UBC Theses and Dissertations

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UBC Theses and Dissertations

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

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ISO  ON NON-LINEAR TIME-LAG EVOLUTION EQUATIONS  by  CHE-BOR LAM B.A., U n i v e r s i t y  o f Hong Kong, 1966.  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE  REQUIREMENT FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  in- -the D e p a r t m e n t of Mathematics  V7e a c c e p t required  THE  this  thesis  as c o n f o r m i n g  to the  standard  UNIVERSITY OF B R I T I S H December, 1972.  COLUMBIA  In p r e s e n t i n g  t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r  an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree t h a t the  L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e  and  study.  I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may by h i s r e p r e s e n t a t i v e s .  be  granted by  permission.  Department o f The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8 , Canada  Date  Department or  I t i s understood t h a t copying or  of t h i s t h e s i s f o r f i n a n c i a l g a i n written  the Head of my  publication  s h a l l not be allowed without  my  ii S u p e r v i s o r : An  Ton  Bui.  ABSTRACT  The theorems spaces  purpose of  and  Banach s p a c e s .  or delayed  Galerkin  approximation  the e x i s t e n c e  • spaces.  shall  prove  fractional  apply basis.  the  the  method w i l l  one,  we  derivatives  be  study  the  derivatives  used  to  of order  t h r e e , we  e q u a t i o n s . H e r e we to prove  approximating  first  a the  establish  in Hilbert  less  than  1/2.  i n Banach s p a c e s .  study use  equations.  chapters.  on  the  solutions.  In  time chapter  Again,  method, b u t w i t h  a  periodic solutions  we  special of  the Schauder-Tychonov  the e x i s t e n c e of p e r i o d i c  of appliactions three  approximate  a solution with  Galerkin approximation  theorem  the  thesis,  the problem  of the  that there exists  In c h a p t e r  examples the  example o f  argument. Throughout  c o n s i d e r the problem  evolution point  Hilbert  -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  fractional  we  A practical  in  theorems.  In chapter  two,  i s to obtain existence  u n d e r 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  time-lag  We  thesis  f o r perturbated e v o l u t i o n equations  perturbations  time  this  In the  last  fixed  solution  c h a p t e r , we  give  o f the v a r i o u s theorems p r o v e d  in  to  iii  TABLE OF CONTENTS  Page  INTRODUCTION  1  CHAPTER  I :  EVOLUTION EQUATIONS  IN HILBERT SPACES  . .  3  CHAPTER  II :  EVOLUTION EQUATIONS  IN BANACH SPACES  . .  30  CHAPTER  III :  CHAPTER  IV :  BIBLIOGRAPHY  PERIODIC SOLUTIONS  APPLICATIONS  .  OF EVOLUTION EQUATIONS  i  .  .  .  .  41  5.7  75  ACKNOWLE D GE f'ENT  I Dr.  am  deeply  A. T. B u i , f o r h i s a d v i c e  preparation  of this  The  gratefully  research  and e n c o u r a g e m e n t  supervisor, during  the  thesis.  financial  C o u n c i l o f Canada is  g r a t e f u l t o my  support  o f the N a t i o n a l Research  and t h e U n i v e r s i t y o f B r i t i s h  acknowledged.  Columbia  1  INTRODUCTION  Differential  equations  with  a delayed  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 has  b e e n made b y f y s h k i s and E l ' s g o l ' t s  the  importance o f t h i s  processes with  with  a delayed  class  mechanics, engineering, theory  o f automatic  bibliography  spaces case  describe  differential  equations  many a p p l i c a t i o n s i n P h y s i c s ,  e c o n o m i c s , b i o l o g y and e s p e c i a l l y , i n t h e  c o n t r o l . We s h a l l  Artola  refer  [l]considered  a r g u m e n t . He s t u d i e d d i f f e r e n t i a l and B a n a c h s p a c e s  of linear  treated tone  with  such  e v o l u t i o n equations  e x t e n s i v e l y by A r t o l a .  t o [ l o ] and i t s  evolution case  equations  o f which  operators  isa  in Hilbert  a r e g u l a r p e r t u r b a t i o n . The i n Hilbert  spaces  was  For the non-linear case,  mono-  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 equations using  direction,  strongly non-linear parabolic  have been s t u d i e d b y Browder  the theory  separable Ton  i s t h a t they  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  delayed  problems  f o r further details.  Recently, with  Therefore  argument h a v e f o u n d  o f these  [ l O j . One r e a s o n f o r  o f equations  "after-effects".  argument have  o f monotone o p e r a t o r s  Banach space.  has extended  operators which  In a s e r i e s  [ 3 ] and b y L i o n s d e f i n e d on a  o f papers  these works t o a l a r g e r i n c l u d e the Navier-Stokes  class  [9]  reflexive  [12,13,14,15], of d i f f e r e n t i a l  e q u a t i o n s . The  2  purpose of t h i s in  of  thesis  i s to study  1  f o r the  class  The  crucial  technique  Artola  the p e r t u r b a t i o n problems  o f o p e r a t o r s s t u d i e d by e m p l o y e d by  a singular perturbation involving  h i s work, A r t o l a s h a l l make use  In  used  the  of both  we  equations  in Hilbert  this  a r e t h e b o u n d s on  case  derivatives two, using  we  study  the  In chapter evolution Artola.  of order  techniques  shall  less  the  1/2  Galerkin approximation  equations. Periodic  In the  last  apriori  c h a p t e r , we  o f the v a r i o u s theorems p r o v e d  thesis.  estimates  in  derivatives  of  fractional  are o b t a i n e d . In i n Banach  method w i t h  first  chapter spaces basis.  of time-lag  are not  g i v e examples i n the  time-  a special  solutions  solutions  We  non-linear evolution  equations  study p e r i o d i c  In  method.  in this  fractional  time-lag evolution  t h r e e , we  study  Solutions with than  use  a d u a l i t y mapping.  spaces. C r u c i a l  the approximate s o l u t i o n s .  i s the  Galerkin approximation  of these  c h a p t e r one,  Ton  Ton.  treated  of  three  by  applications chapters.  3  CHAPTER I  EVOLUTION EQUATIONS  IN HILBERT SPACES  1.1. N o t a t i o n s .  Let product i n  H  H .  be a H i l b e r t Let  Suppose t h e n a t u r a l and and  W  i s dense  W  b e a Banach  injection of  in  H .  We  W  ('f'Jn  ke  space such t h a t into  identify  H  H  t  n  e  W  i c  n  n  e  i s continuous,  with i t s dual  W  Throughout pairing W  .  between If  H  ,  (1.3)  W*  (»,») *  and  W*  and  |•|  w e W  and  h e H ,  then  (h,w) * w  i s a Hilbert  J *  =  u(t)  t h e Banach from  |u(t)|Pdt  will  w  (h,w)  space, then  denotes  of functions  f  will  w  W  general. L P ( 0 , T ; W ) classes  H c  this thesis,  (1.2)  Thus i f W  c  h e W  H  t h e norm  and  .  (h,w)  w  y  (h,w) * w  in  space o f e q u i v a l e n c e  [ O , T ] to  <  denote the  denote  W  with  r  H ,  therefore  (1.1)  in  s p a c e and  4  and w i t h t h e norm  (1.4)  = [ f  |u| L (0,T;W)  than the  p  i s n o t l e s s than  1 ,  p'  following  will  1 .  always  P  dt  ll/P  1  L Jo  P  where  |u(t) \  If  denote  p  is strictly  t h e number w h i c h  larger satifies  equation:  (1.5)  1 + 1 , - 1 .  1.2. F u n d a m e n t a l lemmas.  In  this  lemmas w h i c h w i l l The proof of this  section„, we  shall  s t a t e s e v e r a l well-known  be r e f e r r e d t o f r e q u e n t l y .  following  lemma i s c a l l e d  lemma c a n b e f o u n d  in  Gronwall's  Artola  lemma. A  [ l ] , p . 148.  Lemma 1.1:  Let functions C  l  u  and  integrable  v  over  b e two p o s i t i v e a finite  be a p o s i t i v e c o n s t a n t such  (1.6)  u(t)  <  Ci  real-valued  interval  [0,T]  that  +  |  u ( o ) v ( a ) da  and l e t  5  for  almost  all  t  (1.7)  [0,T]  in  u(t)  $  , t h en  Ciexp  I f o 1  for  almost  a l l  The  Therrem  0  d  B  in  1  [o,TJ.  following  theorem  i s due  to Aubin  [2].  1.1;  Given B  t  v(a)da J  i  c: B i  (1.8)  B -* 0  t h r e e Banach  , and  B  B.^  and  spaces  reflexive  B + B j  B  ,  Q  for  B  i  a r e compact  ,  =  Bj  1,2.  and  , with Moreover,  continuous  respectively.  Let is  G  =  finite  {v: v e L ( O , T ; B ) , P  0  and  1 < p,  r < °° .  v'  e L (0,T;Bi)}  G  i s _ a Banach s p a c e w i t h  r  , where  T the  norm:  G  Then L (0,T;B) p  ' L  1  P  ( O . T ; B  G cz L ( 0 , T ; B )  i s compact.  P  0  )  and  '  '  L  r  ( 0 , T ; B ! )  the i n j e c t i o n o f  G  into  6  Lemma 1 . 2 ; ( C f .  