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UBC Theses and Dissertations

On non-linear time-lag evolution equations Lam, Che-Bor 1972

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I S O ON NON-LINEAR TIME-LAG EVOLUTION EQUATIONS by CHE-BOR LAM B.A., U n i v e r s i t y of Hong Kong, 1966. A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENT FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in- -the Department of Mathematics V7e accept t h i s t h e s i s as conforming t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA December, 1972. In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Department or by h i s representatives. It i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of The University of B r i t i s h Columbia Vancouver 8 , Canada Date i i S u p e r v i s o r : An Ton B u i . ABSTRACT The purpose of t h i s t h e s i s i s to o b t a i n e x i s t e n c e theorems f o r p e r t u r b a t e d e v o l u t i o n equations i n H i l b e r t spaces and Banach spaces. A p r a c t i c a l example o f the p e r t u r b a t i o n s under c o n s i d e r a t i o n i s an o p e r a t o r w i t h a t i m e - l a g or d e l a y e d argument. Throughout the t h e s i s , the G a l e r k i n approximation method w i l l be used to e s t a b l i s h the e x i s t e n c e theorems. In chapter one, we study the problem i n H i l b e r t • spaces. -T-h-e -k-ey i s -to o b t a i n a p r i o r i - - e s t i m a t e s on the time f r a c t i o n a l d e r i v a t i v e s of the approximate s o l u t i o n s . We s h a l l prove t h a t t h e r e e x i s t s a s o l u t i o n w i t h time f r a c t i o n a l d e r i v a t i v e s of o r d e r l e s s than 1/2. In chapter two, we c o n s i d e r the problem i n Banach spaces. Again, we apply the G a l e r k i n approximation method, but w i t h a s p e c i a l b a s i s . In chapter t h r e e , we study p e r i o d i c s o l u t i o n s of e v o l u t i o n e q u a t i o n s . Here we use the Schauder-Tychonov f i x e d p o i n t theorem to prove the e x i s t e n c e of p e r i o d i c s o l u t i o n t o the approximating e q u a t i o n s . In the l a s t chapter, we g i v e examples of a p p l i a c t i o n s of the v a r i o u s theorems proved i n the f i r s t t h r e e c h a p t e r s . i i i TABLE OF CONTENTS Page INTRODUCTION 1 CHAPTER I : EVOLUTION EQUATIONS IN HILBERT SPACES . . 3 CHAPTER I I : EVOLUTION EQUATIONS IN BANACH SPACES . . 30 CHAPTER I I I : PERIODIC SOLUTIONS OF EVOLUTION EQUATIONS 41 CHAPTER IV : APPLICATIONS i . . . . 5.7 BIBLIOGRAPHY . 75 ACKNOWLE D GE f'ENT I am deeply g r a t e f u l to my r e s e a r c h s u p e r v i s o r , Dr. A. T. B u i , f o r h i s a d v i c e and encouragement d u r i n g the p r e p a r a t i o n o f t h i s t h e s i s . The f i n a n c i a l support o f the N a t i o n a l Research C o u n c i l o f Canada and the U n i v e r s i t y o f B r i t i s h Columbia i s g r a t e f u l l y acknowledged. 1 INTRODUCTION D i f f e r e n t i a l equations w i t h a delayed argument have been s t u d i e d e x t e n s i v e l y . A r e c e n t survey of these problems has been made by f y s h k i s and E l ' s g o l ' t s [ l O j . One reason f o r the importance o f t h i s c l a s s o f equations i s t h a t they d e s c r i b e processes w i t h " a f t e r - e f f e c t s " . T h e r e f o r e d i f f e r e n t i a l equations w i t h a dela y e d argument have found many a p p l i c a t i o n s i n P h y s i c s , mechanics, e n g i n e e r i n g , economics, b i o l o g y and e s p e c i a l l y , i n the theory o f automatic c o n t r o l . We s h a l l r e f e r to [lo] and i t s b i b l i o g r a p h y f o r f u r t h e r d e t a i l s . R e c e n t l y , A r t o l a [ l ] c o n s i d e r e d e v o l u t i o n e q u a t i o n s w i t h an a b s t r a c t p e r t u r b a t i o n , a p a r t i c u l a r case o f which i s a de l a y e d argument. He s t u d i e d d i f f e r e n t i a l o p e r a t o r s i n H i l b e r t spaces and Banach spaces w i t h such a r e g u l a r p e r t u r b a t i o n . The case o f l i n e a r e v o l u t i o n equations i n H i l b e r t spaces was t r e a t e d e x t e n s i v e l y by A r t o l a . For the n o n - l i n e a r case, mono-tone p a r a b o l i c o p e r a t o r s was s t u d i e d . In another d i r e c t i o n , s t r o n g l y n o n - l i n e a r p a r a b o l i c equations have been s t u d i e d by Browder [ 3 ] and by L i o n s [ 9 ] u s i n g the theory o f monotone o p e r a t o r s d e f i n e d on a r e f l e x i v e s e p a r a b l e Banach space. In a s e r i e s o f papers [12,13,14,15], Ton has extended these works t o a l a r g e r c l a s s of d i f f e r e n t i a l o p e r a t o r s which i n c l u d e the Navier-Stokes e q u a t i o n s . The 2 purpose o f t h i s t h e s i s i s t o study the p e r t u r b a t i o n problems i n A r t o l a 1 f o r the c l a s s of o p e r a t o r s s t u d i e d by Ton. The c r u c i a l technique employed by Ton i s the use of a s i n g u l a r p e r t u r b a t i o n i n v o l v i n g a d u a l i t y mapping. In h i s work, A r t o l a used the G a l e r k i n approximation method. We s h a l l make use of both of these techniques i n t h i s t h e s i s . I n chapter one, we s h a l l study n o n - l i n e a r e v o l u t i o n equations i n H i l b e r t spaces. C r u c i a l a p r i o r i e stimates i n t h i s case are the bounds on the f r a c t i o n a l d e r i v a t i v e s o f the approximate s o l u t i o n s . S o l u t i o n s w i t h f r a c t i o n a l time-d e r i v a t i v e s of o r d e r l e s s than 1/2 are o b t a i n e d . In chapter two, we study t i m e - l a g e v o l u t i o n equations i n Banach spaces u s i n g the G a l e r k i n approximation method w i t h a s p e c i a l b a s i s . In chapter t h r e e , we study p e r i o d i c s o l u t i o n s o f t i m e - l a g e v o l u t i o n e q u a t i o n s . P e r i o d i c s o l u t i o n s are not t r e a t e d by A r t o l a . In the l a s t c h a p t e r , we g i v e examples of a p p l i c a t i o n s of the v a r i o u s theorems proved i n the f i r s t t h r e e c h a p t e r s . 3 CHAPTER I EVOLUTION EQUATIONS IN HILBERT SPACES 1.1. N o t a t i o n s . L e t H be a H i l b e r t space and ('f'Jn ke t n e i - n n e r product i n H . L e t W be a Banach space such t h a t W c H , Suppose the n a t u r a l i n j e c t i o n o f W i n t o H i s continuous, and W i s dense i n H . We i d e n t i f y H w i t h i t s d u a l H , and t h e r e f o r e (1.1) W c H c W* Throughout t h i s t h e s i s , ( » , » ) w * w i l l denote the p a i r i n g between W and W* f and |•| w w i l l denote the norm i n W . I f w e W and h e H , then h e W and (1.2) (h,w) w* = (h,w) H . Thus i f W i s a H i l b e r t space, then (h,w) w y (h,w) w* i n g e n e r a l . L P ( 0 , T ; W ) denotes the Banach space of e q u i v a l e n c e c l a s s e s o f f u n c t i o n s u(t) from [ O , T ] to W w i t h (1.3) J * | u ( t ) | P d t < 4 and w i t h the norm l l / P (1.4) |u| = [ f |u(t) \ P d t 1 L P(0,T;W) L Jo where p i s not l e s s than 1 . I f p i s s t r i c t l y l a r g e r than 1 , p' w i l l always denote the number which s a t i f i e s the f o l l o w i n g e q u a t i o n : (1.5) 1 + 1 , - 1 . 1.2. Fundamental lemmas. In t h i s section„, we s h a l l s t a t e s e v e r a l well-known lemmas which w i l l be r e f e r r e d t o f r e q u e n t l y . The f o l l o w i n g lemma i s c a l l e d Gronwall's lemma. A p r o o f of t h i s lemma can be found i n A r t o l a [ l ] , p. 148. Lemma 1.1: L e t u and v be two p o s i t i v e r e a l - v a l u e d  f u n c t i o n s i n t e g r a b l e over a f i n i t e i n t e r v a l [0,T] and l e t Cl be a p o s i t i v e c o n s t a n t such t h a t (1 . 6 ) u(t) < Ci + | u(o)v(a) da 5 f o r almost a l l t i n [0,T] , th en (1.7) u(t) $ Ciexp I f v(a)da J 1 i o 1 f o r almost a l l t i n [o,TJ. The f o l l o w i n g theorem i s due t o Aubin [ 2 ] . Therrem 1.1; Given t h r e e Banach spaces B Q , B , Bj , w i t h B 0 d B c: Bi , and B.^  r e f l e x i v e f o r i = 1 , 2 . Moreover, (1.8) B0-* B and B + B j are compact and continuous  r e s p e c t i v e l y . L e t G = {v: v e L P ( O , T ; B 0 ) , v' e L r ( 0 , T ; B i ) } , where T i s f i n i t e and 1 < p, r < °° . G is_ a Banach space w i t h the norm: G 1 ' L P ( O . T ; B 0 ) ' ' L r ( 0 , T ; B ! ) Then G cz L P(0,T;B) and the i n j e c t i o n of G i n t o L p(0,T;B) i s compact. 