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A Theory of Nonlinear Evolution Equations
Trends in the Theory and Practice of NorrLmear Analysis V. Lakshmikantham (Editor) Elsevier Science Publishers B.V.(North-Holland), 1985
17
A THEORY OF NONLINEAR EVOLUTION EQUATIONS Mieczyslaw Altman Department of Mathematics Louisiana State University Baton Rouge, Louisiana U.S.A.
A theory of solving nonlinear evolution equations based on three different methods is presented which is independent of Co-semigroups. From the standpoint of applications, its important feature is the possibility of solving nonlinear evolution equations via linearized evolution equations. INTRODUCTION Kato's [12] theory of quasilinear evolution equations, which he successfully applied to nonlinear partial differential equations of mathematical physics, was the first significant step in the nonlinear direction. His famous theorem which applies only to reflexive Banach spaces was generalized to nonreflexive ones in [2] by a different method. In 1956 Nash [14] introduced the concept of smoothing operators and in 1966 Moser [13] introduced the concept of the degree of approximate linearization. Based on these notions and what is called the Nash-Moser technique constitutes a truely fundamental development in handling of the "loss of derivatives." But so far the related theorems and their generalizations are designed for nonlinear operator equations and do not apply to abstract nonlinear evolution equations (NEE). In a series of investigations [5,7,8] an attempt was made to solve the nonlinear problem for EE. Our theory is based on three global linearization iterative methods (GLIM). Global linearization means that the equation is linearized about a vector running over a nonbounded set B whereas local linearization indicates that B is bounded. The theory also applies to nonlinear operator equations ( s e e 131). Thus a unified approach has been developed to both nonlinear operator and evolution equations. Although the general theory is independent of Co-semigroups, nevertheless from the point of view of applications to nonlinear partial differential equations it is rather important that nonlinear evolution equations can be solved via linearized evolution equations. In this way one can make use of the theory of Co-semigroups which is more powerful and advanced than the theory of nonlinear semigroups which is rather limited in scope. Some highlights of our theory will be presented below.
1. CONVEX APPROXIMATE LINEARIZATION AND GLOBAL LINEARIZATION ITERATIVE METHODS (GLIM) FOR SOLVING NONLINEAR EVOLUTION EQUATIONS
Let
Z C Y C X be Banach spaces with norms
(l.A ) that'
(1.O )
11 -11
z ->
11 * 11
IIXII
<
c
-
-
-x
> 11 11
YWe assume that there exist constants C > 0 and
s
with
0 <
IIxlll-s IIXIIS
s
< 1
such
YX 2' b, denote by C(0, b; X) the Banach space of all continuous functions
Given 0 < x = x(t) defined on the interval
[ O , b]
with values in X
and norm
M.Altman
18 RxR
I n t h e same way a r e d e f i n e d respectively.
Denote by
t h e norms
C1(O,b;
d i f f e r e n t i a b l e f u n c t i o n s from with center
in
xo
t h e c l o s u r e of
in
Vo
and
llzl
the vector
[O,b]
to
ro
Let
X.
>
0.
Put
Y
for
-,Z
space of
and
2,
a l l continuously
be an open b a l l in
Wo
and l e t
Vo = Vo n Z
Y
be
V1
Y.
F : [O,b] x V1 + X
Let
< b].
t
IlyII*,Y
x)
and r a d i u s
2
<
sup[llx(t)!lx : 0
=
-,x
be a n o n l i n e a r
mapping and c o n s i d e r
t h e Cauchy
problem Px(t)
(1.1) where
+
dx/dt
S
be t h e set of f u n c t i o n s
G
-
x ( 0 ) = ~ 0 , Rx
<
X ~ N - , ~
<
f ( t , x ) = 0, 0 < t
b, x ( 0 ) = xo,
i s a l s o a n o n l i n e a r mapping.
f : [O,b] x V 1 + X
Let
+
F(t,x)
r o and
We assume t h a t t h e mapping
x
llxl F
-.
z <
-9
1 n C (0,b; X)
C(0,b; Vo(l.Uz))
E
i s d i f f e r e n t i a b l e i n t h e f o l l o w i n g sense.
For e a c h ( t , x ) c [O,b] x G, there e x i s t s a linear operator F'(t,x) -1 E IIF(*,x) + Eh) - F ( . , x ) - E F ' ( * , x ) h l + 0 as E + O+, where -,x 1 h E C(0,b; Z) n C (0.b; XI. We make t h e f o l l o w i n g assumptions. a
Cauchy sequence
bounded i n -1 E lF(*,xn n
+
(1.A:) t
E
with
(1.A;)
+
with
h)
-
F'(t,u)(u
(l.AZ)
- x i -,Y n
true for
f.
+
Eh)
Rf
(a
,x
E
G
and
-
F,f
implies
be be
vII Ilu 0
C
>
0
such t h a t
- vly, M1
>
for a l l
0
such t h a t x e G,
a r e continuous i n t h e f o l l o w i n g s e n s e ,
- F ( . , X ) E ~ , ~+ 0
IIF(.,xn)
qo
>
0
as
n +
-,
and t h e same i s
such t h a t
f( - , ~ ) l l ~ < ,qOEUhUm,X. ~
with
<
agn,,X
+
m,
F'(t,x)h
if
+
Co
>
h
i s a s o l u t i o n of t h e e q u a t i o n
0
w i t h t h e following p r o p e r t y .
g = 0, 0 < t
C
For
b, h ( 0 ) = 0,
Ilhlla,X < bCOlgUm,X. (1.A4)
(1.2)
-9
{xn} c G
n + -; o r
< M l l h ! ~ ~ ~ I I h l E , Z where , 0 0 < 8 < 1.
There e x i s t s a c o n s t a n t g
-
v)llo < Clllu
and
X),
1
{hn} c C(0,b; Y) n C (0,b; X )
E ~ F ' ( * , X ~ ) ~ , +U 0~ , a~s
- P'(.,x)hll
dh/dt then
-
Let
such t h a t
implies
There exists a c o n s t a n t
(l.A3)
x
0
let
i = 1.
with
There e x i s t s a c o n s t a n t
i = 3.
F(*,x)
The f u n c t i o n s +
and
Y) cn + O+
or
h c C(0,b; Z ) n C1(O,b;
Ilx
(1.A;)
There e x i s t s a c o n s t a n t
i = 2.
