. Math. pures appl. 54, 305-387 (1975). 8. GRISVARDP., Commutativit6 de deux Foncteurs d'Interpolation et Applications. J. Math. pures appl., 45, 143-290 (1966). 9. KATO T., Nonlinear Semigroups and Evolution Equations. J. math. Soc. Japan 19, 508-519(1967). 10. DA PRArO G., Applications Croissantes et Equations d'Evolution dans les Espaces de Banach. Ist. Naz. Alta. Mat. Roma (1975); Academic Press (to appear). 11. FRIEDMANA., Partial Differential Equations.Holt, Rinehart and Winston, New York (1969).
SEMI-LINEAR EVOLUTION EQUATIONS IN INTERPOLATION SPACES EUGENIO SINESTRARI* a n d PAOLA VERNOLE* Istituto Guido Castelnuovo, Universita degli studi--Roma, 00100 Roma, Italy (Received 27 July 1976) Key words: Interpolation spaces, nonlinear evolution equations
INTRODUCTION
Tim OBJECTof this paper is the study of the following semilinear evolution equation: u'(t) = Au(t) + q>(t,u(t))
(0.1)
with initial condition u(0) = x. A: D~, _ E --) E is the infinitesimal generator of an analytic semigroup in the Banach space E. The equation (0.1) has been studied by several authors (see [1]-[4] and references indicated there) under the assumption that q~(t, x) is defined from R+ x DAo tO E where DAO is the domain of 0-th power of - A. We assume that tp(t, x) is defined from R+ x DA(0, ]9) t o E where DA(0, /9) is the real interpolation space between DA and E (with parameters 0e ]0, 1[ and p ~ [1, + ~]), obtained by the Lions-Peetre's method (see [5]). The use of interpolation spaces enables us to give an exact description of DA(O, p) in important applications (for example, when A is an elliptic differential operator with homogeneous Dirichelet conditions, D^(0, p) is a subspace of Sobolev or Besov spaces : see [6] and the example in the last section of this paper) and to give easy conditions to the mild solution in order to be "strong" or classical by using the theory of Da Prato-Grisvard
[7]. We study our problem under rather weak assumptions i.e. when tp is locally Lipschitz continuous (and not necessarily Lipschitz continuous on bounded sets) in the variable x; in this case we establish (see Section 2) a local existence theorem for a mild solution of (0.1) i.e. for a continuous function u: [0, T] ~ DA(O,p) satisfying the integral equation: u(t)
en'x + f'o ea"-s)g°(s'u(s)) ds
0 <~ t <~ T
(0.2)
Section 3 is concerned with the dependence of the solution on all the data (i.e. A, ~a and x); under reasonable assumptions it is shown that the maximal mild solution can be uniformly approximated by the solutions of the approximating problems. In Section 4 we study the regularity properties of the mild solution: in particular we show that the mild solution is classical when q~ is continuous with values into another interpolation space; we also study the problem of the global existence of the solution which is proved under suitable conditions. * Work supported by G.N.A.F.A. of C.N.R. 249
250
E. SINESTRARIAND P. VERNOLE
In Section 5 a simple application is given, The results of this paper can be easily extended to the case when A is dependent on t. The generalization to the case of integro-differential equations with delay will be considered in another paper. I. A N A L Y T I C
SEMIGROUPS
AND INTERPOLATION
SPACES