6 Lemma 1.2;(Cf. Lions [8], p.7) Le t B be a_ Banach space. I f f e L^(0,T;B) and f• e L P ( 0 , T ; B ) , 1 < p < 0 0, then f i_s_ continuous from [0,T] to B. L e t V and H be two H i l b e r t spaces with V c H, For y > 0, we d e f i n e E^. (-°°,co;V,H) as the e q u i v a l e n c e c l a s s e s of f u n c t i o n s u(t) from (-00 , 0 0) to V w i t h u e L 2 (-0O,°°;V) and | T | U ( T ) e L 2 (-00 , 00;H) , where u i s the F o u r i e r t r a n s f o r m of u(t) wi t h r e s p e c t to t . By (-00 , 00;V,H) i s a H i l b e r t space w i t h the norm: U . I O ) l » l | T ( . . f . | V f H ) - £ j u ( t > | ' d t + } _ + J T | ^ | 0 ( T ) | ^ T . The s e t of r e s t r i c t i o n s of elements i n E y (-«>,<»; V,H) to [0,T] i s a l s o a H i l b e r t space, denoted by 5 (0,T;V rH), w i t h the q u o t i e n t norm: (1.11) |u| = i n f jv| E Y(0,T;V,H) v (-<»,«>; V,H) where v e 5 y (-00,00; V,H) and v = u almost everywhere on [ 0 , T ] , We have the f o l l o w i n g theorem which i s due to L i o n s [ 8], p.60. 7 Theorem 1.2: L e t V 0, Vi and H be t h r e e H i l b e r t spaces w i t h Vo cr V c H. Suppose the i n j e c t i o n of V Q i n t o V and  t h a t o f Vj i n t o H are compact and continuous r e s p e c t i v e l y , then the i n j e c t i o n o f 5 (0,T;V Q,H) i n t o L 2 ( 0 , T ; V j ) i s  compact i f y > 0. 1.3. P e r t u r b a t i o n of l o c a l type. L e t H be a H i l b e r t space. For f i x e d t < T, we s h a l l i d e n t i f y L°°(0,t;H) as a c l o s e d subspcce o f L°°(0,T;H) by extending every f u n c t i o n i n L°°(0,t;H) t o be zero o u t s i d e [0,T]. I f u e L°°(0,T;H), we s h a l l d e f i n e s e (o,t) o t h e r w i s e . An o p e r a t o r M e L (L°° (0 , T;H) ,L°° (0 ,T ;H) ) i s s a i d t o be o f l o c a l type i f there e x i s t s a c o n s t a n t u > 0, not depending on t such t h a t f u(s) (1.12) Y u(s) t (1.13) ]ytmlL~(0,T;H) * y | Y t U | L - ( 0 / T ; H ) 8 f o r a l l t e [0,T] . Many examples o f o p e r a t o r s of l o c a l type have been g i v e n by A r t o l a i n [ l ] . The f o l l o w i n g example i s of s p e c i a l i n t e r e s t and w i l l be c a l l e d a t i m e - l a g p e r t u r b a t i o n . Example: L e t t -*• w(t) be a p o s i t i v e measurable f u n c t i o n d e f i n e d on [ O , T ] . For u e L°°(0,T ; H ) , d e f i n e ( u ( t - w(t) ) t - w(t) > o (1.14) W u (t) = o t h e r w i s e . C l e a r l y , M i s an o p e r a t o r of l o c a l type. An o p e r a t o r M e L ( L ° 9 (0,T ;H) ,L°(0,T ;H) ) i s s a i d t o s a t i s f y the convergence c o n d i t i o n i f (i) => ( i i ) , where (i) u e L°°(0,T;H), u (t) u ( t ) i n H a.e. n n ( i i ) Mu (t) Mu(t) i n H a.e. 9 1.4 Assumptions and f o r m u l a t i o n o f the problem. L e t V and H be two H i l b e r t spaces such t h a t V cr H , V i s dense i n H , and the n a t u r a l mapping o f V i n t o H i s compact. D e f i n i t i o n : The f r a c t i o n a l d e r i v a t i v e o f order y , o < y, of <f>(t) i_s d e f i n e d by (1.15) D^Mt) d t f t (t - s)"Yc|>(s)ds whenever the r i g h t hand s i d e makes sense. Suppose t h a t <f> e L 2(0,T;H) and e L 2 (0,T;I1) f o r some y > 0 . Then there e x i s t s a f u n c t i o n $ e L 2 (-00,00;H) w i t h <}>(t) - = $ (t) almost everywhere such t h a t (1.16) f +<» (1 + | t | 2 Y ) |?(T) I 2 dT < «> , H Where $(T) i s the F o u r i e r t r a n s f o r m o f $(t) w i t h r e s p e c t t o t . Moreover, i f <j>n i s a sequence o f f u n c t i o n s d e f i n e d on [o,T] such t h a t | cf> \ + |D Y4> | are n L 2(0,T;H) C n L 2(0,T;H) 10 bounded by a c o n s t a n t , then we can choose <$>n i n such a way . + 0 0 so t h a t ( 1 + | T | 2 Y ) | $ ( T ) l ^ d T are a l s o bounded by a / — 00 c o n s t a n t . I f <J> z L 2 ( 0 , T ; H ) and D <|> e L 2 ( 0 , T ; H ) , then D^ 4> and D^ 4» a l s o b e l o n g t o L 2 ( 0 , T ; H ) i f 0 < a,3 < 1. I f a + 3 = 1 , then ( 1 . 1 7 ) DlDt^ = D t D t ( f ) = Dt*« I f cf> e L 2 ( 0 , T ; H ) and DJ<J> e L 2 ( 0 , T ; H ) f o r some y i n ( 0 , 1 ) , then there e x i s t s c o n s t a n t K ( y ) , depending o n l y on Yr such t h a t ( 1 . 1 8 ) <J)(t) = K(Y) f t(t - s) Y~ 1D Y ;<f>(s)ds. J o The d e f i n i t i o n and p r o p e r t i e s of the f r a c t i o n a l d e r i v a t i v e s s t a t e d above are q u i t e well-known. (Cf. S h i n b r o t [ l l ] . ) In t h i s c h apter, we s h a l l c o n s i d e r n o n - l i n e a r o p e r a t o r s A mapping V i n t o V s a t i s f y i n g the f o l l o w i n g assumption: Assumption I (i) There e x i s t s a c o n s t a n t ' C such t h a t 11 | A U | v * < C,[|u|* + l ] f o r a l l u i n V. ( i i ) There e x i s t c onstants C 2 and 3, o < 3 < l f such  t h a t | ( A u , v ) v * | « C 2 [ | u | ^ | v | v + |u| v|v| v] f o r a l l u , v , i n V. ( i i i ) A i_s_ continuous from the s t r o n g topology o f V to the weak topology o f V*. (iv) ( A u , u ) ^ ijs i n t e g r a b l e on [ O , T ] f o r a l l u i n L 2 (0,T;V) nL r o(0,T;H) . The main r e s u l t o f t h i s chapter i s the f o l l o w i n g theorem. Theorem 1.3: L e t A be a n o n - l i n e a r o p e r a t o r mapping V i n t o V* s a t i s f y i n g Assumption I . Moreover, suppose t h a t (a) (Au,u) v * > c | u | 2 V 12 f o r a l l u i n V and some c > 0 (b) L e t u n e L 5 (0,T;V) , e ( L S (0,T; V) ) *. Suppose t h a t u n u weakly i n L 2(0,T;V) , u n -*• u i n the weak * topology o f L~(0,T;H) , u n ( o ) + u(o)we i n H and D Y u n + D^u weakly i n L 2 (0,T;H) f o r t h a t l i m N i n f {|u n(o)| 2 + jTQ (ui + A u n , u n ) v * d t } fT $ |u(o) | 2 + J .(g,u) v*dt and fT fT (u£ + A u n , v ) v * d t -> (g,v) v*dt Jo J o f o r a l l v e cj(0,T;V) , then we assume t h a t fT fT j ( A u n , v ) v * d t + j ( A u , v ) v * d t f o r a l l v e C(0,T;V). L e t M e L ( L ° ° (0,T;H) ,L°° (0,T;H)) , be o f l o c a l type  and s a t i s f y the convergence c o n d i t i o n . Then f o r each u Q i n H and f i n L 2(0,T;V*) , there e x i s t s u i n L 2(0,T;V) and L° (0,T;H) w i t h D^u e L 2 (0,T;H) f o r some y e (o,|- f) such t h a t 13 (1.19) -J ( u , v ' ) R d t + J (Au.,v) v,*dt + Jr ( ^ , v ) H d t = | ( f f v ) y * d t + ( u Q , v ( o ) ) H o f o r a l l v e C '(0,T;V) with v(T) = 0. T h i s theorem g e n e r a l i z e s a r e s u l t of Ton [12], where M = 0, and a r e s u l t o f Lions [7] , p.216. Remark 1.1: I f c o n d i t i o n ( i i ) of assumption I i s r e p l a c e d by ( j j ) |(Au,v) *| $ C l u l I v l + |u|*|u|*|v|*|v|* f o r a l l u, v i n V, then the c o n c l u s i o n o f theorem 1.3 s t i l l h o l d s , except t h a t we have D£U e L2(0,T;H) only f o r y i n (0,T). 1.5. A p r i o r i i n e q u a l i t i e s . L e t v , v , 1 2 be a b a s i s f o r V, i . e . v , v , 1 2 *"* , v n are l i n e a r l y independent f o r any n and the s e t 0 0 r n U v ,v ,'**,v i s dense i n V. S i n c e V i s dense i n H, 1 L I 2 n J ' so i f u e H, then there e x i s t s a sequence {u„ } w i t h u o ' om om i n [v ,v^, * *' , v n ] and u Q m •*• u i n H as m -»• <». 