-
[O,bl; u , v c Vo;
IIF(*,x
C(0,b; Then
- F(*,xn) -
Enhn)
- F(t,v)
IIF(t,u)
in
C(0,b; Y).
with
dz/dt
For
+
x
E
G,
F'(t,x)z
the linearized equation
+
F(t,x)
- F'(t,x)x
+
f ( t , x ) = 0, 0 < t < b,
Z(0)
=
"0
19
A Theory of Nonlinear Evolution Equations a d m i t s approximate s o l u t i o n s of o r d e r t h e following
with
(p,u,u)
0
< u
<
1
in t h e s e n s e of
Then D e f i n i t i o n 1.1. [9]. L e t p > 0 , v > 0 , u > 0 be g i v e n numbers. if t h e l i n e a r i z e d e q u a t i o n (1.2) admits approximate s o l u t i o n s of o r d e r (U,U,U) t h e r e e x i s t s a c o n s t a n t M > 0 which has t h e f o l l o w i n g p r o p e r t y . For e v e r y < K then t h e r e e x i s t s a r e s i d u a l ( e r r o r ) x E G, K > 1, and Q > 1, if nxll function
z
-9
and a f u n c t i o n
y
such t h a t
z
iizu - 9 2
llyll (1.2')
dz/dt
<
0
+
+
F'(t,x)z
m,X
< MQK" and
s MQ-!-IK',
F(t,x) = F'(t,x)x
+
+y
f(t,x)
= 0,
t C b, z ( 0 ) = xo.
Now f o r x E G, l e t z be a s o l u t i o n of t h e e q u a t i o n ( 1 . 2 ' ) z = x + h. Then o b v i o u s l y h is a s o l u t i o n of t h e e q u a t i o n
(1.3)
dh/dt
+
+
F'(t,x)h
+
Px
and put
y = 0, 0 < t < b, h ( 0 ) = 0 ,
and we g e t
Put
The f o l l o w i n g is a n i t e r a t i v e method of c o n t r a c t o r d i r e c t i o n s and assume t h a t x0,x1 ,...,xn 7: xo , 0, ,..., GLIM-I.
to =
E G; t O , t l
t
[61. are
known and put (1.6) that
x
and
~ =+ xn~ + ~~h~
tn+l = tn + cn,
S
= (1
Xn+l
-
En'Xn
+
EnZn,
which j u s t i f i e s t h e term "convex approximate l i n e a r i z a t i o n , where and
h ,,
being a s o l u t i o n of (1.3).
-
qObCO = q
<
q/2
<
1/2,
and put
, x ) s qlPxollm ,exp(-(l n n 9 e x i s t s 0 < E < 1 s u c h that P(1,h
where
po = XPxoII
-9
x
and put
To detemine
P(E,h,x)
-
q)tn),
E
n
= c-lllP(x
then
put
2F/q
let
E~
+
Eh) E
n
-
(1
= 1.
Otherwise
= E.
The method (1.6) s a t i s f i e s t h e f o l l o w i n g ' i n d u c t i o n
zn =
+ hn
< c < 1, where - E)PXU-,~. I f
assumptions
there
20
M.Altman
Ux It n m,Z
I l F ' ~ ~ l l , , ~G llPxoi~
-9
Theorem 1.1 [91.
u(1
-
-
v)
u
>
N = [(I
/u <
-
q)
a ) :
>
by
has a s o l u t i o n
x
b'
-
(1
+
GLIM-11. The directions. Put
- q)tn),
exp(-(1
0,
a
- q) -
>
with
i = 1, suppose t h a t
[ ~ ( l v)
11,
and
- 01-'
and
i s such t h a t
b'
-
-
- q)AlS < ro.
- q)G)C[b'Co(l + ~ ) p o ] l - s [ a ( l
0
and
and
a Ilxn
_ _ < (1 - s ) / s . - X U -,y + 0
iterative
~ =+ zn~ = xn
+
Then (1.1) as
,
(xn
with
-.
i s independent of
method
hn
n +
b
replaced
contractor
G),
E
as i n (1.6) and i n d u c t i o n assumptions
with
<
IIxn II-,Z where
>
1
following
xo, x
x0 ( t )
(1.7)
-
- q)6l-'exp((l
6
= Kn
0 t o be determined.
Al-v-o/lJ [ a ( l
where
= 1
x
I n a d d i t i o n t o (1.AO-1.A4)
0, a(l
M(2M)l /u(Tpo)-l
>
a,A
f o r some c o n s t a n t s
- q)tn)
exp(a(1
q
<
2-1/a
Theorem 1.2.
and
Aq-m
(C1ro
+
= Kn
and
qO)bCO(l + 27)
I n a d d i t i o n t o (l.AO-l.Aq)
c poq",
IIpXnII,,x
<
<
q
with
1;
4<
i = 2
q.
and
E = 1
suppose t h a t
0 = 1
- F(1 +
a)
>
0, N(1
- q 0) <
ro, N = C[b'CO(l
_
i n (l.Az),
+ ~)poll-s[A(q-u -
-
1)Is,
Then t h e s t a t e m e n t of Theorem 1.1 h o l d s t r u e . GLIM-111. The f o l l o w i n g i s a r a p i d l y convergent i t e r a t i o n method which i s based on t h e e s s e n t i a l technique of MDser [131.
(1.A5)
Let 1
<
a,T;
(11
-
X
<
a)-l(i
LI
be such t h a t
+
a(1
+ X) +
11)
<
'I
<
2
-
-ao
<
2
-
a,
21
A Theory of Nonlinear Evolution Equations
<
0
where
<
.to
i s such t h a t
1
o < B < Remark. 0 < 2A
- n)[(l
uA(ao
-
5
0.
t h e constant
bCo
(1.A;)
=
2.
x
1.
= KT+a n
and
I I P X ~ I I ~< , K ; ’ .~
-
+
<
T
-
A-1
2 )P+1
<
0
<
.
X
+
-,y
< (u +
1
Then
T
>
are s a t i s f i e d w i t h
1 such t h a t :
i = 3
if
-.
t h e n e q u a t i o n (1.1) w i t h as n + L e t us n o t i c e that
2),
+ 0
i n (A3) can be r e p l a c e d by
(1
ul:
-
CO.
1)/2
and
is a number such that
2.
Theorem 1.3 remains v a l i d i f (l.Ag) is r e p l a c e d by
(1.A;)
and
0.