Let E be a Banach space with norm I[" II"Let A(E) denote the set of the infinitesimal generators of analytic semigroups on E. Assume that the linear operator A: D^ c E ~ E has the following property: (Lt) A ~ A(E) and if {eat} is the semigroup generated by A, we have:*
}[e*'l]~,~, ~< M1,
(t >~ 0),
IlAeat}l~tn)~< M2/t,
t > 0
Let us recall, without proof, some results about the real interpolation spaces; we use the notations of [6] and [8] (where the proofs can be found). Definition 1.1. If 0 ~]0, 1[ and p e [1, + oo] we denote by (DA, E)o., the real interpolation space between D a and E (with parameters 0 and p), obtained by the method of Lions-Peetre [5]. For equivalent definitions of (D^.E).. p see appendix of [7]. We set for 0 < 0 < 1 OA(O,p) = (DA, E)t_o, p
if
1 ~< p < +
DA(O, o0) = the completion of DA with the norm of (OA, E)1-0, ® so that DA is dense in DA(O, p) for each p e [1, + oo] and 0 e l 0 , 1[. Let us denote by II " I[0,p the norm Of DA(O,p). The spaces DA(O, p) have the following interpolation property: let F be another Banach space and F E A(F) : if ~fTe £P(D^, Dr) with IITII~,OA,or, <~ a o
and
T~.~(E,F)
with
IITII , , ,
~ al
then T e .~°(DA(0, p), Dr(O, p))and 11Tl[z,o^,o,.,.o^,0.9,, ~< Caoa 0 ~1 - o where c is a constant independent of T. The proof is in [6] for the case 1 ~< p < + ~ ; when p = + ~ , it is a consequence of the fact that (D^, E)1 -0. ® and (Dr, F)t _ 0, ~ohave the interpolation property and if T ~ La(DA,Dr) n ~e(E, F) then T maps D^(0, ~ ) in Dr(0, co). We will now prove some results which play an important role in the sequel: PROPOSITION 1.1. (i) {eAt} is a strongly continuous semigroup on D^(0, p) and IIe^'ll ~e(o^(0, p)) ~< M
(t > 0)
(ii) HeA'11~,~, o^ ,o,,)) ~< M(1 + 1/t) °
(t > O)
where
M = max (M 1, M2)
(iii) I f t o > 0 and x ~E, then lim ]]e^'x - e^'°xH0.. = 0. t--'to
* If E and F are two Banach spaces, C~(E, F) is the Banach space of continuous linear operators from E to F, with the usual norm. We set &e(E, E) = .o~'(E). t DA is regarded as a Banach space with the graph norm.
S e m i - l i n e a r e v o l u t i o n e q u a t i o n s in i n t e r p o l a t i o n s p a c e s
251
Proof. (i) It is easy to see that IleA'l]~,.~> ~< M,: hence by interpolation IleA'l[~(Oa(0,~), ~< e M, ~< M As D A is a dense subset of DA(O, p), for any x E DA(O, p) and ~. > 0, there exists x~ ~ D A such that II~ - ~,11,,~ < ~,: thus for t > 0 we have: lieA'x - xil0, p ~< lieA'x - er'tx~llo, p +
IleA'x~- X~II0,~ + Ilx,- xllo, ~
~< IleA'x~- X~II0,p+ (M + 1)c:
lim IIeA'x~- X~[IOA = O,
as
l--,O +
lim [leA'x -- XlI0,p = 0: hence {eA'} is strongly continuous on DA(0, p). t~0 ~
(ii) It can be checked that
[leA'll~(~,OZ)~
M, +
M2/t when
t > 0. Hence by interpolation
]le A~ ])~
-
)le A('-o'e
- e A"°-''e A xllo .
AS eA'~x~ D^(O, p), we get the thesis from (i). 2. LOCAL EXISTENCE AND U N I Q U E N E S S OF THE MILD SOLUTION
Definition 2.1. Let X r = C(0, T; E) be the Banach space of continuous functions u: [0, T ] -~ E, with the usual n o r m ; similarly we set Xo, p, r = C(O, T; DA(O, p)). Let q0: R ÷ x DA(O, p) ~ E, (t, x) ~ tp(t, x) be a given function, satisfying the assumption: (F 0 F o r any x ~ DA(O, p), t ~ ¢p(t, x) is a continuous function from R+ to E and there exist r~ and K~ > 0 such that
lifo(t, x,) - ~o(t, x~)ll ~< g
llx, - x~ll0,~
for each t ~ R+ and x~, x 2 ~ P(x, r.~)*. It is easy to see that ¢p is continuous from R+ × DA(0, p) to E.