14 Lemma 1.3: L e t u o m be as above. L e t J be the d u a l i t y  mapping from V i n t o V a s s o c i a t e d w i t h the gauge f u n c t i o n Tp(s) = sH. (Cf. L i o n s 8 , pl74.) L e t A and M be as i n theoreml.3. Then g i v e n any u 0 i n H , f i n L 2(0,T;V*) , and e i n (0,1), there e x i s t s m (1.20) "emM = I 9 " e i m ( t ) v i such t h a t (1.21) (u' + eJ(eu ) + Au m + Mu „,,v.) t T* = (f,v.) * , l ^ j ^ m em em em emf j v j V u (o) = u em om Moreover, there e x i s t s a_ c o n s t a n t K , independent of e and m , such t h a t (1.22) sup lu ( a ) | 2 + leu |'«,_ m __« + |u I* , , ae(0,T) e m H em'L 5(0,T;V) 1 em'L 2(0,T; V) < K. For f i x e d e and m , u^.m e ( L 5 (0,T;V)) *. P r o o f : The system (1.21) i s e q u i v a l e n t to the f o l l o w i n g : 15 m I i = i m (1,23) < 7 g' . (t) (v. ,v.) *. + e f j ( e I g (t)v.) ,v.) * f l ^ e i m 1 ' j v v elm 1 D'V m m + ( A ( J ^ £ i m ( t ) v i ) ' v j ) v * + i | l ( M ( 9 c i m ( t , v i , ' v j ) = ( f ( t ) , v _ . ) v * f 1 $ j $ m. H g . (o)v. = u e lm I om. T h i s i s a system of m o r d i n a r y d i f f e r e n t i a l e q u ations w i t h 9" e£ m ( i = = I f 2, •••• , m) as dependent v a r i a b l e s and t as the independent v a r i a b l e . Now ^ e i m ^ v i ' ' ~ 1 /2, , m , span a f i n i t e d i m e n s i o n a l subspace V m of V , hence the e x p r e s s i o n : (1.24) m m i s continuous from t h i s subspace to the weak topology o f V M i s a l i n e a r o p e r a t o r , so i t i s continuous from V m to Vi T h e r e f o r e (1.25) m m (eJ(e I g e i m ( t ) v . ) + A ( l g e i r a ( t ) v . ) , v ) i n 1=1 •m + J l M(g Eim(t)v i),v ) * i = i r v i s continuous i n g e i m i i» j = 1, 2, *••, m. C l e a r l y i t i s measurable i n t i f g . are f i x e d . Since A, J , and eim V- map bounded s e t s o f V i n t o bounded s e t s o f V*, the 16 e x p r e s s i o n (1.25) i s bounded i f there e x i s t s a co n s t a n t b such t h a t |g . I < b f o r i = 1, 2, ••• , m. I t f o l l o w s from Caratheodory's theorem (Cf. [ 4 ], p.43) t h a t there e x i s t s a s o l u t i o n o f (1.21) i n the i n t e r v a l [0,6 m] . The s o l u t i o n i s g l o b a l i f (1.22) h o l d s . M u l t i p l y i n g the j - t h eq u a t i o n o f (1.21) by g £ ^ m ( t ) , t a k i n g the summation from j = l , 2, , m, and i n t e g r a t i n g from o t o t , we o b t a i n : ^ K m ^ l H + T ( e J( £ u em( 0>> + A u e m ( a ) ' u e m ( a ) ) v * d a ' o 1: ( M u e m ( a ) ' U e m ( a ) ) H d a o ( 1- 2 6> s *l uomIS + l f l L M 0 , T ; V * > f K.m«»lv*> u o 2 < i l u I 2 + |f 2 1 om1 1 + n 1 sup as (o,t) 1 "emvu ' 1H L M 0 , T , V * ) ' U e m ( 0 ) l v d a _ u s i n g the assumption on J and on A , and r e a r r a n g i n g , we o b t a i n from (1.26): 17 | f^P J u ( a ) | a + [ * | e u (a) | 5 d a + c f t |u ( a ) | 2 d a * o e ( o , t ) ' em ' H J ' em ' v J 1 emv ' v (1.27) < i | U c J H + | f lLMO fT;V*) | u e m ( a ) | v d a rt u (a) |'da em 1H *• t f m : „ „ „ . „ . , • § [ ! « „ ( » > i > + u T L z (0,T;V") | u e m ( a ) | H d a . T h e r e f o r e T s u p |u ( a ) | 2 + e 5 f | u (a)|'da + ff* |u (a)|'da a e ( o , t ) ' em ' H } 1 e m ' v 2J 1 emv ' v (1.28) X i i 2 2 i i 2 < 2 " + H f * + V i T em' H c 1 ' L 2 ( 0 , T ; V * ) - t |u (a) | 2da 1 em 1 H 0 f t 2 K 2 Iu (a)I da, 2 J 1 em 'v • O where K x and K 2 are co n s t a n t s independent o f e and m. I t f o l l o w s from lemma 1.1 t h a t (1.29) a e ( o ! T ) ' U e m { a ) | H < The i n e q u a l i t i e s (1.2 8) and (1.29) i m p l i e s t h a t 18 (1.30) ^ p m J u (a) |* + e 5 [ T |u (a) |'da + f T |u (a) I 2da ae(0,T)' em H J Q : . ' em 1 v J q 1 em 'v i s bounded by a constant K 3 which i s independent of ra and e . This proves (1.22) . I t also follows that a global s o l u t i o n e x i s t s . Using (1.22) , we can write (1.23) as: (1.31) g* . (t)(v.,v.) * = (h(t),v.) * . 1 * j *m, eim 1 3 V 3 V where h(t) = -feJ(eu (t)) + Au (t) + Mu (t) - f ( t ) l i s i n <- em em em ( L (0,T;V ) J . Since v l f v 2 , ••• , v are l i n e a r l y inde-pendent, the determinant of the matrix (a.•) , where a i j ~ ^ V i / V j ^ v * ^ S n o t z e r o • H e n c e by Cramer's r u l e , we can solve for g*. (t) and m (1.32) g' (t) = I (h(t ) ,B..v) , eim j=i i j j where 3.. are numbers. C l e a r l y , g' e ( L 5 ( 0 , T ; V ) ) * . I t 1D eim follows that m 5 " i s an element of ( L (0,T;V)) for each fi x e d m and e This completes the proof of the lemma. 19 The f o l l o w i n g lemma g i v e s an estimate o f the f r a c t i o n a l d e r i v a t i v e s of s o l u t i o n s t o (1.21). Lemma 1.4: Suppose a l l the hypotheses of lemma 1.3 are  s a t i s f i e d , and l e t u be the s o l u t i o n o f (1.21) of 1 8 lemma 1.3, then f o r any y i n (0 , — - ~) , we have U - 3 4 ) K U E A * < O , T ; H ) S K W K(y) i s independent of m and e. Pr o o f : To prove t h i s lemma, we s h a l l f o l l o w the argument used by Ton i n [12]. From (1.32), we see t h a t g'. i s i n C 1 ( 0 , T m ) and t h e r e f o r e by (1.33) and the p r o p e r t y o f J , A and M, u' i s i n L 2 (0,T ;H) . Hence f o r t < T T, em m m both D 2 Y u £ m and D ^ D J ^ m - uem are i n L 2 (0,T m;H) p r o v i d e d y < ~« * t f o l l o w s from (1.18) t h a t rT (1.35) D 2 Y u (t) = K. (y) I ( t - s ) - 2 Y D u ( s ) d s , t em 1 J s em where Kl (y) depends on l y on y. Th e r e f o r e (D 2 Y U ( t ) , u (t))„ v t em em ; H (1.36) = K , (y) f T (t - s ) - 2 Y ( D U (s) ,u (s)) ds 1 i n s em em = Kj (y) (E 1 + E 2 + E 3 + E j , where J T (t - s ) " 2 Y ( f (s) , u £ m ( t ) ) v * d s (t - s ) - 2 Y ( j ( e u e m ( s ) ) , u c m ( t ) ) v * d £ • r •'o TQ ( t - s ) " " 2 Y K m ( s ) ' u e m ( t ) ) v * d s i : (t - s ) " 2 Y ( M u c . (S) , U ( t ) ) u d s . em ' em ; H Def i n e h m : R -»- R by (1.37) t " 2Y o < t < T, h ^ ( t ) = i m 0 t < o or t » T, m 2.- 8 Then h_. e L p (R) i f o < y < — - —. For any f u n c t i o n 2 4 ^ d e f i n e d on [ o , T ] , we s h a l l denote by g the f u n c t i o n (1.38) g ( t ) = g ( t ) o < t < T 0 t < o or t £ T. Using Young's i n e q u a l i t y f o r c o n v o l u t i o n , we g e t (1.39) d t fTm f+°° ^ < | u em ( t ) l v J h m ( t - s) |£(s) | v * d s d t o ~ °° = l u e m ( t ) l v [ V l ? l v * ] d t * K m L 2 (0,T;V) ^ I J L 1 (R) ' F ' L 2 (0,T;V*) S i m i l a r l y , Lm | E 2 | d t (1.40) £ e = e = e ' O J — oo f T m |u (t) |„[h * | j ( e u (s))|„*ldt I 1 em 'VL m 1 em ' 'V J o ' j " " K m ^ H ^ K J ^ d t em'L 5(0,T;V) 1 m'L 1(R) and ( 1 . 4 2 ) 2 2 fm |E,|dt ' o ( T m |u \Jh * | A U L * l d t J 1 em 1VI m 1 em1V J o a.4D < c2 f™ KJv[v(l5£mi;+B + |aem|v) ' o L £ C 2 | u I „ Ih I l u d t em L 2(0,T;V) m LT~<3 (R) emL 2(0,T;V) + C z l uem'L 2(0,T;V) ^rn^L 1(R)'"em'L 2(O fT;V) f m l E j d t ; o f T * J m l u e m l H [ V l M u e m l H ] d t * , U e i J L 2 ( 0 , T ? H ) ' ^ ' L 1 (R) ' M Uem lL 2(.0 f T;H) * y/T | u £ M | L 2 ( 0 , T ; H ) | h m | L i ( R ) | u e m | L - ( p , T ; H ) • I t can be e a s i l y shown t h a t I I L 1 (R ) A N C ^ |h I ^ 4 - 8 - . v a ^ e bounded by a c o n s t a n t , t h e r e f o r e i t f o l l o w s m L 2 P(R) from lemma 1.3 t h a t I f ™ |E.|dt are bounded by a c o n s t a n t i = i ; o 1 which i s independent o f m and e. 23 A computation as i n S h i n b r o t [ l l ] , p.150, lemma 5.1, shows t h a t (1.43) m|DYu (t) |'dt < sec(YTr) m (D?Yu (t) , U (t))„dt I 1 t em H J v t em em ' H ' o o T h e r e f o r e l D t U e m ( t ) 'H d t * sec(yrr) J ^ j ^ l E j d t (1.44) < K ( y ) , where k*(y) T m = T. i s independent o f m and e, I t f o l l o w s t h a t 1.6. P r o o f of theorem 1.3. L e t e be f i x e d . From the weaknesspactness of the u n i t b a l l i n a r e f l e x i v e Banach space, and from (1.22) and (1.34), we can e x t r a c t subsequences o f u e m (again denoted by u ) such t h a t em (1.45) u em eu em DJU t em u £ weakly i n L (0,T; V) and i n the weak *topo ogy of L°°(0,T;H), e u £ weakly i n L s ( 0 , T ; V ) , D^u^. weakly i n L 2 (0,T;H) , 24 as m -*• °°. App l y i n g theorem 1.2, we can e x t r a c t f u r t h e r subsequences such t h a t (1.46) uem u e i n l 2 ( ° ' t ' h ' a n d I u (t) -»• u (t) i i v. cm c i n H f o r almost a l l t i n [0,T1 era ' ' e ' Because M s a t i s f i e s the convergence c o n d i t i o n , t h e r e f o r e (1.47) Mu ™(t) -> Hu (t) i n H f o r almost a l l t i n [0,T] em e S i n c e lu I », ' < C, and M i s a bounded l i n e a r em L (0, T; H) 00 mapping of L (0,T;H) xnto L (0,T;H), the Lebesque convergence theorem y i e l d s : :i.48) < Mu •+ Mu i n L P(0,T;H) f o r 1 < p <°°. em em / ^ I l - u e m ( t ) ' u e m ^ ^ H d t * £ % u e <t> ,u £ <t> > Hdt. Sets m0 (1.49) <f> (t) = I 6 . ( t ) v , , m o j=i J J 25 where 0. e C 1 (0,T) , 0 . (T) = o.,. j = l f . 2, ••• , m . From (1.21), we have (1.50) f Tm f Tm, . (u ) T T d t + (eJ(eu„ ) + Au„ , <J> ) T T * d t J c em m0 H J 0 em em , Tm 0- 'v fT fT (f,<j> ) * d t - ( M U ,<{> )„dt + (u ,<}> (o) ) J 0 ' X v J Q em'vm0 H »ni' ym ( H p r o v i d e d m > m . I t f o l l o w s t h a t (1.51) - f T ( u ,<j>' ) d t + f T m f e J ( e u ) + Au ,<(>). d t J D em m0 H J Q em e m M m V P ( f , ' n -*• i it.$ ) „ * d t - (Mu ,<}> ) dt + (u ,<f) (o) ) o m0 V J Q e m0 H " -e' m0 II as m -*• o o . I t i s c l e a r t h a t T fT (u )„dt o f 1 - I (u ,<j>' ) . T d t a s m -»• 0 0. we a l s o n o t e t h a t I e m o i i o l i m i n f |u ( o ) I ' + f T (u' ,u ) d t m L em 1H J em em H (1.52) fT + feJ(eu ) + Au ,u 1 * d t i 0 em em em'V » H + r < f<vv* d t- r < , , u E ' u e ' H D T -L e t A(u,v) =s eJ(ev) + Au, then A(u,v) i s a semi-monotone o p e r a t o r mapping L 5(o,T;V) i n t o i t s d u a l . A p r o o f e x a c t l y as i n [13] , (lemma 1, p . 