Smoothing o p e r a t o r s combined w i t h e l l i p t i c r e g u l a r i z a t i o n . The choice of Moser’s d e g r e e
of e l l i p t i c r e g u l a r i z a t i o n . linearization. with
{Xj}
Let
norms such t h a t
o <
<
i n t h e above. Then = 1 + a, and put Kn+l
K~(M,B,LI,X,~)
0 and s < A(A and Exn XU
Suppose t h a t
Theorem 1.4.
a
and
- - - -S T >
x 0 < B <--x+l 1 < ( 1 -$)-’
1,
~ ( 2+ U)I-’
+
Suppose that assumptions (l.AO-l.Ag) Then t h e r e e x i s t s
6 = (I s)A has a s o l u t i o n
where f = 0
CI
!i)
>
a,)
as in (1.7) w i t h i n d u c t i o n assumptions
(xn]
Theorem 1.3. f
+
3) u =
-
ao)/(l
+
+
# x n # m , z < Kn
and
+
(1
1)/(u One can put a. = (u (11 - 1 ) / ( ~+ 1). W e assume
<
define
>
!I
ml
<
m2
<
3
W e assume t h a t
(2.k)
operators
s
0 <
< j implies < 5 < p. i
So,
0
> 1,
p
v
The d e g r e e
k
o f approximate
be a scale of Banach s p a c e s w i t h i n c r e a s i n g X. c Xi
J
there
and
W*Rj
>
II*li
and l e t
e x i s t s a one parameter
( s e e Nash [ 1 4 ] , Moser [ 1 3 ] ) such t h a t
f a m i l y of
linear
22
M. Altman P-9 IISXil < c k l e p
f o r some c o n s t a n t
>
C
0. where
< cnxii~-Aiixiih f o r j
that
P
m2
i s t h e i d e n t i t y mapping.
I
-
(1
=
IIXII
+
A)r
W e a l s o assume
0 c A c 1.
~ p . with
Using t h e same n o t a t i o n a s i n S e c t i o n 1 p u t lixil
m,j
= sup[Ix(t)!l
center
xo = 0
c l o s u r e of
in
F,f
>
ro
and r a d i u s
Vo
Let
0 c t c b],
*
j '
and l e t
0.
Put
Wo
Vo =
Xs.
: [O,b] x
Xs + Xo
be
+
+
two
CI
be an open b a l l w i t h
X,
wO n XP and l e t Vs
nonlinear
mappings
and
be the
consider
the
Cauchy problem (2.1) Let
Px(t) G
%
dx/dt
F(t,x)
be t h e s e t of f u n c t i o n s
x ( 0 ) = 0 , I l ~ l l ~<, r~ o
and
Y
=
with
Xs
s
<
( t , x ) c [O,b]
x
m2
ml
G
Cllhil
+
F'(t,x)z
0 c t < b , z ( 0 ) = 0, solution
z
-
>
C
For
k > 0 (t,x) E
+
0L.z
with s m a l l
0
Xg, 2 =
=
Yp
and
such t h a t
< CMhM
llF'(t,x)hllo
>
0
ml
<
0
L = L(n)
101
<
F'(t,x)x
x
G, l t e
z
C
such t h a t
and the modified l i n e a r i z e d e q u a t i o n
+ F(t,x) -
t o be determind and [O,b]
X
s = s/p.
F'(t,x)x 1
and
+ f(t,x)
>
+
= 0,
( t , x ) E [O,b]
such t h a t
-
with
m.
m2
C
f o r some
= 0.
Vo.
f o r some c o n s t a n t
dz/dt
(2.2)
< b, x ( 0 )
"SP a r e the same as i n S e c t i o n 1, where
There e x i s t s a l i n e a r ( r e g u l a r i z i n g ) o p e r a t o r llLzIIo & CUzll
t
1
t o be determined and
ilF'(t,x)hR
<
C(0,b; V o ( l l ~ l l ) ) n C (0,b; Xo) P
There e x i s t s a c o n s t a n t
(2.Aq)
for all
E
llxll
Assumptions (2.!1-2.A3) and
x
f ( t , x ) = 0, 0
f(t,x)l~ m2
0.
be a s o l u t i o n o f (2.1).
x
G
has
a
23
A Theory of Nonlinear Evolution Equations Lemma 2 . 1 ( 9 1 . llxll
<
'0.P
>
K, K
The f o l l o w i n g holds f o r
M1,M2,M3
>
0
-
and
0
x
G
with
-
+ n ) n -"K;
M1(9 -1
and
z = S,z
[O,bl
(t,x) E
u = T/p.
1,
- S,)dF/dtii,m,O C
[(I
f o r some
0
1 and
<
C
1.
Now put z = x + h. Then z i s an approximate s o l u t i o n of ( 1 . 2 ) of o r d e r w i t h u = a ( t o be determined) and II = m/p-m2 where
(u,u,a)
-
m = min(ml,m2
But t h i s c h o i c e of
ml).
r e q u i r e s s t r o n g e r assumptions f o r
!J
The f o l l o w i n g l e m m a g i v e s a b e t t e r choice.
L.
-
Lemma 2 . 2 [9]. For s u c h t h a t f o r K > 1 and p - 9 -i;; satisfy 9 rl K < QK'
<
0
C
7;
p
>
i?(l
1
- 7.
- k).
Since (2.A4) where
zn
=
and
X = Xo,
[u(l
-
xn
+
vl-'
Theorem 1 . 3 ,
a
< v < 1, > 1 one
and
(gem
implies (l.A4) z
h,,
<
(1
= v = 1
-
+
--
~ ) r l - ~ K "< Q-'Kv
provided
i t f o l l o w s t h a t Theorems 1 . 1 - 1 . 4
a = v, s = s / p
s)/y
+
a
in
case of
with
X(X
s<
with
+
2).
such t h a t
s
Theorems
1.1
But
Note t h a t k = 1 i n Theorem 1 . 3 i s a d m i s s i b l e . II exceeds one and Lemma 2 . 2 h a s t o be modified.
but
Fi
3.
If
F ' = Fi
+
Fi,
then
P'
i s s u b j e c t t o t h e same c o n d i t i o n s of
The case of
L = 0
remain v a l i d ,
= s z n, zn _being a s o l u t i o n of ( 2 . 2 ) s a t i s f y i n g ( 2 . 3 ) ,
5, -
1.4,
Remark 2 . 1 .
- >0
with ~ ( 1 u) v 1 and 8 > 1 which
u
<
n
- -
=
-
there exists can f i n d 0 <
then the last i n e q u a l i t y i s equivalent t o
= F/p,
Y = Ys, 2
-
w)
u
If
v
Q
and
and
a = 0
1.2.
I n case of
i n Theorem 1 . 4 .
I n b o t h Theorems 1 . 3 and
i n (2.A4) can be r e p l a c e d by
Fi
f.
k=0
T h i s i s a s p e c i a l case of S e c t i o n 2 and Lemmas 2 . 1 and 2 . 2 a r e s t i l l v a l i d as w e l l as a m o d i f i c a t i o n of Lemma 2 . 2 which i s needed f o r GLIM-111. Thus Theorems 2 . 1 - 2 . 4 c a n be proved i f t h e assumptions of S e c t i o n 2 are s a t i s f i e d with L = 0 and k = 0.
24
M. Altman Nonlinear e v o l u t i o n e q u a t i o n s v i a l i n e a r i z e d e v o l u t i o n e q u a t i o n s
4.
