Dynamic theory of quasilinear parabolic equations—I. Abstract evolution equations

(9)

S > 0 so that

T1-Or-b e-sro(l-E)

’ (1 - s)-~s-~~~ I0

d,sllkjjca,Bj s p/2.

(10)

Dynamic

Now we deduce

from (7)-(10)

theory

of quasilinear

parabolic

899

equations-I

that

lll~lll6 :=

~uPw4lla~ lll4lls G 1) c PT

provided 6 > 0 is suitably chosen. Hence 1 E p(K) and u := (1 - K)-‘a is the unique solution of F, is open in Z(F) and of (6) in F : = B&r(E). S ince %92(F), the group of automorphisms [B ++ B-l] E Cm@%(F), L?(F)), th e second assertion follows from [k++Kl which is a consequence

COROLLARY quasinilpotent

of (4).

E T(K,((E,

m, P), ~(B,C,(E))),

n

1.2. Suppose that in Z(B,CT(E)).

((u + p) v (p + y) < 1. Then

Proof. This follows from the arbitrariness

of p in (0,l).

the

integral

operator

K is

n

Remark 1.3. The assumption that ((u + p) v (p + y) < 1 in the above theorem is crucial. This is a consequence of the following example, which I owe to Quittner. Let E := [w and k(t, s) := +(t - s)-%~/*, so that k E K(E, (Y, /3) with a: = /? = l/2 and Ilkllca,s,= l/2. Then the integral operator K with kernel k satisfies K E Y(CdE)), by (4), and has the eigenfunction u(t) := t”* to the eigenvalue 1. n

2. Interpolation spaces We denote by % the category Banach spaces and the bounded

of Banach spaces, whose objects and morphisms are the linear operators, respectively. ?A2 denotes the category of densely injected Banach couples, that is, the objects of ?A2are the pairs E := (E,, E,) of Banach d

with E, 4 EO, and the morphisms T : E -+ F are the maps T E X(Eo, FO) satisfying T E X(E1, F,). (We do not distinguish between a map and its restriction if no confusion seems likely.) We write ,?? E ?AA2 to indicate that .)? is an object of s2, and E_:= (E,, E,) for I? E ?&. For each 8 E (0,l) we choose any one of the real interpolation functors (. , .)o,_,. 1 s p < 00, the complex interpolation functor [ *, *lo, or the continuous interpolation functor (. , .)g,, of Da Prato and Grisvard [ll] (cf. also [12]), and denote this fixed choice by B,(E). We refer to [9, 20, 291 for the basic facts of interpolation theory. (If Dd= R we have to complexify the spaces to use the complex interpolation functor [. , .le. Then we get back to the real situation by letting E e := [E& rl EO, endowed with the topology induced by [E&.) Then it follows from [6, (5.3)] that spaces

E,~E,,~E$+EO, where E 0 := B,(E). If 0 c q < 8 < 1, then

EE93*,0
r] and 8 are well related if either I/.x& s cl-#,-“‘ell~]l$

x E Ee,

(1)

8 = 0, or 8 > 0 and 12E

932

3

.(@)g=)Ls 3 v 103 uorvqos pwawepun3 3yoqe.xed auo ~sour be sJs!xa alaql 3.’ uo!lnIos plyI au0 JSOLU$12sassassod ‘iv‘“‘“‘(~~) ‘,+It?ag .‘/‘v’“‘“‘(d3) 30 uoynlos ppzu e s! (s ‘ .)je n + x(s ‘ . )n =: n uaql [J ‘s] uo pauyap IIaM s! (s ‘ . )j * 0 keys duadold ayl sey “g c [J ‘01 : 43! pue ‘02 3 x 3’ ‘(@)x)Js 3 v .103 uognlos lewaurepun3 sgoqezd e SFn 31 2 3 3 3 2 3 s 3 0 ~03 (s ‘l)fl(2 ‘l)n = (s ‘I)fl wql (n) pue LIE s! $1 ‘(0 ‘Og)ly 3 [(I)~w (s ‘I)] ~I!M (0~~)“s 34 L3guap! aM alaqM (!) 30 a3uanbasuoD

Aq3tw3

ha

-wauq aql 30 [J. ‘s] uo uognIos

‘(s ‘ .)l* fl + x(s ‘ .)fl = n uaql ‘(J’v’“‘s)(d3) walqold e s! n pue (Og)Js x OTJx (J ‘01 3 (j‘x‘s) 3! (u) ‘(1 ‘~)Lx

u (Pa)%

‘Yzb

u ((“a)“~

‘V&J

3n (9

(a)~)“3 3 v .mJ uoynlos pwaurvpun~ xgoqvmd e aq 01 pys s! (Q)a; t VJ : n uoyun3 v ‘! ‘Og = A wy$ amnsw ahi ase3 syl UI .(JV‘“‘“)(d3) Icq alouap v 31 ‘Jpue aM y~!y~ ‘utalqoAd riz&mv~ .mau~]e s! (“‘“)(d3~) uaqJ ‘ n 3o 1ua p uadapur a_wJpue v uodn aDuapuadap sl! aw!pu! 01 (Jv‘x’s)(d3~) Aq (“‘“)&)o) alouap ~[cys am IC~~euo~se~~~ .uognlos 1vqo18 c s! n uayl ‘(CO‘s] u I = f 31 .azkw~xa 40 ~vawy ~vu4~xt9tu B s! f pue uognlos ~vu.i~xvtu e S! n uaql n 30 uo!sualxa ladold e sy qzyq~ uognlos e Is!xa IOU saop alaq] pun [ uo uognlos c y n 31 .(Os ‘f-)[3 u (‘2 ‘f-)3 3 n 31 13?.41ss! f uo n uognlos v ‘1 uo asylu!od ‘“‘“‘(dam) rz sy f 3ey~ y9ns SaLJS!WS q3!qM ‘0 # {S}jJ =: [ q)!M ‘S %U!U~EJUO3‘( CO‘s] u 1 30 pzhIalu!qns (‘9‘~-)3

u (04‘~)r3

u GI ‘r)s 3 n

uogmn3 ‘x=

(“‘“)(d30)