Lions  Let  B  b e a_ Banach s p a c e .  e LP(0,T;B),  f•  [0,T]  to  1 < p <  y > 0,  V we  and  of functions  u  0O  the  (- ,°°;V)  2  Fourier  The  with  l»l|  u(t)  f  | V f H )  (1.11)  u(t)  Lions  v e 5  to  00  (- , ;H) , 00  with  00  V  with  where  respect  to  u  is  t.  t h e norm:  }_ JT|^|0(T)|^T. +  +  in  E  y  (-«>,<»; V,H)  space, denoted by  00  and  have t h e f o l l o w i n g  [ 8 ] , p.60.  2  V c  5  to  (0,T;V H), r  norm:  (- , ; V,H) 00  y  00  o f elements  |u| E (0,T;V,H)  We  from  spaces with  (- , )  space w i t h  Y  [0,T],  and  as t h e e q u i v a l e n c e  -£ju(t>|'dt  a Hilbert  the quotient  from  | T | U(T) e L  and  .. .  T (  i s also  where  i_s_ c o n t i n u o u s  co  set of restrictions  [0,T]  f  E^. (-°°, ;V,H)  i s a Hilbert  00  U.IO)  then  f e L^(0,T;B)  b e two H i l b e r t  transform of  By (- , ;V,H) 00  H  define  classes e L  ,  0 0  If  B.  Let For  [ 8 ] , p.7)  =  inf v  v = u theorem  jv| (-<»,«>; ,H) V  almost everywhere which  i s due t o  on  H,  7  Theorem  1.2:  Let Vo cr V  c  V ,  H.  Suppose  that  of  then  the i n j e c t i o n  compact  1.3.  Vj  i f  Vi  0  into  H  and  a r e compact  of  [0,T].  H  identify  If  three Hilbert  5  of  V  spaces  into  Q  V  and c o n t i n u o u s  (0,T;V ,H)  into  Q  with  and  respectively,  L (0,T;Vj) 2  is  y > 0.  Let  by e x t e n d i n g  be  the i n j e c t i o n  Perturbation of l o c a l  shall  H  be  a Hilbert  L°°(0,t;H)  every  type.  function  u e L°°(0,T;H),  space.  as a c l o s e d in we  subspcce  L°°(0,t;H) shall  f (1.12)  For fixed  t o be  t < T, of  we  L°°(0,T;H)  zero o u t s i d e  define  s e (o,t)  u(s)  Y u(s) t otherwise.  An o p e r a t o r t o be o f l o c a l depending  (1.13)  on  type t  i f there e x i s t s  such  t L~(0,T;H)  ]y ml  M e L (L°° (0 , T;H) ,L°° (0 ,T ;H) ) a constant  that  *  y  | Y  t  U |  L-(  0 /  T;H)  i s said  u > 0,  not  8  for  a l l  t  e  [0,T]  .  Many e x a m p l e s o f o p e r a t o r s g i v e n by A r t o l a interest  i n [ l ] . The  and w i l l  be  called  of local  type  f o l l o w i n g example  have  i s of  been  special  a time-lag perturbation.  Example:  Let defined  t -*• w(t)  [O,T]  on  .  be  a positive  u e L°°(0,T;H)  For  ( u(t -  (1.14)  W u (t)  measurable f u n c t i o n ,  define  w(t) )  t -  w(t) > o  = otherwise.  Clearly,  to  satisfy  (i)  M  i s an o p e r a t o r  An  operator  e  u  n (ii)  Mu  L ( L °  9  type.  (0,T ;H) ,L°(0,T ;H) )  the convergence c o n d i t i o n i f  e L°°(0,T;H),  u  M  of local  (t)  u(t)  n (t)  Mu(t)  in  H  a.e.  in  ( i ) => ( i i ) ,  H  a.e.  i s said where  9  1.4 A s s u m p t i o n s a n d f o r m u l a t i o n  Let V cr H , V  into  V H  V  and  H  o f the problem.  b e two H i l b e r t s p a c e s s u c h  i s dense i n  H ,  that  and t h e n a t u r a l mapping o f  i s compact.  Definition:  The <f>(t)  f r a c t i o n a l d e r i v a t i v e o f order  i_s d e f i n e d b y  whenever  dt  (t - s)" c|>(s)ds  t h e r i g h t hand s i d e makes  Suppose t h a t some y > 0 .  with  ft  D^Mt)  (1.15)  for  y , o < y, o f  <}>(t) - =  sense.  < f > e L (0,T;H)  and  2  Then t h e r e  $ (t)  Y  e L (0,T;I1) 2  exists a function  almost everywhere such  $ e L  2  (- , ;H) 00  00  that  f +<»  (1 + | t |  (1.16)  Where  $(T) i s the Fourier  to  t .  Moreover, i f  on  [o,T] such  that  2 Y  ) |?(T) I dT H  transform  <j>  of  «> ,  $(t) with  respect  i s a sequence o f f u n c t i o n s  n  | cf> \ L (0,T;H) n  <  2  2  +  |D 4> Y  C  n  | L (0,T;H) 2  defined are  10  b o u n d e d b y a c o n s t a n t , t h e n we  can choose  <$> n  i n such  a way  .+00  so t h a t  (1  +  | T |  2  Y  ) | $  a r e a l s o bounded by a  (T)l^dT  / — 00  constant. < J > z L (0,T;H)  If and  D^4»  also belong to  a + 3 = 1,  D  cf> e L ( 0 , T ; H ) 2  (0,1), on  such  D  and  <J)(t)  definition  stated  above  In operators  L (0,T;H)  i f  2  0 < a,3 < 1.  A  D  =  DJ<J>  t t D  ( f )  =  D  (i)  D^4>  If  t*«  e L (0,T;H)  constant  y  f o r some  2  K ( y ) , depending  =  K(Y)  and p r o p e r t i e s  f (t Jo t  -  in only  s) ~ D <f>(s)ds. Y  1  Y  ;  of the f r a c t i o n a l  derivatives  are q u i t e well-known.  (Cf. Shinbrot  this  consider  c h a p t e r , we  mapping  V  into  shall V  satisfying  I  There  exists  a constant ' C  such  [ll].)  non-linear  assumption:  Assumption  then  2  that  (1.18)  The  l t^  then there e x i s t s  Yr  D < | > e L (0,T;H),  then  (1.17)  If  and  2  that  the following  11  <  | A U | * v  for  (ii)  a l l  There  u  in  exist  C,[|u|*  + l]  V.  constants  C  and  2  3,  o < 3 < l  f  such  that  |(Au,v) *| v  for (iii)  (iv)  a l l  u ,  «  C [|u|^|v| 2  v ,  in  A  i_s_ c o n t i n u o u s  to  t h e weak t o p o l o g y  (Au,u)^ L  2  from  v  +  |u| |v| ] v  v  V. the strong topology  of  of  V  V*.  ijs i n t e g r a b l e on  [O,T] fora l l  u  in  (0,T;V) n L ( 0 , T ; H ) .  The  ro  main r e s u l t  of this  chapter  i s the f o l l o w i n g  theorem.  Theorem  1.3: Let  V*  satisfying  (a)  A  be a n o n - l i n e a r o p e r a t o r mapping  Assumption  I . Moreover, suppose  (Au,u) * v  >  c|u| V  2  that  V  into  12  for (b)  a l l u  Let  u  that  e L  n  u  H  5  V  and some  (0,T;V)  u  n  t h e weak in  in  and  Y  n  Suppose  S  weakly i n  D u  0  e ( L (0,T; V) ) *.  ,  * topology  c >  L (0,T;V)  ,  u  L~(0,T;H)  ,  u (o) + u(o)we  weakly i n  L  2  of  + D^u  -*• u  n  in  n  (0,T;H)  2  for  that  lim inf  {|u (o)|  N  +  2  n  j  (ui +  T Q  Au ,u ) *dt} n  n  v  fT $  |u(o) |  +  2  J  .(g,u) *dt v  and fT  (u£  Jo for  Let and H  satisfy and  f  n  v  a l l v e cj(0,T;V)  fT j for  + Au ,v) *dt  (Au ,v) *dt n  v  ,  +  ->  fT Jo  (g,v) *dt v  t h e n we  fT j  assume  (Au,v) *dt v  a l l v e C(0,T;V). M e  L(L°°  (0,T;H) ,L°° (0,T;H)) , b e o f l o c a l  t h e c o n v e r g e n c e c o n d i t i o n . Then in  that  L (0,T;V*) 2  and  L° (0,T;H) w i t h  such  that  ,  D^u e L  there exists 2  (0,T;H)  f o r each u  f o r some  in y  type u  Q  in  L (0,T;V) 2  e (o,|- f)  13 -J  (u,v') dt  J  +  R  + J  (Au.,v) ,*dt  (^,v) dt  r  v  H  (1.19) =  | o  (f v) *dt f  f o r a l l v e C'(0,T;V)  This where  If (jj)  condition  a l l  then the c o n c l u s i o n  1  so in  v  n  r L  ( i i ) o f assumption *|  u,  v  $  C  in  lul  ,  Ton  [12],  p.216.  I  2  I i s replaced by  Ivl  +  |u|*|u|*|v|*|v|*  still  holds,  V,  o f theorem  v ,  v ,  1  2  are l i n e a r l y  v ,v , ' * * , v  n n  J  be  1.3  y  in  except  that  (0,T).  a basis  independent  i s dense i n  V.  e H, then there e x i s t s o ' ,v^, * *' , v ] and u •*• u n  f o r V,  i.e. v ,  v ,  1  i f u [v  of  inequalities.  Let  U  [7]  Lions  2  A priori  ,  of  a result  D£U e L (0,T;H) o n l y f o r  we have  00  H  v ( T ) = 0.  with  and a r e s u l t  |(Au,v) for  *"*  Q  1.1:  Remark  1.5.  (u ,v(o))  theorem g e n e r a l i z e s  0,  M =  +  y  Q  m  f o r any Since  V  a sequence in  H  as  n  2  and t h e s e t i s dense i n  H, '  {u„ } w i t h om m -»• <».  u om  14  Lemma  1.3:  u  Let mapping  from  =  Tp(s)  V  s.  be  o m  into  V  Then g i v e n  and  (0,1),  in  8 , pl74.) any  there  J  be  u  in  0  ,  A  and  f  in  =  I  9"eim  be  as  L (0,T;V*) 2  in ,  ( t ) v  i  that  + eJ(eu  em  ) + Au  m  em  + Mu  (1.21)  „,,v.) * = em j v tT  f  u  (o)  (f,v.) * j V  =  Moreover, there e x i s t s e  (1.22)  and  m  ,  such  sup lu ae(0,T) e  fixed  e  and  2  +  H  m  l^j^m  om  a_ c o n s t a n t  K  ,  independent  that  (a)| m  ,  u  em  For  M  exists  "emM  (u' em  of  duality  t h e gauge f u n c t i o n  Let H  the  m  (1.20)  such  above. L e t  associated with  (Cf. Lions  H  theoreml.3. e  as  ,  leu |'«,_ __« em'L (0,T;V) m  5  u^.  m  e  + 1  |u I* , , em'L (0,T; 2  V)  <  ( L ( 0 , T ; V ) ) *. 5  Proof:  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 :  K.  15  m m 7 g' . (t) (v. ,v.) *. + e f j ( e I g ( t ) v . ) ,v.) * fl^eim 1' j v elm 1 D'V i=i m m  I  (  +  (1,23)  <  v  A  (  J^ im  (  t  )  i '  v  )  £  =  j  )v*  i  v  9cim  ( t , v  i ' j) ,  u  ordinary d i f f e r e n t i a l  variables  and  t  as t h e i n d e p e n d e n t v a r i a b l e . Now  ^ e i m ^  ''  ~  1 /2,  v  i  subspace  V  m  of  ==  V ,  I f 2, •••• , m)  , m  ,  is  continuous  M  i s a linear  as d e p e n d e n t  span a f i n i t e  dimensional  hence the e x p r e s s i o n :  m  (1.24)  H  om.  with  m  v  1 $ j $ m.  f  I  m  M (  l  equations  e  (i  | (  . (o)v. =  e lm  i s a system o f 9" £  +  (f(t),v_.) *  g  This  v  m  from t h i s  s u b s p a c e t o t h e weak t o p o l o g y  operator,  so i t i s continuous  from  V  of  V  to V  i  m  Therefore  (1.25)  m (eJ(e I g i n  e i m  m + A(l g  (t)v.) •m  M  E  i  i= i continuous  in  g  i s measurable i n  t  e  (t)v.),v )  Jl (g im(t)v ),v ) *  +  is  e i r a  1=1  i  m  i  i» j  i f g  V- map b o u n d e d s e t s o f  =  r v  1, 2, *••, m.  . are fixed. Since eim V i n t o bounded s e t s o f  Clearly i t A,  J,  V*, t h e  and  16  expression such t h a t  (1.25)  i s bounded i f t h e r e  |g . I < b  from C a r a t h e o d o r y ' s exists  solution  i sglobal i f  g ^ (t) £  m  ,  taking  i n the i n t e r v a l  (1.22)  t h e summation f r o m  ^ K m ^ l H  o  [0,6 ]  . The  m  of  (1.21)  j = l , 2,  by , m,  t o t , we o b t a i n :  T ( 'o  +  there  holds.  the j - t h equation  and i n t e g r a t i n g from  I t follows  (Cf. [ 4 ] , p.43) that  (1.21)  Multiplying  a constant b  f o r i = 1, 2, ••• , m.  theorem  a solution of  exists  1:  ( M u  o  (  e J  em  £ u  ( a )  m( >> 0  e  ' em U  +  ( a ) )  m  A u e  H  ( a )  ' em u  ( a )  )v*  d a  d a  2 ( 1  -  2 6  >  *l omIS  s  u  l lLM0,T;V*>  +  f  f  u  <  i l u I om 2 1  1 + n  using  the assumption  we o b t a i n  from  1  2  +  1  |f  LM0,T,V*)  sup as (o,t)  on  (1.26):  J  1  "em  vu  and on  K.m«»lv*> o  ' em U  ( 0 )  lv  d a  _  'H 1  A  ,  and r e a r r a n g i n g ,  17  | f^P J u (a)| +[*|eu (a) | d a + c f * o e ( o , t ) ' em ' H J ' em 'v J a  <  i  |  U  c J  +  5  | f l  H  |u  LMO T;V*)  e m  t 1  |u em  (a)| da 'v 2  v  (a)| da v  f  rt (1.27)  u  *•  (a) |'da H  em  1  t f m : L„ „(0,T;V") „.„.,•§[  !«„(»>i>  z  + u T  |u  e m  (a) | da . H  Therefore  T  |u ( a ) | + e 5 f | u a e ( o , t ) ' em ' H } e s  u  p  2  1  (1.28)  <  2  X i "  2i + H  i 2  c  H  2  *  where It  K  x  and  f o l l o w s from  (1.29)  K  2  U  The  inequalities  ( 1 . 2 8)  |u (a)|'da em 'v v  |u (a) | d a em H 2  1  0  1  I u ( a ) I da, em 'v  independent o f  that  ae(o! )' em T  -t  1  2 1  are constants  lemma 1.1  ViT  L  J • O  2 2  +  ' 2(0,T;V*)  1  ft K  ( a ) | ' d a + ff* 'v 2J  i  f  em'  m  { a ) |  H  and (1.29)  <  implies that  e  and  m.  18  (1.30)  ^ J u (a) |* + e [ |u (a) |'da + f ae(0,T)' em H J Q : . ' em v J p  5  T  m  T  1  1  q  i s bounded by a c o n s t a n t and  e .  T h i s proves  K  |u em  (a) I d a 'v 2  which i s independent o f ra  3  (1.22) .  I t a l s o follows that a  global solution exists. Using  (1.31)  where (L  can w r i t e  =  (1.23)  as:  (h(t),v.) * . 1 3 V  *m,  * j  h(t) = -feJ(eu (t)) + Au (t) + Mu (t) - f ( t ) l i s i n <em em em  (0,T;V)J  ij  we  g* . ( t ) ( v . , v . ) * eim 1 3 V  pendent, a  (1.22) ,  .  v  Since  l f  v ,  ••• , v  2  the determinant o f the m a t r i x  ~  ^ i V  / V  j^v*  can s o l v e f o r  g*.  ^  S  n  o  t  (t)  z e r o  •  H  e  n  c  e  are l i n e a r l y (a.•) ,  inde-  where  by Cramer's r u l e ,  we  and m  (1.32)  g'  (t) =  eim where  (h(t),B..v)  I j=i  ,  i j j  3.. are numbers. C l e a r l y , D follows that 1  g' e (L (0,T;V))*. It eim 5  m  5  is  an element o f  "  ( L (0,T;V))  T h i s completes the p r o o f  f o r each f i x e d  o f the lemma.  m  and  e  19  The fractional  following  derivatives  lemma g i v e s  an e s t i m a t e o f t h e  of solutions  to  (1.21).  Lemma 1.4:  Suppose satisfied,  a l l the hypotheses  and l e t u  be the s o l u t i o n o f  lemma 1.3, t h e n f o r any  U  -  3  4  K  )  K(y)  i s independent  o f lemma 1.3 a r e  U E  y  of  m  1 8 (0 , — - ~) ,  in  A * < O , T  ;  and  (1.21)  H )  S  K  of  we h a v e  W  e.  Proof:  To p r o v e used by  Ton  in  C (0,T )  J,  A  in  m  both  D  provided  2 Y  u  M,  £ m  y  lemma, we s h a l l  [ 1 2 ] . From  and t h e r e f o r e  1  and  this  u' em and  <  isin  (1.32),  by L  2  D ^ D J ^ m  ~«  * t follows  (1.33) (0,T ;H) . m -  u  from  T ( t ) = K. (y) I em J  em  follow  t h e argument  we s e e t h a t  g'.  and t h e p r o p e r t y Hence  of  for t < T m  are i n (1.18)  is  L (0,T ;H) 2  m  that  r  (1.35)  D  2 Y  t  u  1  ( t - s ) -  2  Y  D u (s)ds, s em  T,  where  K  l  (y)  depends  (D U 2  (1.36)  Y  t  v  =  em  o n l y on  (t),u  i  1  =  H  ;  (t - s ) -  T  Therefore  (t))„  em  K , (y) f  y.  2 Y  (D  n  Kj (y) ( E  1  + E  + E  2  U (s) ,u ( s ) ) ds s em em  + E j ,  3  where  JT  (t - s ) "  •r •'o T  i: Define  h : m  R -»- R  (1.37)  Then  h^(t)  h_. e L  2.p  (t - s ) -  ( t Q  (R)  -  s )  ( f (s) , u  2 Y  2  Y  ""  (j(eu  2 Y  Km  (t - s ) " ( M u .  m  ( s )  2 Y  c  e  em  (t)) *ds  £ m  v  (s)),  ' em u  u  ( t )  (S) , U  ' em  c m  (t)) *d v  )v*  £  d s  (t)) ds. H ;  u  by  =  i  t " Y  o < t < T,  2  0  m  t < o  8 i f o < y < — - —. 2  or  t » T,  m  F o r any f u n c t i o n 4  ^  defined  on  [o,T],  we s h a l l  d e n o t e by  g(t) g(t)  (1.38)  the function  o < t < T  = 0  Using  g  t < o  or  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 ,  t £ T.  we g e t  dt  fTm | u  <  e  m  ( t )  o  (1.39)  =  l  *  K m L  u e  m  f+°° ^ h ( t - s) |£(s) | * d s d t ~ °°  lvJ  ( t )  m  v  l [Vl lv*] ?  v  (0,T;V) ^ I J L  2  d t  (R) ' ' L  1  F  2  (0,T;V*)  Similarly,  m  L  |E |dt 2  £  e  J  ' O  (1.40)  f I o  T m  |u ( t ) |„[h * | j ( e u em 'VL m em  =  e  =  e ' j " "  1  — oo  1  (s))|„*ldt ' 'V J  K m ^ H ^ K J ^ d t  em'L (0,T;V) 5  1  m'L (R) 1  22  f 'o  |E,|dt  m  ( Jo  T m  1  |u \Jh em V I m 1  1  * | A U L*ldt em V J 1  a.4D < c f™ KJ[v(l5 i; |a|) +B  2 'o  £  C  +  |u  2  C  z  v  L  £m  dt  +  em v  I „ Ih I lu em L ( 0 , T ; V ) m L ~< (R) e m L ( 0 , T ; V ) 2  T  3  2  l em'L (0,T;V) ^rn^L (R)'"em'L (O T;V) u  2  1  2  f  and  f  lEjdt  m  o  ;  fT (1.42)  *  J  *  , U  *  It |h  I ^4-8-. m L (R)  from  2  P  v  lemma  m  l emlH[Vl u  y/T  |u  £ M  | 2 L  ( 0  d t  H  2  1  (R) '  , ;H)|h | i T  m  L  shown t h a t  M U  em L (.0  ( R )  I  l  |u  2  e m  1.3  that  I f ™ i=i o of m  i s independent  |E.|dt 1  and  e.  f T  ;H)  | -(p,T;H)• L  I L (R) 1  ^ e bounded by a c o n s t a n t , t h e r e f o r e  ;  which  eml ]  eiJL (0,T?H) ' ^ ' L  can be e a s i l y a  M u  A N C  ^  i t follows  a r e b o u n d e d by a c o n s t a n t  23  A c o m p u t a t i o n as i n lemma 5.1,  (1.43)  shows  Shinbrot  that  |DYu ( t ) | ' d t < sec(YTr) t em H J 'o o m  m  1  I  [ l l ] , p.150,  v  (D?Y t  u  em  (t) , U (t))„dt em 'H  Therefore  l t em D  U  'H  ( t )  d t  *  J^j^  sec(yrr)  lEjdt  (1.44) < K(y),  where T  m  is  k*(y)  independent o f  and  e,  It  follows  that  = T.  1.6. P r o o f o f t h e o r e m  Let unit b a l l (1.34), by  m  u  em  b e f i x e d . From t h e w e a k n e s s p a c t n e s s  i n a r e f l e x i v e Banach we  )  e  1.3.  s p a c e , and f r o m  can e x t r a c t subsequences  such  of  u e  m  of the  (1.22)  (again  and  denoted  that  u  em  u  weakly  £  in  t h e weak  L  (0,T; V)  * t o p o ogy o f  (1.45) eu  em DJU t em  eu  £  D^u^.  weakly  in  L (0,T;V),  weakly  in  L  s  2  (0,T;H) ,  and i n L°°(0,T;H),  24  as  m  -*• °°.  Applying such  theorem  1.2, we c a n e x t r a c t  further  subsequences  that  u  (1.46)  em  Iu v.  u  e  i  (°' ' ' t  ii H  c ' e  Because  l 2  (t) i n  ( t ) -»• u  cm ' ' era  n  M  h  a  n  d  f o r almost a l l  t  s a t i s f i e s t h e convergence  in  [0,T1  condition,  therefore  (1.47)  Mu ™ ( t )  -> Hu  em Since  (t) i n H  f o r almost a l l t i n  [0,T]  e  lu I », '< em L (0, T; H)  mapping o f  L  convergence  theorem  Mu  (0,T;H)  and  M  i s a bounded  linear  00  xnto  L (0,T;H),  t h e Lebesque  yields:  •+  em  C,  Mu  i n L (0,T;H) P  em  f o r 1 < p <°°.  :i.48) < /^  I l  - em u  ( t )  ' em^^H u  d t  * £ % u  e  Sets  m (1.49)  < f > (t) = o m  0  I 6.(t)v, j=i J  J  ,  <t> , u <t> > d t . £  H  25  where  0. e C (0,T) ,  0 . (T) = o.,.  1  (1.21),  T  J  c  T  f  J  em  m  f m, . (eJ(eu„ ) + A u „ , <J> ) *dt J em em m -'v T T  ,T  0  0  f  0  I t follows  0  P  I tis clear  (u ,<}> (o) ) H »ni' m  v  y  0  (  that  -  0  'n  ( M U ,<{> ) „ d t + em' m H  Q  feJ(eu ) + Au ,<(>). d t em e m m V  T m  M  Q  i (it.$ )„*dt f, o m V  -*•  I  H  dt +  T  (1.51)  - I  0  T T  f ( u ,<j>' ) d t + f J em m H J D  m -*• o o .  )  T (f,<j> ) * d t' X v J  m > m .  -  f  From  T  (u  (1.50)  as  . 2, ••• , m .  f  we h a v e  f m  provided  j = l  that  J  (Mu ,<}> m  Q  e  ) d t + (u ,<f) H "e' -m  0  0  (o) ) II  fT (u  )„dt  o  T  1  (u ,<j>' ) . d t a s m -»• . e mo ii 0 0  T  o  lim  i n f  |u ( o ) I ' + f L em H J  m  +  »H  fT i  feJ(eu  em  0  note  A(u,v)  =s  operator  proof  as i n  mapping  d t  ) + Au ,u 1 * d t em em'V  dt  eJ(ev)  that  ( u ' ,u ) e m em H  r < <vv* - r f  +  semi-monotone exactly  T  1  (1.52)  Let  we a l s o  + Au,  <  ,  ,  u  ' e'H u  E  then  L (o,T;V) 5  [13] , (lemma 1, p . 