Denote by semigroups
t h e set of a l l n e g a t i v e i n f i n i t e s i m a l g e n e r a t o r s of
G(XO)
{U(t)}
an
-A.
i s t h e semigroup g e n e r a t e d by assume
that
A(t,x(t))
For
x
i s a f u n c t i o n from
A
is s t a b l e i n
a l s o assume t h a t s t a b i l i t y of
-A E G(XO),
If
Xo.
being dense 2and
put
G
E
0 < t
{ U ( t ) } = {e-tA},
Co-
<
the
[O,b] x V0
in
Xo
w,
A(t,x(t)) = F'(t,x(t)). into
G(Xo),
and
A(t)
(see Kato [ l o - 1 2 1 , Yosida [ 1 6 ] , Tanabe 1151).
Xo
Xm
A(t)
then
i s a d m i s s i b l e and
evolution operators
U(t,s;
We =
We
preserves the
x)
generated
by
A ( t , x ( t ) ) e x i s t and
for all
x E G
and some L = 0
( 2 . 3 ) hold w i t h
and
>
MO,MZ
k
=
Tnese assumptions imply t h a t ( 2 . 2 ) and
0.
0.
Example 1.
consider the nonlinear equation
For
w i t h a p p r o p r i a t e c h o i c e of
x E G
s , i t f o l l o w s from K a t o ' s
K
> 2,
since
It
u
follows
-9s
+
t
u xxx
from
and
Example 3.
(4.3) where
ro.
s
+
F(t,u,u
Kato's
e q u a t i o n s f o r (4.3) u E G
<
b r t e w e g - d e Vries n o n l i n e a r e q u a t i o n
Example 2 .
(4.3)
full
[ I 2 1 argument
= Hr(-m,m), X m2
t h a t the linearized equations f o r ( 4 . 2 ) s a t i s f y ( 4 . 1 ) with
) = 0, 0
[12]
c
argument
satisfy (4.1) i f
i s p r o p e r l y chosen.
X
m2
t
< b,
-m
<
and Remark 4
= H (--,-)
x
<
m.
that
2.1 with
K
the
> 3,
linearized
where
Consider t h e n o n l i n e a r system du/dt
+
F(t,x,u,ux)
u = u(t,x) = u,(t.x)
=
0, 0 < t < b,
,...,u N ( t , x ) ) ,
x E Rm, F
=
(F1
,...,FN).
Suppose t h a t
t h e l i n e a r i z e d systems f o r ( 4 . 3 ) s a t i s f y Kato's ( s e e ( 1 2 ) assumptions f o r s y m m e t r i c h y p e r b o l i c systems, t h e n (4.1) is s a t i s f i e d w i t h x r e p l a c e d by u provided that s i s p r o p e r l y chosen.
25
A Theory of Nonlinear Evolution Equations The e v o l u t i o n o p e r a t o r s U(t,s;x) i n (4.1) can b e r e p l a c e d by Remark 4.1. Um(t , s ; x) g e n e r a t e d by t h e a p p r o p r i a t e s t e p f u n c t i o n s t h e i r approx i m a t i o n s (see [31). Remark 4.2.
I n a d d i t i o n t o t h e a s s u m p t i o n s made above, s u p p o s e t h a t
-
F(t,u)
for all
F(t,v)
-
F'(t,v)(u
t : [O,b]; h , u , v : V s
u n i q u e f o r small
b.
-
and some
v)!l0 < C : I U
c
>
0.
-
vII
xO
!lu
-
vllX
Then t h e s o l u t i o n of (1.1)
is
(see [31).
Re f er ences Altman, C o n t r a c t o r s and C o n t r a c t o r D i r e c t i o n s , Theory and A p p l i c a t i o n s , L e c t u r e Notes i n Pure and App. Math. (M. Dekker, N e w York, 1977).
M.
Altman, Q u a s 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 i n n o n r e f l e x i v e s p a c e s , J. I n t e g r a l Equ. 3 (1981), 153-164.
Banach
M.
M. Altman, N o n l i n e a r e v o l u t i o n I n t e g r a l Equ. 4 (1982), 307-322.
equations
in
Banach
M. Altman, G l o b a l l i n e a r i z a t i o n i t e r a t i v e methods p a r t i a l d i f f e r e n t i a l e q u a t i o n s I,II,III, t o a p p e a r .
sapces,
and
J.
nonlinear
Altman, N o n l i n e a r e v o l u t i o n e q u a t i o n s and smoothing o p e r a t o r s i n Banach s p a c e s , J. Nonlin. Analys. 8 ( 1 9 8 4 ) , 481-490.
M.
Altman, I t e r a t i v e methods o f Analys. 4 (1980), 761-722.
M.
contractor directions,
J.
NOnlin.
Altman, N o n l i n e a r e q u a t i o n s o f e v o l u t i o n and convex a p p r o x i m a t e l i n e a r i z a t i o n i n Banach s p a c e s I , I I , J. NOnlin. Analys. 8 (1984), 4 57-4 70.
M.
M. Altman, N o n l i n e a r e q u a t i o n s of Nonlin. Analys. 8 (1984), 491-499. Altman, G l o b a l l i n e a r i z a t i o n evolution equations, t o appear.
M.
T. Kato,
S c i . Univ.
e v o l u t i o n i n Banach iterative
mehtods
L i n e a r e v o l u t i o n e q u a t i o n s of " h y p e r b o l i c " Tokyo, Sec. I. 17 (1970), 241-258.
spaces,
for type,
J.
nonlinear J.
Fac.
T. Kato, L i n e a r e v o l u t i o n e q u a t i o n s o f " h y p e r b o l i c t y p e 11, J. Math SOC. J a p a n , 25 (1973),
648-666.
Kato, Q u a s i - l i n e a r e q u a t i o n s f o e v o l u t i o n w i t h a p p l i c a t i o n s t o p a r t i a l d i f f e r e n t i a l e q u a t i o n s , i n L e c t u r e Notes i n Math. No. 448 ( S p r i n g e r - V e r l a g , New York, 19751, pp. 25-70.
T.
26
M. Altman
[I31
J. Moser, A r a p i d l y convergent i t e r a t i o n method and non-linear p a r t i a l d i f f e r e n t i a l e q u a t i o n s -I, Ann. Scuola Norm. Sup. F'isa 20 (1966), 265-315.
[I41
J. Nash, The embedding problem f o r Riemannian manifolds, Ann. Math. 63 (1956), 20-63.
[I51
H. Tanabe, Equations of Evolutions, (Pitman, 1979).
[I61
K.
Yosida, Functional Analysis, 1968).
2nd ed.