(s)n

‘(CO‘S)~131 ~qqo~d

O8 x (2% hkuauou

Xy3nq

+A

‘(n‘l)J=

n(n‘l)v

muy!svnb

B wmu

aM

+ 12

aql30

1 uo n uotlnlos e ICY .n x z 3 (x‘s) Ia1 pw

X z : (S‘v)

e aq A Ial pue ‘0 BU~U~WIO~ +H 30 leAlawqns

ley~yuou

wql asoddns +J 30 lasqns e aq 1 Ial ‘ZE 3 TJ 137

stualqo.td Xy3nv3 ‘E

q8noql ‘IWaUa8 alow e u! 710~ aM ‘suoyen$rs ala.woD 01 s$Insal Iscesqe aqJ30 uope3gdde aql 103 rluyq’xag Iawal8 e uyqo 01 lap10 u! ‘IaAaMoH ‘asm srql u! paysyes li~~e~~~ewo~ne aq 111~ ssaupavqal IIaM 30 suogdtunsse aql airs Inoq8nolq) ‘Iopun3 uoyqodlalul xalduro3 aql $?.a ‘poqlaur uoyqodlawr paxy I? sasn au0 3! $eqMawos lCJgdw!s II!M uoyeluasald 8u~~ojlo3 aqL .0/h luauodxa 30 louun3 uogtqodlalu! due ST@I’%pue ,,SUIIOU waIeA!nba,, sueazu F araqM ‘(By ~OyJ)BPfg = L"g

‘%3g

3! pavIal

IIa.4 am 8 pue 0 wql amasqo

‘1 3 m 3 0 ‘ng u! WIOU aql salouap

NVlWIV

nil. 11alaqM 006

‘H

Dynamic theory of quasilinear parabolic equations-1 It is well known that a linear operator there exist constants M and o so that

901

to S?(E) iff A is closed and

A : El C E0 + E,, belongs

(i) {h E @; Re /J 2 o} C &-A), G M/(1 + IA]), Re il 2 w. (4 I@ + A)-‘~~~~cE,~ A subset SB of x(E) is said to be regularly bounded if it is bounded in LZ(EJ, if there exis’ constants M and o so that (i) and (ii) are true for all A E SQ, and if {(w + A)-‘; A E Se} is bounded in 9(E). LEMMA 4.1. “de(E) is open neighbourhood.

in JZ(EJ and

A E X(E)

each

possesses

a regularly

bounded

Proof. By the open mapping theorem, D(A) = E, for A E X(E), where D(A) is given the graph norm._Hence it follows that (A + A)-’ E 2!?(E) for A E p(-A). Thus B(L + A)-’ E Y(E,) for AEX(E), BEJ.??(EJ and A E p(-A). Using this fact, the assertion is now an easy consequence of the proof of [25, theorem 3.2.1.1, the smoothness of the inversion map B-+ B-‘, and the above characterization of the elements of Z(E). n For T > 0, 0 < p < 1 and a nonempty

subset

M of a Banach

C+(M) := P([O,

space F we put

T], 44).

Then a subset d of Ct(SY(E)) is regularly bounded if {A(t); 0 s t G T, A E Se} is a regularly bounded subset of X(E) and there exists a constant L such that ]]A(s) - A(~)]~z~E) s J$ - ?, Of course,

(1) means

that d is equi-Holder

PROPOSITION 4.2. C+(Z(E)) bounded neighbourhood.

is open

Proof. This is an easy consequence

$9 t E LO, Z-1, A E d.

continuous

in C+(LE(E))

of lemma

of exponent

(1)

p.

and each A E C$(X(E))

4.1 and the compactness

has a regularly

of [0, 7’j.

n

1s If X is a metric space, then a subset fi of X x C$(%(E)) . said to be regularly bounded if there exist a bounded subset B of X and a regularly bounded subset Se of C$(SY(E)) with JM C B x ~4. Finally a map 91 : X+ C+(%(E)) is said to be (locally) regularly bounded if q,(X) [each x E X has a neighbourhood V so that q(V)] is regularly bounded in C$(‘X(E)). 5. Linear Cauchy problems Let ,?? E S3,, 0 < p < 1 and T > 0. Then we have the basic THEOREM

C$(X(E)),

5.1. There and u,

exists

EK,(E,,Q,~

a unique

nRr(~,

parabolic

fundamental

Elyi - 6) nwEEa

solution,

U,,

for each

n wAdwE)>

A E

(1)

forO
K,(E~,

E,,

E) n K,(E,,

E,, 1 - E) n K,(E,,

01,

A ++UUA

(2)

‘/‘v’“‘&J) tualqold

‘X = (rj)n Aqme3

leatry n

‘J S 1 > 0 aql laprsuos .uogeIodIau.r!

‘(1)s = nO)v am (Q)L3@g Lq (z) pw

aAold aM uaqL + n 3 4 pue ((g&)$3 _

3 v .IO~

(I) ~11013SMOIIOJ s!y~

.slas papunoq

@e@?aJ

$oo.q

uo papunoq

s!

Vn++v ‘W- 2 ‘%I‘ W% +&%e)+i3 dmu ayl pue

uaqL

.alqets

s! (2 ‘0) ~eyl pue 1 > 2 > 0 > 0 ley~ asoddns

‘2.5 AWTIOXO~

-raw! xaldwo3 aql +.a ‘poylaw uogyod~a$u! paxy e asn aM 31 alqels IC~pmymo~ne s! (2 ‘0) ley~ amasqo ‘0 luauodxa 30 0% lolcmn3 uoyeIodla$u! amos 103 (12 ‘l’g)(‘-I)l(‘-~)~ y ~TJ _IO (%J ‘Q)U’“k + %J layl!a pm 1 3 2 3 b 3 0 31aIqe$s aq OJ p!es s! (2 ‘0) .yed IZ~U!MOIIO~ayl UI n ‘([9] OSIE .33) [E’S euu.ua~ pm Z’S umoayl ‘z] 30 samanbasuo3 am suoyasse 8u~u~etua~ ayL ‘([gz ‘g ‘61 ‘~11 OSIB .3r~) [,y] aqeutg+ pue [97J gsAaIoqoS 30 s$Insal umouy IjaM u10.13SMOIIOJvn 30 ssauanb!un put amaysyxa aql $ooAd

NVPIPiV

‘H

206

Dynamic

Proof: 5.2, and

theory

of quasilinear

parabolic

Let k(t, s) := UA(t, s), so that k E Kr(E,,

U.-,*Of, 0) =

903

equations-1

E,, E - c), by theorem

I’W, s)f(s) h,

5.1 and corollary

O
0

It follows

from (1.4) and (1) that u := U,(.

,0)x + U, *f(.