7 i ) , g i v e s : 26 (1.53) A(u ,u ) = eJ(eu ) + Au — em em em em -*• eJ(eu ) + Au e e weakly i n ( L 5 (0,T; V)) * as m ~. Thus (1.54) - f (u ,4/ )„dt + ( T (eJ(eu ) + Au ,4> ) T 7 * d t e'rmo H k e e: m/V* ' o ' o T f o r a l l (J>m of the form (1.49). I f cj> e C 1 (0,T;V) and (J> (T) = 0, then there e x i s t s a sequence { l } of the above 0 form such t h a t (1.55) m„ 4>' i n Lz(0,T;V) i n L (0,T;V) T h e r e f o r e f o r a l l 4> e C l (0,T;V) v;ith 4> (T) = 0, (1.56) "J ( ue ' * , , H d t + J ( e J { e u e ) + A u e ' * ) v * d t = I ( f f < r ) v * d t - j (Mu e,<M Hdt + (u ,<f>(o)) H In p a r t i c u l a r , (1.56) holds f o r a l l <J> i n C„(0,T;V). T h e r e f o r e 27 (1.57) u' - e J ( e u ) - Au - Mu + f i n D'(0,T;V*). S i n c e the r i g h t hand s i d e of t h i s i s i n ( L 5 ( 0 , T ; V ) ) * , the e q u a l i t y h o l d s a l s o i n ( L 5 ( 0 , T ; V ) ) * and u' e ( L 5 ( 0 , T ; V ) j * . The bounds i n (1.22) and (1.34) are independent o f e, hence we o b t a i n by t a k i n g subsequences i f necessary: (1.58) u £ -*• u weakly i n L 2(0,T;V) and i n the weak *topology of 1/° (0 ,T;H) , eu •*• o weakly i n L s (O rT;V) Y l D t u D Yu weakly i n L 2(0,T;H) as e -> o. Again, theorem 1.2 and the Lebesque convergence theorem y i e l d : (1.59) Mu £ -»• Mu i n l/(0,T;H) f o r 1 < p < f T (Mu ,u ) d t + f T (Mu,u) d t . On the o t h e r hand, l i m i n f ^ + A u £ , u e ) v * d t fT (1.60) = l i m i n f ( f - eJ(eu ) - Mu , u ) v * d t G * o £ G £ ^ (f - Mu,u) v*dt. 28 I t i s a l s o c l e a r from ( 1 . 5 7 ) t h a t u ' + A u -*• f - u e e weakly i n ( L 5 ( 0 , T ; V ) ) * . T h e r e f o r e i t f o l l o w s from the assumption on A t h a t ( 1 . 6 1 ) f (Au ,<M*dt - f (Au,<}>) * d t f o r any <f> e C 1 (O fT;V) w i t h <})(T) = 0 i f we assume t h a t u £ ( o ) = u(o) = u Q . T h e r e f o r e -J (u,cJ>')Hdt + | (Au,cj>)v*dt + | (Mu,u) Rdt ( 1 . 6 2 ) ° T ° ° • = f (f ,cj>) * d t + ( u 0 ,4>(o-) ) ' o v H f o r any <J> e C 1 (0,T;V) v;ith <f> (T) = o. I t remains to show t h a t u (o) = u(o) = u . S i n c e e o u £ e ( L 5 ( 0 , T ; V ) ) * , ( 1 . 5 7 ) y i e l d s ( 1 . 6 3 ) ° j"T rT J (u^f«|))v*dt + j ( J ( e u e ) + Au £,(J)) v*dt fT ° f T = J o ( f , + ) v * d t - J ( M u £ , * ) H d t f o r any <j> e C 1 (0,T;V) wi t h <J> (T) = o. Comparing t h i s w i t h ( 1 . 5 6 ) , we have: (T rT ( 1 . 6 4 ) J . ( u £ ,<M v *d t + j ( u £ , ( f ) ' ) v * d t = - ( u £ ( o ) ,(()(o)) H 29 f o r any <j> e C 1 (0,T;V) w i t h <j> (T) = o. But i t i s c l e a r t h a t (1.65) J (u' e,(f.) v*dt + j ( u £ ,*•) v * d t = - (u e(o) , * ( o ) ) H o ' o i f 4> e C l (0,T;V) and <j> (T) = o. T h e r e f o r e ( 1 . 6 6 ) ( u 0 , < M o ) ) H = ( u e ( o ) , c t > ( o ) ) H . T h i s shows t h a t u (o) = u f o r any e. e o From (1.57), we see t h a t u^ l i e s i n a bounded s e t of ( L 5 ( 0 , T ; V ) ) * . S i n c e the i n j e c t i o n o f V i n t o V* i s compact, we have by t a k i n g subsequences i f necessary: (1.67) u (o) u(o) i n V*. e But u £ ( o ) = u Q f o r any e, t h e r e f o r e u(o) = u . T h i s completes the p r o o f of theorem 1.3. 30 CHAPTER I I EVOLUTION EQUATIONS IN BANACH SPACES. L e t H and W be two s e p a r a b l e H i l b e r t spaces, and V a s e p a r a b l e r e f l e x i v e Banach space w i t h W c v c H, L e t W and V be dense i n V and i n H r e s p e c t i v e l y , fbreover, suppose the n a t u r a l i n j e c t i o n o f W i n t o V and t h a t o f V i n t o H are continuous and compact r e s p e c t i v e l y . L e t F, Y, and X be the Banach spaces LP(0,T:V), L r(0,T;V) and L P (0,T;V) D L 0 0 (0,T;H) r e s p e c t i v e l y , where 2 £ p «? r < 0 0. In t h i s c h a p t e r , we s h a l l c o n s i d e r the i n i t i a l v a l u e problem f o r n o n - l i n e a r o p e r a t o r s mapping X^Y i n t o Y and s a t i s f y i n g the f o l l o w i n g assumption. Assumption I I (i) A i_s continuous from f i n i t e d i m e n s i o n a l subspaces •k of Y i n t o the weak topology of Y . ( i i ) I f u -»• u weakly i n X, u i n Y, u' -*- u' — n —*- — n — ' n weakly i n Y*, A u n -»• h weakly i n Y* w i t h h + u' i n F" , u(o) i n H and £ (h + u' , u ) p * + |u(o) | 2 31 then Au = h. ( i i i ) A maps bounded s e t s of X and bounded s e t s of Y i n t o bounded s e t s of Y (iv) There e x i s t s c o n s t a n t c > o such t h a t ( A u ( t ) f u ( t ) ) * £ c | u ( t ) | P w v f o r a l l u i n Y and f o r almost a l l t i_n[o,T] In t h i s c hapter, we s h a l l e s t a b l i s h the f o l l o w i n g theorem. Theorem 2.1; L e t A b_e an o p e r a t o r mapping X(JY i n t o Y* and  s a t i s f y i n g assumption I I . Suppose Me L (L°°(0,T;H) ,L°° (0,T;H)) , i s of l o c a l type and s a t i s f i e s the convergence c o n d i t i o n . Then  f o r each f i n F and u 0 i n H, t h e r e e x i s t s u i n X w i t h u' i n Y* such t h a t f u' + Au + Mu = f (2.1) u(o) 32 Remark 2.1; Theorem 2.1 g e n e r a l i z e s a r e s u l t of A r t o l a [ l ] . The semi-monotone o p e r a t o r s c o n s i d e r e d by Browder [ 3 ] , L i o n s [ 9 ] , •k and a l l weakly continuous o p e r a t o r s from F i n t o F s a t i s f y p a r t s (i) and ( i i ) of assumption I I . The theorem w i t h M = 0 was e s t a b l i s h e d by Ton [14]. 2.2. S p e c i a l b a s i s . Because of l e s s s t r i n g e n t c o n d i t i o n s on A, we can no l o n g e r o b t a i n the f r a c t i o n a l d e r i v a t i v e s as i n Chapter I . I n s t e a d , we s h a l l prove the theorem by making use o f a s p e c i a l b a s i s , an i d e a i n t r o d u c e d by L i o n s i n [ 8 ] . Lemma 2.1; There e x i s t s an orthonormal b a s i s w , w2, w3, * * * o f W, and a_ sequence of numbers X l , X 2 , X 3 , • • • w i t h I X j < |X2| < j X 3 J < • • • -»• °° such t h a t (2.2) ( W j , v ) w = X j ( W j , v ) H f o r a l l v i n W and f o r j = 1, 2, 3, 3 3 P r o o f : For a l l u and v i n W, l e t ( 2 . 3 ) b(u,v) = ( u , v ) . w Then b: W x W -*• R i s a b i l i n e a r form on W x W such t h a t ( 2 . 4 ) |b(u,v)| $ | u | w|v| w u, v e W and ( 2 . 5 ) b ( u , u ) > I u I w u e W. L e t N <=. W be the s e t o f a l l elements u f o r which the l i n e a r form ( 2 . 6 ) v •> b(u,v) i s continuous on W w i t h the topology induced by H. Then s i n c e W i s dense i n H, t h i s l i n e a r form can be exteneed to a unique continuous l i n e a r form of H. T h e r e f o r e by R e i s z ' s r e p r e s e n t a t i o n theorem, there e x i s t s h i n H such t h a t ( 2 . 7 ) b(u,v) (h#v) H , v e w a H . 3 4 : Hence we can d e f i n e a l i n e a r map A:W H w i t h D(A) = N-,. and f o r a l l u e N and v e W, (2.8) b(.u,v) = ( A u , v ) H . By p r o p o s i t i o n 1.2, p.11 of L i o n s [7], A i s one-one onto H, D(A) i s dense i n H and A i s symmetric. T h e r e f o r e A 1 :H D(A) c: W c: H i s a mapping from H to H. Furthermore, f o r a l l u i n N, |u|^ = b(u,u) = (Au,u) R " U u | H | u | H £ c l A u l l u l . H W T h e r e f o r e we have O (2.9) | | u | w $ |Au| H, T h i s shovjs t h a t A""1 i s continuous from H i n t o D(A) cz w. But the i n j e c t i o n of W i n t o H i s compact, so as a mapping of H i n t o i t s e l f , A"1 i s a l s o compact. I t f o l l o w s from a well-known theorem on compact symmetric o p e r a t o r s t h a t t h e r e 35 e x i s t and ortho-normal b a s i s u t , u 2 , u 3 , *•* i n H and numbers X l t X2, X3, ••• such t h a t | X t |_1 > \XZ\'X > \X3\'1 •*• and (2.10) A " 1 u n = l „ n n Note t h a t u = X A _ 1 u i s i n f a c t an element of n n n D(A) cz w. So we have ( V v , w = b < v v ) (2.11) - ( A u n € v ) H f o r a l l v e W and t h e r e f o r e (2.12) (u v)„ = X n ( u n , v ) H f o r a l l v e V7 and n = l , 2, 3, ••• . Hence u^ i s a l s o o r t h o g o n a l to each o t h e r i n W and so by n o r m a l i z i n g , we get an orthonormal b a s i s i n W s a t i s f y i n g (2.12). 2.3. A - p r i o r i e s t i m a t e s o f the approximate s o l u t i o n s . S i n c e W i s dense i n H, th e r e e x i s t s a sequence (u t i n W with u i n fw,. w,. ••• , w 1 and such 1 0nJ on L i ' 2 ' ' n J t h a t u u i n H. on o 36 Lemma 2.2: L e t n 0 n be_ as above and w t , w. w be a b a s i s of W as i n lemma 2.1. L e t g i v e n i n theorem 2.1. Then f o r each f i r i F A and H be_ o p e r a t o r s * there e x i s t s u (t) such t h a t m (2.13) m um < t ) I 9. (t)w. 111 i=i im l 1 £ j $ m, u (o) = u m om oreover, ( 2 - 1 4 ) oelo^T) | U ™ ( 0 ) | H + « • K -where K is_ independent of m. For each f i x e d m, and 1 ^ i $ m, g! (t) e L r (0,T) and hence u' e Y . 