(Springer-Verlag,
New York,
T h i s paper i s i n f i n a l form a n d n o v e r s i o n of i t w i l l be submitted f o r p u b l i c a t i o n elsewhere.
17
A THEORY OF NONLINEAR EVOLUTION EQUATIONS Mieczyslaw Altman Department of Mathematics Louisiana State University Baton Rouge, Louisiana U.S.A.
A theory of solving nonlinear evolution equations based on three different methods is presented which is independent of Co-semigroups. From the standpoint of applications, its important feature is the possibility of solving nonlinear evolution equations via linearized evolution equations. INTRODUCTION Kato's [12] theory of quasilinear evolution equations, which he successfully applied to nonlinear partial differential equations of mathematical physics, was the first significant step in the nonlinear direction. His famous theorem which applies only to reflexive Banach spaces was generalized to nonreflexive ones in [2] by a different method. In 1956 Nash [14] introduced the concept of smoothing operators and in 1966 Moser [13] introduced the concept of the degree of approximate linearization. Based on these notions and what is called the Nash-Moser technique constitutes a truely fundamental development in handling of the "loss of derivatives." But so far the related theorems and their generalizations are designed for nonlinear operator equations and do not apply to abstract nonlinear evolution equations (NEE). In a series of investigations [5,7,8] an attempt was made to solve the nonlinear problem for EE. Our theory is based on three global linearization iterative methods (GLIM). Global linearization means that the equation is linearized about a vector running over a nonbounded set B whereas local linearization indicates that B is bounded. The theory also applies to nonlinear operator equations ( s e e 131). Thus a unified approach has been developed to both nonlinear operator and evolution equations. Although the general theory is independent of Co-semigroups, nevertheless from the point of view of applications to nonlinear partial differential equations it is rather important that nonlinear evolution equations can be solved via linearized evolution equations. In this way one can make use of the theory of Co-semigroups which is more powerful and advanced than the theory of nonlinear semigroups which is rather limited in scope. Some highlights of our theory will be presented below.
1. CONVEX APPROXIMATE LINEARIZATION AND GLOBAL LINEARIZATION ITERATIVE METHODS (GLIM) FOR SOLVING NONLINEAR EVOLUTION EQUATIONS
Let
Z C Y C X be Banach spaces with norms
(l.A ) that'
(1.O )
11 -11
z ->
11 * 11
IIXII
<
c
-
-
-x
> 11 11
YWe assume that there exist constants C > 0 and
s
with
0 <
IIxlll-s IIXIIS
s
< 1
such
YX 2' b, denote by C(0, b; X) the Banach space of all continuous functions
Given 0 < x = x(t) defined on the interval
[ O , b]
with values in X
and norm
M.Altman
18 RxR
I n t h e same way a r e d e f i n e d respectively.
Denote by
t h e norms
C1(O,b;
d i f f e r e n t i a b l e f u n c t i o n s from with center
in
xo
t h e c l o s u r e of
in
Vo
and
llzl
the vector
[O,b]
to
ro
Let
X.
>
0.
Put
Y
for
-,Z
space of
and
2,
a l l continuously
be an open b a l l in
Wo
and l e t
Vo = Vo n Z
Y
be
V1
Y.
F : [O,b] x V1 + X
Let
< b].
t
IlyII*,Y
x)
and r a d i u s
2
<
sup[llx(t)!lx : 0
=
-,x
be a n o n l i n e a r
mapping and c o n s i d e r
t h e Cauchy
problem Px(t)
(1.1) where
+
dx/dt
S
be t h e set of f u n c t i o n s
G
-
x ( 0 ) = ~ 0 , Rx
<
X ~ N - , ~
<
f ( t , x ) = 0, 0 < t
b, x ( 0 ) = xo,
i s a l s o a n o n l i n e a r mapping.
f : [O,b] x V 1 + X
Let
+
F(t,x)
r o and
We assume t h a t t h e mapping
x
llxl F
-.
z <
-9
1 n C (0,b; X)
C(0,b; Vo(l.Uz))
E
i s d i f f e r e n t i a b l e i n t h e f o l l o w i n g sense.
For e a c h ( t , x ) c [O,b] x G, there e x i s t s a linear operator F'(t,x) -1 E IIF(*,x) + Eh) - F ( . , x ) - E F ' ( * , x ) h l + 0 as E + O+, where -,x 1 h E C(0,b; Z) n C (0.b; XI. We make t h e f o l l o w i n g assumptions. a
Cauchy sequence
bounded i n -1 E lF(*,xn n
+
(1.A:) t
E
with
(1.A;)
+
with
h)
-
F'(t,u)(u
(l.AZ)
- x i -,Y n
true for
f.
+
Eh)
Rf
(a
,x
E
G
and
-
F,f
implies
be be
vII Ilu 0
C
>
0
such t h a t
- vly, M1
>
for a l l
0
such t h a t x e G,
a r e continuous i n t h e f o l l o w i n g s e n s e ,
- F ( . , X ) E ~ , ~+ 0
IIF(.,xn)
qo
>
0
as
n +
-,
and t h e same i s
such t h a t
f( - , ~ ) l l ~ < ,qOEUhUm,X. ~
with
<
agn,,X
+
m,
F'(t,x)h
if
+
Co
>
h
i s a s o l u t i o n of t h e e q u a t i o n
0
w i t h t h e following p r o p e r t y .
g = 0, 0 < t
C
For
b, h ( 0 ) = 0,
Ilhlla,X < bCOlgUm,X. (1.A4)
(1.2)
-9
{xn} c G
n + -; o r
< M l l h ! ~ ~ ~ I I h l E , Z where , 0 0 < 8 < 1.
There e x i s t s a c o n s t a n t g
-
v)llo < Clllu
and
X),
1
{hn} c C(0,b; Y) n C (0,b; X )
E ~ F ' ( * , X ~ ) ~ , +U 0~ , a~s
- P'(.,x)hll
dh/dt then
-
Let
such t h a t
implies
There exists a c o n s t a n t
(l.A3)
x
0
let
i = 1.
with
There e x i s t s a c o n s t a n t
i = 3.
F(*,x)
The f u n c t i o n s +
and
Y) cn + O+
or
h c C(0,b; Z ) n C1(O,b;
Ilx
(1.A;)
There e x i s t s a c o n s t a n t
i = 2.