The boundedness of the map (3) is corollary 5.2. Suppose now that 5 + 6 < 1 + 5‘. arbitrary,puty:=u(s),anddefinegE Then [2, theorems 4.1 and 4.21 imply Since u(t) = U(t, s)y + u *f(t,

, 0) E S&(ES)

II B1_&(E1).

an easy consequence

of (1.3),

(1,4),

theorem

5.1 and

Then u E C_XE& by (1) and (1.5). Let s E (0, 7’) be Cr(E&bygl[s, T]:=fl[s, T]andg(t):=f(s),OGtGs. that (CP)(s,y,A,g) has a unique strict solution u on [s, T].

s) = U(t, s)U(s, 0)x + u(t, s)U *f@, 0) + u *f(t,

s) = u(t)

for s G t < T, by the properties of the fundamental solution U : = U,, we see that u is a solution on ]O, Tl of (Cp)~,f,~ which is strict if 5 = 1 and f E CAEJ. Since every solution is apparently a mild solution (if g + 6 < 1 + f), u is the unique solution of (CP)(,,A,fj. (0, T], we infer from (3) that IIt’-%(t)](, < c Suppose now 5 + 6 < 1. Since zi = -Au +fon for all (x, A,f) in a regularly bounded subset of Eg x C+(X(E)) x B&(E,). Hence Ilu(t) - u(s)IIo G j-’ Ilt’-%(r)llo(r s for 0 < s < t s T. Hence

u E Ct(E,),

and the remaining

Remarks 5.4. (a) If we drop the assumption solution u E BCAE,) and the map Eg x CPT(WE))

is bounded

on regularly

bounded

are now obvious.

&(E&

has still a unique

(x,A,~)H

n mild

u

5 + 6 < 1, but assume, instead that fE B60cr(EO) for other hypotheses), then the above proof shows that

sets, provided

THEOREM~.~. Supposethat0<5cl,O< and that (g, 5) and (5,~) are stable.

assertions

sets, provided 5 < 1. above proof, since 5‘> 0 has only been used for proving

x &&(Eo)-,

The following continuity theorem, 0, 1, is now almost trivial.

dt s (c/c)@ - s)~

that 5 > 0, then (CP),,,A,f,

x &&(Eo)+

is still bounded on regularly bounded This follows from the first part of the the assertions about B1 _ ,&dE,). (b) If we drop the assumption that some 6, E [0,1 - 51 (and retain all u E C$-“(E,) and that the map E, x CP,(W@)

- s)c-l

C;-“(E,), 5 < 1.

in which ui denotes

(~A,f)e

u

n the mild solution

of (CP) (X,,Ai,fjj, j =

[==r]
Then

904

there

AMMAN

H.

exists a constant

c, so that the following (xj,Aj,fj)

estimates

are true for all j = 0,l.

E EC X GJ X BeCr(

(i) If q G E, then

(ii) If &‘< r~ and E, + %(rl-5),(1_5)(E5,E,), then

IlkI - 4B~_C”~(E~) =z4llAo- AIIIc@?(J-%IIE + IlfOllB&fE~)) + llfo-fIIIB&fEg)+ 11x0 -fill& Proof. This is an easy consequence Ug-Ur where

= (Ucl - U,)x,

of

+ (Uo - U,) *fo + Ul * (fo -f1)

r/j := UA,, i = 0, 1, and of theorems

It should be remarked that theorems and 5.4 of [2], respectively.

EXISTENCE

AND

CONTINUITY

5.1 and 5.3 and corollary

5.3 and 5.5 are sharpened

FOR

QUASILINEAR

+ U,(xo -x1), n

5.2. versions

CAUCHY

of theorems

4.2

PROBLEMS

6. A local existence result Let E. and El be Banach spaces with El+ Eo. Then E, is said to be relatively complete with respect to E. if El coincides with the set of all x E E, which are limits in E, of bounded sequences in E, (cf. [20, paragraph I.11 for more details). Clearly, El is relatively complete with respect to E. if El is reflexive. We assume now that

(01)

are stable.

V is open in E,, and E, is relatively IfMisasubsetofEoandO<~
complete

with respect

to Eg.

A4 n E,, endowed with the topology induced in E, if rl s c s 1 and M, is open in E,, as follows from

E,=+ E,. Let X, Y and 2 be metric spaces. Then we write f E Cp*“(X X Y, Z), p, (7 E (0, l), if each point in X x Y has a product neighbourhood U x V so that

d(f(xo,Yo)>f(x1>Yd)

~44xo>xl)lP + ]~(YO>Yl)l”~~

(1)

for (xi, yi) E U x V, j = 0, 1. If (1) is true with o = 1, we writef E CP,‘-(X X Y, Z). (In general l- stands for (locally) “Lipschitz continuous”.) We write f E C’*“(X X Y, Z), o E (0, 1) U + Z is uniformly a-Holder continuous on {l-},iff(.,y)E C(X,Z)foryEYandf(x;):Y V, uniformly with respect to x E U. Similar notations are employed for product spaces with

905

Dynamic theory of quasilinear parabolic equations-I

more than two factors. Finally, we write fE properties are uniform. Using these conventions we assume that A is a metric

I!JCP,~(X x Y, Z) if the indicated

continuity

space.. A E CP*‘-s”([O, T] x V, x A, X(E))

for some p E (0, l), (T E [0,1) U {l-} The map

and T > 0.

(Q2)

[(Y>~)-~.,Y,~)];

is locally

regularly

V, x A-

bounded. fE

co,1-,o

(PA

Then we prove the following local existence, uniqueness quasilinear parameter dependent Cauchy problem

we consider

first the special

Tl

x

VP

x

A,

&>.

and continuity theorem for the

s
ti + A(t, u, A)u = f(t, u, A>, where

C$.X@))

u(s) = x,

(QW(s,x,q

case that s = 0,

6.1. Let S x K C V, x A be compact and nonempty. Then there exist bounded closed neighbourhoods U and W of S in E, and 2 of K in A, respectively, and a constant t E (0, T] with the following properties:

PROPOSITION

(i) S C U C ti C W C V,. (ii) (QCP)(O,,J) has for each (x, A) E U x 2 a unique u(. ) 0,. ) *) E w~‘-q[o,

solution

u(. , 0, x, A) on [0, r] so that

t] x u x 2, W).

Proof. Let U and W be the closed I- and 2.s-neighbourhoods of S in E,, respectively, for some E > 0. Then the compactness of S and the openness of V, imply that we can choose E so that (i) is true. Moreover, by (Q2) we can also assume that there are a regularly bounded subset % of Cg(x(E)) x C&E,) an d a closed neighbourhood C of K so that (y, A) E w x X} c %.

{(A(.,y,A),f(.‘,y,;i));

(2)

For 0 < t =z T let se, := {c][O, t]; c E %} and Wt := {w : [O, t]-+

w; j/w(s) - w(t)llp =s 1s- tip>,

endowed with the topology induced by C,(E&. Then we deduce from the boundedness in E, and (01) that Wr is a complete metric space. ForwEW,andAEZletA,,,:=A(.,w(,),A)E C~(%?(E))andf,,,:=f(.,w(.),~)~ C,(E,). We can obviously assume that % has been chosen in such a way that {(Aw,n,fw,*);wE”Mr,,~E~:)C(e.,

O
of W

H. AMMAN

906

By theorem

5.3, the linear

U x W, x 2 a unique

Cauchy

solution

problem

(CP)

possesses, for each (x,w, A) E

(x,Aw,L,fw,l)

u( . ,x, w, A) on [0, t] with

u(. ,x, w, A) E C,(E,)

f-l C:-a(&),

(3)

and the map u x cMrt x z-, is uniformly

bounded.