Definition 2.2. Set for each u ~ X0, p, r and t ~ [0, T] f (u)(t) = ~o(t, u(t)); then f(u) ~ X r. Given x ~ DA(0, p) we shall study the p r o b l e m of finding a function u from R+ to DA(O, p) satisfying:
u'(t) = Au(t) + ¢p(t, u(t)) u(0) = x
t >t 0 (P*)
We call a mild solution of the p r o b l e m (P*) in [0, T ] a function u ~ Xo, p, r satisfying the integral * Let B be a Banach space, x ~ B and r > 0: we set P(x, r) = {y e B, IIY - xll < r} and/5(x, r) = {y ~ B, IlY - xll < r}.
E. SINESTRARI AND P. VERNOLE
252
equation: u(t)
eA'x + Jo eA"-s)q°(s'u(s))ds
(P)
In this section we prove the local existence and uniqueness of the mild solution under the assumptions (L,) and (Ft). W e must study the integralon the right hand side of (P) when the integrand is considered as a DA(O , p)-valued function. Definition 2.3. If u ~ U(0, T; E)* with 1/(1 - 0) < r ~ + oo we define
(Qu)(t)=f'oeA"-s)u(s)ds
O<~t<~T
Q has the following p r o p e r t y : PROPOSITION 2.1. Q u e Xo, p, T a n d if u e X T then IIQ II.~(x~, xo . . . . ) ,N< fl(T) where fl(T) =
M ( T + 1)°T 1 - ° 1 - 0
Proof. F o r a given t e ]0, T], let us set : v: [0, t[ -+ DA(O, p), v(s) = e A"-')u(s); it is easy to see that v is B o c h n e r - m e a s u r a b l e ; if u • X T we can also p r o v e that v is c o n t i n u o u s : in fact, let 0 ~< s, s + h ~< t - 6 < t: there exists M6 such that IleA~ll ~et~, o, (o, p)) <~ M v Vs >1 6 then
llv(s + h) - v(s)Ila,,, <~ {leA('-s-h~u(s + h) - eA(t-s-h)u(s)llo,p + {leA(t-~-h)U(S) -- eA('-~)U(S)IIO,p <~ M,
Ilu(s +
h) - u(s)l} + [leA"-s-h)u(s) -- eA('-%(S)[10, /
as h -+ 0 the thesis follows using (iii) of P r o p o s i t i o n 1.1. Let us p r o v e t h a t s --+ is integrable on [0, t]. F o r 0 ~< s < t we h a v e a.e.
II*)llo,,
Ile'*-"u(s)ll0, ~ <. M(1 +
1/t - s)°llu(s)l]:
hence, using H f l d e r ' s inequality, v is integrable (owing to 1/1 - 0 < r). Let us p r o v e that Qu • X o p T: we can s u p p o s e 1/1 -- 0 < r < + oo. Let us put iT(t) = u(t) for t e [0, T ] ; a(t) = 0 for t e R/[O, T]. If 0 < t, t + h ~< T we h a v e :
II(eu)(t+ h)-
= IIf2 e'[a(t + h - s)- a(,- s)]dsll0.
~< M
(1 + 1/s) ° ' / ' - * ds
IlEt(t + h - s) - Et(t - s)l I d s )
As r < + oo we get the c o n t i n u i t y o f Qu. Letu•X randO
II(Qu)(t)[Io,~ < Ilullx.
;
M(1 + 1/t - s)° ds < IlUllx~M(t + I) ° o
;o
s - ° ds
< II.ll~ ( T + l) ° T ' - ° 1 --
0
M = HU]IxTfl(T)
* U(0, T; E) is the Banach space of{classes of) functions u: [0, T] -+ E, Bochner-measurable and such that I1~(')11~ I:(0, T) with u s u a l n o r m .
Semi-linear evolution equations in interpolation spaces
253
hence
IIo,II o.p.T -<
l (r)IlUllx .