7 i ) ,  D  T  -  A(u,v)  i s a  into i t s dual. A gives:  26  A(u ,u ) = e J ( e u ) + Au — em em em em (1.53) -*• e J ( e u  weakly  ( L (0,T; V ) ) *  in  as  5  m  e  ) + Au  ~.  e  Thus  -f (u , 4 / ) „ d t + ( ( e J ( e u ) + Au ,4> ) * d t e' mo H e e: m/V* 'o 'o T T  r  (1.54)  for  a l l (J>  o f t h e form  m  (J> (T) = 0,  then  T 7  k  (1.49).  there exists  If  c j > e C (0,T;V) 1  a sequence  {l  }  and  o f t h e above  0 form  such  that  4>'  m„  in  L (0,T;V)  in  L  z  (1.55)  Therefore  f o r a l l 4> e C (0,T;V)  "J  (1.56)  =  In C„(0,T;V).  v;ith  l  (  I  u  e  ' *  (f  ,  f < r  ,  H  t  +  J  ) *dt - j  particular, Therefore  d  v  (1.56)  (  e  J  {  e  u  (0,T;V)  4> (T) = 0,  e  )  +  A  u  e  ' * )  v  *  d  t  ( M u , < M d t + (u ,<f>(o)) e  holds  H  for a l l < J > in  H  27  (1.57)  in  D'(0,T;V*).  (L (0,T;V))*,  Since  the r i g h t  - Mu  hand  + f  side of this  also i n  is in  (L (0,T;V))* 5  and  (L (0,T;V)j*. 5  The b o u n d s i n of  ) - Au  the e q u a l i t y holds  5  u' e  -eJ(eu  u'  e,  h e n c e we  u  (1.22)  o b t a i n by  -*• u  £  and  taking  weakly weak  (1.34)  are independent  subsequences  in  L (0,T;V) of  necessary:  and i n t h e  2  *topology  i f  1/° (0 ,T;H) ,  (1.58) eu  •*• o  Y  l  as  D  e -> o.  theorem  t  D u Y  u  Again,  weakly  in  L  weakly  in  L (0,T;H)  and  the Lebesque  theorem  1.2  s  (O T;V) r  2  convergence  yield:  Mu  -»• Mu  £  in  l/(0,T;H)  for  1 < p <  (1.59) f  On  the other  (Mu  T  ,u ) d t +  f  (Mu,u) d t .  T  hand,  lim i n f  ^ + Au ,u ) *dt £  e  v  fT (1.60)  =  lim i n f  G (f -  (f - eJ(eu  *o  ) - Mu  £  Mu,u) *dt. v  , u  G  ) *dt v  £ ^  28  It  i s also  clear  (1.57)  from  u'  that  +  Au  e weakly  (L (0,T;V))*.  in  assumption  on  A  any  u (o) £  <f> e C  = u(o)  f  = u .  H  °  =  any  with  It  i f we  (Au,cj>) *dt +  |  v  °  f 'o  1  assume  £  e  •  ( u ,4>(o-) ) H 0  v  (0,T;V)  R  °  ( f ,cj>) * d t +  remains  v;ith  <f> (T) =  t o show t h a t (1.57)  (L (0,T;V))*, 5  j"T J (1.63)  u  o.  (o) = u(o)  = u  for  any  (1.56),  (1.64)  f  J  <j> e C we  (T J.  Since  rT f  =  . o  yields  (u^ «|)) *dt + j °  that  (Mu,u) dt  e u  the  (Au,<}>) * d t  <})(T) = 0  |  T  <J> e C  f  ,<M*dt -  (u,cJ>') dt +  (1.62)  i t f o l l o w s from  Therefore  Q  -J  for  (Au  (O T;V)  1  u  that  f  (1.61)  for  Therefore  5  -*• f e  T o  1  (J(eu )  v  (f,+) *dt v  (0,T;V)  + Au ,(J)) *dt  e  ° T J  £  v  f  with  (Mu ,*) dt £  H  < J > (T) = o.  Comparing  this  with  have:  rT ( u £ ,<M *dt + j v  ( u , ( f ) ' ) * d t = - ( u ( o ) ,(()(o)) £  v  £  H  29  for  any  <j> e C  = o.  But i t i s c l e a r  (u' ,(f.) *dt + j ( u ,*•) * d t ' o  = - (u (o) , * ( o ) )  1  (0,T;V)  with  <j> (T)  that  J  (1.65)  e  o if  4> e C  l  v  (0,T;V)  and  (1.66)  This  £  <j> (T) = o.  (  shows t h a t  u  u ,<Mo)) 0  (o) = u e  From set is  of  (1.57),  H  Therefore  =  (u (o) e  f o r any  e.  we  see t h a t  u^  Since  5  have by  (1.67)  H  e  ,ct>(o)) . H  o  (L (0,T;V))*.  c o m p a c t , we  v  the i n j e c t i o n of  taking  u  lies  (o)  i n a bounded V  into  V*  subsequences i f necessary:  u(o)  in  V*.  e  But  u (o) £  completes  =  u  Q  the proof  f o r any  e,  o f theorem  therefore 1.3.  u(o) = u  . This  30  CHAPTER I I  EVOLUTION EQUATIONS IN  Let  H  and  and  V  a separable  Let  W  and  V  be  W  be  two  reflexive dense i n  the n a t u r a l  that  of  V  into  H  are  Let  F,  Y,  and  X  be  L (0,T;V)  and  r  2  £ p  «? r  <  00  .  value problem Y  and  L  P  (i)  (ii)  and  in  H  and  (0,T;H)  I n t h i s c h a p t e r , we  W  into  compact  H,  V  and  respectively.  LP(0,T:V),  respectively,  where  s h a l l c o n s i d e r the  f o r n o n - l i n e a r o p e r a t o r s mapping following  v c  respectively,  the Banach spaces 00  spaces,  W c  with  i n j e c t i o n of  continuous  (0,T;V) D L  s a t i s f y i n g the  Assumption  separable Hilbert  Banach space V  f b r e o v e r , suppose  BANACH SPACES.  X^Y  initial into  assumption.  II  A  i_s c o n t i n u o u s  of  Y  If —  u  n  weakly in  from  finite  dimensional  into  t h e weak t o p o l o g y  -»• u  weakly i n —*- —  in  F" ,  Y*, u(o)  £  Au in  X,  -»• h  n  H  u  of  n  weakly  Y  in — in  and  (h + u' , u ) * + p  |u(o) |  2  subspaces  •k  .  Y, ' Y*  u' -*- u' n with  h +  u'  31  then (iii)  (iv)  Au = h .  A  maps b o u n d e d  sets  of  X  Y  i n t o bounded s e t s  of  Y  There e x i s t s  constant  (Au(t) u(t))  c > o  *  f  £  In  a l l u  this  in  Y  such  that  P  and f o r a l m o s t a l l  c h a p t e r , we  sets of  c|u(t)| v  w  for  and b o u n d e d  shall  establish  t  the  i_n[o,T]  following  theorem.  Theorem  2.1;  Let satisfying is  each  with  u'  b_e an o p e r a t o r m a p p i n g  a s s u m p t i o n I I . Suppose  of l o c a l  for  A  t y p e and s a t i s f i e s  f in  in Y*  F  and such  f  u  0  Me  X(JY  into  Y*  L (L°°(0,T;H) ,L°° (0,T;H)) ,  the convergence c o n d i t i o n . in  H,  there exists  that  u' + Au + Mu  (2.1) u(o)  =  and  f  u  in  Then X  32  Remark  2.1;  Theorem  2.1  generalizes  semi-monotone o p e r a t o r s  a result  considered  by  of A r t o l a  Browder  [ l ] . The  [3], Lions  [9],  •k  and  a l l weakly continuous  satisfy with  ( i ) and  M = 0 was  2.2.  no  parts  Instead,  F  assumption  e s t a b l i s h e d by  Ton  into  F  I I . The  theorem  [14].  Because of  less stringent conditions  on  obtain  fractional  i n Chapter  we  basis,  an  Lemma  2.1;  W,  idea  IXj  <  and |X | 2  the  shall  the  e x i s t s an  a_ s e q u e n c e o f <  by  in  Lions  and  for  X ,  numbers  w  l  such  3  W  in  =  j =  X , 2  that  2,  of  we  can I.  a special  [8].  Xj(Wj,v)  1,  A,  m a k i n g use  orthonormal b a s i s  (Wj,v)  v  d e r i v a t i v e s as  t h e o r e m by  j X J < • • • -»• °°  (2.2)  all  prove  introduced  There  for  ( i i ) of  from  Special basis.  longer  of  operators  3,  H  w  , w,  X , ••• 3  2  w, 3  with  ** *  33  Proof:  For  a l l  u  and v  (2.3)  Then  in  b(u,v)  b : W x W -*• R  W,  =  (u,v). w  is a bilinear  (2.4)  |b(u,v)|  $  let  form on  |u| |v| w  W x W  such  that  u, v e W  w  and  Let which  b(u,u) > I Iw  ue  be  u  u  (2.5)  N  <=. W  the l i n e a r  form  the s e t o f a l l elements  (2.6)  is  c o n t i n u o u s on  since to  v  W  i s dense  W  in  H,  this  representation  linear  form o f  theorem,  for  b(u,v)  w i t h the t o p o l o g y i n d u c e d by  a unique continuous l i n e a r  Reisz's  •>  W.  there  H.  f o r m c a n be H.  Then  exteneed  T h e r e f o r e by  exists  h  in  H  such  that  (2.7)  b(u,v)  (h#v) , H  v  ew  a  H.  3 : 4  Hence we and  can d e f i n e  for a l l  a l i n e a r map  u e N  and  (2.8)  v e  one  onto  H,  Therefore H.  A  D(A) 1  Furthermore,  =  1.2,  with  D(A)  i s dense  in  N-,.  H  of Lions  H  W  c: H  for a l l  u  in  |u|^  and  A  [7], is  A  i s one-  symmetric.  i s a mapping  from  H  to  N,  =  b(u,u)  =  (Au,u)  R  " Uu| |u| H  £  clAul H  T h e r e f o r e we  =  (Au,v) .  p.11  D(A) c:  :H  H  W,  b(.u,v)  By p r o p o s i t i o n  A:W  H  lul . W  have  O (2.9)  This  ||u|  shovjs t h a t  But  the i n j e c t i o n  of  H  into  well-known  A""  1  of  itself, theorem  on  w  $  |Au| , H  i s c o n t i n u o u s from W A"  into 1  H  i s also  H  into  i s compact, compact.  compact s y m m e t r i c  D(A)  cz  w.  s o as a m a p p i n g  I t follows  operators that  from there  a  35  exist  a n d ortho-normal X  numbers •*•  X,  l t  basis  X,  2  u ,  •••  3  u ,  t  u , *•*  2  such  3  that  |X |  _1  i n H and  > \X \'  X  Z  t  >  \X \'  1  3  and  (2.10)  A" u  = l „  1  n  n  n Note t h a t D(A)  cz w.  u = X A n n  _  1  u n  i s i n fact  an e l e m e n t o f  So we h a v e  (  V  v ,  w  =  b  < v  v  )  (2.11)  for  a l l  v e W  (u  a l l  v e V7  orthogonal an  A-priori  1  0  t nJ  that  in W u  on  estimates  W  with u o  =  X (u ,v)  n €  2,  in W  basis i n W  Since (u  v)„  and n = l ,  t o each o t h e r  orthonormal  2.3.  (Au v)  H  and t h e r e f o r e  (2.12)  for  -  n  3,  n  ••• .  i n H.  in  L  u^  i s also  (2.12).  approximate  i s d e n s e i n H, u on  Hence  a n d s o b y n o r m a l i z i n g , we g e t  satisfying  o f the  H  fw,. i '  w,.  2 '  solutions.  