-
[O,bl; u , v c Vo;
IIF(*,x
C(0,b; Then
- F(*,xn) -
Enhn)
- F(t,v)
IIF(t,u)
in
C(0,b; Y).
with
dz/dt
For
+
x
E
G,
F'(t,x)z
the linearized equation
+
F(t,x)
- F'(t,x)x
+
f ( t , x ) = 0, 0 < t < b,
Z(0)
=
"0
19
A Theory of Nonlinear Evolution Equations a d m i t s approximate s o l u t i o n s of o r d e r t h e following
with
(p,u,u)
0
< u
<
1
in t h e s e n s e of
Then D e f i n i t i o n 1.1. [9]. L e t p > 0 , v > 0 , u > 0 be g i v e n numbers. if t h e l i n e a r i z e d e q u a t i o n (1.2) admits approximate s o l u t i o n s of o r d e r (U,U,U) t h e r e e x i s t s a c o n s t a n t M > 0 which has t h e f o l l o w i n g p r o p e r t y . For e v e r y < K then t h e r e e x i s t s a r e s i d u a l ( e r r o r ) x E G, K > 1, and Q > 1, if nxll function
z
-9
and a f u n c t i o n
y
such t h a t
z
iizu - 9 2
llyll (1.2')
dz/dt
<
0
+
+
F'(t,x)z
m,X
< MQK" and
s MQ-!-IK',
F(t,x) = F'(t,x)x
+
+y
f(t,x)
= 0,
t C b, z ( 0 ) = xo.
Now f o r x E G, l e t z be a s o l u t i o n of t h e e q u a t i o n ( 1 . 2 ' ) z = x + h. Then o b v i o u s l y h is a s o l u t i o n of t h e e q u a t i o n
(1.3)
dh/dt
+
+
F'(t,x)h
+
Px
and put
y = 0, 0 < t < b, h ( 0 ) = 0 ,
and we g e t
Put
The f o l l o w i n g is a n i t e r a t i v e method of c o n t r a c t o r d i r e c t i o n s and assume t h a t x0,x1 ,...,xn 7: xo , 0, ,..., GLIM-I.
to =
E G; t O , t l
t
[61. are
known and put (1.6) that
x
and
~ =+ xn~ + ~~h~
tn+l = tn + cn,
S
= (1
Xn+l
-
En'Xn
+
EnZn,
which j u s t i f i e s t h e term "convex approximate l i n e a r i z a t i o n , where and
h ,,
being a s o l u t i o n of (1.3).
-
qObCO = q
<
q/2
<
1/2,
and put
, x ) s qlPxollm ,exp(-(l n n 9 e x i s t s 0 < E < 1 s u c h that P(1,h
where
po = XPxoII
-9
x
and put
To detemine
P(E,h,x)
-
q)tn),
E
n
= c-lllP(x
then
put
2F/q
let
E~
+
Eh) E
n
-
(1
= 1.
Otherwise
= E.
The method (1.6) s a t i s f i e s t h e f o l l o w i n g ' i n d u c t i o n
zn =
+ hn
< c < 1, where - E)PXU-,~. I f
assumptions
there
20
M.Altman
Ux It n m,Z
I l F ' ~ ~ l l , , ~G llPxoi~
-9
Theorem 1.1 [91.
u(1
-
-
v)
u
>
N = [(I
/u <
-
q)
a ) :
>
by
has a s o l u t i o n
x
b'
-
(1
+
GLIM-11. The directions. Put
- q)tn),
exp(-(1
0,
a
- q) -
>
with
i = 1, suppose t h a t
[ ~ ( l v)
11,
and
- 01-'
and
i s such t h a t
b'
-
-
- q)AlS < ro.
- q)G)C[b'Co(l + ~ ) p o ] l - s [ a ( l
0
and
and
a Ilxn
_ _ < (1 - s ) / s . - X U -,y + 0
iterative
~ =+ zn~ = xn
+
Then (1.1) as
,
(xn
with
-.
i s independent of
method
hn
n +
b
replaced
contractor
G),
E
as i n (1.6) and i n d u c t i o n assumptions
with
<
IIxn II-,Z where
>
1
following
xo, x
x0 ( t )
(1.7)
-
- q)6l-'exp((l
6
= Kn
0 t o be determined.
Al-v-o/lJ [ a ( l
where
= 1
x
I n a d d i t i o n t o (1.AO-1.A4)
0, a(l
M(2M)l /u(Tpo)-l
>
a,A
f o r some c o n s t a n t s
- q)tn)
exp(a(1
q
<
2-1/a
Theorem 1.2.
and
Aq-m
(C1ro
+
= Kn
and
qO)bCO(l + 27)
I n a d d i t i o n t o (l.AO-l.Aq)
c poq",
IIpXnII,,x
<
<
q
with
1;
4<
i = 2
q.
and
E = 1
suppose t h a t
0 = 1
- F(1 +
a)
>
0, N(1
- q 0) <
ro, N = C[b'CO(l
_
i n (l.Az),
+ ~)poll-s[A(q-u -
-
1)Is,
Then t h e s t a t e m e n t of Theorem 1.1 h o l d s t r u e . GLIM-111. The f o l l o w i n g i s a r a p i d l y convergent i t e r a t i o n method which i s based on t h e e s s e n t i a l technique of MDser [131.
(1.A5)
Let 1
<
a,T;
(11
-
X
<
a)-l(i
LI
be such t h a t
+
a(1
+ X) +
11)
<
'I
<
2
-
-ao
<
2
-
a,
21
A Theory of Nonlinear Evolution Equations
<
0
where
<
.to
i s such t h a t
1
o < B < Remark. 0 < 2A
- n)[(l
uA(ao
-
5
0.
t h e constant
bCo
(1.A;)
=
2.
x
1.
= KT+a n
and
I I P X ~ I I ~< , K ; ’ .~
-
+
<
T
-
A-1
2 )P+1
<
0
<
.
X
+
-,y
< (u +
1
Then
T
>
are s a t i s f i e d w i t h
1 such t h a t :
i = 3
if
-.
t h e n e q u a t i o n (1.1) w i t h as n + L e t us n o t i c e that
2),
+ 0
i n (A3) can be r e p l a c e d by
(1
ul:
-
CO.
1)/2
and
is a number such that
2.
Theorem 1.3 remains v a l i d i f (l.Ag) is r e p l a c e d by
(1.A;)
and
0.