3im ' m Proof; The p r o o f o f t h i s lemma i s i d e n t i c a l t o t h a t o f lemma 1.3. 37 2.4 Pro o f of theorem 2.1, From lemma 2.2, we know t h a t there e x i s t s u. m such t h a t (2.15) ( um + A um + Mum'wjV = <f'wjV 1 < j < m u ^ o ) = u om L e t P m be the p r o j e c t i o n o f H i n t o [w 1 ( w2, ••• , w ] d e f i n e d by m (2.16) P h = .Y (h,w.)„w. , h e H. m j h i 3 H 3 The r e s t r i c t i o n o f P m to W i s an element of L(W,W) We s h a l l f i r s t prove t h a t (2.17) 'Pm'L(W,W) * C * ' where Ci i s independent of m. I f h e W, then (2.18) 2 m m l pm h lw = ( I (h,Wj) HWj, I (h,Wj) HWj) w j=i j=i 2 1 1 2 m , I |(h,w.) H| 2|w 3=1 j 1 W 38 Using lemma 2.1, we have (2 19) 3 " 1 j 1 « T r F J . l ( h ' w i > « | , | w i l » But w, , w , w, , 0 * * i s an orthonormal b a s i s o f W, so < 2' 2 0> l p m h l w * ] 7 ^ j T M w by the B e s s e l ' s i n e q u a l i t y . T h i s proves (2.17) w i t h |X A| 2 equal to c, which i s independent o f m. > . * * * Now c o n s i d e r the mapping p m * w w d e f i n e d by m (2.21) P*w* = | 4 (w*,w.)^*w. f o r a l l w* IN W*. T h i s i s an e x t e n s i o n o f P m t o W*. For any w i n W, we have (2.22) l ( P m w * ' w ) W * l = ' j l i ( w * , w . ) w * ( w j , w ) w * * m = I ( w ,.| W j ( w j , w ) w * ) w * | 1 'W*1 m W * l w * l W * l p m l L ( W , W ) l w l w 39-T h e r e f o r e (2.23) |Pm w*l W* < l Pml L(W,W) |w*|w and so (2.24) | P * L ,„* „ * . * | P L , M M , « 1 (2.25) m'L(W*,W") v 1 m'L(W,W) v | X | 2 ' From (2.15), we have "m = u' = P * f - P*Au - P * M U . m m m m Sinc e Au , Mu are i n a bounded s e t o f Y*. i t f o l l o w s irr m from (2.24) and (2.25) t h a t u^ are a l s o i n a bounded it t s e t o f Y . So by t a k i n g i n t o account the r e s u l t o f lemma 2.2, we g e t : < 2- 2 6> l u m l y * + K I L - ( 0 , T , H ) + K ' F K K ' K i s independent o f m. Using (2.26), we can e x t r a c t subsequences o f u m such t h a t : 40 U ' m -»• u' weakly i n 1 ( 2 .27 ) ( -»• u weakly i n F and i n the weak *topology of L ° ° ( 0 / T ; H ) weakly i n Y . Si n c e the i n j e c t i o n o f V i n t o H i s compact, i t f o l l o w s from theorem 1.1 t h a t we can e x t r a c t subsequences of u^ such t h a t L 2 ( 0 , T ; H ) and H f o r . a l m o s t a l l t i n [ O , T ] . ->• u i n (2.28) 1 u ( t ) i n Hence by the assumption on M, (2.29) Mu -*• Mu i n H f o r almost a l l t i n [o,T] . m m Ap p l y i n g the Lebesque convergence theorem, we have Ku -»• Mu i n L S(0,T;H) f o r any s > 1, So l e t t i n g m -> ~ i n (2.15), we get ( 2 . 3 0 ) 4 0 a (2.31) (u' + h + Mu,w.)__* = (f,w.) * 3 w 3 v f o r j = 1, 2, 3, ••• . Wj i s a b t . j i s o f W, t h e r e f o r e (2.32) u» + h + Mu = f i n D'(0,T;W). But f e F*, hence (2.32) a l s o holds i n F* and u 1 + h e F*. Since u 1 -*• u' weakly i n i* and m ft u -*- u weakly i n F, we have u (o) u(o) i n W . On m m the other hand, u (o) = u u„ i n H. T h e r e f o r e ' m m o (2.33) u(o) = u 0 I t remains to prove t h a t h = Au, From (2.15), we have Lim sup |, ( o ) |* + ( u . + A u ^ u ^ * } m oo . l i n k u p { | U O M , 2 + ( F . M U ^ U J Y * } (2.34) < I UOIH + ( F "" M U ' U ) F * • | u ( o ) | ^ + (u' + h , u ) p * . I t f o l l o w s from the assumption on A t h a t Au - h and the theorem i s proved. 41 CHAPTER I I I PERIODIC SOLUTIONS OF EVOLUTION EQUATIONS 3.1. P e r i o d i c s o l u t i o n s f o r monotone o p e r a t o r s . L e t H be a H i l b e r t space and V a s e p a r a b l e r e f l e x i v e Banach space w i t h V c= H and V dense i n H. Suppose the i n j e c t i o n of V i n t o H i s compact. We s h a l l c o n s i d e r n o n - l i n e a r monotone o p e r a t o r s mapping L P(0,T;V) i n t o lik (0,T;V ) s a t i s f y i n g the f o l l o w i n g assumption. Assumption I I I : (i) A maps bounded s e t s o f L p(0,T;V) i n t o bounded  s e t s o f LP'(0,T;V*). ( i i ) A i_s continuous from l i n e s i n L p (0,T; V) t o the  weak topology of L p ' ( 0 , T ; V * ) . ( i i i ) (Au(t) , u ( t ) ) v # £ c | u ( t ) | P f o r a l l u i n L p(0,T;V) a_t almost a l l p o i n t s t i n [ 0 , T ] . (iv) (Au - Av,u - v ) v * ^  0 f o r a l l u and v i l l LP(0 fT;V) a t almost a l l p o i n t s t i n [ 0 , T ] , 42 The main r e s u l t o f t h i s c h a p t e r i s the f o l l o w i n g theorem. Theorem 3.1; L e t A be an o p e r a t o r mapping L p(0,T;V) i n t o L r (0,T;V ) s a t i s f y i n g assumption I I I w i t h p > 2. Suppose M i s an o p e r a t o r i n L(L°°(0,T;H) , l T ( 0 , T ; H ) ) , o f l o c a l type, and s a t i s f i e s the convergence c o n d i t i o n . Then f o r any f i n iP (0,T;V*), t h e r e e x i s t s u i n L P(0,T;V) DL^tOjTfH) w i t h u' i n L P'(0,T;V*) such t h a t (3.1) u 1 + Au + Mu = f u(o) u(T) 3.2. Approximate s o l u t i o n s . As b e f o r e , we s h a l l f i r s t o b t a i n a - p r i o r i e s t i m a t e s of the approximate s o l u t i o n s . T h i s i s done i n the f o l l o w i n g lemmas. Lemma 3.1: L e t w} , w2, w be a b a s i s of V, m a f i x e d 43 i n t e g e r and u 0 e |w , w , * * * , w | . L e t A, 11 and f be as i n theorem 3.1. Then there e x i s t s a_ unique f u n c t i o n m " m ^ J = i l 1 g i m ( t ) w i s u c h t h a t (3.2) (u'.w.) * + (Au ,V7.) * + (Mu ,w.) m' j V * mf 3 V* m' j H = (f,w.) * 1 £ j < m j V and (3.3) V L ~ ( 0 , T ; H ) + 1^m'L P(0,T;V) < K j * Ki depends on u 0 and i s independent o f m. Moreover, u' e L p (0,T;V*) and there e x i s t s R > o, independent of m such t h a t i f | u 0 | H £ R, then (3.4) | u m ( T ) | H < R. Proof: The p r o o f of the e x i s t e n c e o f u m s a t i s f y i n g (3.2) and (3.3) i s s i m i l a r t o t h a t o f lemma 1.3. Suppose t h a t both u and v s a t i s f i e s the 44 system (3.2). L e t w = u - v, then w s a t i s f i e s (w*,w) * + (Au - Av,u - v) * = - (Hw,w) V V n (3.5) w (o) = 0. I n t e g r a t i n g from o t o t , we g e t f T ( o ? t ) l w<°> IH « J I (" w' w>Hl a o 5 i ^ p t > | w ( o , | H + "M! i » « o > i > -1 2 ae. . . , Q (3.6) So i t f o l l o w s from lemma 1.1.that (3.7) 7 0 ^ 0 % IH < °' T h e r e f o r e w(a) = 0 almost everywhere, and u i s unique. I t remains to show t h a t there e x i s t s R > 0, independent o f m, such t h a t i f |u 0|„ < R, then |u ( T ) L £ R. M u l t i p l y i n g the j-*th e q u a t i o n o f (3.2) by g m j ( t ) , summing up from j = 1 to j = m, and i n t e g r a t i n g from 0 t o t , we g e t : 2 K ( t > H + c ^ ^ ( a J l ^ d a ' o | | U , I H + { [ } f ^ ) l v * l u m ( 0 ) l v + l<Mu f f i(a),u m(o)> H|]da. (3.8) ° < ~ l u 45 By Young's i n e q u a l i t y , we can f i n d a number K 2, depending on c and p o n l y , such t h a t (3.9) |f <o) L J - u J o ) |„ ^ K 2 | f ( a ) \ l l + f | u j a ) |?t. 'V*' m 'V ' ' v * * ' m ' V T h e r e f o r e < | | u 0 | 2 + i a £ ^ P t ) K ( a ) |H + y 2TJ F C | u m ( a ) | * d a + K f T P' K 3 = K 2 | f ( a ) | v * d a and i s independent o f m and | u „ | H . The n a t u r a l i n j e c t i o n of V i n t o H i s c o n t i n u o u s , so there e x i s t s a c o n s t a n t c 0 > 0 such t h a t f o r a l l u i n V, (3.11) |u| H < c 0 | u | v . Using t h i s f a c t and r e a r r a n g i n g (3.10), we have 4* o U o ^ t ) K ^ H H + T 0 / K ( O ) l ^ o 4 o 3 • (3.12) <• 2 l u o l H + y T C o rt |u t a ) | 2 d a + K . J 1 m 1V 3 46 S i n c e p > 2, we can use the Holder's i n e q u a l i t y and the Young's i n e q u a l i t y to show t h a t 2 M 2 y T c 0 (3.13) |u ( a ) | 2 d a $ K |u ( a ) | P d a 2/p f o r some con s t a n t s and K 5, which depend o n l y on y, T, c, c c , p, and T, and are independent o f m and l u 0 ! H -S u b s t i t u t e t h i s i n t o (3.12) and put t = T, then (3.14) + V where K g = K 3 + K g. In p a r t i c u l a r , < 3- 1 5> 0e(S PT>!