Definition 2.4. F o r each u ~ Xo, p, T we set O(U)(t) = J t ° eA(t-S)~o(s, u(s))ds
0 <<. t <<. T;
from P r o p o s i t i o n 2.1 we get ~(u) e Xo.p, r. R e m a r k 2.1. • has the following p r o p e r t y : F o r each x o ~ DA(O, p) and T > 0, there exist r o and K o (depending on Xo) such that
ll*(.,)(t) - *(.2)(t)L.
ko (r)[I.1 - "2[Ixe,.,T
for each u, ~ Xo. ,, r, Ilui(t) - xoIIo, p ~< ro (i = 1, 2) and t e [0, T ] In fact from (FI) setting K o = Kxo and r o = rxo, we get:
II
;o
eA(t-s)Fcp(s , lgl(S))
-
-
~0(S, //2(S))] dsl[0, p ~< g o M
i
(1 + 1/t - s)°Hul(s) - u2(s)Ho, p dS o
THEOREM 2.1. (Local existence and uniqueness of the mild solution). Let 0 ~ ] 0 , 1[, p c [1, + ~ ] , A verify (L1) and q~ verify (F 0. F o r each x o ~ DA(O, p), there are positive n u m b e r s 6 o and TO (depending on Xo) such that: if x ~ DA(O, p) and Hx - Xollo ' p <~ 6 o there exists a unique u ~ Xo, p, r mild solution of (P*) (with initial d a t u m x) P r o o f Given x o ~ DA(O , p) let r o and K o be the constants of the preceding remark. Let us choose 6 o such that 0<6
(1)
o < ro/M
Let us define for every T >~ 0 the following functions
o(r) = too(T)=
sup Ile*'xo- Xoll,..
te[O, Yi
sup Hq~(t,Xo)H re[0, TI
7o(T) = ~o(T) + mo(T)[3(T) + r o K o B ( T )
It is easy to see (using (i) of P r o p o s i t i o n 1.1) that 7o: R+ ~ R+ is continuous, strictly increasing and 7o(0) = 0, 7o(+ oo) = + oo. Let us choose n o w To > 0 such that : 7o(To) ~< r o - M6 o
(2)
from this follows K o f l ( T o ) < 1. Let us take X ~ D A ( O , p ) such that Ilx -Xo[Io, p <. 6 o and set uo(t) = x o, v(t) = cA'x, Vt ~ [0, To]. By virtue of R e m a r k 2.1, ~ : P(u o, to) ~_ Xo, p, ro ~ Xo.p, To is a contraction m a p with Lipschitz constant a ~< Kofl(To) < 1. The mild solutions of (P*) in [0, To] are the solutions of the equation u - @(u) = v
in
Xo, v. to"
254
E. SINESTRARIAND P. VERNOLE
It is well known, [see also L e m m a 3.1 and R e m a r k 2.2] that we have a unique solution if
(3)
I1~ - no + OtUo)ll~o,, ~o ~ rot1 - ~)holds. The inequality (3) is verified since }Iv - Uollxo,.,~o = sup
IleA'x - Xollo, v -%< sup
t~[O, To]
t~[O, To]
IleA'(x - Xo)ll0,, +
sup
re[0, To]
I1:%-
Xok,
<~ M 6 o + % ( T o)
and IIO(Uo)Lo ,.To ~
~(To)mo(To)
so that IIv - u o + (1)(Uo)llxs,~,ro ~< M 6 o + % ( T o) + mo(To)fl(To) = M 6 o + ?o(To) - roKofl(To)
~< rol-1 - K o f l ( T o ) ] <~ ro(1 - cO. PROPOSITION 2.2. Let x 1, x 2 ~P(xo, 30). If u I and u 2 ~ X o , p. ro are the mild solutions of (P*) corresponding to the initial values x 1 and x 2 we have
where M CO
l -- Kofl (To)
In fact, if we set v i = u i - O(ui) we have
1
1 sup leAtx1 -- eAtx2}l I1~, -~211xo,~,~o ~< i - K o f l ( T o ) ,~t0,ro]
Ilu~ - uzllxo,~,To "%<~
REMARK 2.2. Let u ~ Xo. p, r be a solution of (P) in [0, T ] and v e Xo, v, rl verify
eA,
/)(t)
+ fl °
+
, ts))ds
0. t
.<
/4)
If we set fu(t) w(t) = ( v ( t - T)
O <.%t <<. T T <~ t <~ T +
T1
(5)
it can be checked that w e X0,p, r + r , and verifies (P) in [13, T + 7~]. Conversely ifw is a solution of (P) in [0, T + TI] and if we define u and v by (5), it is easy to see that v verifies (4). where u(T) = w(T). The uniqueness on the large of solutions to (P) now follows. 3. D E P E N D E N C E