there exists ••• , w 1 ' n J  a sequence and such  36  Lemma  2.2:  Let a basis of given  W  0  u (t) m  n  w ,  2.1. T h e n f o r e a c h  such  A  and  f  iri F  be  w  w.  t  as i n lemma 2.1. L e t  i n theorem  exists  be_ as above and  n  H  be_ o p e r a t o r s  *  there  that  u  m  <t)  I  i=i  m 111  9.  im  (t)w. l  (2.13) 1 £ j $ m, u  m  (o)  =  u  om  oreover,  ( 2  -  oelo^T)  1 4 )  where  K  |  U  ™  (  0  )  |  H  i s _ i n d e p e n d e n t o f m.  1 ^ i $ m, g ! 3  im  (t) e L  r  (0,T) '  +  F o r each and hence  «•  fixed  K  -  m, and  u' e Y . m  Proof;  The lemma  1.3.  proof  of this  lemma i s i d e n t i c a l  to that of  37  2.4 P r o o f o f t h e o r e m 2.1,  From lemma 2.2, we know t h a t such  there  exists  u. m  that  (u  m  +  A u  m  +  Mu  =  m' w jV  <f'wjV  <j  1  < m  (2.15) u^o)  Let  P  m  defined  be t h e p r o j e c t i o n o f  P h m  The r e s t r i c t i o n We  of  shall  m .Y  =  P  into  [w  to  m  (h,w.)„w. 3 H 3  W  1 (  w,  ••• , w ]  2  P  i s independent of  2  l m lw p  h  =  (  j=i m I  3=1  L(W,W)  that  *  m.  C  *'  If  m  I  h e H.  i s an e l e m e n t o f  prove  ' m'L(W,W)  Ci  ,  jhi  first  (2.17)  (2.18)  H  om  by  (2.16)  where  = u  h e W,  then  m  I  (h,Wj) Wj, H  (h,Wj) Wj) H  j=i , 2 1 1 2 |(h,w.) | |wj W 2  H  1  w  38  Using  lemma 2.1, we  have  (2 19)  3  "  1  j  1  «TrFJ. But  w, , w  , w, ,  **  0  by  l m lw p  20  the Bessel's  equal  to  c,  h  For  any  w  >  IN in  W*. W,  This proves  This we  m |  =  w  W,  so  m  (2.17)  with  |X |  2  A  m.  * *  p  * w  * w  d e f i n e d by  (w*,w.)^*w.  4  of  P  to  m  W*.  have  w )  'jli (w*,w.) *(w ,w) *  =  w  *  (2.22)  l  i s an e x t e n s i o n  l m *' W*l ( P  i »  | , | w  M w  ]7^jT  . t h e mapping  P*w*  w*  w  which i s independent o f  (2.21)  all  *  inequality.  > Now c o n s i d e r  for  ' i «  i s an o r t h o n o r m a l b a s i s o f  < ' > 2  l ( h  m  = I ( w ,.| W j ( w , w ) * ) * | j  'W*  1  *  m  1  l *l *l w  W  p m  w  w  W  lL(W,W)  l lw w  j  w  39-  Therefore  (2.23)  |Pm *l *  < l m l ( W , W ) |w*|  w  P  W  L  w  and s o  | P * L ,„* „ * . * | P L , , « m'L(W*,W") m'L(W,W) |  (2.24)  v  From  MM  1  1  v  X  |  '  2  ( 2 . 1 5 ) , we h a v e  "m  =  u' m  =  (2.25)  Since from  Au , irr (2.24)  of  Y .  m  a r e i n a bounded  and  (2.25)  that  -  m  Mu m  it  set  P * f - P*Au  P * M U  m  set of  u^  .  Y*.  arealso  i t follows  i n a bounded  t  So b y t a k i n g  i n t o account  ther e s u l t o f  lemma 2.2, we g e t :  < - > 2  K  l ml *  26  u  +  y  i sindependent o f  Using such  that:  KI -(0,T,H) L  +  K ' F  K  K  '  m.  (2.26),  we c a n e x t r a c t  subsequences o f  u  m  40  U'  m  (2.27)  -»•  u'  weakly  in  1  -»•  u  weakly  in  F  ( weak  *topology  weakly  Since  the i n j e c t i o n  from  theorem  such  that  (2.28)  of  V  into  ->•  1  (2.29)  Applying  Mu m  u  in  L  u(t) i n  H  -*• Mu m  t h e Lebesque  Ku  L°°(0 T;H) /  in Y .  H  i s compact,  in  2  ( 0 , T ; H )  i t follows  m -> ~  of  u^  and  for.almost  a l l  t  in  [O,T].  f o r almost a l l  t  in  [o,T] .  M,  H  c o n v e r g e n c e t h e o r e m , we h a v e  -»• Mu  in  L (0,T;H)  i  (2.15),  S  (2.30)  letting  of  1.1 t h a t we c a n e x t r a c t s u b s e q u e n c e s  Hence b y t h e a s s u m p t i o n on  So  and i n t h e  n  we g e t  f o r any  s > 1,  40a  (2.31)  ( u ' + h + Mu,w.)__* =  for  j = 1, 2, 3, ••• .  (2.32)  D'(0,T;W).  F*  and  m the  -*- u  3  i s a bt.jis o f  u» + h + Mu  in  u  Wj  u  But  f e F*,  + h e F*.  1  weakly i n  o t h e r hand, '  F,  =  u  Lim m  (2.15),  we  sup |,  u(o) H.  in  i*  and  ft W .  On  Therefore  h = Au,  have  |*  ( o )  a l s o holds i n  0  remains t o prove t h a t  From  therefore  weakly i n  u (o) m u„ in o  u (o) = u m m  W,  (2.32)  1  we h a v e  v  f  u -*• u' m  u(o)  It  =  hence  Since  (2.33)  (f,w.) *  w  3  +  ( u  .  +  (  F  Au^u^*}  oo  .  linkup  <  I OIH  {  |  (  F  U  O  M  ,  2 +  .  MU^UJY*}  (2.34) U  +  |u(o)|^ +  It  follows  ""  M  U  '  U  )  F  •  (u' + h , u ) *  f r o m t h e a s s u m p t i o n on  theorem i s p r o v e d .  *  p  A  .  that  Au - h  and t h e  41  CHAPTER I I I  PERIODIC SOLUTIONS OF EVOLUTION EQUATIONS  3.1. P e r i o d i c  solutions  Let reflexive Suppose  H  Banach  f o r monotone o p e r a t o r s .  be a H i l b e r t space w i t h  the i n j e c t i o n  of  space  V c= H  V  into  H  and  V  a separable  and  V  dense i n  i s c o m p a c t . We  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 m a p p i n g into  li  k  (0,T;V )  satisfying  the f o l l o w i n g  H. shall  L (0,T;V) P  assumption.  Assumption I I I :  (i)  A  maps b o u n d e d  sets (ii)  A  of  sets  i_s c o n t i n u o u s f r o m  (Au(t) , u ( t ) ) L (0,T;V) p  (iv)  v #  f  p  lines  in  L  into  bounded  (0,T; V)  t o the  p  £  v  p  L '(0,T;V*). c|u(t)|  P  fora l l u  a_t a l m o s t a l l p o i n t s  (Au - A v , u - v ) LP(0 T;V)  L (0,T;V)  LP'(0,T;V*).  weak t o p o l o g y o f (iii)  of  *^ 0  fora l l  a t almost a l l points  t u  in and  t  in  in [0,T] . vi l l [0,T],  42 The m a i n r e s u l t  of this chapter  i s the following  theorem.  3.1;  Theorem  Let L  r  (0,T;V )  Suppose local for  A  b e an o p e r a t o r m a p p i n g  satisfying  i s an o p e r a t o r  type,  and s a t i s f i e s  f  iP  in  in  with  P  u  1  p >  2.  L(L°°(0,T;H) , l T ( 0 , T ; H ) ) ,  of  t h e c o n v e r g e n c e c o n d i t i o n . Then  (0,T;V*),  L (0,T;V) DL^tOjTfH)  into  p  assumption I I I with  M  any  L (0,T;V)  there exists u'  in  + Au + Mu  u  in  L '(0,T;V*) P  =  such  that  f  (3.1) u(o)  3.2.  Approximate  u(T)  solutions.  As b e f o r e , we  shall  first  obtain a-priori  o f t h e a p p r o x i m a t e s o l u t i o n s . T h i s i s done  estimates  i n the following  lemmas.  Lemma  3.1:  Let  w  }  , w, 2  w  be a b a s i s o f  V,  m  a  fixed  43  integer be  and  u  e  0  as i n t h e o r e m  |w  , w  , *** , w  3.1. T h e n  there  | .  Let  exists  11  A,  a_ u n i q u e  and  f  function  m " m ^ J  = i  l  g 1  i m  (  t  )  w  i  s  u  c  h  t  (u'.w.) * + m' j V *  h  a  t  (Au ,V7.) * + m 3 V* f  (3.2)  =  (Mu ,w.) m' j H  (f,w.) * j V  1 £ j < m  and  (3.3)  Ki  depends  u' e L m  VL~(0,T;H)  p  on  u  (0,T;V*)  such t h a t  i f  +  1  ^m'LP(0,T;V)  <  and i s i n d e p e n d e n t o f  0  and t h e r e |u | £ 0  (3.4)  H  R,  exists  j  m.  R > o,  *  Moreover,  independent o f  then  |u (T)| m  K  H  <  R.  Proof:  The (3.2)  and  proof  (3.3)  Suppose  o f the e x i s t e n c e  of  i s s i m i l a r to that  that both  u  and  u  m  satisfying  o f lemma  v  1.3.  s a t i s f i e s the  44  (3.2).  system  L e t w = u - v,  (w*,w) * +  then  w  satisfies  (Au - A v , u - v) * = - (Hw,w)  V  V  n  (3.5) w (o)  Integrating  from  o  1 ( o ? t ) l <°> 2 ae. . . w  to  IH  I (" ' >Hl w  w  p t  >  |  w  remains  i f  that  ,  |  H  | u | „ < R,  up f r o m  to  we g e t :  2K  (  then  0  summing  t  >H  < |~| U l ,uI H  the  j = 1  +  c  "M!  +  i»« >i>o  IH < °'  to  and  R > 0,  there e x i s t s  Multiplying  t,  o  w(a) = 0 a l m o s t e v e r y w h e r e ,  t o show  that  (  1.1.that  7 0^0%  Therefore  a o  Q  i ^  f r o m lemma  (3.7)  (3.8)  « J , 5  i t follows  such  t , we g e t  fT  (3.6)  So  0.  =  u  i s unique. I t  independent o f  m,  |u ( T ) L £ R.  j-*th e q u a t i o n j = m,  of  (3.2)  and i n t e g r a t i n g  by  g j(t), m  0  from  ^ ^ ( a J l ^ d a 'o  °  + {  [}f^)lv*l m u  ( 0 )  lv  +  l<Mu  ffi  (a),u (o)> |]da. m  H  45  By Y o u n g ' s on  c  inequality,  and  p  only,  we  can  such  find  a number  depending  2  that  | f <o) L J - u J o ) |„ ^ K | f 'V*' m 'V '  (3.9)  K ,  2  (a)\ll ' v*  f  +  *  | u j a ) |? . ' m 'V t  Therefore  <  f K3 The  = K  ||u |  |f(a)| *da  2  a  £  ^  v  )  injection  a constant  y TJ 2  c  0  i s independent of  of  V  > 0  fact  and  H  into such  |u|  this  +  and  (3.11)  Using  t  |H  K(a)  P  FC  |u (a)|*da+K m  P'  T  natural  exists  + i  2  0  H  that  <  for a l l  0  K^HH+T /  u  in  v  K  0  4  ( O )  we  have  l^o  o  rt  <•  2  |u t a ) | d a m V 2  l ol u  + H  y  T C  o  J  1  |u„| . H  i s c o n t i n u o u s , so t h e r e  (3.10),  (3.12)  and  c |u| .  rearranging  4* o U o ^ t )  m  1  + K 3  .  V,  3 •  46  Since  p > 2,  we  can use t h e H o l d e r ' s  Young's i n e q u a l i t y  2  y Tc M  t o show  2  and t h e  that  2/p  (a)| da 2  |u  0  inequality  $ K  (a)| da  |u  P  (3.13)  for y,  some c o n s t a n t s  and  K ,  w h i c h depend o n l y on  5  T, c , c , p , and T, and a r e i n d e p e n d e n t o f  m  c  Substitute  this  into  (3.12)  and p u t  t = T,  and  + K  g  = K  < - > 3  15  On t h e o t h e r  3  + K .  