Smoothing o p e r a t o r s combined w i t h e l l i p t i c r e g u l a r i z a t i o n . The choice of Moser’s d e g r e e
of e l l i p t i c r e g u l a r i z a t i o n . linearization. with
{Xj}
Let
norms such t h a t
o <
<
i n t h e above. Then = 1 + a, and put Kn+l
K~(M,B,LI,X,~)
0 and s < A(A and Exn XU
Suppose t h a t
Theorem 1.4.
a
and
- - - -S T >
x 0 < B <--x+l 1 < ( 1 -$)-’
1,
~ ( 2+ U)I-’
+
Suppose that assumptions (l.AO-l.Ag) Then t h e r e e x i s t s
6 = (I s)A has a s o l u t i o n
where f = 0
CI
!i)
>
a,)
as in (1.7) w i t h i n d u c t i o n assumptions
(xn]
Theorem 1.3. f
+
3) u =
-
ao)/(l
+
+
# x n # m , z < Kn
and
+
(1
1)/(u One can put a. = (u (11 - 1 ) / ( ~+ 1). W e assume
<
define
>
!I
ml
<
m2
<
3
W e assume t h a t
(2.k)
operators
s
0 <
< j implies < 5 < p. i
So,
0
> 1,
p
v
The d e g r e e
k
o f approximate
be a scale of Banach s p a c e s w i t h i n c r e a s i n g X. c Xi
J
there
and
W*Rj
>
II*li
and l e t
e x i s t s a one parameter
( s e e Nash [ 1 4 ] , Moser [ 1 3 ] ) such t h a t
f a m i l y of
linear
22
M. Altman P-9 IISXil < c k l e p
f o r some c o n s t a n t
>
C
0. where
< cnxii~-Aiixiih f o r j
that
P
m2
i s t h e i d e n t i t y mapping.
I
-
(1
=
IIXII
+
A)r
W e a l s o assume
0 c A c 1.
~ p . with
Using t h e same n o t a t i o n a s i n S e c t i o n 1 p u t lixil
m,j
= sup[Ix(t)!l
center
xo = 0
c l o s u r e of
in
F,f
>
ro
and r a d i u s
Vo
Let
0 c t c b],
*
j '
and l e t
0.
Put
Wo
Vo =
Xs.
: [O,b] x
Xs + Xo
be
+
+
two
CI
be an open b a l l w i t h
X,
wO n XP and l e t Vs
nonlinear
mappings
and
be the
consider
the
Cauchy problem (2.1) Let
Px(t) G
%
dx/dt
F(t,x)
be t h e s e t of f u n c t i o n s
x ( 0 ) = 0 , I l ~ l l ~<, r~ o
and
Y
=
with
Xs
s
<
( t , x ) c [O,b]
x
m2
ml
G
Cllhil
+
F'(t,x)z
0 c t < b , z ( 0 ) = 0, solution
z
-
>
C
For
k > 0 (t,x) E
+
0L.z
with s m a l l
0
Xg, 2 =
=
Yp
and
such t h a t
< CMhM
llF'(t,x)hllo
>
0
ml
<
0
L = L(n)
101
<
F'(t,x)x
x
G, l t e
z
C
such t h a t
and the modified l i n e a r i z e d e q u a t i o n
+ F(t,x) -
t o be determind and [O,b]
X
s = s/p.
F'(t,x)x 1
and
+ f(t,x)
>
+
= 0,
( t , x ) E [O,b]
such t h a t
-
with
m.
m2
C
f o r some
= 0.
Vo.
f o r some c o n s t a n t
dz/dt
(2.2)
< b, x ( 0 )
"SP a r e the same as i n S e c t i o n 1, where
There e x i s t s a l i n e a r ( r e g u l a r i z i n g ) o p e r a t o r llLzIIo & CUzll
t
1
t o be determined and
ilF'(t,x)hR
<
C(0,b; V o ( l l ~ l l ) ) n C (0,b; Xo) P
There e x i s t s a c o n s t a n t
(2.Aq)
for all
E
llxll
Assumptions (2.!1-2.A3) and
x
f ( t , x ) = 0, 0
f(t,x)l~ m2
0.
be a s o l u t i o n o f (2.1).
x
G
has
a
23
A Theory of Nonlinear Evolution Equations Lemma 2 . 1 ( 9 1 . llxll
<
'0.P
>
K, K
The f o l l o w i n g holds f o r
M1,M2,M3
>
0
-
and
0
x
G
with
-
+ n ) n -"K;
M1(9 -1
and
z = S,z
[O,bl
(t,x) E
u = T/p.
1,
- S,)dF/dtii,m,O C
[(I
f o r some
0
1 and
<
C
1.
Now put z = x + h. Then z i s an approximate s o l u t i o n of ( 1 . 2 ) of o r d e r w i t h u = a ( t o be determined) and II = m/p-m2 where
(u,u,a)
-
m = min(ml,m2
But t h i s c h o i c e of
ml).
r e q u i r e s s t r o n g e r assumptions f o r
!J
The f o l l o w i n g l e m m a g i v e s a b e t t e r choice.
L.
-
Lemma 2 . 2 [9]. For s u c h t h a t f o r K > 1 and p - 9 -i;; satisfy 9 rl K < QK'
<
0
C
7;
p
>
i?(l
1
- 7.
- k).
Since (2.A4) where
zn
=
and
X = Xo,
[u(l
-
xn
+
vl-'
Theorem 1 . 3 ,
a
< v < 1, > 1 one
and
(gem
implies (l.A4) z
h,,
<
(1
= v = 1
-
+
--
~ ) r l - ~ K "< Q-'Kv
provided
i t f o l l o w s t h a t Theorems 1 . 1 - 1 . 4
a = v, s = s / p
s)/y
+
a
in
case of
with
X(X
s<
with
+
2).
such t h a t
s
Theorems
1.1
But
Note t h a t k = 1 i n Theorem 1 . 3 i s a d m i s s i b l e . II exceeds one and Lemma 2 . 2 h a s t o be modified.
but
Fi
3.
If
F ' = Fi
+
Fi,
then
P'
i s s u b j e c t t o t h e same c o n d i t i o n s of
The case of
L = 0
remain v a l i d ,
= s z n, zn _being a s o l u t i o n of ( 2 . 2 ) s a t i s f y i n g ( 2 . 3 ) ,
5, -
1.4,
Remark 2 . 1 .
- >0
with ~ ( 1 u) v 1 and 8 > 1 which
u
<
n
- -
=
-
there exists can f i n d 0 <
then the last i n e q u a l i t y i s equivalent t o
= F/p,
Y = Ys, 2
-
w)
u
If
v
Q
and
and
a = 0
1.2.
I n case of
i n Theorem 1 . 4 .
I n b o t h Theorems 1 . 3 and
i n (2.A4) can be r e p l a c e d by
Fi
f.
k=0
T h i s i s a s p e c i a l case of S e c t i o n 2 and Lemmas 2 . 1 and 2 . 2 a r e s t i l l v a l i d as w e l l as a m o d i f i c a t i o n of Lemma 2 . 2 which i s needed f o r GLIM-111. Thus Theorems 2 . 1 - 2 . 4 c a n be proved i f t h e assumptions of S e c t i o n 2 are s a t i s f i e d with L = 0 and k = 0.
24
M. Altman Nonlinear e v o l u t i o n e q u a t i o n s v i a l i n e a r i z e d e v o l u t i o n e q u a t i o n s
4.