F o r every n e N let An:
D A --~ E --~
OF SOLUTION
ON THE DATA
E be a linear o p e r a t o r verifying the condition (L1) with the
Semi-linear evolution equations in interpolation spaces
255
same constants M 1 a n d M 2. Let us also suppose that the following condition holds: (A) lim IleA'X --
e^'xll0., = 0 uniformly in t e I, a c o m p a c t interval of R+, and x ~ K, a com-
n .-~ oo
pact set of E. F o r every n ~ N, let tpn: R+ x D^(O,p) ~ E be a continuous function, verifying the condition (F 0 with the same constants r x and K~. M o r e o v e r let the following conditions hold : (B) lim II~o.(t,x) - ~(t, x)ll -- o uniformly in t ~ I, a c o m p a c t interval of R+, and x e K, a comtl--~ oO
pact set OfDA(0, p). The following t h e o r e m describes the dependence of the solution to (P) from the data A, ¢p and x. THEOREM 3.1. G i v e n x o • DA(0, p), let 60, TO be the positive constants of T h e o r e m 2.1. If we choose x. D^(O.p). IIx. - xollo., ~< 6o and T0 ~ ]0, To[,, then for each n t> n o (depending on x o and To) there exists a unique u. e X0, p, rb such that
u.(t)=eA"'xn+f'oe^""-S)q~.(s,u.(s))ds
O<~t<~To
(P)n
If in addition lim x. = x
DA(O,p)
in
(3.1)
n---b o o
and u ~ Xo, p, r6 is the mild solution to (P) with initial value x, then we have lim u. = u
in
Xo, p, T6
M--* o o
Proof. Let ro, Ko, 60, To be the constants of T h e o r e m 2.1. As in the p r o o f of T h e o r e m 2.1 let us define for each n e N and T > / 0 0t(o")(T) =
sup
Ile^-'Xo - xoll0.,
sup
![¢pn(t,Xo)[[
le[O, To]
m(on~(T)=
te[O, 7"]
T(o")(T) = a(on)(T) +
m(o")(T)~(T)+ roKofl(T)
F r o m (A) a n d (B) it follows that lira 7(o")(T) = ?o(T) for each T >i 0 and x o a
DA(O,p). If we choose
R~oo
T Oe
]0, To[- we have lira ~,(o")(To)-- yo(T0) < yo(To) ~< r o - M6 o
hence there exists n o (depending on x o and T0) such that ),to")(T0) <~ r o - M 6 o
for
n ~> n o
(3.2)
Let us suppose n >/no and x. eD (0, p) such that [Ix. - Xoll0.p ~< 6 o. Set v.(t) = eA'x., o each t e [0, To]; the solutions to the equation (P). are the solutions u. ~ X0,p. r~ of the equation
Uo(t) = x o for
u. - q).(u.) = v.
(3.3)
256
E.
SINESTRARI A N D
P.
VERNOLE
where ~ . : P(uo, ro) ~- Xo, p, To ~ Xo. o. T6
¢.(u)(t)=f'oeA""-s'~,(s,u(s))ds
is a contraction map with Lipschitz constant a. ~< Kofl(Td) < 1 because of(3.2). The problem (3.3) admits a unique solution if we have ]l~.
-
Uo + *.(Uo)[]xo, p,~.o <~ ro (1 - ~.)
As in the proof of the T h e o r e m 2.1 we get
I[~, - Uollxop,r; < M 6 o + a(o"'(To) ]l(I).(Uo)]lxo,.,.,; ~< B(Vo)m(o~'(T'o) hence because of(3.2)
[J~. - Uo + *.(Uo)]lxo,.,T, < to[1 - KoB(T~)] < r o 0 - a.) T o prove the last part of the theorem we need a simple lemma: LEMMA 3.1. Let a be a Banach space with n o r m II" 11and for each n ~ N let * , : P(xo, r) ~_ B ~ B be an application such that: (1) HtI).(xl) - (I).(x2) H ~< allx 1 - x2{ [ Vxl, x 2 e P ( x o , r ) , n e N w i t h O (2) there exists for any x e P(x o, r), lira q).(x) = q)(x).