« M< O , ,H < / 2 l u o l H + 2 / K e « On the o t h e r hand, (3.16) fT o |u m(0) | HdO S K, < K. < K. T o f T |u n(a)|Pda r r1 |u ( a ) I U „ m d a V P + 2K. < KXO|IUOIH + ( 2 K e ) P 47-f o r some cons t a n t s ( i = 7 t o 10) which are independent o f m and |u | H . T h e r e f o r e there e x i s t numbers cx and c 2 such t h a t •T — (3.17) |u m(a) | R d a * c j u j ^ + J o f o r a l l m. Returning to (3.2), we have (3.18) < j f ( t ) | v , | u m ( t ) | v + | ( H u m ( t ) , u m ( t ) ) H | < K , | f ( t ) | l j ; + f | „ B ( t ) | | J + K ^ m < t ) , U r a ( t ) ) H | . . P Because p > 2 , s o i f x > 0 , then x z < x + 1 . T h e r e f o r e (3.19) 0 « I < M u ( t ) | 2 + £ . | u ( t ) | 2 2 d t m H 2 m V * I l - l u ( t ) | 2 + l \ u ( t ) | P + £ 2 d t n H 2 m v 2 $ K | f (t)| * + - + | (Mu (t) ,u (t) ) 2 ' ' i v * 2 1 K m ' m H M u l t i p l i c a t i o n by 2 e x p ( — ) , and i n t e g r a t i o n from o to T c o 48 g i v e s •o (3.20) f T < K M + 2 « c p @ y t e « P T ) | u n ( t ) | H f | u n ( t ) | H d t ( where K i s independent of m and |u.|„. Using (3.15) (3.17) and (3.20), we have (3.21) e x p ( p | u m ( T ) | - < |u,|» + K i ; | u o l H 2 + K ] > 0 f o r some con s t a n t s K and K . Set: 1 2 1 3 (3.22) 2e * exp(-S|) - 1. e cT'x C l e a r l y e > 0 and exp [ — j ) > 1 + e. Because p > 2, there e x i s t s c o n s t a n t K such t h a t 1 <• 1+2' 2 (3.23) K |u I 2 $ e I u I + K The r e f o r e (3.24) exp(^|)|u ( T ) | 2 * ( l + e ) | u | 2 + K , c : ' m H 0 H i s -^-A > 1, so we 1 O 13 11 C can put: 49 K ( 3 . 2 5 ) R 2 _ 1 5 e x p ( - — ) 1 - ( 1 + e ) e x p ( - £ y ) I f | u 0 | S R , t h e n i t f o l l o w s f r o m ( 3 . 2 4 ) t h a t H I V T ) ! ' « e x p ( - c f ) ( 1 + e ) | u J H + K 1 5 ( 3 . 2 6 ) c 2 ' ( 1 + e ) K e x p ( - ^ 1 - ( 1 4 e ) e x p - K 1 s = R z . T h i s c o m p l e t e s t h e p r o o f o f t h e l e m m a . R e m a r k 3 . 1 : I n t h e p r o o f o f t h e a b o v e l e m m a , i t i s e s s e n t i a l t h a t p > 2 , i n o r d e r t o o b t a i n ( 3 . 2 3 ) a n d ( 3 . 2 4 ) . H e n c e t h e m e t h o d a b o v e o n l y p r o v e s t h e l e m m a i n t h e c a s e w h e n p > 2 . I f p = 2 , w e h a v e t o m o d i f y t h e o p e r a t o r t h e o p e r a t o r A . W e s h a l l d o t h i s i n s e c t i o n 4 . 50 Lemma 3.2: L e t f w , w , ••• be a b a s i s i n V. L e t A, •M and f be the same as i n theorem 3.1. Suppose R i_s the  number d e f i n e d i n lemma 3.1. Then f o r each g i v e n m, there m I i= i e x i s t s a f u n c t i o n u „ ( t ) = T g . ( t ) w . s u c h t h a t — m . L , ^ i m l (3.27) (u',w.) * + (Au ,w.) * + (Mu , W . ) m } V m j V m ;j H ( f , w , ) v * , 1 « j « m u (0) = m u (T) m | u m ( 0 ) | H « R . Proof; S i n c e the s o l u t i o n o f (3.2) i s unique, and s i n c e g m i ( t ) are continuous on [0,T] f o r i = 1, 2, ••• m, the mapping (3.28) u 0 •> T (u 0) = u (T) u m 0 m i s w e l l - d e f i n e d f o r any u n i n W. I f um and v are J 0 m m two s o l u t i o n s o f (3.2) w i t h u (0) = u „ and v (0) = v., m 0 m 0 we can deduce from Gronwall's lemma t h a t 51 (3.29) |u m(T) - v m ( T ) | H |u o - v J H , where K i s independent of u and v . This shows that o o T m i s continuous on W with the topology induced by H.. From (3.26), we see that T maps the closed ' m H-ball m , (3.30) R H^ m = {w: w = ^ a .w . , |w|R < R} into i t s e l f . Hence by the Schauder-Tychonov fixed point theorem, there e x i s t s u„ i n IL m such that (3.31) u = T (u ) . om m om Therefore there e x i s t s u s a t i s f y i n g (3.27). m 3.3. Proof of theorem 3.1. Let W be a separable H i l h e r t space with W c V c: H. Suppose the i n j e c t i o n of W into V i s continuous and W i s dense i n H. By lemma 2.1, there exists an orthonormal basis w , w , w , ••• of W and numbers X , X , X , ••• 1 2 3 1 2 3 with .|X I < | X ] < | X | <•••-•«> such that 1 2 3 52 (3.32) (w ., v) = X . (w . , v) „ f o r a l l v i n W. By lemma 3.2, there e x i s t s u (t) such t h a t J ' m (3.33) (u',w.).T + (Au ,w. V + (Mu ,w.)tT m' 3 V m' 3 V m' 3 H (f,Wj) , 1 £ j < m um ( o ) = um ( T ) w i t h |u (0)I £ R, which i s independent of m. Using m H (3.32) and (3.33), and a p p l y i n g an argument s i m i l a r to t h a t of theorem 2.1, we can show t h a t ( 3 * 3 4 ) l UmllP'(0,T;W*) + > Um ' I? (0 ,T; V) + K 'L" (0 ,T;H) K K ' where K i s independent of m. S i n c e the i n j e c t i o n o f V i n t o H i s compact, we can e x t r a c t subsequences such t h a t : (3.35) P * u^ -> u' weakly i n L (0,T;W ), u u weakly i n L (0,T;V) and i n the weak topology of L°°(0,T;H), u^(0) ->- u„ weakly i n H, A u m + x weakly i n L (0,T;V*) 53 Using the same argument as i n the p r e v i o u s c h a p t e r s , we have (3.36) I Mu -*• Mu i n L' (0,T;H) i f 1 < s < °°, m r f T (Mu,u) Hdt. ' n Nov; lemma 1.2 and theorem y i e l d : (3.37) u (t) •*• u ( t ) i n W* f o r a l l t i n [ 0 , T ] , But ^ ( T ) = ^ ( 0 ) converges t o u 0 weakly i n H, t h e r e f o r e (3.38) u(T) = u(0) Hence i f we l e t m •* °°, then we have (3.39) u' + X + Mu = f u(0) = u ( T ) . u' i s i n L p (0,T;W ). But s i n c e f - Mu - x e ^ (0,T;V ), p' * u' i s i n f a c t i n L (0,T;V ) . I t remains t o show t h a t X = Au. From (3.33), we g e t : 54 (3.40) l i m sup m T (Au ,u ) * d t m m V = l i m sup m r T ( f , u f f l ) v * d t -rT T (Mu ,u ) T,dt m m H = f (f,u) * d t . - f (Mu,u). dt = { (X,u }V * d t ' L e t $ be an a r b i t r a r y element i n L p ( 0 , T ; V ) , then s i n c e A i s monotone, (3.41) 0 < lim^sup T ( A U m _ A K U m . d t  J o < J (x - A ,u - * ) v * d t . Now put <J> = u + \ty, where X > 0, Again, i s an a r b i t r a r y element o f L p ( 0 , T ; V ) . From (3.41), we have (3.42) 0 < j (x - A(u + Xi|0 * d t . I f X •-»- 0, then (3.43) 0 < I - Au, d t . S ince i s an a r b i t r a r y element o f L l (0,T;V), i t f o l l o w s t h a t (3.44) Au = X. 55 3•4". The case when p = 2. In t h i s s e c t i o n , we c o n s i d e r the case when p = 2. We s h a l l prove the f o l l o w i n g theorem. Theorem 3.2: L e t A be an o p e r a t o r mapping L 2(0,T;V) i n t o L 2(0,T;V*) s a t i s f y i n g assumption I I I . L e t M be_ as b e f o r e . Then f o r any g i v e n f i n L 2 (0,T;V*) , there e x i s t s u in. L 2 (0,T;V) OL 0 0 (0,T;H) w i t h u 1 i n L 2(0,T;V*) such t h a t I u' + Au + Au + Mu = f (3.45) | ( u(0) = u ( T ) , where X i s a s u f f i c i e n t l y l a r g e p o s i t i v e number. Pr o o f : S i n c e X i s l a r g e , the p r o o f of lemma i s v a l i d up to (3.19). (3.20) becomes: (3.46) e x p ( ^ ) | u ( T ) | 2 - | u j 2 + Xf |u ( t ) | 2 d t ' c 2 / m 0 II J c m H 56 Let e be the number defined by (3.22), then (3.47) for some constant K, . Since X i s large, (3.24) can 1 6 be obtained. The rest of the argument i n theorem 3.1 i s s t i l l v a l i d . Therefore the theorem i s proved. Remark 3.2: Theorem 3.2 has been proved by Gaultier 5 when A i s l i n e a r , X - 0 and c = 1. In that paper, i t i s assumed that 57 CHAPTER IV APPLICATIONS In t h i s c h a p t e r , we g i v e some a p p l i c a t i o n s o f the a b s t r a c t theorems proved i n the p r e v i o u s c h a p t e r s . L e t ft be a bounded open subset o f R n w i t h a smooth boundary 3ft. The p o i n t s o f ft w i l l be denoted by x = ( x j f x 2 , ••• , x n ) . 3 Set D.= ——- f o r j = 1, 2, ••• , n; and f o r each n - t u p l e i 3 x . 3 (a , a , , a ) of non-negative i n t e g e r s , we w r i t e n a. n (4.1) D a = n D.3 and |a| = £ a. . j=l D j=l 3 L e t k be a p o s i t i v e i n t e g e r . Denote by W ' (ft) the Banach space (4.2) w ' P ( f t ) = {u: D au e L P ( f t ) , |a| < k} w i t h the norm (4.3) |u| = { I | D a u | P p } P, 1 < p < k,p M ^ LMfi) k,P WQ (ft) i s the c l o s u r e o f C Q ( f t ) , the f a m i l y o f a l l i n f i n t e l y 58 d i f f e r e n t i a b l e f u n c t i o n s w i t h compact support i n ft, w i t h r e s p e c t to the norm | • |, K ,p 4.1. A g e n e r a l n o n - l i n e a r p a r a b o l i c o p e r a t o r . L e t m be a p o s i t i v e i n t e g e r and N be the number of d e r i v a t i v e s of o r d e r l e s s than or equal to m - 1. We s h a l l c o n s i d e r r e a l f u n c t i o n s A _(x,t,A,n) and A (x,t,A,n) ap a d e f i n e d on ft X [o,T~J X R X R N f o r |a| £ m, J 31 £ m, having the f o l l o w i n g p r o p e r t i e s : r ( i ) f o r almost a l l (x,t) e ft X [ 0 , T ] , the f u n c t i o n s (A,n) -*• A ag(x,t,A,n) and (A,n) •*• A a(x,t,A,n) are N (4.4) continuous on R X R . i i ) f o r a l l (A,n) e R X R , the f u n c t i o n s (x,t) A a g ( x / t , A , n ) and (x,t) A^ (x , t , A ,n) are a measurable on ft X [ 0 , T ] . Set: D ku = {D 3u: | 3 | = k} 6u {Du, D 2 U , ••• , D i n~ 1u} A (x,t,A ,n) : (x,t) •*• A ( x , t , u ( x , t ) ,<Su(x,t) ) a8 aB 59 Suppose th e r e e x i s t s a c o n s t a n t C such t h a t 1 (i) A ^ (x,t,X,n) * (1 + JX | ) ( i i ) A (x,t,x,n) * c ( l + | x | ) ( l + |X| + |n|). a I L e t V = W^' 2(fi) and W = W^'' 2(ft) w i t h m' > m + 1 + K I t f o l l o w from the Sobolev imbedding theorem t h a t W cz Cm(ft~) The p r o o f o f the f o l l o w i n g p r o p o s i t i o n i s s t r a i g h t -forward and t h e r e f o r e w i l l be o m i t t e d . P r o p o s i t i o n 4.1t Suppose t h a t A A ^ ( x , t , X , n ) and A a ( x , t , X , n ) s a t i s f y (4,4) and (4,5). Set H = L 2 (P.) , F = L 2 ( 0 , t : v ) , X = L 2 (0 , T ;V ) fl L°° (0 ,T;H) and Y = L"(0,T;W). L e t •a (t;u,v) = 1 1 A s (x,t,u,6u)D euD avdx, 1 \r*<m \Pi<m Jo P (4.6) 2 |a|«m laK IBK 'ft a (t;u,v) = I rT 0 A (x,t,u,6u)D avdx, Q a a. (u,v) = f a . ( t ; u , v ) d t i = 1, 2, 2 2 a(u,v) = I a. (u,v) = £ ( A . u , v ) v * = ( A U , V ) _ _ * . i = l 1 i = l 1 * Y Then A i s a mapping from X U Y i n t o Y . I t maps bounded s e t s o f X and bounded s e t s o f Y i n t o Y*. 