Denote by semigroups
t h e set of a l l n e g a t i v e i n f i n i t e s i m a l g e n e r a t o r s of
G(XO)
{U(t)}
an
-A.
i s t h e semigroup g e n e r a t e d by assume
that
A(t,x(t))
For
x
i s a f u n c t i o n from
A
is s t a b l e i n
a l s o assume t h a t s t a b i l i t y of
-A E G(XO),
If
Xo.
being dense 2and
put
G
E
0 < t
{ U ( t ) } = {e-tA},
Co-
<
the
[O,b] x V0
in
Xo
w,
A(t,x(t)) = F'(t,x(t)). into
G(Xo),
and
A(t)
(see Kato [ l o - 1 2 1 , Yosida [ 1 6 ] , Tanabe 1151).
Xo
Xm
A(t)
then
i s a d m i s s i b l e and
evolution operators
U(t,s;
We =
We
preserves the
x)
generated
by
A ( t , x ( t ) ) e x i s t and
for all
x E G
and some L = 0
( 2 . 3 ) hold w i t h
and
>
MO,MZ
k
=
Tnese assumptions imply t h a t ( 2 . 2 ) and
0.
0.
Example 1.
consider the nonlinear equation
For
w i t h a p p r o p r i a t e c h o i c e of
x E G
s , i t f o l l o w s from K a t o ' s
K
> 2,
since
It
u
follows
-9s
+
t
u xxx
from
and
Example 3.
(4.3) where
ro.
s
+
F(t,u,u
Kato's
e q u a t i o n s f o r (4.3) u E G
<
b r t e w e g - d e Vries n o n l i n e a r e q u a t i o n
Example 2 .
(4.3)
full
[ I 2 1 argument
= Hr(-m,m), X m2
t h a t the linearized equations f o r ( 4 . 2 ) s a t i s f y ( 4 . 1 ) with
) = 0, 0
[12]
c
argument
satisfy (4.1) i f
i s p r o p e r l y chosen.
X
m2
t
< b,
-m
<
and Remark 4
= H (--,-)
x
<
m.
that
2.1 with
K
the
> 3,
linearized
where
Consider t h e n o n l i n e a r system du/dt
+
F(t,x,u,ux)
u = u(t,x) = u,(t.x)
=
0, 0 < t < b,
,...,u N ( t , x ) ) ,
x E Rm, F
=
(F1
,...,FN).
Suppose t h a t
t h e l i n e a r i z e d systems f o r ( 4 . 3 ) s a t i s f y Kato's ( s e e ( 1 2 ) assumptions f o r s y m m e t r i c h y p e r b o l i c systems, t h e n (4.1) is s a t i s f i e d w i t h x r e p l a c e d by u provided that s i s p r o p e r l y chosen.
25
A Theory of Nonlinear Evolution Equations The e v o l u t i o n o p e r a t o r s U(t,s;x) i n (4.1) can b e r e p l a c e d by Remark 4.1. Um(t , s ; x) g e n e r a t e d by t h e a p p r o p r i a t e s t e p f u n c t i o n s t h e i r approx i m a t i o n s (see [31). Remark 4.2.
I n a d d i t i o n t o t h e a s s u m p t i o n s made above, s u p p o s e t h a t
-
F(t,u)
for all
F(t,v)
-
F'(t,v)(u
t : [O,b]; h , u , v : V s
u n i q u e f o r small
b.
-
and some
v)!l0 < C : I U
c
>
0.
-
vII
xO
!lu
-
vllX
Then t h e s o l u t i o n of (1.1)
is
(see [31).
Re f er ences Altman, C o n t r a c t o r s and C o n t r a c t o r D i r e c t i o n s , Theory and A p p l i c a t i o n s , L e c t u r e Notes i n Pure and App. Math. (M. Dekker, N e w York, 1977).
M.
Altman, Q u a s 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 i n n o n r e f l e x i v e s p a c e s , J. I n t e g r a l Equ. 3 (1981), 153-164.
Banach
M.
M. Altman, N o n l i n e a r e v o l u t i o n I n t e g r a l Equ. 4 (1982), 307-322.
equations
in
Banach
M. Altman, G l o b a l l i n e a r i z a t i o n i t e r a t i v e methods p a r t i a l d i f f e r e n t i a l e q u a t i o n s I,II,III, t o a p p e a r .
sapces,
and
J.
nonlinear
Altman, N o n l i n e a r e v o l u t i o n e q u a t i o n s and smoothing o p e r a t o r s i n Banach s p a c e s , J. Nonlin. Analys. 8 ( 1 9 8 4 ) , 481-490.
M.
Altman, I t e r a t i v e methods o f Analys. 4 (1980), 761-722.
M.
contractor directions,
J.
NOnlin.
Altman, N o n l i n e a r e q u a t i o n s o f e v o l u t i o n and convex a p p r o x i m a t e l i n e a r i z a t i o n i n Banach s p a c e s I , I I , J. NOnlin. Analys. 8 (1984), 4 57-4 70.
M.
M. Altman, N o n l i n e a r e q u a t i o n s of Nonlin. Analys. 8 (1984), 491-499. Altman, G l o b a l l i n e a r i z a t i o n evolution equations, t o appear.
M.
T. Kato,
S c i . Univ.
e v o l u t i o n i n Banach iterative
mehtods
L i n e a r e v o l u t i o n e q u a t i o n s of " h y p e r b o l i c " Tokyo, Sec. I. 17 (1970), 241-258.
spaces,
for type,
J.
nonlinear J.
Fac.
T. Kato, L i n e a r e v o l u t i o n e q u a t i o n s o f " h y p e r b o l i c t y p e 11, J. Math SOC. J a p a n , 25 (1973),
648-666.
Kato, Q u a s i - l i n e a r e q u a t i o n s f o e v o l u t i o n w i t h a p p l i c a t i o n s t o p a r t i a l d i f f e r e n t i a l e q u a t i o n s , i n L e c t u r e Notes i n Math. No. 448 ( S p r i n g e r - V e r l a g , New York, 19751, pp. 25-70.
T.
26
M. Altman
[I31
J. Moser, A r a p i d l y convergent i t e r a t i o n method and non-linear p a r t i a l d i f f e r e n t i a l e q u a t i o n s -I, Ann. Scuola Norm. Sup. F'isa 20 (1966), 265-315.
[I41
J. Nash, The embedding problem f o r Riemannian manifolds, Ann. Math. 63 (1956), 20-63.
[I51
H. Tanabe, Equations of Evolutions, (Pitman, 1979).
[I61
K.
Yosida, Functional Analysis, 1968).
2nd ed.
(Springer-Verlag,
New York,
T h i s paper i s i n f i n a l form a n d n o v e r s i o n of i t w i l l be submitted f o r p u b l i c a t i o n elsewhere.