< a < 1.
n~ct3
(3) y. e P ( x o - ~.(Xo), r(1 - a)) and there exists lira y. = y.. Then there exist unique x, x. e P(x o, r) such that "~ ~ x. - (l).(x.) = y.
(3.4)
x - ¢(x) = y
(3.5)
M o r e o v e r we have lim x. = x n---b oO
P r o o f The first part of the lemma follows from the contraction mapping principle, applied to the applications of P(x o, r) into itself: x ~ y. + ~.(x)
and
x ~ y + ~(x)
F r o m (3.4) and (3.5) we get
I1~. - ~ll = II0.(~.) - ¢(x) + y. - yll < IfO.(x.) - ¢.(x)H + II¢.(x) - ¢(x)rJ + [ly. - y[f ~< allx,, - xll + ]l(1),,(x) - (I)(x)ll + Ily. - yll; hence we have 1
11~. - xll < -i-~_ ~ (ll¢.(x) from this the thesis follows.
-
¢(x)l[ + []y.
-
Y[I);
Semi-linear evolution equations in interpolation spaces
257
Let us complete the p r o o f of t h e o r e m 3.1 by using the preceding lemma. Let t e [0, To]; we have
Ile^-,x. - e ,x O,p <~ IleA"tXn -- e^°,x O,p + Ile^-'x < Mllx.-
xllo,.
+
O,p
Ile "'x - eat. II
from (A) we get lim v. = v in Xo, p, r6 Let us p r o v e that
lira n~oo •
f:
;o
e ^"~'- ~)q).(s, u(s)) ds =
e ^<'- ~)~o(s, u(s))ds
uniformly for t e [0, To]. As
f'oe^"('-~)q~.(s,u(s))ds-f'oeA"-~)q~(s,u(s))ds =
u s)) -
u(s))] ds +
-
s
u(t -
s)) d s -
-
s
u(t -
s))
s
T h e first integral can be majorized by
fl(To) m a x ]lq~.(t, u(t)) - ~o(t, u(t))ll re[O, Tb]
hence it goes to zero for n ~ ~ (because of (B)) ; the last difference converges to zero uniformly on [0, To] for the a s s u m p t i o n (A). T h e next t h e o r e m shows that a solution of (P) can be uniformly a p p r o x i m a t e d by solutions of (P). in every c o m p a c t interval [0, T]. THEOREM 3.2. F o r each n e N, let A., q~. and x. verify the assumptions of T h e o r e m 3.1. Let u e Xo,v, + be a mild solution to (P*) in [0, T]. F o r sufficiently large n there exists u. e Xo, p, t, a solution of (P). in [0, T], and we have lira u. = u in Xo, p, ~. n--+ co
Proof We proved that given y e DA(O,p) there exist 6 r, Tr and ny such that if x, x . ~ P(y, 6y) and n >~ n r there exist u and u. in Xo,., ry, solutions of (P) and (P). respectively and !irn~ u. = u in Xo, p,r.; we can suppose that if we change ~o with (t, x) ~ q~(t + h, x) and ~o. with (t, x) ~ % (t + h, x) we can use the same constants 6~. Tr and n~ for each h e [0, ~ ] " it is sufficient to set in the proofs of T h e o r e m s 2.1 and 3.1, t o o ( T ) = sup [Iq~ (t, Xo)ll and m(o")(T)= sup IIq~. t E [ 0 , ~ + T]
t ~ [ 0 . ~ + T]
(t, Xo)II. Let F be the range of u: [0, ~ ] ~ DA(O, p) and H a finite covering of F m a d e by 1~ = P(y, 6y), y ~ DA(O, p). Let x o be such that u(0) ~ Ixo ~ H ; there exists n o such that if n >~ n o then x. ~ lxo and (P). has solution u. in [0, Txo] a n d t i m u. = u in Xo, p, rxO" If Txo < ~ let x 1 be such that u(T~) I~, ~ I1 ; there exist nl such that if n ~> n o + n 1 then u.(T~o) e Ix, and (P.) has solution u. in [0, T~o + T~,] and lira u. = u in Xo,.,r~ +r~ (see R e m a r k 2.2). After k times the solutions of (P). are defined (for sufficiently large n) on an interval of amplitude 1 >t zk where r = min {Ty, Ix ~ H} : so that when k >t T/T the thesis follows.