In p a r t i c u l a r ,  g  0e(S T>!« < H O,,  P  M  <  /  2  l  u  o l  +  2 / K  H  e«  hand, fT  T | u ( 0 ) | d O S K, m  H  |u (a)|Pda n  o  o  r fr  T  < K.  1  |u  U„  ( a )V d Ia m  P  (3.16)  + 2K.  < K.  < XO|I OIH K  U  +  u 0  ! H  then  (3.14)  where  l  (  2  K  e  )  P  V  47-  for  some c o n s t a n t s  of  m  c  such  2  and  |u  | . H  ( i = 7 t o 10) Therefore  which  there e x i s t  are  independent c  numbers  and  x  that •T |u (a) | da  *  — c j u j ^ +  to  we  have  (3.17)  m  R  J o  for  all  m. Returning  (3.18)  <  <  Because  p>2,  (3.2),  jf(t)| ,|u (t) | v  K  m  ,|f(t)|lj;  s o i f  +  x>0,  +  v  | (Hu (t),u (t)) | m  f|„ (t)||J B  then  x  z  m  H  K^ <t),U  +  m  < x  P  +1.  r a  (t)) |.. H  Therefore  0  « I2< Mdu t ( mt ) | +H £ . | u m( t ) | V * 2I ld -t l un ( t )H| 2l m\ u ( tv) | 2+ £ 2  2  2  (3.19)  2  P  +  $  K 2  Multiplication  by  | f (t)| * + - + ' ' i * 2  2exp(—), c o  v  and  1 K  | (Mu m  ( t ) ,u ' m  integration  H  (t) )  from  o  to  T  48  gives  •o (3.20)  f  <  where  K  (3.17)  K  M  +  2«cp@y  t e  i s independent and  (3.21)  (3.20),  we  «P  of  T )  T  |u (t)| f n  m  |u (t)| dt  H  and  n  H  |u.|„. U s i n g  (  (3.15)  have  exp(p|u (T)|-  <  m  |u,|» + K  i ;  |u l o  2 +  H  K  ]  >  0  for  some c o n s t a n t s  K  and  K  12  (3.22)  2e  Clearly exists  e > 0  *  K  Set:  exp(-S|)  e cT'x exp[—j) > 1 +  and  constant  . 13  such  - 1.  e.  Because  p > 2,  there  that  1 <•  (3.23)  1+2' |u I $  K  2  eIu I  2  + K  Therefore  (3.24)  exp(^|)|u c:'  (T)| m  H  2  *  ( l + e ) | u | 0  -^-A 1 O  can p u t :  13  11  C  H  2  + K  , is  > 1,  s o we  49  K exp(-—) 1 - (1 + e)exp(- y) 1 5  (3.25) If  R  2  _  £  |u | S R, then it follows from H 0  IVT)!'  (3.24)  « e x p ( - c f ) (1 + e) |uJ (1 + e)K  (3.26)  that H  + K  exp(-^  c ' 1 - (1 4 e)exp 2  =  1 5  -K  1  s  R. z  This completes the proof of the lemma. Remark 3.1: In the proof of the above lemma, it is essential that p > 2, in order to obtain (3.23) and (3.24). Hence the method above only proves the lemma in the case when p>2. If p=2, we have to modify the operator the operator A. We shall do this in section 4.  50  Lemma 3.2:  Let •M  and  f  f  w  ,  w  , •••  be a b a s i s  b e t h e same as i n t h e o r e m  number d e f i n e d  i n lemma 3.1.  Then  in  V.  3.1. S u p p o s e  f o r each g i v e n  Let  A,  R  i_s t h e  m,  there  m exists  a  function —  u„(t) m  I  =  T g. ( t ) w . . , ^im l  such  L  that  i= i  (u',w.) * + m } V  (Au ,w.) m j V  (Mu , W . ) m ;j H  * +  (f,w,) *,  1 « j « m  v  (3.27) u  m  |u (0)| m  u (T) m  (0) = «  H  R.  Proof;  Since g i(t) m  the  the s o l u t i o n o f  are continuous  [0,T] f o r  on  i s unique, i = 1,  u  •>  0  T (u ) m 0  u  is well-defined  0  f o r any J  solutions of  (3.2)  u  n  in  W.  =  •••  m,  u (T) m u  If  0  with  u  (0) = u „ m lemma t h a t 0  we  2,  and s i n c e  mapping  (3.28)  two  (3.2)  can deduce from G r o n w a l l ' s  m  and v are m m a n d v (0) = v . , m 0  51  (3.29)  where  |u (T) - v ( T ) | m  K  m  i s independent  of  |u - v J  H  u  o  and  v .  o  T  m  i s continuous on From  W  H  ,  T h i s shows t h a t  o  w i t h the topology induced by  (3.26), '  we see t h a t  T  m  H..  maps the c l o s e d  H-ball  (3.30)  R ^ H  m  = {w: w =  m ^  a .w . , |w|  R  , < R}  i n t o i t s e l f . Hence by the Schauder-Tychonov f i x e d p o i n t theorem, t h e r e e x i s t s  (3.31)  u„  in  u  =  u m  such  m  that  T (u ) . m om  om Therefore there e x i s t s  IL  satisfying  (3.27).  3.3. Proof o f theorem 3.1.  Let  W  be a s e p a r a b l e H i l h e r t space w i t h  Suppose the i n j e c t i o n o f i s dense i n basis  H.  V  i s continuous  of  W  and numbers  3  2  2  and  | <•••-•«> 3  such  that  X ,  X , 1  w i t h .|X I < | X ] < | X 1  into  W  By lemma 2.1, there e x i s t s an orthonormal  w , w , w , ••• 1  W  W c V c: H.  2  X , ••• 3  52  (3.32)  for  (w ., v)  a l l v  in By J  =  X . (w . , v) „  W.  lemma 3.2, t h e r e e x i s t s '  (u',w.). m' 3 V  +  T  (Au ,w. V m' 3 V  (3.33)  |u ( 0 ) I £ R, m H  (3.32)  and  *34)  into  2.1, we  U  K H  =  ( o )  +  (t)  (Mu  c a n show  +  i s c o m p a c t , we  ->  u  m.  m  tT  ,  1 £ j < m  ( T )  (0 ,T; V)  Since  can e x t r a c t  m.  Using  +  K 'L"  to  (0 ,T;H)  the i n j e c t i o n  in  L  u  weakly  in  L  P  (0,T;V)  u„  weakly  in  H,  +  x  weakly  in  L  and i n t h e  L°°(0,T;H),  (0,T;V*)  K  V  that:  * (0,T;W ) ,  topology of  ->-  K  of  subsequences such  weakly  weak  Au  ,w.) m' 3 H  that  > m ' I? U  that  an a r g u m e n t s i m i l a r  u'  (3.35)  u^(0)  m  such  i s independent of  i s independent of  u^  u  and a p p l y i n g  l mllP'(0,T;W*)  where  m  which  (3.33),  t h a t o f theorem  (3  m  (f,Wj)  u  with  u  '  53  Using  t h e same argument  Mu (3.36)  -*• Mu m  I  as i n t h e p r e v i o u s  c h a p t e r s , we  in  1 < s < °°,  L' (0,T;H)  r  f  But  u  ^(T)  (Mu,u) dt. H  ( t ) •*• u ( t )  ^(0)  =  in  W*  u(T)  l e t m •* °°,  u'  +  u  =  t h e n we  X  +  yield:  for a l l t  converges t o  (3.38)  Hence i f we  T  'n  Nov; lemma 1.2 and t h e o r e m  (3.37)  i f  have  0  weakly  in  in  [0,T],  H,  therefore  u(0)  have  Mu  =  f  (3.39) u(0)  u'  is in  L  u'  i s i n fact  p  (0,T;W ) .  p' in  L  But since  f - Mu - x e ^  (0,T;V ) ,  * (0,T;V ) .  X = Au. From  = u(T).  (3.33),  we g e t :  I t r e m a i n s t o show  that  54 T  lim  sup  m  =  l i m sup m  (Au  m  rT  ,u ) * d t m V T (f,u  f f l  (3.40)  Let A  $  ) *dt  (Mu  -  v  m  ,u ) , d t m H T  rT  be  =  f  (f,u) * d t .-  =  {  (X,u }  *  V  an a r b i t r a r y  d t  f  (Mu,u). d t  '  element i n  L (0,T;V), p  then  since  i s monotone,  0  < lim^sup  (3.41)  (x - A  < J  Now  put  <J> = u + \ty,  element o f  X •-»- 0,  (3.43)  Since  where  L (0,T;V). p  ( A U m  _  K  U  m  .  d  t  o  *) *dt.  X > 0,  Again,  v  (3.41),  we  (x - A ( u + Xi|0  i s an  arbitrary  have  *dt.  then  0 <  I  i s an a r b i t r a r y  -  Au,  dt.  element o f  that  (3.44)  A  ,u -  From  0 < j  (3.42)  If  T J  Au  =  X.  L  l  (0,T;V), i t f o l l o w s  55  3•4".  The c a s e when p = 2.  In We  shall  Theorem  this  prove  L (0,T;V*)  2  t h e c a s e when  A  b e an o p e r a t o r m a p p i n g  f o r any g i v e n  (0,T;V) O L  0 0  f  (0,T;H)  I  (3.45)  assumption in  L  with  u'  +  2  u  III.  (0,T;V*) , in  1  Au  +  Au  L (0,T;V) 2  Let  M  there  exists such  2  +  Mu  =  into  be_ a s b e f o r e .  L (0,T;V*)  u in. that  f  |  (  where  p = 2.  the f o l l o w i n g theorem.  satisfying  2  L  consider  3.2:  Let  Then  s e c t i o n , we  X  u(0)  i s a sufficiently  large positive  = u(T),  number.  Proof:  Since to  (3.19).  X  (3.20)  i s l a r g e , the proof becomes:  exp(^)|u (T)| 'c / m 2  2  (3.46)  o f lemma i s v a l i d  | u j + Xf II J 2  0  c  |u m  (t)| dt H 2  up  56  Let  e  be the number d e f i n e d by  (3.22),  then  (3.47)  f o r some constant  K, .  Since  X  i s large,  (3.24)  can  1 6  be o b t a i n e d . The r e s t o f the argument i n theorem 3.1 i s still  v a l i d . T h e r e f o r e the theorem i s proved.  Remark 3.2:  Theorem 3.2 has been proved by G a u l t i e r A  i s linear,  assumed t h a t  X -  0  and  c  5  when  = 1. In t h a t paper, i t i s  57  CHAPTER I V  APPLICATIONS  In t h i s abstract be  c h a p t e r , we g i v e  theorems proved  i n the previous  a b o u n d e d open s u b s e t  The  points of  ft  Set  3 D.= — — i 3x .  for  some a p p l i c a t i o n s o f t h e  of  R  with  n  w i l l be denoted by  chapters.  L e t ft  a smooth b o u n d a r y x =  (x  j f  x , 2  j = 1, 2, ••• , n; and f o r e a c h  3ft.  ••• , x  n  ).  n-tuple  3  (a  , a ,  , a )  o f non-negative  a.  n  (4.1)  D  =  a  i n t e g e r s , we  n D.3 j=l  and  |a|  n £  =  j=l  D  Let the Banach  be a p o s i t i v e  a. . 3  i n t e g e r . Denote by  W  ' (ft)  space  w ' ( f t ) = {u: D u  (4.2)  with  k  write  P  e L (ft),  a  |a| < k}  P  t h e norm  (4.3)  |u| k,p  =  { I |D u| } , ^ LMfi) a  P  P  p  1 < p <  M  k,P W  Q  (ft) i s t h e c l o s u r e o f  C (ft), Q  the family of a l l i n f i n t e l y  58  differentiable  functions with  respect  norm  to  the  A general  m  be  consider  defined having  on the  r  (i)  ft  real  X  less  ft,  with  [o,T~J  X R  integer  X R  A  equal  N  |a|  be  to  _(x,t,A,n) ap for  N  and  m and  the  number  1.  