C,(E,)

n c;-B(Efi),

that p < LY- p, it follows that

Since we can assume

(X, w, n) E u x “Mr, x c,

U(.,X,W,A)EWr, provided r is sufficiently small. Suppose first that (T > 0. Then

we deduce

114.>x0 wo>A) - 4. 9

(&W,h)HU(.,X,W,A)

from theorem

5.5 that

,x1> w17 PL)llC,(E&q)

c w-~llwo - W1lIC,(Eo) + 11x0for (xi, wj, A) E V x ‘IF, x Z:, j = 0,l. that c~~~fi c l/2, that is, that

Il4.A

Hence,

wo,a> - 4.,x>

for wo, w1 E “Mrr, h E E and x E ZJ. Thus theorem that the map

by making

t smaller,

it follows

from

PI”>

if necessary,

(5)

we can assume

(6)

(4), (6) and Banach’s

fixed point

w H U( * ) x, w, A)

has for each (x, A) E U x C a unique fixed point U( I, x, A). By applying again theorem 5.5 (with E = n = a), we obtain

the estimate

- 4*~YAIC,(E,) s 44

7X?A) - 4.) YJ Pu)llC,(Efi) + Ik - Ylla + 4k

for (x, A), (y, 1~) E U x 2. On the other

hand,

Ilu(* 9x9A) - 4.2 Y, P)IIC,(EB) =s4x Hence the assertion the above argument

Xllln + 4L

4IIC,@)6 dlwo - WllIC,(Ep)

w17

wX+Wr,

lM*.%~>

(4)

PI”1

(7)

(5) with CV-@ < f implies

- Ylla + 44 P)“),

x,yEU,AEX

follows from (3), (7) and (8). If o = 0, it is obvious to obtain the assertion in this case also. n

(8)

how one has to modify

Remark 6.2. It is clear that proposition 6.1 remains valid if S is an arbitrary nonempty of VInwhich possesses a bounded closed neighbourhoodbW in V, so that (2) is satisfied. case U can be any neighbourhood of S satisfying UC W. n

subset In this

7. General existence, uniqueness and continuity theorems Given assumptions uniqueness.

(Ql)

and (Q2) it is now easy to prove the following

global existence and

THEOREM 7.1. The quasilinear Cauchy probhem (QCP),,,,,,, possesses for each (s, x, A) E [0, T) X V X A a unique maximal solution u( -, s, x, A). The maximal interval of existence,

Dynamic theory of quasilinear parabolic equations-I

J(s, x, A), is open

907

in [s, 7’1 and

u(. , s, x, A) E C(J, V,) r-l c-q.&

Ep) n C(j, E,) n C’Q,

E,),

where J := J(s, X, A). Proof. nontrivial

Proposition 6.1 implies the existence of a unique solution u0 of (QCP)(,,,,,, on some interval [s, to]. If to < T, we can apply proposition 6.1 again to problem to find a unique solution ui on some nontrivial interval [to, ti]. Since u(ta) E E,, (QW(w(w)) it follows from theorem 5.3 that ui is a strict solution on [to, ti]. Hence we can piece us and ui together to get a solution of (QCP)(,,,,n) on [s, ti]. Let J(s, x, A) := U {[s, t] c [s, T]; (QCP)(,,,,J)

has a solution

on [s, t]}.

Then J(s, X, A) is a nontrivial subinterval of [s, T] containing s. Moreover J(s, x, A) is open in [s, T], since, otherwise, we could apply proposition 6.1 at its right end point, which would give a contradiction. Clearly J(s, x, A) is a maximal interval of existence of a solution u( +, s, x) of (QCP),s,X,k). Finally the uniqueness part of proposition 6.1 implies that there is only one maximal solution. LetA,:=A(.,u(., s,x,il),A) andf,:=f(., u(. , s, x, A), A). Then u(. , s, x, A)l[s, z] is, for each fixed r E J(s, x, A), the unique solution of the linear Cauchy problem k + A.(+

= fU(t),

Since A, E CP([s, t], %?(I?)), we obtain theorem 5.3. n For each s E [0, T) we denote

S
the asserted

by 9(s)

the domain

u(s) = x.

regularity

we prove

THEOREM

the following

7.2. 9(s)

is open

of

u( . , s, x, A) from

of u(. , s, *, a), that is,

9?(s) := {(t, x, A) E [s, T] x V x A; Then

properties

t E J(s, x, A)}.

global continuity. [s, T]

x

V, X A and u( a, s; ;)

E ~~‘-2”(9(s),

V,).

Proof. Let (to, x0, A,) E 9(s) be given and put S := u( [s, to], s, x0, il,) and K := {A,}. Let D,(S, E) be the closed c-neighbourhood of S in E,, and suppose first that o > 0. Then proposition 6.1 implies the existence of positive constants E, t and K and a neighbourhood E of K so that u(. , r, x, A) is defined on J,,, := [r, T] II [r, r + z], for each (r, x, A) E [s, to] x B,(S, 2~) x C, and satisfies I]@, r, x, A) - u(t, r,y, MII,

6 d/IX - Ylla + w,

Y)“),

for r E [s, to] and (x, A), (y, cl) E B,(S, 2~) x Z. The fact that t can of r E [s, to] follows by observing that the number t in proposition set % in (6.2) (and upon T). Wefixnowpointss=:r,
t E Jr,,

be choosen 6.1 depends

(2)

independently only upon the

j = 0, . . . , m + 1, so that , rj, x, A) exists on J,,,, and

H.

908

AMMAN

satisfies U(f,Tj,X,jl)EB,(u(r,s,xo,~),&j+l),

t E

Jr,.,

(3)

>

and IIu(t,rj,X,A)

- ~(t,rj,Y,p)II,

-YII~ +d(h,~u)“,

src{IIx

1 E Jr,., 7

forx, y E j = 0, 1,. . . , m, and A, Jo E Z. From the J(s, x, A) 3 [s, (to + z) A Tj and that B,(U(rj,

S,

X0,

A,),

Ej),

(4)

(3) and (4) we deduce

u(t, s, X, A) E B,(S, 2~) fort E [s, (to + z) A T] and (x, A) E K(S, provided

Z is small enough.