258
E. S[NESTRAmAND P. VERNOLE 4. C L A S S I C A L
SOLUTIONS
AND
A PRIORI
ESTIMATES
In the present section we use some results of [7] for the solutions in X r to the problem: u'(t) = Au(t) + 9(0
0 <~ t <~ T (4.1)
u(0) = 0
where 9 E X r. PROPOSITION 4.1. Let u e Xo, p, r be a mild solution to (P*). Set g(t) = q~(t, u(t)) t ~ [0, T ] ; there exist u. ~ Xo, p, r, continuously differentiable from ]0, T ] to E such that lim u. = u in X r and n ' - * O0
g. E X r such that lim 0. = 0 in X r and n.--~ oo
u'.(t) = Au.(t) + O.(t)
0 < t <~ T
(4.1n)
u.(0) = x
Proof We can write u = u o + u I where Uo(t) = eAtX and ul is a strong solution (according to [7]) of the problem (4.1) where O(t) = q~(t, u(t)) t ~ [0, T]. The conclusion follows from the theorem 4.5of[7]. PROPOSITION 4.2. Let ~0: R+ x DA(O, 19)--~ DA(O', p), O' E l 0 , 1[, p'~ [1, + oo] be a continuous function and u e Xo.v.r a mild solution to (P*): then u is a classical solution to (P*), i.e. for each t > O, u(t) ~ O^, u is continuously differentiable from ]0, T] to E, and (P*) holds.
Proof. As u E Xe, p, r, we have g(. ) = ~0(., u(. )) e Xe; v', r - C(0, T; (DA, E ) I _ 0', oo) thus setting
ugt)=foe^"-"O(s)ds from theorem 4.7 of [7] we know that u x e C(0, T, D^) n C~(0, T; E) and verifies (4.1) in [0, T]. Suppose now that A and q~verify the assumptions of Theorem 2.1: it is easy to see that for each x ~ D^(O, p) there exists a unique maximal mild solution u*: [0, T * [ - * DA(O, p) where T* ~< + oo. F r o m theorem 3.2 it follows easily: PROPOSITION 4.3. F o r each n ~ N, let A., ~o. and x. verify the assumptions of Theorem 3.1. Let us choose T < T*: for sufficiently large n, there exist u. ~ Xe, p, r, solutions to (P). such that lim u. = u* in Xo, p, r. n---* oo
PROPOSITION 4.4. Let ~o verify the following condition for each T > 0: if u: [0, T [ --* DA(O, p) is continuous and bounded then q~ (-, u(.)) e E(0, T; E) with r > 1/(1 - 0). Then if u*: [0, T*[ is a maximal mild solution to (P*) and T* < + oo, we have sup IIu*(t)IIo, p t ~ [o. T*[
=+00.
Proof. Suppose that u* ~ U°(O, T*, DA(O, p)); from (F2) it follows that (o(., u*(')) ~ E(O, T*, E). Hence, by proposition 2.1, Qu* E Xo, p,r.- Since u* can be extended to a solution of(P) in [0, T*] we get a contradiction by applying Theorem 2.1.
Semi-linear evolution equations in interpolation spaces
259
THEOREM 4.1. Let ~p verify the following condition:
(F3)
D+llxll~(t,x) <. 0
x ~ D^(O,p),
t >10.
Let {eAt} be a contraction semigroup i.e.?
(t~)
Ile^'ll~,,~, .< 1.
Let u: [0, T [ ~ DA(O, p) (0 < T ~< + oo) be a solution to (P)in [0, T[; then Ilu(t)ll <. [lu(O)l[ for each t e [0, 7~[.