60 We s h a l l now e s t a b l i s h the f o l l o w i n g theorem, Theorem 4.1: L e t A be _a n o n " l i n e a r o p e r a t o r mapping X y Y i n t o Y* as i n P r o p o s i t i o n 4.1, Suppose f u r t h e r t h a t (4.7) ( i ) I I l A g(x,t,u,6u)DBuD<*udx > c I f |D au| 2dx, jal^m |3UmJ ft | a k n r ft ( i i ) I [ A a(x,t,u,6u)D audx > 0. latenr ft L e t B e L (0,T) and T ^ > 0. Then f o r a l l f i n F and u 0 i n H, there e x i s t s u iri X w i t h u' _in Yi such t h a t u' (x,t) + Au(x,t) + B (t)u(x,t-T„) = f ( x , t ) u (x, 0) = u (x) x e ft, (4.8) < u(x,t) = 0 t > 0 , u(x,t) = 0 (x, t) e3ftX[0,T] . Proof To prove t h i s theorem, we s h a l l apply theorem 2.1 w i t h : 61 ' B ( t ) u ( t - T ) i f t e [T , T ] (4.9) M u ( t ) = i f t e [0 , T ) . o H i s c l e a r l y o f l o c a l type and s a t i s f i e s the convergence c o n d i t i o n . I t remains to check assumption I I . P a r t s (i) and (iv) are c l e a r , and p a r t ( i i i ) f o l l o w s from p r o p o s i t i o n 4.1. T h e r e f o r e i t i s s u f f i c i e n t to show t h a t p a r t ( i i ) i s t r u e . I f u -*• u weakly i n X, u* -> u' weakly i n Y*, n n i s then s i n c e the i n j e c t i o n o f W ™ ' 2 ^ ) i n t o Wm~1'2 (ft) o 0 compact, i t f o l l o w s from theorem 1.1 t h a t there e x i s t s sub-sequence of u n such t h a t -*• u i n L 2 [0 , T; Wm*"^  ' 2 (ft) ) , and hence D a u n -> D au i n L 2 (ft X [0,T]) f o r a l l |a| m-1. C l e a r l y , i f u eX, v e Y, then A (x,t,u,6u)D av a 3 l i e s i n L 2 ( f t X [0,T]) f o r |a| S m, | 3 | S m. I t f o l l o w s from theorem 2.1, p»22 of K r a s n o s e l 1 s k i i [6] t h a t A a j 3 ( x , t , u n , 6 u n ) D a v A a g (x,t,u,6u)D av i n L 2 (ft X [ 0 , T ] ) . But D' • r au n D mu weakly i n L 2 (ft X [0,T~J), t h e r e f o r e T Uft ^ a 3 ( X ' f c ' U n ' 6 U n } D 6 u n D a v ~ A a 3 ( x ' f c' U n ' 6 u n } D BuD«v] d x d t rT (4 '10) = lJn[^ B(x't'Un'5VDav-AaB(3C't'un'6un>Dav D^u dxdt n = | | A a B ( x , t , u , 6 u ) D a v [ D 3 u n - D Bu]dxdt 62 converges to 0 as n °°. Hence A u -*• A u weakly in i n i * 3f . S i m i l a r l y , i f u E X, v e Y, then A a(x,t,u,6u)D av i s i n L 1 (ft X [0,TJ). So by the same theorem of [6], we have [ f A (x,t,u ,6u )D°vdxdt (4.11) A (x,t,u,6u)D vdxdt for a l l |a| < m. Hence A u A u weakly i n Y . There-1 1 2 n 2 y fore, part ( i i ) of assumption II i s v e r i f i e d and the theorem i s proved. Remark 4.1: The above theorem remains true i f we replace every function u by an r-vector function u = (u , u , ••• , 1 2 In t h i s case, WJ0c'P(ft) w i l l be replaced by (V7^' p(ft)) r with the norm (4.12) u k,p " r P m=l |oKk L 63 Remark 4,2: L e t T (j = 0, 1, * * * , m - 1) be m g i v e n o p e r a t o r s i n L (w™' 2 (ft) ,W~ ^ " ^ ' 2 Oft)) . Set: (4.13) a(u,v) = a(u,v) + { T _ . U , Y . . V } , where Y-v = and {•,•) the. p a i r i n g b e b w e n 3 ^ v / n - j - ^ , 2 ( 3 0 ) a n d i t s d u a l W " ( m"3 _ 35) ' 2 ( 3 f t ) . The theorem i s s t i l l v a l i d w i t h a(u,v) r e p l a c e d m o m1 2 by a(u,v) and w i t h V = W ^ M f t ) , W = W ' (ft) . The boundary c o n d i t i o n s i n v o l v e d are o f the type: (4.14) S^u = TjU on 9ft, j = 0, 1, ••• , m-1. S_. are d i f f e r e n t i a l o p e r a t o r s o f order 2m - j - 1 d e f i n e d by: (4.15) m-1 Au"vdx = a(t;u,v) + £ {S.u,Y-v} , ft j=0 3 3 where Au i s g i v e n by: (4,16) I (-l)lalDar I A (x,t,u,6u)D 3u + A (x,t,u,6u) la^m L |3km a t i a 64 (4.14) i s c a l l e d the Neuman's type of boundary c o n d i t i o n i f T_. = 0, j = 1, 2, , m - 1. For T j ^ °» w e have the s o - c a l l e d t h i r d boundary - v a l u e d e r i v a t i v e s o r " o b l i q u e " d e r i v a t i v e s . Remark 4.3: Using the Sobolev imbedding theorem, we may improve the r e s u l t o f the above theorem and i n c r e a s e the exponents of |X| and |n| i n p a r t ( i i ) o f (4.5). L e t N denotes the number of d e r i v a t i v e s o f o r d e r j , j = 0, 1, •**, m-1. Then p a r t ( i i ) o f (4.5) can be r e p l a c e d by: m-1 | A f Y ( x , t f x 0 , x l , - " , x ) |< C (1 + | X 0 | ) ( 1 + I |x I J ) u j = 0 J (4.16) n e j n+2(j-m) , e. = 0 3 i f n+2(j-m) > 0 i f n+2(j-m) ^ 0 in 2 I t f o l l o w s from the Sobolev theorem t h a t i f u e W0' (ft), then (4.17) m-1 . e. 1 + I iD^ul ^ j = l ^ 2W" C | U ' ^ ' 2 ( f t ) C i s independent of u, 64a 4.2. A pseudo monotone operator. L e t m be a p o s t i v e i n t e g e r and p £ 2. L e t N be the number o f d e r i v a t i v e s of o r d e r l e s s than o r equal to m - 1 and N be the number of d e r i v a t i v e s of o r d e r m. 2 C o n s i d e r the f a m i l y o f r e a l f u n c t i o n s A a ( x , n , 5 ) / | ot| $ m, N i N ? d e f i n e d on ft X R X R s a t i s f y i n g c o n d i t i o n s s i m i l a r t o those g i v e n by (4.4). Set: u = {u, Du, •*• / D m u ) . Suppose there e x i s t s a c o n s t a n t C > 0 such t h a t (4.18) |A a(x,n,S)| < c f j n l ^ 1 + U ^ " 1 + l ] . L e t V = W m ' p ( f t ) , H = L 2 ( f t ) , F = L P(0,T;V) and X = FnL°°(o,T;H) . I f u, v e V, then the f u n c t i o n s A a(x,Su,D mu) are i n L P ' ( f t ) . D efine a:V X V -*• R by: (4.19) a(u,v) = f I A (x,6u,D mu)D avdx. jft laKrn a The form v ->• a(u,v) i s l i n e a r and continuous on V. We w r i t e (5.20) a(u,v) = ( A u , v ) v * . it Au e V . T h e r e f o r e a mapping A:V -> V i s d e f i n e d , 65 We s h a l l now s t a t e the f o l l o w i n g p r o p o s i t i o n , the p r o o f of which i s s t r a i g h t - f o r w a r d and i s t h e r e f o r e o m i t t e d . P r o p o s i t i o n 4.2: L e t A be as_ above. Suppose t h a t (i) t here e x i s t s c o n s t a n t c > 0 such t h a t I A ( x , n , O X > c I \x \ lal<m |a|<m a f o r a l l X = (X : |a| < m) e RN* X R N 2. ( i i ) I (A Q(x fn,5) - A a(x,n,5)) (5 a - 5 a) > 0 |aj=m f o r almost a l l x e ft, a l l n and K ¥ £, hi o r e o v e r , suppose B e L (0 ,T) and x 0 > 0. Then for, a l l f i n F* and u. i n H, t h e r e e x i s t s u i n X w i t h u' i n F* such t h a t u'(x,t) + Au(x,t).+ B ( t ) u ( x , t - x 0 ) = f ( x , t ) u (x, 0) = u (x) x e ft (4.21) u(x,t) = 0 t < 0 u(x,t) = 0 (x,t) e 9ft X [0,T] 66 Remark 4.4: In theorem 5.1, the o p e r a t o r c o n s i s t s of two p a r t s . In b o t h of t h e s e , the o p e r a t o r i s l i n e a r w i t h r e s p e c t to the d e r i v a t i v e s of the h i g h e s t o r d e r . In p r o p o s i t i o n 4.2, the o p e r a t o r i s n o n - l i n e a r w i t h r e s p e c t to d e r i v a t i v e s of any o r d e r . T h i s i s accomplished a t the expense o f o m i t t i n g the second p a r t of the o p e r a t o r c o n s i d e r e d i n theorem 4*1. 4.3. Time dependent d e l a y . We s h a l l now c o n s i d e r an example w i t h a d e l a y which, depends on t . L e t H = ( L 2 ( f t ) ) n , V = ( W o ' 2 ( f t ) ) n , and W = (W?' ' 2 (ft) ) n ; where m» > 1 + |. Suppose F = L 2 ( 0 , T ; V ) , Y = L"*(0,T;W) and X = F O L°° (0,T;H). L e t B e L°°(0,T) and w(t) be a r e a l measurable f u n c t i o n d e f i n e d on (0,T). Suppose there e x i s t s t e (0,T) such t h a t t - w(t) > 0 f o r a l l t e ( t Q , T) and t - w(t) > 0 f o r a l l t e ( 0,t )'. S e t : We s h a l l e s t a b l i s h the = t E ( S / r ) It " ««t)|. f o l i a t i n g p r o p o s i t i o n . 