Hence

9(s)

Eo) x 2,

is open in [s, T] X V, X A. Moreover

IW s, x, A)- 4t, s, Y, ~>lla s 4lx for s 6 t s (to + z) A T and

(x, A), (y, p) E fi,(S,

~11, + d(& CL)?

.so) X C. Now the assertion

u(. , s, x, A) E C(J(s, x, A), V,). If u = 0 we have to modify the above proof in the obvious We strengthen that

now the boundedness

requirement

(4) implies

way to obtain

follows

from n

the assertion.

in (Q2) slightly by assuming,

additionally,

the map : V,+

[Y ~(A(.,y,~),f(.,y,A))l

G(WE))

is, for each A E A, regularly bounded on bounded bounded away from the boundary of V,. Then

we can prove

the following

x CT(&) subsets

03) of V, which

are

useful

THEOREM 7.3.Let assumption (B) be satisfied, let (s, x, A) E [0, T) x V, x A be fixed, and suppose that u(J, s, x, A) is bounded in E, and bounded away from the boundary of V,, where J := J(s, x, A). Then / = [s, T], that is, u(. , s, x, A) is a global solution of (QCP)(,,,,,,. Proof. Since A is fixed, we can assume that A = {A}. Let S := u(J, s, X, A). Then it follows from (B), remark 6.2 and proposition 6.1 that we can find a bounded closed neighbourhood U of S in V, and a positive number T so that (QCP)(,,,,A) has for each (a, y) E [s, T) x U a unique solution on [ 0, o + t] n [0, T]. Thus, if f+ : = sup J < T, we can solve (QCP)(,,,,A, for o := (b+ - t/2) v s and y := ~(a, s, x, A) on [a, (a + r) A T]. Consequently, we can extend u( . , s, x, A) beyond J, which contradicts the maximality of J. This shows that t+ = T and, also, that J = [s, T]. W In the special case that VW= E,, theorem 7.3 shows that u(. , s, x, A) is a global provided we can find an a priori estimate of the form IIU(t)lln where

u(t) := u(t, s, x, A).

QC
9

t E

J,

solution,

Dynamic theory of quasilinearparabolic equations-I

8. Autonomous problems Let X be a metric space and t+ : X+

909

(0, m]. Put

9 := u

[O, t+(x))

x

{x}

XEX

and, for a map 9, : 9 +X, let q’(x) := q(t, x). Then Q, is said to be a (local) semifrow on x provided (i) 9 is open in 58, X X, (ii) q” = id,; (iii) @I(X) = Q?“Oq’(x) whenever 0 G s < t’(@(x)) and s + t < t’(x). If t’(x) = cc,for all x E Xthen ~1 is aglobal semifow on X. The set y+(x) : = {@(x); 0 s t < t’(x)} is the positive (semi-) orbit through x and t’(x) is the (positiue) exit time ofx. (We refer to [l, 101 for the basic properties of semiflows.) We consider now the autonomous quasilinear Cauchy problem ti + A(u)u = f(u), Thus we suppose that assumptions oft and (for simplicity) of A. Then (1) possesses for each (s, x) E R, on some open subinterval J(s, x)

s
u(s) = x.

(1)

(Ql) and (Q2) are satisfied and that A andfare independent the results of Section 7 are true for every T > 0. Consequently X V a unique maximal solution u(. , s, x), which is defined of [s, m), containing s. We put

~(t,x):=u(t,O,x),t+(x):=supJ(O,x) and 9:={(t,x)ER+ Then

the following

THEOREM 8.1.

theorem

x v;O=st
is an obvious

~1 is a semiflow

consequence

of theorems

on V,, and 91 E Co.‘-@,

V,).

In the following theorem q([O, T], x) has to be interpreted one does not know, a priori, that t’(x) = 00. THEOREM 8.2. Suppose

as cp([O, T] fl [0, t’(x)),

x), since

that the map (A,f)

is regularly Then t’(x)

7.1 and 7.2.

: V, -+ x(E)

x E,

bounded on bounded subsets, which are bounded = 00,provided q([O, T], x ). is b ounded and bounded

Proof. This is an easy consequence

of theorem

7.3.

(2) away from the boundary of V,. away from aV, for every T > 0.

n

COROLLARY 8.3. Let the boundedness condition (B) for the map (2) be satisfied that y+(x) is bounded and bounded away from av,.Then t’(x) = 30.

and suppose

It should be noted that q has also a smoothing property, namely y+(x)\(x) C El for each x E V. Moreover theorem 7.2 shows that 47 depends continuously upon (A,f) in appropriate topologies.

H. AMMAN

910

The theorems in [2].

and proofs

of Sections

SMOOTHNESS

6-8 are extensions

PROPERTIES

OF THE

of corresponding

GENERAL

results

obtained

SOLUTION

9. Additional regularity properties In Section 11 below we shall show that the map (x, A) H u(t, s, x, A) is differentiable, provided A and f satisfy appropriate regularity conditions. In the present section we introduce some of these conditions and derive “higher regularity” results. The additional hypotheses introduced below have been motivated by applications to quasilinear parabolic systems. In such a concrete setting these assumptions are satisfied by rather natural choices of the underlying spaces, as will be shown elsewhere. Let E and F be Banach spaces with E 4 F, and let B be a linear operator in F. Then the E-realization, BE, of B is the linear operator in E, defined by D(B,) := {x E E f~ D(B);

Bx E E}, B,x := Bx. Let assumptions

(Ql)

and (Q2) be satisfied.

E,(t,y,

A) := &@(t,y,

We put

A))>

(t>y, A) E 10,

Z-1x Vx A,

where these spaces are given the respective graph norms (with respect to the closed operators A2(t, y, A) in E,). Observe that (E,, E2(t, y, A)) E 9Q2.Then we let E,+o(t,.v,A)

0<8<1,

:= %(E,,E,(r,y>4),

fixed (y,,, A,,) E V x A,

and, for an arbitrarily

E 1+0

:=

EI+o(~>YO, U

Finally, E, := (E,, E1+o), 0 < 0 s 1. Observe We assume that there

linear

exists E E (0, r//p)

E,+&>y, with uniformly

that E. E &!&,by (2.1).

so that,

a>&EI+~>

equivalent

E, G +,,/cc-n(E,,

0<8Gl.

for each E E (0, E],

(t, Y, A) E [O,Tl x

norms,

v x A,

(Q3>

and oGEGn=GcGl+&.

EC),

We denote by A&t, y, A), 0 < E s E, the Eg-realization of the linear operator A(t, y, )3) in EO. Then it follows from [6, theorem 6.11 that D(Ag(t, y, I.)) = E, +$.Thus the following assumption is meaningful. A, E CJ’+,O([O,

T] x V, x A, %(&))

and the map

[(YJ)-A~LYAI: is locally It follows

bounded

that even more is true.