Proof. We need the following lemma: (see [9]). LEMMA 4.1. Let v e Cl(a, b; E): then t ~ IIv(t)ll is Lipschitz continuous and we have
IIv(t)ll --V(a)II + ~' DUv(s)ll,~'(s)ds
t~[a,b].
We give now the proof of the theorem. If we choose T e ]0, ~[, u is the mild solution to (P*) in [0, T]: let u n and fin be the functions of the Proposition 4.1. Let 0 < ~, < t ~< T: if we apply the Lemma 4.1 to the functions u, e C1(~, T; L0 from (4.1n) of Proposition 4.1 we get
Ilu~(t)ll- II..(*)ll =
ollu#( ))lu~(s)ds
<- f' D +llu,,(s)llAu,,(s)d~ + ~' D +li,,,,,(s)li,,,(s)ds <~
ds
by virtue of (L2). Using Theorem 11.3 of [10] and taking max lim we get n-¢oo
II~t)l[- I~011 -< ~<
;
max lim n~oo
max
lim D+ Ilu.(s)llo.(~)ds < n -,-~ oo
f" O+llu.(s)llg~(s)d~
;o+
Ilu(s)llq,(s, ~(s))ds .< 0.
Letting e ~ 0 +, the conclusion follows. THEOREM 4.2. Let (F3) and (L2) hold and 9 verify the following assumption: (see [11] p. 175) (F,). There are continuous functions a(.), b(.) from R÷ to R÷ such that
Ilk(t, x)ll -< a(llxll) + b(llxll)llxlle,~t >1 0, x
~
Do(O, p).
Then if T < + oo and u: [-0, TI- ~ DA(0, p) is a solution to (P) we have sup Ilu(t)lk~ < + ~ .
te[O, T[
f If x, y ~ E, D]]xl]y(D + ]]xHy) denotes the derivative (right derivative) of the function t -* Ux + ty]l at zero.
260
E. SINESTRARIANDP. VERNOLE
Proof From Theorem 4.1 it follows that for any t ~ [0, T[Hu(t)H <<. I]u(0)]l and from (P) we get ]Ju(t)lJ°'P "< Mllxl]°'P+
f'o M(1
+
1/t- s)°(a(llu(s)]l)+ b(lJu(s)llu(s)Jlo.) ds
As a(.)and b(.)are continuous they are bounded on the interval [0,
Ilu(t)J]o,. <. Co + cl
(t
s)-°llu(s)llo,
-
Ilxll], hence we have
ds
From a lemma (similar to Gromwall's Lemma) proved in [4] we get
Ilu(t)ll0, .<
c s e r'
hence the conclusion follows. F r o m the Proposition 4.4 and from Theorem 4.2 we get the following theorem about the global existence of the mild solution to (P*). THEOREM 4.3. If A verifies the assumptions (L1), (L2) and ~p verifies the assumptions (Ft), (F2), (F3), (F4); then for any x ~ D^(O, p) there exists a unique mild solution to the problem (P*) in
[0, + 5. APPLICATION We give now a very simple application of the preceding theory. Let f~ be a bounded domain in R* with smooth boundary df~. Let 0 ~ ] 1/2, 1[ and p > n/20 - 1. We put E = LP(~). We consider the following problem
c~u(t, x) ~ c~2u(t,x) = ~ ~x-----~i + Z., (u(t, x))"°(D,ju(t, x)) nj ~t i=1 j=l u(O, x) = h(x) u(t,x) = 0
x ~ x ~R~t
~ [0, T]
where no, ni are some non-negative integers, h is an assigned function in W 20"p (f~). It is known that the operator A defined by D A = W2'/'(~) n WI'P(~)
Au = Au verifies (L1), (L2). Moreover it can be proved that DA(O, p) = I/Vo2O'P(~'))~ W 1' P(~')) (see [8]). Setting
~p(u) = ~ un°(Dx~u) ~j j=l
and using Sobolev's lemma, we can verify the assumption (F1). Conditions (F2), (Fa) , (F, 0 hold for instance when p is even and n o = nj = 1.
Acknowledaements The authors would like to thank ProfessorsG. Da Prato and P. Grisvard for their helpful advice.
Semi-linear evolution equations in interpolation spaces
261
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