67 P r o p o s i t i o n 4.3 L e t w be p o s i t i v e . L e t g be a f u n c t i o n i n L 2 ( - T 0 , 0 ; H ) . Then f o r any f i n F * and u Q i_n H , t h e r e * e x i s t s u i n X w i t h u' i n Y such t h a t (4.22) u'(x,t) - Au(x,t) + B (t ) u ( x , t - t o ( t ) ) '+ I D j U j ( x , t ) u ( x , t ) + 2 U J ( x , t ) D j u ( x , t ) j = l = f ( x , t ) , u (x,0) = u 0 (x) X £ ft, u(x,t) = g(x,t) t e (-T ,0), u(x,t) = 0 (x,t) e 3ft X [ 0 , T ] . Proof; w i t h To prove t h i s theorem, we s h a l l apply theorem 2.1 (4.23) Mu(t) = B ( t ) u ( t - to(t) ) i f t e ( t o ,T] i f t e [ 0 , t j S i n c e B e L°°(0,T), Me L (L (0 ,T; H ) ,L°° (0 ,T ; H ) ), i s of l o c a l 68 type and s a t i s f i e s the convergence c o n d i t i o n . Nov; d e f i n e f i by: (4.24) f|(t) = B ( t ) g ( t - w ( t ) ) i f t e [ 0 , t ), o i f t e [ t , T ] . Because g i s i n L 2 (-T 0 ,0)H) , t h e r e f o r e f i s i n L 2(0,T;H) <= L 2 ( 0 , T ; V * ) . I f u(t) s a t i s f i e s ' u ' ( t ) + Au(t) + Ku(t) = f ( t ) - f j (t) (4.25) u ( t ) = g ( t ) i f t e [-T 0,0) u(0) = u 0 , n where Au = - A u + J. [(D.u.u) + 2u.D.u] , then u(t) a l s o j = l 3 J J D s a t i s f i e s (4.22). T h e r e f o r e the theorem i s proved i f we show t h a t A s a t i s f i e s the hypotheses of theorem 4.1, so t h a t theorem 2.1 can be a p p l i e d . I t i s c l e a r t h a t (4.5) i s s a t i s f i e d . I f u i s i n ( W ^ ' 2 ( ^ ) ) n , then 1 2 n where {«} i s a norm e q u i v a l e n t t o the (WQ' (ft)) norm. 69* Moreover, i f u e (w™ ' ' 2 ( n ) ) , then n n I I k=l j = l n n r (4 .27 ) = 1 1 -u..D. (u, ) 2dx k=l j = l J f i 3 3 K n n r = -2 I I u -D u -u dx. k=l j = 3 D K K T h i s means t h a t (4 .7 ) i s s a t i s f i e d and so the theorem i s proved. In the above p r o p o s i t i o n , we may add to M another o p e r a t o r M d e f i n e d by: ( 4 .28 ) M u(t) = | (t - s ) P u ( s ) d s 1 ' o where p > 0 and u e L°°(0,T;H). I t i s c l e a r t h a t M i s i n L(L°°(0,T;H) ,L°°(0,T;H)) and i s of l o c a l type, we can a l s o deduce from the Lebesque's convergence theorem t h a t M l s a t i s f i e s the convergence c o n d i t i o n . T h e r e f o r e we have the f o l l o w i n g p r o p o s i t i o n . L DJ Uj' Uk , uk d x 70 P r o p o s i t i o n 4.4: L e t M b e as a b o v e and s u p p o s e t h a t a l l t h e h y p o t h e s e s o f p r o p o s i t i o n 4.3 a r e s a t i s f i e d . T h e n f o r any g i v e n f i n F* and u 0 i n H, t h e r e e x i s t s u i n X w i t h u' i n Y* s u c h t h a t (4.29) u' ( x , t ) - A u ( x , t ) n + D j U j ( x , t ) u ( x , t ) + 2 U J ( x , t ) D j U ( x , T ) + B ( t ) u ( x , t-u>(t) ) + f ( t - s ) P u ( x , s ) d s ' o f ( x , t ) , u ( x , 0 ) = u (x) 0 u ( x , 0 ) = g ( x , t ) u ( x , t ) = 0 x e ft, t e (-T0 ,0) , ( x , t ) e 3ft X [0,T] 4.4. S o l u t i o n s w i t h f r a c t i o n a l d e r i v a t i v e s . I n t h i s s e c t i o n , we g i v e e x a m p l e s o f n o n - l i n e a r p a r a b o l i c e q u a t i o n s w h i c h h a v e s o l u t i o n s w i t h f r a c t i o n a l t i m e d e r i v a t i v e s o f o r d e r y, 0 < y < H, i n L 2 ( 0 , T ; L 2 ( ^ ) ) . 71 P r o p o s i t i o n 4.5: L e t H = L 2 (ft) , V = W1 '^2 (ft) , F = L 2 (0, ;V) and X = FnL°°(0,T;H) . D e f i n e A:V V* as i n p r o p o s i t i o n 4.2 w i t h p = 2. Then the o p e r a t o r A a l s o s a t i s f i e s  assumption I w i t h 6 = 0 . The p r o o f of the above p r o p o s i t i o n i s s t r a i g h t -forward. The f o l l o w i n g theorem i s an immediate consequence o f theorem 1.3 and p r o p o s i t i o n 4.5. Theorem 4.2: L e t a l l the hypotheses of p r o p o s i t i o n 4.5 be s a t i s f i e d . Then f o r any f i n F* and u i n H, t h e r e o e x i s t s u i n X with u' i n F* such t h a t o (4.30) u(x,0) = u (x) x e ft, o u(x,t) 0 (x,t) e 8ft X [0,T] , DY u(x,t) e L 2(0,T;H) 0 < Y < 3s. 72 P r o o f : I t i s c l e a r t h a t M d e f i n e d by rT (4.31) H u ( t ) = / F s u ( s ) d s , u e L°°(0,T;H), ' o i s an o p e r a t o r i n L (L°° (0 ,T;H) ,L°° (0 ,T;H)) , i s of l o c a l type and s a t i s f i e s the convergence c o n d i t i o n . T h e r e f o r e the theorem f o l l o w s from theorem 1.3 and p r o p o s i t i o n 4.5. When A i s l i n e a r , theorem 4.2 has been o b t a i n e d by A r t o l a [l] w i t h 0 < y < h. 4.5, P e r i o d i c s o l u t i o n s . In t h i s s e c t i o n , we g i v e some a p p l i c a t i o n s of theorem 3.1. Theorem 4.3: L e t I-I = L 2 (ft) , V = wj-* p(ft) w i t h p > 2, Then  f o r any g i v e n f i n If (0,T; V ) , there e x i s t s u i l l L p(0,T;V) nL°°(0,T;H) w i t h u' i n L p'(0,T;V*) such t h a t n P-2 u ' ( x , t ) + c I -D. (|D . u ( x , t ) | D _ j U ( x , t ) ) j = l J J J (4.32) I + C>|u(X, t ) | P" 2U+ u ( X , t - ( j ) ( t ) ) = f ( X , t ) , u ( x , 0 ) = u ( x , T ) x e ft, u ( x , t ) = o ( x , t ) e an x [ O , T ] P r o o f i I t i s w e l l - k n o w n t h a t t h e o p e r a t o r A:W^' P(Q) W _ 1 / P ' ( f t ) d e f i n e d b y (4.33) Au = c n _ l - D j ( i D . u j ^ V u ) + |u| P" 2u w i t h c > 0 i s mo n o t o n e and (4.34) ( A u , u ) v * P c l u l v T h e r e f o r e t h e t h e o r e m f o l l o w s i m m e d i a t e l y f r o m t h o o r e m 3.1 I n t h e c a s e w h e r e p = 2, i t i s n e c e s s a r y t o m o d i f y t h e o p e r a t o r A. We s h a l l s t a t e t h e f o l l o w i n g p r o p o s i t i o n w h i c h i s an i m m e d i a t e c o n s e q u e n c e o f t h e o r e m 3 74, Proposition 4 . 6 : 1 2 Let H = L z (ft) , V = V70' (ft) and c > 0. Suppose X i s a s u f f i c i e n t large number. Then for any given f i n L 2(0,T;V*), there exists u i n L(0,T;V) nL°°(0,T;H) with u' i n L 2(0,T;V*) such that u' (x,t) - cAu(x,t) + Xu(x,t) + u(x,t-w(t)) = f ( x , t ) , ( 4 . 3 5 ) u(x,0) = u(x,T) X G ft, u(x,t) = 0 (x,t) e 8ft X [ O , T ] 75 BIBLIOGRAPHY M. A r t o l a , Sur l e s p e r t u r b a t i o n s des Equations d ' e v o l u t i o n , a p p l i c a t i o n s a des problemes de r e t a r d . Ann, S c i e n t . E c o l e Normale Sup., 4 e s e r i e , t.2, 19 69, p.137 a p,253. J , Aubin, Un theordme de compacite, C. R. Acad. Sc. P a r i s . S e r i e A256(1963), 5044 - 5047. F. E. Brow.der, S t r o n g l y N o n - l i n e a r P a r a b o l i c Boundary Value Problems, Amer. J . Math. 86 (1964), p.339 - p.357. E. Coddington and N. L e v i n s o n , Theory of O r d i n a r y D i f f e r e n t i a l E q u a t i o n s , Mcgraw-Hill, New f o r k 19 55. M. G a u l t i e r , S o l u t i o n s f a i b l e s p e r i d i q u e s d'equations d'Evolution du premier ordre p e r t u r b d e s , C. R. Acad. Sc. P a r i s , s e r i e A272 (1971), p.118 a p.120. M. K r a s n o s e l ' s k i i , T o p o l o g i c a l Methods i n the Theory o f N o n - l i n e a r I n t e g r a l E q u a t i o n s , Pergamon, 19 64. J . L. L i o n s , Equations d i f f e r e n t i a l l e s o . p e r a t i o n e l l e s , e t probldmes aux l i m i t e s , S p r i n g e r - V e r l a g , B e r l i n , 19 61. , Quelques Methods de r e s o l u t i o n des p r o b l probilmeseaux l i m i t e s n o n - l i n d a i r e s , Dunod, P a r i s , 19 69. , Sur c e r t a i n e s equations p a r a b o l i q u e s n o n - i i n g a i r e s , B u l l . Soc. Math. France, 93, (19 65) p.155 a p.175. A. I^yskis and E l ' s g o l ' t s , Some R e s u l t s and Problems i n the Theory of D i f f e r e n t i a l E q u a t i o n s , Russian Math. Surveys, V o l . 22, No. 2, 1967, p l 7 to p.57. M. S h i n b r o t , F r a c t i o n a l D e r i v a t i v e s of S o l u t i o n s of the N avier-Stokes E o u a t i o n s , Arch. Rat. Mech. A n a l . 40 (1971) , p.139 to p.154. B. A. Ton, F r a c t i o n a l D e r i v a t i v e s of S o l u t i o n s of Non-l i n e a r E v o l u t i o n Equations i n Banach Spaces (to appear) , N o n - l i n e a r P a r a b o l i c I n i t i a l - v a l u e Problems. I n d i a n a U. Math. J o u r . V o l . 20, No. 1, (1970), p.69 to p.80. 76 j14| B. A. Ton, On Strongly Non-linear Parabolic Equations. J . Func. Anal., Vol. 1, No. 1, (1971), p.147 to p.155. |15| , Periodic Solutions of Non-linear Evolution Equations i n Banach Spaces,,Can. J . of Math. 23, 19 71, p.189 to p.196. 

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