V, x A* ‘3(~e(&>)

for each 5 E (0, as is seen from

E]

(Q4)

911

Dynamic theory of quasilinearparabolic equations-I LEMMA 9.1. A&t, y, A) E X(&)

and the map

A)1: V, x A+ G(W&))

[(Y, A) -A&Y, is locally

regularly

bounded

for each 5 E (0, E].

Proof. This is a consequence

of [6, theorem

For (s, x, A) E [0, T) x V x A we consider

ci + A&, in Eg, 0 < 5 G E. Then

n

6.11.

the quasilinear

Cauchy

s < r s T,

u, n)u = f(t, u, A),

problem

u(s) = x

(QW,s,x,m

we prove the regularity.

PROPOSITION 9.2. (QCP)(,,,,Aj,S

U*(.,&

p assesses

., .) = u(.,s,

a unique

maximal

., *),

s E [O, T),

u& * , s, x, A), and

solution

E E (0, cl.

(Ql) and (Q2) are Proof. We deduce from (Q3), (Q4) and lemma 9.1 that assumptions satisfied, if we replace there E, a, p, y and A by EE, (Y- 5, /3 - E, y - 6 and Ag, respectively. has a unique maximal solution u5 : = CL&., s, x, A) Thus theorem 7.1 shows that (QCP)+kj,S on Jg : = J&s, x, A). It is obvious that u5 is also a solution of (QCP)(,,,,A). Hence JE C J := J(s, x, I_) and u5 = u( . , s, x, k)lJ6. Let A(t) := A(t, u(t, x, ,I), )3) and f(t) := f(t, u(t, x, A), A). Then we can apply the regularity theorem 8.2 of [6] (cf. also [6, section 9.13) to the linear Cauchy problem s < t =Gt,

zi + A(t)u = f(t), for each fixed t E j, to deduce

that J, > J.

u(s) = x

n

COROLLARY 9.3. Let J : = J(s, x, A). Then

u( - , s, x, A) E C(j, E, + E) fl C’(j, E,).

10. Auxiliary considerations Throughout this section we presuppose

assumptions

A is a one-point

(Ql)-(Q4)

and assume

that

space.

Clearly, this means that A and f are parameter independent. So we omit any reference Moreover, we restrict ourselves to the case that s = 0 and assume that J(0, x) = [0, T]

to A.

for all x E V.

Then we let u(t, x) := u(t, 0, x). Finally, we assume that O<&
we derive now some technical

LEMMA 10.1. Suppose

that g E C(V x W, B,+,_,

d + A y (t, u(t, X)>U = g(t, x, Y>,

results,

kdE,>).

where k\(E,) Then

0 < t =S T,

the linear u(0) = z

: = C’((0, Cauchy

T], E,). problem (1)

H. AMMAN

912

possesses

for

(x, y, z) E V x W x W

each

CT(Ei + J n Cl, (E,),

Proof. It follows

a

unique

solution

u(. ,x, y, z) E C+(.E,) n

and

from (Q4),

theorem

[.x.-A&

7.2 and lemma

u(. >4)l

9.1 that

E C(Yx> C&W”)))

and that each x0 E V has a neighbourhood U in V, so that {A,(. , u( . , x)); x E CJ}is a regularly bounded subset of CPT_(X(EU)). (Observe that u(. ,x) E Cp([O, T], VP).) Replace now E by E, and put E := IX- u, n := 1 + 6 - u, 5 := p - u and 6 := 1 + p - a. Then it follows from theorem 5.3 and (Q3) that (1) possesses a unique mild solution, u(. ,x, y, z), and (Q3) and theorem 5.5 imply easily the asserted continuity as a function of (x, y, z). On the other hand, if we choose any &‘IE (IL, cu) and replace 6 in the above consideration by 5’ := a+ - Y so that t’ + 6 = 1 + ,D - v - (LY- cu’) < 1 + p - v = 1 + 5, we deduce from theorem .5.3 that u(. , x,y, z) is, in fact, a solution of (l), so that u(. >x, Y>z> E CT(&)

n CT(G + y> n %Ev>.

w

COROLLARY 10.2. [x H u(. ,x)] E C(V,, B1+6_n&E1+6))

for every 6 E (0, E).

Proof. Let g(t, x, y) : = f(t, u(t, x)) for (t, x) E [0, T] x V and y E W : = V. Then we deduce from theorem 7.2 that [(x9 Y) ++ g( . >~3 ~11 E C(V x W, WE,)). Hence,

letting

,U :=

E

and y := z := x in lemma

10.1, we see that the linear 0 < t s T,

d + A v(t, u(c, x>)u = f (6 ~(4 x)), possesses

a unique

solution

from proposition

For the remainder

problem

u(O) = x

u(. , x) so that [x++~(.,x)l

But it follows

Cauchy

E C(K

&+b-&(G+h)).

9.2 that u( *, x) = u( . , x), which proves

of this section

we assume

the assertion.