We  A  (x,t,A,n) a  £ m,  J 31  £  m,  properties:  f o r almost  a l l  (x,t)  e ft X  -*• A g ( x , t , A , n )  and  a  continuous  operator.  than or  functions  following  (A,n)  parabolic  a positive  of d e r i v a t i v e s of order shall  in  ,p  non-linear  Let  support  | • |, K  4.1.  compact  on  R  X R  N  [0,T], (A,n)  the  functions  •*• A ( x , t , A , n ) a  .  (4.4) ii)  for a l l  (A,n)  e R  A g(x t,A,n)  and  m e a s u r a b l e on  ft  a  /  X R  ,  the  (x,t) X  functions  A^ (x , t , A ,n) a  (x,t) are  [0,T] .  Set: D u k  6u  A  =  {D u:  {Du,  ( x , t , A ,n) : ( x , t ) a8  |3| =  3  D  2  U , •••  •*• A aB  k}  , D ~ u} in  1  ( x , t , u ( x , t ) ,<Su(x,t) )  are  59  Suppose t h e r e  exists  a constant  C  such  that  1 A ^ (x,t,X,n) *  (i)  A  (ii)  (x,t,x,n) * c a  V = W^' (fi)  Let It  follow  from The  forward  JX | )  |x|)(l +  |X| + | n | ) .  I  and  2  (l+  (1 +  W = W^'' (ft)  with  2  the Sobolev  imbedding  proof o f the following  and t h e r e f o r e  w i l l be  m'  theorem  > m + 1 + that  proposition  K  W cz C (ft~) m  i s straight-  omitted.  P r o p o s i t i o n 4.1t  Suppose satisfy X=  L  2  (4,4)  A ^(x,t,X,n)  that  and  (4,5).  (0 , T ; V )flL°° (0 ,T;H)  •a  (t;u,v)  =  1  Set  and  A (x,t,X,n)  and  A  H = L  a  (P.) ,  F =  Y = L"(0,T;W).  Let  1 1 A laKm \r*<m IBKm \Pi<m 'ft Jo  2  e  a  a  I  Then  A  sets  of  i s a mapping X  a  P  (t;u,v) =  2  (4.6)  2  (x,t,u,6u)D uD vdx,  s  I A (x,t,u,6u)D vdx, Q |a|«m rT a. (u,v) = f a.(t;u,v)dt i = 1, 2, 0 2 2 a(u,v) = a. (u,v) = £ (A.u,v) * = i=l i=l * a  L (0,t:v),  v  1  from  and b o u n d e d s e t s  1  XU Y of  into Y  into  Y  . Y*.  (AU,V)__*. Y  I t maps b o u n d e d  60  We  Theorem  shall  establish  t h e f o l l o w i n g theorem,  4.1:  Let into  now  Y*  A  be _a n o n " l i n e a r o p e r a t o r m a p p i n g  as i n P r o p o s i t i o n 4.1,  I  ( i )  I l A g(x,t,u,6u)DBuD<*udx  jal^m |3UmJ  (4.7)  Suppose  further  that  > c I f  |D u| dx,  ft  I  (ii)  [  XyY  a  2  | a k n r ft  A (x,t,u,6u)D udx  > 0.  a  a  latenr ft Let  B e L  and  u  such  that  (0,T)  in  0  H,  and  T ^ > 0.  there e x i s t s  Then u  for a l l  iri X  f  with  u ' ( x , t ) + A u ( x , t ) + B (t)u(x,t-T„)  =  u'  F  _in  Y  i  f(x,t)  u (x, 0) = u (4.8)  in  (x)  x e ft,  < u(x,t)  = 0  u(x,t) = 0  t  >  (x, t ) e3ftX[0,T] .  Proof  To p r o v e t h i s with:  t h e o r e m , we  shall  apply  0,  theorem  2.1  61  'B(t)u(t - T ) (4.9)  Mu(t)  i f  t e  [T  , T ]  if  t e  [0, T  = ) .  o  H  i s clearly  condition. (iv)  of local  I t remains  t y p e and s a t i s f i e s t o check  a r e c l e a r , and p a r t  Therefore  If then s i n c e  u  -*• u  n  sequence  of  and h e n c e  u  D u  t o show  weakly  the i n j e c t i o n  compact, i t f o l l o w s  assumption  ( i i i ) follows  i t i s sufficient  in  of  from  -> D u  a  n  Clearly,  X,  in  L  i f  u eX,  2  4.1.  weakly  in  W ~ ' (ft) m  1  2  Y*,  is  0  1.1 t h a t  (ft X  ( i ) and  ( i i ) i s true.  into  -*• u  a  part  u* -> u' n  2  theorem  I I . Parts  from p r o p o s i t i o n  that  W™' ^) o  such t h a t  n  the convergence  in  L  there 2  [0,T])  v e Y,  exists  [0 , T; W*"^  ' (ft) ) ,  m  for a l l  then  A  sub-  2  |a|  m-1.  (x,t,u,6u)D v a  a3  lies  in  L (ft  from  theorem  X [0,T])  2  2.1, p » 2 2  A (x,t,u ,6u )D v D' • u D u weakly  A in  a  a j 3  n  ra  for  n  m  n  |a| S m,  | 3 | S m.  of Krasnosel skii  [6]  1  (x,t,u,6u)D v L (ft X [0,T~J),  I t follows that  i n L (ft X therefore  a  2  a g  2  [0,T]). But  T Uft (4  ^  a  3 ( X  '  f c  ' n' U  6 U  n  } D  6  u  n  D  a  v  ~ a3 A  (  x  ' 'n' fc  U  6  u  n D uD«v] }  B  '10) = lJn[^B(x't'Un'5VDav-AaB(3C't'un'6un>Dav rT  = | |  A  (x,t,u,6u)D v[D u a  a B  3  n  -  D u]dxdt B  d  D^u  x  n  d  t  dxdt  62  converges t o  0  as  n  °°.  Hence  if  u E X,  A u i  * 3f . Similarly, is in  L (ft X [0,TJ).  n  v e Y,  -*• A u  weakly i n  i  then  A (x,t,u,6u)D v a  a  So by the same theorem o f  1  [6],  we  have  [ f  A (x,t,u ,6u  )D°vdxdt  (4.11) A (x,t,u,6u)D vdxdt  f o r a l l |a| < m. 1  Hence  1  fore, part  A u 2  n  A u  weakly i n  Y  y  .  There-  2  ( i i ) o f assumption I I i s v e r i f i e d and the theorem  i s proved.  Remark 4.1:  The above theorem remains t r u e i f we r e p l a c e every function  u  by  an  r-vector function  u  =  (u , u , ••• , 1  In t h i s case,  W 'P(ft) J c 0  w i l l be r e p l a c e d by  the norm "  r  (4.12)  u k,p  m=l |oKk  P L  2  (V7^' (ft)) p  r  with  63  Remark  4,2:  Let operators  T  in  L (w™' (ft) ,W~  a(u,v)  Y-v  =  /n-j-^,2  a(u,v)  conditions  a  n  d  i  t  s  d  u  {•,•)  a  theorem  +  {T_.U,Y..V} ,  the. p a i r i n g  l  W  bebwen  " "3 5) ' (3ft) . _ 3  is still m  and w i t h  2  valid with  o  V = W^Mft),  W = W  m  1  a(u,v)  replaced  2 ' (ft) .  The boundary  i n v o l v e d are o f the type:  (4.14)  S_.  a(u,v)  ( m  ( 3 0 )  The by  given  ^  3  v  m  2  =  and  be  ^ " ^ ' Oft)) . S e t :  2  (4.13)  where  ( j = 0, 1, * * * , m - 1)  S^u  are d i f f e r e n t i a l  =  TjU  operators  on  9ft, j = 0, 1, ••• , m-1.  of order  2m - j - 1  defined  by:  (4.15)  Au"vdx  =  a(t;u,v)  ft where  (4,16)  Au  +  m-1 £ j=0  {S.u,Y-v} , 3  3  i s given by:  I (-l)  la^m  lal  Dr a  L  I  |3km  A a t i  (x,t,u,6u)D u + A a 3  (x,t,u,6u)  64  (4.14)  i s called  t h e Neuman's t y p e o f b o u n d a r y  if  T_. = 0,  j = 1, 2,  , m -  1.  the  s o - c a l l e d t h i r d boundary - v a l u e  For  j ^  T  condition °»  w  derivatives or  have  e  "oblique"  derivatives.  Remark  4.3:  Using  t h e S o b o l e v i m b e d d i n g t h e o r e m , we may  the  r e s u l t o f t h e above  of  |X|  the  number o f d e r i v a t i v e s o f o r d e r  and  Then p a r t  |n|  theorem  i n part  (ii) of  (4.5)  and i n c r e a s e  ( i i ) of  the exponents  (4.5). j ,  improve  Let  N  denotes  j = 0, 1, •**, m-1.  can be r e p l a c e d  by: m-1  |A (x,t x ,x ,-",x f Y  f  0  ) |< C  l  (1 +  |X |)(1 + 0  u  (4.16)  n n+2(j-m)  if  n+2(j-m) > 0  0  if  n+2(j-m)  ^ 0  from t h e S o b o l e v theorem t h a t  i f  e  ,  j  e. 3  =  I |x I ) j = 0 J J  in 2 It  follows  (4.17)  1 +  m-1 . e. I iD^ul ^ ^ W" j=l 2  C i s independent of  u,  C  | U  '^' (ft) 2  u e W' 0  (ft), t h e n  64a  4.2. A p s e u d o monotone  Let be  m  operator.  be a p o s t i v e  integer  t h e number o f d e r i v a t i v e s o f o r d e r  m-1  and  N  and  p £ 2.  Let  N  l e s s than o r equal  b e t h e number o f d e r i v a t i v e s o f o r d e r  to  m.  2  Consider  the family  of real  N i defined  X R  satisfying  (4.4).  Set:  Suppose t h e r e  (4.18)  Let  a  m  p  X = FnL°°(o,T;H) . A (x,Su,D u)  (4.19)  a(u,v)  u,  V.  We  form  e V  f  I  ft laKrn  +  1  such  U ^ "  +l ] .  1  .  Therefore  and  a:V X V -*• R  by:  (x,6u,D u)D vdx. m  a  a  v ->• a ( u , v )  a mapping  that  then the functions  Define  A  D u). m  /  C > 0  P  P  =  cfjnl^  v e V,  a(u,v)  it  $ m,  similar to  i s linear  and c o n t i n u o u s  write  (5.20)  Au  •*•  F = L (0,T;V)  L '(ft).  j  The  <  2  If  conditions  u = {u, Du,  H = L (ft),  are i n  m  a  | ot|  a  e x i s t s a constant  |A (x,n,S)|  V = W ' (ft),  A (x,n,5)/  N?  on ft X R  those g i v e n by  functions  =  (Au,v) *  A:V -> V  v  .  i s defined,  on  65  We proof  shall  now  state  the  f o l l o w i n g p r o p o s i t i o n , the  o f w h i c h i s s t r a i g h t - f o r w a r d and  Proposition  omitted.  4.2:  Let  (i)  i s therefore  A  be  as_ a b o v e . S u p p o s e  there exists  constant  A  (x, ,OX  I  n  that  c > 0  >  such  c  I  lal<m for  a l l  (ii)  \x \  |a|<m  X =  I  that  (X  : |a| < m)  a  e R*  X  N  R . N 2  (A (x n,5) - A (x,n,5)) ( 5 Q  f  a  - 5)  a  a  >  0  |aj=m for  almost  a l l  hi o r e o v e r ,  suppose  B  f  and  in  in  in F*  F*  such  u.  e L H,  x  e ft, a l l  (0 ,T)  n  and  x  K ¥ £,  and > 0.  0  there exists  u  Then in  for, a l l  X  with  u'  that  u'(x,t)  + Au(x,t).+  B(t)u(x,t-x ) = 0  u (x, 0)  =  u  u(x,t)  =  0  u(x,t)  =  0  (x)  f(x,t)  x  e ft  t  <  (4.21)  (x,t)  0 e 9ft X  [0,T]  66  Remark  4.4:  In In b o t h  of  these,  derivatives operator order.  theorem  of  This  (W?' '  2  shall t.  and  w(t)  Suppose a l l  be  now  at the  considered  an  2  m»  X=  i n theorem  exists  t e  ( t , T)  establish  the  the any the  4*1.  t  e  and  2  n  Suppose  F =  (0,T;H).  Let  B  such  w(t)  >  that 0  t ( S / r ) It " E  L (0,T;V), 2  e L°°(0,T)  the f o l i a t i n g  t -  for a l l  ««t)|.  proposition.  