n

that

~~~sKs&andOaf3 s)F(x, y)(s)W) 0 possesses

for each (x, y) E V x W a unique (S, F) : V x W+

is (separately)

continuous,

&,&(W,,

then R : V x W+

solution

OstaT,

h,

R(x, y) E B,C,(T(E,,

E,)) B,~~Z(E,,

x W$(.W,, E,))

E,)).

If

E,))

is (separately)

continuous.

Dynamic theory of quasilinear parabolic equations-I

Proof.

Observe

that [x -

by (Q3) and corollary follows that

r

Ux] E C(v,

5.2. Hence,

letting

theorem

KT(Ex,

for (t,s) E f~,

it

K - ,M,0).

1.1 implies

r(t) = Sk

(2)

E,, K - ~1)

WE,,

k(x, y)(t, s) := U,(t, s)F(x, y)(s)

k : V x W-+

Consequently,

913

that the Volterra integral I W, YW, +(t) dr, Y)(4h + I0

equation OctsT,

possesses for each (x, y, h) E V x W x E, a unique solution r(~, y, h) E B,e:T(EJ. The proof of theorem 1.1 shows that T(X, y, h) = (1 - K(x, y)))‘S(x, y)h, where K(x, y) is the Volterra integral operator in B,c:,(E,) with kernel k(x, y). Hence T(X, y, h) = R(x, y)h, where R := (1 - K)-‘S: Now the assertion

is an obvious

V x W-+

consequence

B&(Lf(E,,

,x,

n

of (2) and (1.4).

Denote by IE!, the open unit ball in E,. We assume positive number E, and functions [h -Fo(.

E,)).

that, for each x E V, there

h)l E C(dLu, W%W,>

are given a

&A>> (3)

[A -F,(..

x, h)l E C(eA,

&&(~(EK,

E,)))

and [hi

G(x, h)l E C(Mn,

T(E,,

ES)),

(4)

so that >x) := Fo(. ,x, O>l E C(V, k&(.Wn,

[x -F,,(.

E,))), (5)

[xc,F,(.,~):=F~(.,~,~)IEC(V,B~~~:T(~~(EK,E~))) and [X I-+ G(x) := G(x, O)] E C(V, 2(E,, Then

we consider

the linear

Cauchy

ES)).

(6)

problem

ti + A(& u(t, X))U = F,(t, x, h)k + Fl(t,x,

h)u,

0 < t s T,

u(O) = G(x, h)k

(7)(x.h,k)

LEMMA 10.4. The Cauchy problem (7)(X,h,kj p ossesses for each x E V, h E E$, solution u(. ,x, h, k). There exists a map R(x, h) E B,_5~&%(E,, E,))

and k E E, a with

for x E V, h E &,lEb,and k E E,.

unique

[h-,

R(x, h)l E C(&JL,‘%@(E,,

E,))),

x E v,

A

x (J

‘(y ‘x ‘s)r suraIqold ?I] 3 (y

3 1

‘(y ‘(1)n ‘1)SZe+ (1>n[fl(r‘(1)n ‘l)VZ@-1 = “(U ‘(1)n ‘1)v +

1Cymc3 paz!~eaug

‘y ‘y ‘x’s)

paxy

vea

‘J,>Sz=O

R

aq~ 30 (y ‘x ‘s)[ uo suoyyos anbyn aql ‘2 x “g x v x y(y ‘x ‘s ‘ . )n* p pue y(y ‘x ‘s ‘ . )nEp pm ~03 ‘ale ‘(“A ‘(s)@I‘O3

3 (* ‘. ‘s‘*)n uayL

(qa x (mK=

(I)

‘(v x %I) x 11 ‘Ol),Q

.(3 ‘0)

pamnouue

3 9

amos 103

3 (S‘v) ~eyl asoddns

~~~y~qvyua@ip

‘1.11. KRIO~HJ,

ay$ aAo.rd pm

‘(J ‘01 3 s 103

{s < 3 f(S)Q 3 (y ‘x ‘1)) =: (f)Q

D wd a&

LJ amds

ym.~e~

atuos 30 lasqns

uado

asoddnsald am uogcm s!yl moy%?nolyL ue s! v 3cq1 amnsse pue (t7O)-_(H3) suoydumsse 49~ laldew pug LIT ‘91~ svws ‘~11.33) x 3 x 01 IDadsal ~I!M &.uro3lun rl~pz~o~‘“fl uy yICpx~e s! ( . ‘x)h pa ~eyl pur! (M ‘fl)r5d 3 (0 sa!lduy (M ‘fl)l.~ 3 cb uayl ‘3 = ~1 31 ‘fl uo snonuyIo~ am ‘3 ‘. . . ‘0 = [ ‘h$@ saAgeA!Jap legmd ayl 31 pue ‘0 # {n 3 (A ‘x) :B 3 k} =: “0 qw x 3 x pa ~03 (d ‘“n)g 3 (. ‘x)h J! ‘(A ‘0)s 3 ~4 9 ‘{CO}n N 3 y ‘(A ‘n)Iled 3 h apM aM UaqL .dpqxdsa~ ‘,J px.w 3 x x

n

u! uado M pue fl pue ‘sands

‘~‘01 euu_ual pue c’s waloayl30 .(((“a

‘“8k@?~-“8

yxue~

.J pue TJ ‘axds

amanbasuo3 ‘n).z)

3 [(o

sno+qo ‘xk

zy)atu e aq x )a7 4fpqtyua~aJja ‘1 I

ue s! uoyasse

aql

MON

-xl Pue

‘A 3 X

‘((YYB ‘“3)?pp8

wq~ (~‘1) pul! E’S d.n?~~o~o~ ‘(~0) ‘Jz=ls=O

‘sp (y

‘x ‘s)Od(s

3 [(Y ‘X)S”

‘“@3)3

uogdumsse ‘1)“fl

I

+

‘1’s uraloayl

Y]

‘(9)-(c)

uro.13 SMOI[OJ

$1

‘Vn =: MY ‘~1s

(z/ ‘x)3(0

Ia7 $‘O.Jd

‘y(y ‘.qg = (T ‘q ‘x ‘ .)a ‘(((“3

‘“3)X)‘?

‘-“8

‘A)3

Jayl

OS

3 [(O ‘X)8 =: (x)8 e-ix]

Pue NVWrUV

‘H

PI6

Dynamic theory of quasilinear parabolic equations-I

915

and ti + A@, u(t), +v

= - [&A@, u(t), Q+(r)

tE.i(s,x,A), w(s) = k, respectively,

where

u(t) : = u(t, s, x, A).

Proof. Let ~:=(E,xF,E,xF)E!~~~, V:=VxA, A:=diag(A,O), u:=(u,jl) f(t, u) := (f(t, u), 0). Th en it is easily verified that assumptions (Ql)-(Q4) are satisfied parameter space A = F := (0). Moreover (QCP)(,,,,kj is apparently equivalent to s < t =s T,

u + A@, u)u = f(t, u),

and for the

u(s) = x

for (s, x) E [0, T) X V. Hence we can assume, without loss of generality, that A = F = (0) and omit any reference to A. Moreover we can also assume that s = 0 and put 9 := 9(O) and u(*,x):=u(.,O,x). Let (t, x) E 5%be given. Then we can find a number r > 0 and a convex neighbourhood W of x in V,, so that [0, t + r] x W C ‘3. Hence, by replacing T by t + r and V by W, respectively, we can assume that u E Co*‘-([O, T] x V,, V,),

(2)

and that V, is convex. Forx,x+hEV,andOGt
:= u(.,x+h)

- U(.,X),Wt(.,X,h)

:= U(.,X)

+ tw(.,x,h)

and (azA> a,f)(.,x,h)

:=

I0

’ (a,A,a,f)(.,w,(.,x,h)))dt,

as well as F1(.,x,

h)u := -[d,A(.

for u E E,. It follows from (l), so that

,x,h)ulu(.,x+h)+a,f(.,x,h)u

(2) an d corollary

(3)

10.2 that, for each x E V, there exists .sX> 0

and that

[XHF~(.,X):=F~(.,X,O)IEC(V~Y,BI+G-~~:T(~~(EB,E~))). Since w( a, x, h) is obviously

a solution

on [0, T] of the Cauchy

ti + A(& u(t, x))w = F,(t, x, h)w,

0 < t s T,

problem w(0) = h,

(4)

916

H.

we deduce

from lemma

AMMAN

10.4 (where now 6 plays the role of p, and where K := g := /3 and Es denoting the injection map) that w( . , x, h) = R(x, h)h, where

G(x, h) : = j, with j := E,h

Rk

.) E C(@,,

&(g(E,,

x E v,

Qr))).

and R: = R( *, 0) E C(V,,

S&(Z(En,

E@))).