which,  and  f u n c t i o n d e f i n e d on  (0,T) t -  a delay  (Wo' (ft)) ,  V=  > 1 + |.  a r e a l measurable  there  example w i t h  n  F O L°°  = shall  to  expense o f o m i t t i n g  Set:  We  respect  to d e r i v a t i v e s of  (L (ft)) ,  H=  and  Q  parts.  I n p r o p o s i t i o n 4.2,  respect  consider  where  n  two  delay.  Let  (ft) ) ;  c o n s i s t s of  i s linear with  order.  operator  Time d e p e n d e n t  = L"*(0,T;W)  for  highest  the  operator  operator  i s accomplished  d e p e n d s on  Y  the  i s non-linear with  We  W =  the  the  second p a r t of  4.3.  5.1,  (0,T).  w(t) t  e  >  0  ( 0 , t )'.  67  Proposition  4.3  Let L (-T ,0;H). 2  0  w  be  positive.  T h e n f o r any  Let  g  in  F*  f  be  a function in  and  u  i_n  Q  H ,  there  * exists  u  in  X  with  u'(x,t) -  '+  I  u'  in  Au(x,t)  Y  such  that  + B (t)u(x,t-to(t) )  DjUj (x,t)u(x,t)  +  2UJ (x,t)Dju(x,t)  j=l =  f(x,t),  u (x,0)  =  u  u(x,t)  =  g(x,t)  u(x,t)  =  0  (4.22) 0  (x)  X £  ft,  t e  (-T  ,0),  ( x , t ) e 3ft X  [0,T].  Proof;  To p r o v e  this  t h e o r e m , we  shall  apply  theorem  2.1  with  B ( t ) u ( t - to(t) ) (4.23)  Since  Mu(t)  B e L°°(0,T),  i f  t e ( t ,T]  if  t e [0,tj  o  =  Me  L (L  (0 ,T; H ) ,L°° (0 ,T ; H ) ), i s o f  local  68  type fi  and s a t i s f i e s  the convergence c o n d i t i o n .  Nov;  define  by:  B(t)g(t-w(t)) (4.24)  i f  t e [0,t ), o  if  t e  f|(t) =  Because  g  L (0,T;H) 2  is in <=  '  L  2  ( - T ,0)H) , 0  L (0,T;V*). I f 2  u'(t)  therefore  f  .  isin  u(t) s a t i s f i e s  + Au(t) + Ku(t) = f ( t ) -  (4.25)  [t , T ]  u(t)  = g(t)  u(0) = u  0  f j (t)  i f  t e  [-T ,0) 0  ,  satisfies  n J. [ ( D . u . u ) + 2u.D.u] , t h e n u(t) also J D 3 J j=l (4.22). T h e r e f o r e t h e t h e o r e m i s p r o v e d i f we  show t h a t  A  where  that  Au = - A u +  theorem  It in  satisfies 2.1  i s clear  (W^' (^)) ,  where  2  {«}  can be  n  t h e hypotheses o f theorem  4.1, s o  applied.  that  (4.5)  i s satisfied.  If  u  is  then  i s a norm e q u i v a l e n t  t o the  12 n (W ' (ft)) Q  n  orm.  69*  u e  Moreover, i f  (w™'' (n)) 2  n  ,  n  I I L j=l  D  k=l (4.27)  J j' k k U  U  , u  d x  n n r 1 1 -u..D. (u, ) d x k=l j = l f i  =  2  J  n = -2  (4.7)  3  3  K  n r  I I  k=l  T h i s means t h a t  then  j=  u -D u -u dx. 3 D K  i s satisfied  K  and s o t h e t h e o r e m i s  proved.  In operator  t h e above p r o p o s i t i o n , we may  M  defined  (4.28)  =  1  where  p  is  L(L°°(0,T;H) ,L°°(0,T;H))  in  a l s o deduce  0  and  M  another  by:  M u(t)  >  add t o  | (t 'o  u e L°°(0,T;H).  s) u(s)ds P  I t i s clear that  and i s o f l o c a l  M  t y p e , we c a n  from t h e Lebesque's convergence theorem  that  M l  satisfies  t h e c o n v e r g e n c e c o n d i t i o n . T h e r e f o r e we h a v e t h e  following  proposition.  70  Proposition  4.4:  Let hypotheses given with  of  M  be  as  above  and  suppose  p r o p o s i t i o n 4.3  are  satisfied.  H,  there  f  in  F*  and  u  in  u'  in  Y*  such  that  0  u' ( x , t )  + +  n  -  +  a l l the Then  exists  u  for  any  in  X  Au(x,t)  DjUj ( x , t ) u ( x , t ) +  B ( t ) u ( x , t-u>(t) )  that  2UJ  f  (x,t)DjU(x,T)  (t -  s)Pu(x,s)ds  ' o (4.29)  f (x,t), u(x,0)  =  u  (x)  x  e ft,  t  e  (x,t)  e  0  4.4.  u(x,0)  =  g(x,t)  u(x,t)  =  0  Solutions with  In parabolic  this  time d e r i v a t i v e s of  ,0) ,  0  3ft X  [0,T]  fractional derivatives.  s e c t i o n , we  equations  (-T  give  examples  of  which have  solutions with  order  0  y,  < y  < H,  in  non-linear fractional L (0,T;L (^)). 2  2  71  Proposition  4.5:  Let  H = L  X = FnL°°(0,T;H) . with  p  =  assumption  2.  2  (ft) ,  V = W^' 1  Define  (ft) ,  2  A:V  V*  Then the o p e r a t o r  I with  F = L  2  ( 0 , ;V)  as i n p r o p o s i t i o n A  also  and 4.2  satisfies  6 = 0 .  The p r o o f o f t h e above p r o p o s i t i o n  i s straight-  forward.  The of  theorem  Theorem  following  1.3  and p r o p o s i t i o n  i s an i m m e d i a t e  consequence  4.5.  4.2:  Let satisfied. exists  theorem  u  a l l the hypotheses of p r o p o s i t i o n  Then in  X  f o r any with  f u'  in in  F* F*  and such  u  o that  4.5  in  be  H,  there  o (4.30)  u(x,0)  =  u(x,t)  DY (x,t) u  u  o  0  e  (x)  x (x,t)  L (0,T;H) 2  e ft, e 8ft X  0 <  Y  [0,T] ,  <  3s.  72  Proof:  It  i s c l e a r that  M  defined  by  rT (4.31)  Hu(t)  =  / F s  u(s)ds,  u e L°°(0,T;H),  ' o is and  an  operator  satisfies  follows  from  in the  4.5,  [l] with  Periodic  In theorem  3.1.  Theorem  4.3:  Let for  any  A  given  and  0 < y <  theorem  i s of  local  Therefore  proposition  is linear,  the  type theorem  4.5.  4.2  has  been obtained  h.  solutions.  this  s e c t i o n , we  I-I = L f  in  L ( 0 , T ; V ) nL°°(0,T;H) p  ,L°° (0 ,T;H)) ,  convergence c o n d i t i o n .  t h e o r e m 1.3  When Artola  L (L°° (0 ,T;H)  2  some a p p l i c a t i o n s  V = wj-* (ft)  (ft) , If  give  p  (0,T; V  with  u'  with  p  ),  there  in  L '(0,T;V*) p  exists  of  > 2, u  Then ill  such  that  by  n  u'(x,t)  + c I -D. (|D . u ( x , t ) | j=l J  P-2  J  + C>|u(X, t ) | " U + u ( X , t - ( j ) ( t ) ) P  (4.32)  D_jU(x,t))  J  f(X,t),  =  2  I u(x,0)  =  u(x,T)  u(x,t)  =  o  x  e ft,  ( x , t ) e an x [ O , T ]  Proofi  It W  _ 1 / P  '(ft)  Au = c  c > 0  i smonotone  (4.34)  n  _l-  D j  (iD.uj^Vu)  A:W^' (Q) P  In  c  P  2  l lv u  follows immediately  t h e case where  the operator  |u| " u  P  v  t h e theorem  +  and  (Au,u) *  Therefore  modify  that the operator  defined by  (4.33)  with  i swell-known  p = 2,  A. We s h a l l  p r o p o s i t i o n w h i c h i s an immediate  from thoorem  i ti s necessary  3.1  to  state thefollowing consequence  o f theorem 3  74,  Proposition 4 . 6 :  1 2 Let X  H = L (ft) ,  V = V7 ' (ft) and c > 0.  z  i s a s u f f i c i e n t l a r g e number.  L (0,T;V*), 2  u'  in  there e x i s t s  L (0,T;V*) 2  such  Suppose  0  u  Then f o r any g i v e n  in  L(0,T;V) nL°°(0,T;H)  f  in  with  that  u' (x,t) - cAu(x,t) + Xu(x,t) + u ( x , t - w ( t ) ) =  f(x,t),  (4.35)  u(x,0)  =  u(x,T)  u(x,t)  =  0  X G ft, (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 E q u a t i o n s d ' e v o l u t i o n , a p p l i c a t i o n s a des p r o b l e m e s de r e t a r d . Ann, S c i e n t . E c o l e N o r m a l e Sup., 4 s e r i e , t . 2 , 19 69, p.137 a p,253. e  J, Aubin, Un theordme de c o m p a c i t e , C. P a r i s . S e r i e A 2 5 6 ( 1 9 6 3 ) , 5044 - 5047.  R.  Acad.  Sc.  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 V a l u e P r o b l e m s , Amer. J . Math. 86 ( 1 9 6 4 ) , p.339 - p . 3 5 7 . E . C o d d i n g t o n and N. L e v i n s o n , T h e o r y o f D i f f e r e n t i a l E q u a t i o n s , M c g r a w - H i l l , New  Ordinary f o r k 19 55.  M. G a u l t i e r , Solutions f a i b l e s peridiques d'equations d ' E v o l u t i o n du p r e m i e r o r d r e p e r t u r b d e s , C. R. A c a d . S c . P a r i s , s e r i e A272 ( 1 9 7 1 ) , 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 M e t h o d s i n t h e T h e o r y 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 , e t probldmes , probilmeseaux  Equations d i ffe r e n t i a l l e s o.perationelles, 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. Q u e l q u e s Methods de r e s o l u t i o n des p r o b l 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 e q u a t i o n s paraboliques n o n - i i n g a i r e s , B u l l . S o c . Math. F r a n c e , 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 P r o b l e m s i n t h e T h e o r y o f D i f f e r e n t i a l E q u a t i o n s , R u s s i a n Math. S u r v e y s , V o l . 22, No. 2, 1967, p l 7 t o 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 t h e N a v i e r - S t o k e s E o u a t i o n s , A r c h . R a t . Mech. A n a l . 40 (1971) , p.139 t o 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 o f S o l u t i o n s o f Nonl 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 Banach S p a c e s ( t o a p p e a r ) , Non-linear 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, ( 1 9 7 0 ) , p.69 to p.80.  76  j14|  B. A. Ton, On 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 E q u a t i o n s . J . Func. A n a l . , V o l . 1, No. 1, (1971), p.147 t o p.155.  |15|  , P e r i o d i c S o l u t i o n s of N o n - l i n e a r E v o l u t i o n Equations i n Banach Spaces,,Can. J . o f Math. 23, 19 71, p.189 t o p.196.  

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