(5)

Thus u(. ,x + h) - u(x) - R(x)h = [R(x, h) - R(x, O)]h

(6)

for x E V and h E E,B,. Hence x H u(t, x) is differentiable as a map from V, to Ep, uniformly with respect to t E [0, T], and a+(. , x)h = R(x)h. Let to E (0, T) be arbitrarily fixed and consider the linear Cauchy problem d + A(t, u(t, X))U = F,(t, x, h)u, It follows

from corollary

to < t s T,

4to)

= R(x, h)(t,)h.

(7)

10.2 that

and ]x ~Fi(.,x)

E C(V,,

It is clear that we can apply lemma G(x, h) := R(x, h)(t,)) to deduce that ]h ‘+R(x,

h)(to +

C(Po,

Tl, %E,, Ed).

10.4 to (7) (with

K

:=

a,

,u :=

(8) 6,

6

:=

0, 5 := /3 and

-11E C(QL L&kt,(W,)))

and ]x*R(x)(to

+

*>IE C(L f&C,-t,,)W,))).

Hence we see from (6) that x H u(t, x) is continuously uniformly with respect to t in any compact subinterval

(9)

differentiable as a map from V, to E,, of (0, T]. H

COROLLARY 11.2Let s E [0, T) and (to, x0, A,) E 53(s) be fixed. Then there exists a neighbourhood Uof (x0, A,) in V, x A and a number r E J(s, x0, A,) with to -C z, so that t E J(s, x, A) for (x, A) E U and [(x, A) ++ (~su, a,~)(. Moreover the derivative subintervals of (s, t]. We extend

+ $7 s, x,

(a-,~, d,u)(t, s, x, A) exists uniformly

now the above differentiability

THEOREM 11.3.Suppose

A>1 E C(K 6,-s (x(Ea x F, CA)).

theorem

with respect

to t in compact

to the case of higher order derivatives.

that K = R and

(A,f)

E C”,‘Y[O, Tl x [(V,

x A)], T(&)

x Es)

(10)

Dynamic

theory

for some k E N U {m} and every

of quasilinear

parabolic

6 E (0, E). Then *, *) E CO,$J(s),

U(*,&

917

equations-I

O=ss
V,),

and the derivatives of u with respect to x and A are the solutions of the linearized Cauchy repeatedly with respect problems which are obtained by differentiating (CP) (S,X,ijappropriately to x and A. Proof We can again assume that F = {0}, s = 0 and 9 := 9(O) = [0, T] X V. Moreover we can choose a sequence (SJ SO that E > Sj > Sj+l >Oand6j-6j+,
O
T,

u(O) =h,

where F,(+,x)u by (3). Hence

:= -[d,A(y(.

+ bf(Y4*,x))u,

we see from (4) and (5) that lx++

Consequently,

,x))ul4*,x)

lemma

F1(.,XNz4.,X>hlE C(V>&+cs-c$‘~&~)).

10.1 implies

, x)hl E C(K &++&(EI+&

[x ~a+(.

(11)

Let u(.,x, for k E E,, and observe

k) := &u(.,x

+ k)h - a&.,x)h

that u(. ,x, k) is a solution

of the linear

Cauchy

d + A(t, u(t, x))u = F,(t, x, k)k + F,(t, x)u,

problem

O
u(0) = 0, where 1

Fo(. ,x, k)l := - [a2A(. ,x, k) I0

d,u(.,x+

rk)ldr]a2u(.,x

+ k)h

1

a2F1(.,x

+ I

+ rk)ldta2u(.,x

+ k)h

0

for 1 E E,. Moreover, a,F,(.,yY=

- (~~A(.,u(.,Y))[~~u(.,Y)~,

*I)~*,Y)

- [~,A(.,u(.,Y)).]~,u(.,Y)~+

for (y, 1) E V, x E,. We see from (5), (10) and corollary [k ++Fo(.,x,

a~f(.,~(.,y))[a2~(.,~)1,

*I

10.2 that

k)l E ‘+x&n ~~++c&P’e(Ea> .&I))

(12)

918

H. AMMAN

and that

[x-Fo(.,x, Moreover,

011 E

c(v,, &+6*-n~T(~(J%~ Es,))).

(13)

(4) implies

Hence we deduce from a1 > 8* and (1.1) and from lemma 10.4 (with K : = p, p : = b2 and 6 : = 1 + 61 - a) that u( *, x, k) = R(x, k)k, where R(x, *) E C(cAu,

&?Cr(W,,

Eg))),

X E v,

and

Thus 8zu( *,x + k)h - a$( * ,x)h - R(x, 0)k = [R(x, k) - R(x, O)]k for x E V and k E .$E8,. Consequently, x H a 2~(t, x)h is differentiable as a map from V, to Es and a:u(. , x)[k, h] = R(x, 0)k. (Ob serve that R(x, 0) is a linear function of h-which we suppressed in our notation.) Let t, E (0, T) be arbitrarily fixed and consider the linear Cauchy problem d + A(t, u(t, X))ZI = F,(t, x, k)k + F, (t, x)u, Go)

to < t s T,

(14)

= R(x, k)(t,)k.

From (12) and (13) we see that

VI E C(dL

[k++Fo(.,x,

C([t,, Tl, WE,, Q)))

and Ix -Fo(‘,x,

O)] E C(V,, C([t,,

Tl, ge(E,, &J)).

By taking into consideration K := cv, p := &,

6 :=

(8) and Eg,‘+E62, and by applying lemma 10.4 to (14) (with 0, 5 := p and G(x, k) := R(x, k)(t,); i.e. with h = k), we find that

[k -RX,

W,

+

[x -R(x,

O)(t, +

.>IE c(~,~,,B,-~c~-ro(~(E,)))

and

011E C&n L&kr,,(~e(&)))~

Hence x H a&t, x) is differentiable as a map from V, to E,, uniformly with respect to t in compact subintervals of (0, T]. This proves the assertion in the case that k = 2. The general case, k E N U {m}, follows by n an obvious induction argument.

Dynamic theory of quasilinear parabolic equations-I

919

The assumption that (10) be true for every 6 E (0, E) has only been made for simplicity. It is clear that it suffices to assume that (Ayf) E

($C”qj([O,‘1 X [VP X A], ‘e(E,i) X ‘6;)

for some sequence .5> 6 1 > ~3’: 2

. . . >

6,

>

0 with 4 - Sj+ I< LY- /3.

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